Finite Group Databasebeta

← #33138136

Subgroups — #33138136

Order 18688.

648 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620480C211yesmaximal, characteristic, complemented
2#22619728C211yesmaximal, characteristic
3#22618874C211yesmaximal, characteristic, complemented
4#22611442C211yesmaximal, characteristic, complemented
5#22611437C211yesmaximal, characteristic, complemented
6#22611289C211yesmaximal, characteristic
7#22611289C211yesmaximal, characteristic
8#22610925C211yesmaximal, characteristic, complemented
9#22610714C211yesmaximal, characteristic, complemented
10#22610709C211yesmaximal, characteristic, complemented
11#22610707C211yesmaximal, characteristic, complemented
12#22610590C211yesmaximal, characteristic
13#22610590C211yesmaximal, characteristic
14C4×#14556508C211yesmaximal, characteristic
15C2×#478×C73C211yesmaximal, characteristic
16#17989144C2211yescharacteristic
17#17988997—21no—
18#17988997—21no—
19#17988986—21no—
20#17988986—21no—
21#17988887—21no—
22#17988887—21no—
23#17988877—21no—
24#17988867—21no—
25#17988716—21no—
26#17988716—21no—
27#17988714—21no—
28#17988711—21no—
29C2×#14556527C2211yescharacteristic
30C2×#14556527C2211yescharacteristic
31#17988627C2211yescharacteristic
32#17988627C2211yescharacteristic
33#17988627C2211yescharacteristic
34#17988627C2211yescharacteristic
35#17988627C2211yescharacteristic
36#17988627C2211yescharacteristic
37#17988613C2211yescharacteristic
38#17988613C2211yescharacteristic
39#17988611C2211yescharacteristic
40#17988592C2211yescharacteristic
41#17988592C2211yescharacteristic
42C2×#14556508C2211yescharacteristic
43C2×#14556508C2211yescharacteristic
44#17988542C2211yescharacteristic
45#17988542C2211yescharacteristic
46C4×#11924012C2211yescharacteristic
47C4×#11924012C2211yescharacteristic
48#17988356C2211yescharacteristic
49#17988356C2211yescharacteristic
50#17988356C2211yescharacteristic
51#17988356C2211yescharacteristic
52#17988356C2211yescharacteristic
53#17988356C2211yescharacteristic
54#17988352C2211yescharacteristic
55#17988352C2211yescharacteristic
56C2×#125×C73C2211yescharacteristic
57C2×C4×D4×C73C2211yescharacteristic
58#478×C73—21no—
59#478×C73—21no—
60#478×C73—21no—
61#478×C73—21no—
62C2×#107×C73C2211yescharacteristic
63C2×#106×C73C2211yescharacteristic
64C2×C4⋊C8×C73C2211yescharacteristic
65C2×#102×C73C2211yescharacteristic
66C2×#102×C73C2211yescharacteristic
67C2×#11924012C2311yescharacteristic
68C2×#11924012C2311yescharacteristic
69C2×#11924012C2311yescharacteristic
70C2×#11924012C2311yescharacteristic
71C2×#11924012C2311yescharacteristic
72C2×#11924012C2311yescharacteristic
73#14556591D411yescharacteristic
74#14556591D411yescharacteristic
75#14556583D411yescharacteristic
76#14556583D411yescharacteristic
77#14556570—21no—
78#14556570—21no—
79C4×Dic146—21no—
80C4×Dic146—21no—
81#14556550C2311yescharacteristic
82#14556550C2311yescharacteristic
83#14556527—22no—
84#14556527—22no—
85#14556524—21no—
86#14556524—21no—
87#14556524—21no—
88#14556524—21no—
89#14556524—21no—
90#14556524—21no—
91#14556522—21no—
92#14556522—21no—
93#14556522—21no—
94#14556522—21no—
95#14556522—21no—
96#14556522—21no—
97C4⋊#10596831—21no—
