Finite Group Databasebeta

← #29686533

Subgroups — #29686533

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839992C211yesmaximal, characteristic
2C2×C4×#11758190C211yesmaximal, characteristic, complemented
3C2×#16839716C211yesmaximal, characteristic, complemented
4#21448513C211yesmaximal, characteristic
5#21448502C211yesmaximal, characteristic, complemented
6#21446671C211yesmaximal, characteristic, complemented
7#21446548C211yesmaximal, characteristic, complemented
8#21446517C211yesmaximal, characteristic, complemented
9#21446510C211yesmaximal, characteristic, complemented
10#21446256C211yesmaximal, characteristic
11#21446251C211yesmaximal, characteristic, complemented
12#21445841C211yesmaximal, characteristic, complemented
13#21445652C211yesmaximal, characteristic, complemented
14#21445647C211yesmaximal, characteristic, complemented
15C4×#106×C61C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic
19C2×#14519757C2211yescharacteristic
20C2×#14519752C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×#14519734C2211yescharacteristic
24#16839992C418yescomplemented
25C4×#11758190—24no—
26C2×#14519709C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33#16839716—24no—
34C2×#14519688C2211yescharacteristic
35C2×#14519687C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839664—21no—
39#16839663C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839661C2211yescharacteristic
42#16839660—21no—
43#16839654—21no—
44#16839653—21no—
45#16839653—21no—
46#16839647—21no—
47#16839640—21no—
48#16839636—21no—
49#16839619C2211yescharacteristic
50#16839619C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C4×C4⋊C4×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244—21no—
64C22×D244C2311yescharacteristic
65C22×C4×D61—21no—
66C22×C4×D61C2311yescharacteristic
67C2×#11758190—22no—
68C2×#11758190—22no—
69C2×#11758190C2×C414yes—
70C2×#11758189—22no—
71C2×#11758189—22no—
72C2×#11758189—22no—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75C2×C8×D61—22no—
76C2×C8×D61—22no—
77C2×C8×D61C2×C414yescomplemented
78#14519757—22no—
79#14519757C2×C414yescomplemented
80#14519752—22no—
81#14519752C2×C414yes—
82C4×Dic122—22no—
83C4×Dic122—22no—
84C4×D244—22no—
85C4×D244—22no—
86#14519734—22no—
87#14519734—22no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×#11758196—21no—
94C2×#11758196D411yescharacteristic
95C2×#11758196D411yescharacteristic
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×#11758198D411yescharacteristic
104C2×#11758198D411yescharacteristic
105C2×C4×Dic61—21no—
106C2×C4×Dic61C2311yescharacteristic
107#14519709—22no—
108#14519702—22no—
109#14519700—22no—
110#14519700C2×C414yescomplemented
111#14519699—22no—
112#14519697—22no—
113#14519697C2×C414yes—
114#14519688—22no—
115#14519687—22no—
116C8×Dic61—22no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C42×C61—21no—
130C2×C42×C61—21no—
131C2×C42×C61C2311yescharacteristic
132#106×C61—22no—
133#106×C61C2×C414yescomplemented
134C4×C8×C61D414yescomplemented
135#95×C61—21no—
136#95×C61—21no—
137C23×D61—21no—
138C23×D61C2×D411yescharacteristic
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—24no—
158C2×D244—81no—
159C2×D244—81no—
160C2×D244C22×C412yes—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—42no—
166C2×C4×D61—81no—
167C2×C4×D61—81no—
168C2×C4×D61C22×C412yes—
169C2×C4×D61#4112yes—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122C22×C412yes—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61C22×C412yes—
179C2×C8×C61C2×D412yes—
180C2×C8×C61C2×D412yes—
181C2×C8×C61#4112yes—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61C22×C412yes—
189C4⋊C4×C61C22×C412yes—
190C42×C61—81no—
191C42×C61—81no—
192C42×C61C2×D412yes—
193C42×C61#4112yes—
194#11758198—24no—
195#11758198—22no—
196#11758198—22no—
197#11758198—24no—
198#11758198—42no—
199#11758198—81no—
200#11758198—81no—
201#11758198—81no—
202#11758198—81no—
203C4⋊Dic61—24no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—81no—
209C4⋊Dic61—81no—
210#11758196—22no—
211#11758196—22no—
212#11758196—42no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—42no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418C22×C412yes—
