Finite Group Databasebeta

← #29709489

Subgroups — #29709489

Order 15616.

978 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×#16839847C211yesmaximal, characteristic, complemented
2C2×#16839814C211yesmaximal, characteristic, complemented
3#21448357C211yesmaximal, characteristic, complemented
4#21448327C211yesmaximal, characteristic
5#21447321C211yesmaximal, characteristic, complemented
6#21447258C211yesmaximal, characteristic, complemented
7#21447206C211yesmaximal, characteristic, complemented
8#21447134C211yesmaximal, characteristic, complemented
9#21447021C211yesmaximal, characteristic, complemented
10#21447003C211yesmaximal, characteristic
11#21446821C211yesmaximal, characteristic
12#107×Dic61C211yesmaximal, characteristic, complemented
13#103×Dic61C211yesmaximal, characteristic, complemented
14#21445633C211yesmaximal, characteristic, complemented
15C61×#2068C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519753C2211yescharacteristic
18C2×#14519752C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519739—21no—
21C2×#14519739C2211yescharacteristic
22C2×C42×D61C2211yescharacteristic
23#16839847—24no—
24#16839814—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×C4⋊#92418C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33C2×#14519689C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35C2×C8×Dic61C2211yescharacteristic
36#16839664C2211yescharacteristic
37#16839660C2211yescharacteristic
38#16839652—21no—
39#16839649—21no—
40#16839640—21no—
41#16839636—21no—
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46#16839613—21no—
47#16839609—21no—
48#16839609—21no—
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×D244—21no—
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C22×C4×D61D411yescharacteristic
65C22×C4×D61D411yescharacteristic
66C22×C4×D61C2311yescharacteristic
67C2×#11758201—22no—
68C2×#11758201—22no—
69C2×#11758201—22no—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519753—22no—
73#14519753—22no—
74#14519752—22no—
75#14519752—22no—
76Q8×Dic61—22no—
77Q8×Dic61—22no—
78#14519739—22no—
79#14519739—22no—
80#14519739—44no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61D411yescharacteristic
98C2×C4×Dic61D411yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C42×D61—22no—
101C42×D61—22no—
102#14519711—22no—
103#14519708—22no—
104C4⋊#92418—22no—
105#14519701—22no—
106#14519699—22no—
107#14519699—22no—
108#14519697—22no—
109#14519689—22no—
110#14519689—22no—
111C8×Dic61—22no—
112C8×Dic61D414yescomplemented
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118C22×Q8×C61C2311yescharacteristic
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126#107×C61—22no—
127C4⋊C8×C61—22no—
128#103×C61—22no—
129#103×C61—22no—
130#95×C61—21no—
131#95×C61—21no—
132#95×C61—21no—
133C23×D61—21no—
134C23×D61C2×D411yescharacteristic
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61#4112yes—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—24no—
159C2×D244—21no—
160C2×D244—42no—
161C2×D244—42no—
162C2×D244#4112yes—
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—22no—
173C2×C4×D61—22no—
174C2×C4×D61—22no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61C2×D412yes—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—42no—
190C4⋊C4×C61#4112yes—
191C42×C61—21no—
192C42×C61—21no—
193#11758201—28no—
194#11758198—24no—
195#11758198—24no—
196#11758198—42no—
197#11758198—42no—
198#11758198—42no—
199#11758198—42no—
200#11758198—42no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—42no—
207C4⋊Dic61#4112yes—
208#11758196—24no—
209#11758196—42no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61—42no—
213C4×Dic61—42no—
214C4×Dic61—42no—
215C4×Dic61C2×D412yes—
216C4×Dic61#4112yes—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418C2×D412yes—
223C23×C61C22×D411yescharacteristic
