Finite Group Databasebeta

← #29686534

Subgroups — #29686534

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×#16839988C211yesmaximal, characteristic
2C2×SD16×Dic61C211yesmaximal, characteristic, complemented
3C2×#16839716C211yesmaximal, characteristic, complemented
4#21448515C211yesmaximal, characteristic
5#21448504C211yesmaximal, characteristic, complemented
6#21446547C211yesmaximal, characteristic, complemented
7#21446519C211yesmaximal, characteristic, complemented
8#21446512C211yesmaximal, characteristic, complemented
9#21446254C211yesmaximal, characteristic
10#21446249C211yesmaximal, characteristic, complemented
11#21445841C211yesmaximal, characteristic, complemented
12#106×Dic61C211yesmaximal, characteristic, complemented
13#21445651C211yesmaximal, characteristic, complemented
14#21445646C211yesmaximal, characteristic, complemented
15C61×#1968C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519753C2211yescharacteristic
19C2×#14519748C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22C2×#14519734C2211yescharacteristic
23#16839988C418yescomplemented
24SD16×Dic61—24no—
25C2×#14519709C2211yescharacteristic
26C2×#14519709C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519698C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31#16839716—24no—
32C2×#14519688C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839662—21no—
36#16839660C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839659—21no—
39#16839654—21no—
40#16839651—21no—
41#16839650—21no—
42#16839644—21no—
43#16839639—21no—
44#16839634—21no—
45#16839619C2211yescharacteristic
46#16839619C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48C4⋊C4×Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C22×SD16×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×C4×C8×C61C2211yescharacteristic
60C61×#391C2211yescharacteristic
61C61×#390C2211yescharacteristic
62C22×C22⋊D61—21no—
63C23×Dic61—21no—
64C22×C4×D61C2311yescharacteristic
65C2×#11758189—22no—
66C2×#11758189—22no—
67C2×#11758189—22no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C2×C8×D61—22no—
71C2×C8×D61—22no—
72C2×C8×D61C2×C414yescomplemented
73#14519753—22no—
74#14519753C2×C414yescomplemented
75#14519748—22no—
76#14519748C2×C414yes—
77Q8×Dic61—22no—
78Q8×Dic61—22no—
79D4×Dic61—22no—
80D4×Dic61—22no—
81#14519734—22no—
82#14519734—22no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196D411yescharacteristic
92C2×#11758196D411yescharacteristic
93C2×C22⋊Dic61—21no—
94C2×C22⋊Dic61—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198D411yescharacteristic
99C2×#11758198D411yescharacteristic
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61C2311yescharacteristic
104#14519709—22no—
105#14519709C2×C414yescomplemented
106#14519701—22no—
107#14519701C2×C414yescomplemented
108#14519698—22no—
109#14519698C2×C414yes—
110#14519688—22no—
111#14519687—22no—
112C8×Dic61D414yescomplemented
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C22×Q8×C61C2311yescharacteristic
120C22×D4×C61C2311yescharacteristic
121C2×SD16×C61—22no—
122C2×SD16×C61—22no—
123C2×SD16×C61C2×C414yes—
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C22⋊C4×C61—21no—
129C2×C22⋊C4×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#106×C61—22no—
132#103×C61—22no—
133#102×C61—22no—
134C4×C8×C61—22no—
135#95×C61—21no—
136#95×C61—21no—
137C24×C61—21no—
138C23×D61C2×D411yescharacteristic
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61C22×C412yes—
145C2×D4×C61—21no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C22×C412yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C2×C22⋊D61—24no—
158C2×C22⋊D61—81no—
159C2×C22⋊D61—81no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61—42no—
165C22×Dic61—81no—
166C22×Dic61—81no—
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61C22×C412yes—
175C2×C4×D61#4112yes—
176SD16×C61—28no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61C22×C412yes—
180C2×C8×C61C2×D412yes—
181C2×C8×C61#4112yes—