221C2×#92418#4112yes—
222#11758190—28no—
223#11758189—28no—
224C8×D61—28no—
225C23×C61C22×D411yescharacteristic
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—22no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61C4×D412yes—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61#12112yes—
257C2×C4×C61#12112yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61C4×D412yes—
282D244—22no—
283D244—22no—
284D244—41no—
285D244—41no—
286D244—84no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—84no—
292Dic122—22no—
293Dic122—22no—
294Dic122—41no—
295Dic122—41no—
296C8×C61—22no—
297C8×C61C4×D414yes—
298#92418—22no—
299#92418—22no—
300#21523—611nomaximal, Sylow
301C22×C61C2×C4×D411yescharacteristic
302C22×C61C2×#12111yescharacteristic
303C22×C61#53211yescharacteristic
304C22×C61#57711yescharacteristic
305C22×C61#57711yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57711yescharacteristic
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—42no—
318C2×D61—81no—
319C2×D61—81no—
320C2×D61—81no—
321C2×D61—81no—
322C4×C61—22no—
323C4×C61—22no—
324C4×C61—42no—
325C4×C61—42no—
326C4×C61—81no—
327C4×C61—81no—
328C4×C61C2×C4×D411yescharacteristic
329C4×C61C4×#4111yescharacteristic
330C4×C61#52511yescharacteristic
331C4×C61#52911yescharacteristic
332Dic61—22no—
333Dic61—41no—
334Dic61—41no—
335Dic61—42no—
336Dic61—81no—
337Dic61—81no—
338C2×#465—611no—
339C2×C4×SD16—611no—
340C2×#405—611no—
341#2324—611no—
342#2323—611no—
343#1968—611no—
344#1941—611no—
345#1936—611no—
346#1935—611no—
347#1895—611no—
348#1894—611no—
349#1807—611no—
350C4×#106D6111yescharacteristic, complemented
351#1763—611no—
352#1762—611no—
353C2×C61C4×#12111yescharacteristic
354C2×C61#294911yescharacteristic
355C2×C61#295211yescharacteristic
356C2×C61#305711yescharacteristic
357C2×C61#306011yescharacteristic
358C2×C61#317311yescharacteristic
359C2×C61#317411yescharacteristic
360D61—42no—
361D61—82no—
362C22×SD16—611no—
363C22×M16—611no—
364C23×C8—611no—
365C2×C22⋊Q8—611no—
366C2×#121—611no—
367C2×C4×Q8—611no—
368C2×C4×D4—611no—
369C2×#117—611no—
370#465—618no—
371C4×SD16—1224no—
372C2×#106—611no—
373C2×#106C2×D6111yescharacteristic
374C2×#106C2×D6111yescharacteristic
375C2×#103—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×#102—611no—
381#405—1224no—
382C2×#97—611no—
383C2×#97—611no—
384C2×C4×C8—611no—
385C2×C4×C8C2×D6111yescharacteristic
386#391—611no—
387#390—611no—
388#390—1221no—
389#389—611no—
390#389—611no—
391#389—1221no—
392#388—1221no—
393#388—1221no—
394#388—1221no—
395#387—1221no—
396#386—1221no—
397#385—1221no—
398#380—611no—
399#380—611no—
400C4×C4⋊C4C2×D6111yescharacteristic
401C4×C4⋊C4C2×D6111yescharacteristic
402#340—611no—
403#340—611no—
404#340—611no—
405#340—611no—
406#340C2×D6111yescharacteristic
407#340C2×D6111yescharacteristic
408C61#2152311yescharacteristic, complemented, Sylow
409C22×Q8—611no—
410C22×D4—611no—
411C22×D4—1221no—
412C23×C4—611no—
413C23×C4—1221no—
414C2×SD16—614no—
415C2×SD16—1222no—
416C2×SD16—1222no—
417C2×M16—1222no—
418C2×M16—1222no—
419C2×M16—1222no—
420C22×C8—611no—
421C22×C8—611no—
422C22×C8—614no—
423C22×C8—1222no—
424C22×C8—1222no—
425C22×C8C22×D6111yescharacteristic
426C22×C8C22×D6111yescharacteristic
427C22⋊Q8—614no—
428C22⋊Q8—1222no—
429#121—614no—
430#121—1222no—
431C4×Q8—1222no—
432C4×Q8—1222no—
433C4×D4—1222no—
434C4×D4—1222no—
435#117—1222no—
436#117—1222no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C22⋊C4—611no—
450C2×C22⋊C4—611no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C42—21no—
459C2×C42—21no—
460C2×C42—611no—
461C2×C42—1221no—
462C2×C42C22×D6111yescharacteristic
463#106—22no—
464#106—1222no—
465#106C4×D6114yescomplemented
466#103—614no—
467#103—1222no—
468#103—1222no—
469#102—614no—
470#102—1222no—
471#102—1222no—
472#97—1222no—
473#97—1222no—
474C4×C8—1222no—
475C4×C8D24414yescomplemented
476#95—21no—
477#95—21no—
478#95—1221no—
479#95—1221no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484C24—611no—
485C24—1221no—
486C2×Q8—612no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×D4—612no—