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—22no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61#13512yes—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—41no—
241Q8×C61—41no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
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—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61#13512yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61#12112yes—
281C2×Dic61#12112yes—
282C2×Dic61AGL(1,ℤ/8)12yes—
283C2×Dic61#13712yes—
284D244—22no—
285D244—22no—
286D244—48no—
287D244—41no—
288D244—41no—
289C4×D61—28no—
290C4×D61—41no—
291C4×D61—41no—
292C4×D61—41no—
293C4×D61—41no—
294C4×D61—44no—
295C4×D61—84no—
296C8×C61—22no—
297C8×C61—41no—
298#92418—22no—
299#92418—41no—
300#23701—611nomaximal, Sylow
301C22×C61C2×#12111yescharacteristic
302C22×C61#52511yescharacteristic
303C22×C61#52511yescharacteristic
304C22×C61C2×#13511yescharacteristic
305C22×C61C2×#13511yescharacteristic
306C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
307C22×C61C2×#13711yescharacteristic
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—42no—
324C4×C61—42no—
325C4×C61—41no—
326C4×C61—81no—
327C4×C61—81no—
328C4×C61#52511yescharacteristic
329C4×C61#52911yescharacteristic
330C4×C61#52911yescharacteristic
331C4×C61#53211yescharacteristic
332Dic61—22no—
333Dic61—42no—
334Dic61—41no—
335Dic61—81no—
336Dic61—81no—
337Dic61#46814yes—
338C2×C4×SD16—611no—
339C2×#431—611no—
340C4×C22⋊Q8—611no—
341C4×#121—611no—
342#2088—611no—
343#2077—611no—
344#2068D6111yescharacteristic, complemented
345#2067—611no—
346#2055—611no—
347#2036—611no—
348#2033—611no—
349#2000—611no—
350C4×#107—611no—
351C4×#103—611no—
352C4×#102—611no—
353C2×C61#239211yescharacteristic
354C2×C61#305011yescharacteristic
355C2×C61C2×#46811yescharacteristic
356C2×C61#310511yescharacteristic
357C2×C61#310611yescharacteristic
358C2×C61#310711yescharacteristic
359C2×C61#310811yescharacteristic
360D61—42no—
361D61—82no—
362C22×SD16—611no—
363C2×C22⋊Q8—611no—
364C2×#121—611no—
365C2×C4×Q8—611no—
366C2×C4×D4—611no—
367C2×C4×D4—1221no—
368C22×C42—611no—
369C4×SD16—1224no—
370#431—1224no—
371C2×#107—611no—
372C2×#107C2×D6111yescharacteristic
373C2×#106—611no—
374C2×C4⋊C8—611no—
375C2×C4⋊C8C2×D6111yescharacteristic
376C2×#103—611no—
377C2×#103C2×D6111yescharacteristic
378C2×#103C2×D6111yescharacteristic
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×C22⋊C8—611no—
383C2×C22⋊C8—611no—
384C2×C4×C8—611no—
385C2×C4×C8—611no—
386#391C2×D6111yescharacteristic
387#390—611no—
388#389—611no—
389#389C2×D6111yescharacteristic
390#388—1221no—
391#387—1221no—
392#386—1221no—
393#385—1221no—
394#380—611no—
395#380—611no—
396C4×C4⋊C4—611no—
397C4×C4⋊C4—611no—
398C4×C4⋊C4—1221no—
399C4×C22⋊C4—1221no—
400C4×C22⋊C4—1221no—
401#340—611no—
402#340—611no—
403#340—611no—
404#340—611no—
405#340—611no—
406#340C2×D6111yescharacteristic
407C61#2370111yescharacteristic, complemented, Sylow
408C22×Q8C22×D6111yescharacteristic
409C22×D4—611no—
410C22×D4—1221no—
411C23×C4—611no—
412C23×C4—611no—
413C23×C4—611no—
414C23×C4—1221no—
415C2×SD16—1222no—
416C2×SD16—1222no—
417C2×SD16—1222no—
418C22×C8—611no—
419C22×C8—611no—
420C22×C8C22×D6111yescharacteristic
421C22×C8C22×D6111yescharacteristic
422C22⋊Q8—1222no—
423C22⋊Q8—1222no—
424#121—1222no—
425#121—1222no—
426C4×Q8—1222no—
427C4×Q8—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C4×D4—2444no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—1222no—
452C2×C42—1222no—
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42C22×D6111yescharacteristic
457#107—22no—
458#107—1222no—
459#106—1222no—
460C4⋊C8—22no—
461C4⋊C8—1222no—
462#103—22no—
463#103—22no—
464#103—1222no—
465#102—1222no—
466#102—1222no—
467#102—1222no—
468C22⋊C8—1222no—
469C22⋊C8—1222no—
470C4×C8—614no—
471C4×C8—1222no—
472#95—21no—
473#95—21no—
474#95—21no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480C24—611no—