182C2×C8×C61#4112yes—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C22⋊C4×C61—81no—
188C22⋊C4×C61—81no—
189C42×C61—21no—
190C42×C61—21no—
191C22⋊Dic61—24no—
192C22⋊Dic61—42no—
193#11758198—24no—
194#11758198—22no—
195#11758198—22no—
196#11758198—81no—
197#11758198—81no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—42no—
205C4⋊Dic61C22×C412yes—
206C4⋊Dic61C22×C412yes—
207#11758196—24no—
208#11758196—22no—
209#11758196—22no—
210#11758196—81no—
211#11758196—81no—
212C4×Dic61—42no—
213C4×Dic61—81no—
214C4×Dic61—81no—
215C4×Dic61C2×D412yes—
216C4×Dic61#4112yes—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418C22×C412yes—
222C2×#92418C2×D412yes—
223#11758189—28no—
224C8×D61—28no—
225C23×C61—22no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61C22×D411yescharacteristic
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—21no—
238C22×D61—21no—
239C22×D61C4×D412yes—
240Q8×C61—22no—
241Q8×C61—22no—
242Q8×C61—41no—
243Q8×C61—41no—
244D4×C61—22no—
245D4×C61—22no—
246D4×C61—41no—
247D4×C61—41no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C22⋊D61—84no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—84no—
289C2×Dic61C4×D412yes—
290C2×Dic61#12112yes—
291C2×Dic61#12112yes—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—22no—
295C4×D61—22no—
296C8×C61—22no—
297C8×C61C4×D414yes—
298#92418—22no—
299#92418—22no—
300#21523—611nomaximal, Sylow
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61C2×C4×D411yescharacteristic
309C22×C61C2×#12111yescharacteristic
310C22×C61#53211yescharacteristic
311C22×C61#57711yescharacteristic
312C22×C61#57711yescharacteristic
313C22×C61#57711yescharacteristic
314C22×C61#57711yescharacteristic
315C2×D61—22no—
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—41no—
319C2×D61—41no—
320C2×D61—41no—
321C2×D61—41no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—42no—
325C4×C61—81no—
326C4×C61—81no—
327C4×C61C2×C4×D411yescharacteristic
328C4×C61C4×#4111yescharacteristic
329C4×C61#52511yescharacteristic
330C4×C61#52911yescharacteristic
331Dic61—22no—
332Dic61—22no—
333Dic61—22no—
334Dic61—42no—
335Dic61—42no—
336Dic61—81no—
337Dic61—81no—
338C2×#465—611no—
339C2×C4×SD16—611no—
340C2×#405—611no—
341#2324—611no—
342#2323—611no—
343#1968D6111yescharacteristic, complemented
344#1941—611no—
345#1936—611no—
346#1935—611no—
347#1895—611no—
348#1894—611no—
349#1807—611no—
350C4×#106—611no—
351#1763—611no—
352#1762—611no—
353C2×C61—82no—
354C2×C61C4×#12111yescharacteristic
355C2×C61#294911yescharacteristic
356C2×C61#295211yescharacteristic
357C2×C61#305711yescharacteristic
358C2×C61#306011yescharacteristic
359C2×C61#317311yescharacteristic
360C2×C61#317411yescharacteristic
361D61—42no—
362C22×SD16C2×D6111yescharacteristic
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×#106—611no—
374C2×#106C2×D6111yescharacteristic
375C2×#103—611no—
376C2×#103—611no—
377C2×#103C2×D6111yescharacteristic
378C2×#102—611no—
379C2×#102—611no—
380C2×#102C2×D6111yescharacteristic
381#405—1224no—
382C2×#97—611no—
383C2×#97—611no—
384C2×C4×C8—611no—
385C2×C4×C8C2×D6111yescharacteristic
386#391C2×D6111yescharacteristic
387#390—1221no—
388#390C2×D6111yescharacteristic
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⋊C4—611no—
401C4×C4⋊C4—611no—
402#340—611no—
403#340—611no—
404#340—611no—
405#340—611no—
406#340—611no—
407#340—611no—
408C61#2152311yescharacteristic, complemented, Sylow
409C22×Q8C22×D6111yescharacteristic
410C22×D4—1221no—
411C22×D4C22×D6111yescharacteristic
412C23×C4—611no—
413C23×C4—1221no—
414C2×SD16—22no—
415C2×SD16—22no—
416C2×SD16C4×D6114yes—
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—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—611no—
452C2×C22⋊C4—611no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C22⋊C4—1221no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C42—611no—
459C2×C42—1221no—
460C2×C42—1221no—
461C2×C42—1221no—
462C2×C42C22×D6111yescharacteristic
463#106—22no—
464#106—614no—
465#106—1222no—
466#103—22no—
467#103—614no—
468#103—1222no—
469#102—22no—