493C2×D4—1224no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—4881no—
500C2×D4—4881no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—611no—
506C22×C4—612no—
507C22×C4—612no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—4881no—
518C22×C4—4881no—
519C22×C4—2442no—
520C22×C4D4×D6111yescharacteristic
521C22×C4D4×D6111yescharacteristic
522C22×C4C23×D6111yescharacteristic
523SD16—1228no—
524M16—1228no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—612no—
528C2×C8—612no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8—1228no—
534C2×C8C2×C4×D6112yes—
535C2×C8C2×D24412yes—
536C2×C8C2×D24412yes—
537C2×C8#1175822212yes—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—1224no—
545C4⋊C4—1222no—
546C4⋊C4—1222no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—4881no—
552C4⋊C4—4881no—
553C4⋊C4—2442no—
554C4⋊C4C2×C4×D6112yes—
555C4⋊C4C2×C4×D6112yes—
556C22⋊C4—1224no—
557C22⋊C4—1224no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—4881no—
561C22⋊C4—4881no—
562C22⋊C4—4881no—
563C22⋊C4—4881no—
564C22⋊C4—2442no—
565C42—81no—
566C42—81no—
567C42—1221no—
568C42—1221no—
569C42—2442no—
570C42C2×D24412yes—
571C42#1175822612yes—
572C23—612no—
573C23—1221no—
574C23—1221no—
575C23—1221no—
576C23—1221no—
577C23—1221no—
578C23—1221no—
579C23—1222no—
580C23—2441no—
581C23—2441no—
582C23—2441no—
583C23—2441no—
584C23—2441no—
585C23—2441no—
586C23C2×D4×D6111yescharacteristic
587Q8—1222no—
588Q8—1222no—
589Q8—2441no—
590Q8—2441no—
591D4—1222no—
592D4—1222no—
593D4—4884no—
594D4—2441no—
595D4—2441no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—612no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—4884no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4#1451975212yes—
636C2×C4#1451975212yes—
637C2×C4C22×C4×D6111yescharacteristic
638C2×C4C22×D24411yescharacteristic
639C2×C4#41×D6111yescharacteristic
640C2×C4#41×D6111yescharacteristic
641C2×C4C2×#1175822211yescharacteristic
642C2×C4C2×#1175822611yescharacteristic
643C8—22no—
644C8—1222no—
645C8—1222no—
646C8C4×D24414yes—
647C22—1222no—
648C22—1222no—
649C22—1222no—
650C22—4881no—
651C22—4881no—
652C22—4881no—
653C22—4881no—
654C22—2442no—
655C22—2442no—
656C22—2442no—
657C22—2441no—
658C22—2441no—
659C22—2441no—
660C22—2441no—
661C22C4×D4×D6111yescharacteristic
662C22C2×#1451975211yescharacteristic
663C22#1684030911yescharacteristic
664C22#1684056811yescharacteristic
665C22#1684056811yescharacteristic
666C22#1684056911yescharacteristic
667C22#1684056911yescharacteristic
668C4—22no—
669C4—22no—
670C4—42no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4—1222no—
675C4—4881no—
676C4—4881no—
677C4—2441no—
678C4—2441no—
679C4—2442no—
680C4C2×C4×D24411yescharacteristic
681C4C4×#1175822611yescharacteristic
682C4#1684027411yescharacteristic
683C4#1684029911yescharacteristic
684C2—4882no—
685C2—2442no—
686C2C4×#1451975211yescharacteristic
687C2#2145467311yescharacteristic
688C2#2145469711yescharacteristic
689C2#2145545911yescharacteristic
690C2#2145548211yescharacteristic
691C2#2145643211yescharacteristic
692C2#2145644311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#106×C61C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
cosocleC22×C4×C61C24
hypercenterC4×#106D61
hypocenterC61#21523
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#106

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C4×#11758190C2semidirect
3. C2×#16839716C2semidirect
5. #21448502C2semidirect
6. #21446671C2semidirect
7. #21446548C2semidirect
8. #21446517C2semidirect
9. #21446510C2semidirect
11. #21446251C2semidirect
12. #21445841C2semidirect
13. #21445652C2semidirect
14. #21445647C2semidirect
15. C4×#106×C61C2semidirect
24. #16839992C4semidirect
77. C2×C8×D61C2×C4semidirect
79. #14519757C2×C4semidirect
110. #14519700C2×C4semidirect
133. #106×C61C2×C4semidirect
134. C4×C8×C61D4semidirect
350. C4×#106D61semidirect
408. C61#21523semidirect
465. #106C4×D61semidirect
475. C4×C8D244semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168399922
C2×C4×#117581902
C2×#168397162
#214485132
#214485022
#214466712
#214465482
#214465172
#214465102
#214462562
#214462512
#214458412
#214456522
#214456472
C4×#106×C612
#2152361