98C4⋊#10596831—21no—
99#14556520—21no—
100#14556520—21no—
101#14556520—21no—
102#14556520—21no—
103#14556512—21no—
104#14556512—21no—
105#14556512—21no—
106#14556512—21no—
107#14556508—22no—
108#14556508Q814yescomplemented
109#14556502—21no—
110#14556502—21no—
111#14556501—21no—
112#14556501—21no—
113C4×#10596831—21no—
114C4×#10596831—21no—
115C8×Dic73—21no—
116C8×Dic73—21no—
117C22×D4×C73C2311yescharacteristic
118C23×C4×C73—21no—
119C22×C8×C73C2311yescharacteristic
120C22×C8×C73C2311yescharacteristic
121#125×C73—21no—
122#125×C73—21no—
123#125×C73—21no—
124#125×C73—21no—
125C4×D4×C73—21no—
126C4×D4×C73—21no—
127C4×D4×C73—21no—
128C4×D4×C73—21no—
129C2×C4⋊C4×C73—21no—
130C2×C4⋊C4×C73—21no—
131C2×C4⋊C4×C73C2311yescharacteristic
132C2×C4⋊C4×C73C2311yescharacteristic
133C2×C4⋊C4×C73C2311yescharacteristic
134C2×C22⋊C4×C73—21no—
135C2×C42×C73C2311yescharacteristic
136#107×C73—21no—
137#107×C73—21no—
138#106×C73—21no—
139#106×C73—21no—
140C4⋊C8×C73D411yescharacteristic
141C4⋊C8×C73D411yescharacteristic
142C4⋊C8×C73D411yescharacteristic
143C4⋊C8×C73D411yescharacteristic
144#102×C73—21no—
145#102×C73—21no—
146#102×C73—21no—
147#102×C73—21no—
148C2×Dic146—21no—
149C2×Dic146—21no—
150#11924012—21no—
151#11924012—21no—
152#11924012—21no—
153#11924012—21no—
154#11924012—21no—
155#11924012—21no—
156#11924012—21no—
157#11924012—21no—
158#11924012—21no—
159#11924012C2×Q812yes—
160#11924012C2×Q812yes—
161#11924012#4112yes—
162C2×#10596831—21no—
163C2×#10596831—21no—
164C2×#10596831—21no—
165C2×#10596831—21no—
166C2×#10596831—21no—
167C2×#10596831—21no—
168C4⋊Dic73—21no—
169C4⋊Dic73—21no—
170C4⋊Dic73C2×D411yescharacteristic
171C4⋊Dic73C2×D411yescharacteristic
172C4⋊Dic73C2×D411yescharacteristic
173C4⋊Dic73C2×D411yescharacteristic
174#11924007—21no—
175#11924007—21no—
176#11924007—21no—
177#11924007—21no—
178C22⋊Dic73—21no—
179C22⋊Dic73—21no—
180C4×Dic73—21no—
181C4×Dic73—21no—
182C24×C73—21no—
183C2×D4×C73—21no—
184C2×D4×C73—21no—
185C2×D4×C73—21no—
186C2×D4×C73—21no—
187C2×D4×C73—21no—
188C2×D4×C73C2×Q812yes—
189C22×C4×C73—21no—
190C22×C4×C73—21no—
191C22×C4×C73—21no—
192C22×C4×C73—41no—
193C22×C4×C73—41no—
194C22×C4×C73—82no—
195C22×C4×C73#4111yescharacteristic
196C22×C4×C73#4111yescharacteristic
197C22×C4×C73C2411yescharacteristic
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—21no—
202C2×C8×C73C2×D411yescharacteristic
203C2×C8×C73C2×D411yescharacteristic
204C2×C8×C73C2×D411yescharacteristic
205C2×C8×C73C2×D411yescharacteristic
206C4⋊C4×C73—21no—
207C4⋊C4×C73—21no—
208C4⋊C4×C73—21no—
209C4⋊C4×C73—21no—
210C4⋊C4×C73—21no—
211C4⋊C4×C73—21no—
212C4⋊C4×C73—41no—
213C4⋊C4×C73—41no—
214C4⋊C4×C73—41no—
215C4⋊C4×C73—41no—
216C22⋊C4×C73—41no—
217C22⋊C4×C73—41no—
218C42×C73C2×D411yescharacteristic
219C42×C73C2×D411yescharacteristic
220C42×C73C2×D411yescharacteristic
221C42×C73C2×D411yescharacteristic
222C23×C73—22no—
223C23×C73—41no—
224C23×C73—41no—
225C23×C73—81no—
226C23×C73—81no—
227C23×C73C2×#4111yescharacteristic
228D4×C73—22no—
229D4×C73—22no—
230D4×C73—41no—
231D4×C73—41no—
232C2×C4×C73—41no—
233C2×C4×C73—41no—
234C2×C4×C73—41no—
235C2×C4×C73—41no—
236C2×C4×C73—41no—
237C2×C4×C73—82no—
238C2×C4×C73—82no—