470#102—614no—
471#102—1222no—
472#97—1222no—
473#97—1222no—
474C4×C8—22no—
475C4×C8—614no—
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—21no—
485C24—611no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8C2×C4×D6112yes—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—21no—
496C2×D4—21no—
497C2×D4—1224no—
498C2×D4—4881no—
499C2×D4—4881no—
500C2×D4C2×C4×D6112yes—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—612no—
508C22×C4—612no—
509C22×C4—1221no—
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—28no—
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×C8D4×D6112yes—
536C2×C8#1175822412yes—
537C2×C8#1175822412yes—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—612no—
543C4⋊C4—612no—
544C4⋊C4—1222no—
545C4⋊C4—1222no—
546C4⋊C4—1224no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—4881no—
554C4⋊C4—4881no—
555C4⋊C4—2442no—
556C22⋊C4—81no—
557C22⋊C4—81no—
558C22⋊C4—1222no—
559C22⋊C4—1222no—
560C22⋊C4—1224no—
561C22⋊C4—1224no—
562C22⋊C4—4881no—
563C22⋊C4—4881no—
564C22⋊C4—2442no—
565C42—21no—
566C42—21no—
567C42—612no—
568C42—612no—
569C42—4881no—
570C42—4881no—
571C42—2442no—
572C23—22no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—612no—
580C23—1221no—
581C23—1221no—
582C23—1221no—
583C23—1221no—
584C23—1221no—
585C23—1221no—
586C23C2×D4×D6111yescharacteristic
587Q8—22no—
588Q8—22no—
589Q8—41no—
590Q8—41no—
591D4—22no—
592D4—22no—
593D4—41no—
594D4—41no—
595D4—4884no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
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—612no—
612C2×C4—612no—
613C2×C4—612no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1222no—
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—2441no—
636C2×C4—2441no—
637C2×C4C22×C4×D6111yescharacteristic
638C2×C4C2×D4×D6111yescharacteristic
639C2×C4#41×D6111yescharacteristic
640C2×C4#41×D6111yescharacteristic
641C2×C4C2×#1175822411yescharacteristic
642C2×C4C2×#1175822411yescharacteristic
643C8—22no—
644C8—1222no—
645C8—1222no—
646C8#1451973714yes—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—1222no—
655C22—1222no—
656C22—1222no—
657C22—2441no—
658C22—2441no—
659C22—2441no—
660C22—2441no—
661C22C4×D4×D6111yescharacteristic
662C22#121×D6111yescharacteristic
663C22#1684031011yescharacteristic
664C22#1684056811yescharacteristic
665C22#1684056811yescharacteristic
666C22#1684056911yescharacteristic
667C22#1684056911yescharacteristic
668C4—41no—
669C4—41no—
670C4—42no—
671C4—81no—
672C4—81no—
673C4—1222no—
674C4—1222no—
675C4—1222no—
676C4—4881no—
677C4—4881no—
678C4—2442no—
679C4—2442no—
680C4C2×#1451973711yescharacteristic
681C4C4×#1175822411yescharacteristic
682C4#1684027211yescharacteristic
683C4#1684029111yescharacteristic
684C2—82no—
685C2—2442no—
686C2#2144833511yescharacteristic
687C2#2145467311yescharacteristic
688C2#2145469711yescharacteristic
689C2#2145545311yescharacteristic
690C2#2145547611yescharacteristic
691C2#2145643311yescharacteristic
692C2#2145644411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1968C2
socleC23×C61C22×D4
prime radicalC2×#14519748C22
cosocleC22×C4×C61C24
hypercenter#1968D61
hypocenterC61#21523
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. C2×SD16×Dic61C2semidirect
3. C2×#16839716C2semidirect
5. #21448504C2semidirect
6. #21446547C2semidirect
7. #21446519C2semidirect
8. #21446512C2semidirect
10. #21446249C2semidirect
11. #21445841C2semidirect
12. #106×Dic61C2semidirect
13. #21445651C2semidirect
14. #21445646C2semidirect
15. C61×#1968C2semidirect
23. #16839988C4semidirect
72. C2×C8×D61C2×C4semidirect
74. #14519753C2×C4semidirect
105. #14519709C2×C4semidirect
107. #14519701C2×C4semidirect
112. C8×Dic61D4semidirect
343. #1968D61semidirect
408. C61#21523semidirect

Maximal subgroups

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

subgroupindex
C2×#168399882
C2×SD16×Dic612
C2×#168397162
#214485152
#214485042
#214465472
#214465192
#214465122
#214462542
#214462492
#214458412
#106×Dic612
#214456512
#214456462
C61×#19682
#2152361