481C24—1221no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8#1175822612yes—
488C2×D4—612no—
489C2×D4—1224no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—2442no—
496C2×D4—2442no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—1222no—
505C22×C4—1222no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—2442no—
520C22×C4—2442no—
521C22×C4—2442no—
522C22×C4D4×D6111yescharacteristic
523C22×C4D4×D6111yescharacteristic
524C22×C4C23×D6111yescharacteristic
525SD16—1228no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—612no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8D4×D6112yes—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—42no—
544C4⋊C4—612no—
545C4⋊C4—1224no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—2442no—
552C4⋊C4—2442no—
553C4⋊C4#1175822412yes—
554C22⋊C4—1224no—
555C22⋊C4—1224no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C22⋊C4—2442no—
560C22⋊C4—2442no—
561C42—21no—
562C42—21no—
563C42—612no—
564C42—612no—
565C42—1222no—
566C42—1222no—
567C42—2442no—
568C42—2442no—
569C42—2442no—
570C23—612no—
571C23—1221no—
572C23—1221no—
573C23—1221no—
574C23—1221no—
575C23—1221no—
576C23—1221no—
577C23—1222no—
578C23—2441no—
579C23—2441no—
580C23—2441no—
581C23—2441no—
582C23—2441no—
583C23—2441no—
584C23C2×D4×D6111yescharacteristic
585Q8—22no—
586Q8—22no—
587Q8—41no—
588Q8—41no—
589D4—1222no—
590D4—1222no—
591D4—2448no—
592D4—2441no—
593D4—2441no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—41no—
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—612no—
612C2×C4—612no—
613C2×C4—612no—
614C2×C4—612no—
615C2×C4—1228no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1222no—
621C2×C4—1221no—
622C2×C4—4884no—
623C2×C4—2444no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
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#1451980012yes—
637C2×C4C2×D4×D6111yescharacteristic
638C2×C4#41×D6111yescharacteristic
639C2×C4#41×D6111yescharacteristic
640C2×C4#41×D6111yescharacteristic
641C2×C4C2×#1175822411yescharacteristic
642C2×C4C2×#1175822611yescharacteristic
643C8—22no—
644C8—41no—
645C8—1222no—
646C8—2441no—
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—
661C22#121×D6111yescharacteristic
662C22#1684027211yescharacteristic
663C22#1684027511yescharacteristic
664C22#135×D6111yescharacteristic
665C22C2×#1451980011yescharacteristic
666C22AGL(1,ℤ/8)×D6111yescharacteristic
667C22#137×D6111yescharacteristic
668C4—22no—
669C4—41no—
670C4—42no—
671C4—42no—
672C4—81no—
673C4—81no—
674C4—614no—
675C4—1222no—
676C4—4881no—
677C4—4881no—
678C4—2442no—
679C4—2441no—
680C4#1684027511yescharacteristic
681C4#1684029511yescharacteristic
682C4#1684029811yescharacteristic
683C4#1684030711yescharacteristic
684C2—4882no—
685C2—2442no—
686C2#2144906411yescharacteristic
687C2#2145540211yescharacteristic
688C2#468×D6111yescharacteristic
689C2#2145586511yescharacteristic
690C2#2145588211yescharacteristic
691C2#2145588911yescharacteristic
692C2#2145590411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2068C2
socleC23×C61C22×D4
prime radicalC2×#14519752C22
hypercenter#2068D61
hypocenterC61#23701
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2068

Complements

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

NH ≅ G/Ntype
1. C2×#16839847C2semidirect
2. C2×#16839814C2semidirect
3. #21448357C2semidirect
5. #21447321C2semidirect
6. #21447258C2semidirect
7. #21447206C2semidirect
8. #21447134C2semidirect
9. #21447021C2semidirect
12. #107×Dic61C2semidirect
13. #103×Dic61C2semidirect
14. #21445633C2semidirect
15. C61×#2068C2semidirect
112. C8×Dic61D4semidirect
344. #2068D61semidirect
407. C61#23701semidirect

Maximal subgroups

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

subgroupindex
C2×#168398472
C2×#168398142
#214483572
#214483272
#214473212
#214472582
#214472062
#214471342
#214470212
#214470032
#214468212
#107×Dic612
#103×Dic612
#214456332
C61×#20682
#2370161