239C2×C4×C73—81no—
240C2×C4×C73—81no—
241C2×C4×C73—81no—
242C2×C4×C73#12111yescharacteristic
243C2×C4×C73#12111yescharacteristic
244C2×C4×C73#12111yescharacteristic
245C2×C4×C73#12111yescharacteristic
246C2×C4×C73C22×D411yescharacteristic
247C2×C4×C73C22×D411yescharacteristic
248C2×C4×C73C22×Q811yescharacteristic
249C2×C4×C73C2×#4111yescharacteristic
250C2×C4×C732−1+411yescharacteristic
251C2×C4×C732−1+411yescharacteristic
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256Dic146—42no—
257Dic146—81no—
258C8×C73—21no—
259C8×C73—21no—
260C8×C73—21no—
261C8×C73—21no—
262#10596831—22no—
263#10596831—22no—
264#10596831—22no—
265#10596831—41no—
266#10596831—41no—
267#10596831—41no—
268C22×C73—21no—
269C22×C73—21no—
270C22×C73—42no—
271C22×C73—82no—
272C22×C73—82no—
273C22×C73—81no—
274C22×C73—81no—
275C22×C73C2×#12111yescharacteristic
276C22×C73#53311yescharacteristic
277C22×C73#54111yescharacteristic
278C4×C73—21no—
279C4×C73—21no—
280C4×C73—21no—
281C4×C73—21no—
282C4×C73—81no—
283C4×C73—81no—
284C4×C73—81no—
285C4×C73#54711yescharacteristic
286C4×C73#54811yescharacteristic
287C4×C73D4×Q811yescharacteristic
288C4×C73D4×Q811yescharacteristic
289Dic73—41no—
290Dic73—41no—
291Dic73—41no—
292Dic73—41no—
293#25493—731nomaximal, Sylow
294C2×C73—21no—
295C2×C73—21no—
296C2×C73—82no—
297C2×C73#242911yescharacteristic
298C2×C73#306711yescharacteristic
299C2×C73#306811yescharacteristic
300#3166—731no—
301#3077—731no—
302C2×#478D7311yescharacteristic
303#2962—731no—
304#2090—731no—
305#2089—731no—
306#2063—731no—
307#2063—731no—
308#1998—731no—
309#1952—731no—
310#1951—731no—
311#1950—731no—
312#1931—731no—
313#1931—731no—
314C4×#100—731no—
315C73#2549311yescharacteristic, complemented, Sylow
316#533—731no—
317C2×#125C2×D7311yescharacteristic
318C2×C4×D4C2×D7311yescharacteristic
319#501—1461no—
320#501—1461no—
321#499—1461no—
322#499—1461no—
323#478—21no—
324#478—21no—
325#478—21no—
326#478—21no—
327#478—1461no—
328#478—1461no—
329#476—1461no—
330#474—1461no—
331#446—1461no—
332#446—1461no—
333#446—1461no—
334C8×Q8—1461no—
335C2×#108—731no—
336C2×#108—731no—
337#428—731no—
338#428—731no—
339#428—731no—
340#428—731no—
341#428—731no—
342#428—731no—
343C2×#107C2×D7311yescharacteristic
344C2×#106C2×D7311yescharacteristic
345#424—731no—
346#424—731no—
347#423—731no—
348C2×C4⋊C8C2×D7311yescharacteristic
349#418—731no—
350#418—731no—
351C2×#102C2×D7311yescharacteristic
352C2×#102C2×D7311yescharacteristic
353C2×#100—731no—
354C2×#100—731no—
355#405—731no—
356#405—731no—
357C4×M16—731no—
358C4×M16—731no—
359#343—731no—
360#343—731no—
361#343—731no—
362#343—731no—
363#343—731no—
364#343—731no—
365#341—731no—
366#341—731no—
367C22×D4C22×D7311yescharacteristic
368C23×C4—21no—
369C2×M16—731no—
370C2×M16—731no—
371C2×M16—731no—
372C2×M16—731no—
373C2×M16—731no—
374C2×M16—731no—
375C22×C8C22×D7311yescharacteristic
376C22×C8C22×D7311yescharacteristic
377C4⋊Q8—731no—
378C4⋊Q8—731no—
379#125—21no—
380#125—21no—
381#125—21no—
382#125—21no—
383#125—731no—
384#125—731no—
385C22⋊Q8—1461no—
386C22⋊Q8—1461no—
387C4×Q8—1461no—
388C4×Q8—1461no—
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393#117—731no—
394#117—731no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4C22×D7311yescharacteristic
398C2×C4⋊C4C22×D7311yescharacteristic
399C2×C4⋊C4C22×D7311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C42C22×D7311yescharacteristic
402#108—1462no—
403#108—1462no—
404#107—21no—
405#107—21no—
406#107—1461no—
407#107—1461no—
408#107—1461no—
409#107—1461no—
410#107—1461no—
411#107—1461no—
412#106—21no—
413#106—21no—
414#106—1461no—
415#106—1461no—
416#106—1461no—
417#106—1461no—
418#106—1461no—
419#106—1461no—
420C4⋊C8—1461no—
421C4⋊C8—1461no—
422C4⋊C8—1461no—
423C4⋊C8—1461no—
424C4⋊C8—1461no—
425C4⋊C8—1461no—
426C4⋊C8C22⋊D7311yescharacteristic
427C4⋊C8C22⋊D7311yescharacteristic
428C4⋊C8C22⋊D7311yescharacteristic
429C4⋊C8C22⋊D7311yescharacteristic
430#102—21no—
431#102—21no—
432#102—21no—
433#102—21no—
434#102—1461no—
435#102—1461no—
436#102—1461no—
437#102—1461no—
438#100—734no—
439#100—1462no—
440#97—1461no—
441#97—1461no—
442#97—1461no—
443#97—1461no—
444C4×C8—1461no—
445C4×C8—1461no—
446C4×C8—1461no—
447C4×C8—1461no—
448C24—21no—
449C2×Q8—1461no—
450C2×Q8—1461no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4Q8×D7312yes—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—82no—
463C22×C4#1192402711yescharacteristic
464C22×C4#1192402711yescharacteristic
465C22×C4C23×D7311yescharacteristic
466M16—732no—
467M16—732no—
468M16—732no—
469M16—1461no—
470M16—1461no—
471M16—1461no—
472M16—1461no—
473M16—1461no—
474M16—1461no—
475M16—1461no—
476M16—1461no—
477M16—1461no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8C2×C22⋊D7311yescharacteristic
489C2×C8C2×C22⋊D7311yescharacteristic
490C2×C8C2×C22⋊D7311yescharacteristic
491C2×C8C2×C22⋊D7311yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—731no—
503C4⋊C4—731no—
504C4⋊C4—731no—
505C4⋊C4—731no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4—1461no—
511C4⋊C4—1461no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—1461no—
515C22⋊C4—1461no—
516C42—1461no—
517C42—1461no—
518C42D4×D7311yescharacteristic
519C42D4×D7311yescharacteristic
520C42C2×C22⋊D7311yescharacteristic
521C42C2×C22⋊D7311yescharacteristic
522C23—22no—
523C23—41no—
524C23—41no—
525C23—81no—
526C23—81no—
527C23C2×#1192402711yescharacteristic
528Q8—2922no—
529Q8—5841no—
530D4—22no—
531D4—22no—
532D4—41no—
533D4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—1461no—
545C2×C4—1461no—
546C2×C4—1461no—
547C2×C4—1461no—
548C2×C4#1455656211yescharacteristic
549C2×C4#1455656211yescharacteristic
550C2×C4#1455656211yescharacteristic
551C2×C4#1455656211yescharacteristic
552C2×C4C2×D4×D7311yescharacteristic
553C2×C4C22×C22⋊D7311yescharacteristic
554C2×C4C2×Q8×D7311yescharacteristic
555C2×C4#41×D7311yescharacteristic
556C2×C4#1455663811yescharacteristic
557C2×C4#1455663911yescharacteristic
558C8—21no—
559C8—21no—
560C8—21no—
561C8—21no—
562C8—2921no—
563C8—2921no—
564C8—2921no—
565C8—1462no—
566C8—1462no—
567C8—1462no—
568C22—21no—
569C22—21no—
570C22—42no—
571C22—82no—
572C22—82no—
573C22—81no—
574C22—81no—
575C22C2×#1455656211yescharacteristic
576C22#1798913911yescharacteristic
577C22#1798919711yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—81no—
583C4—81no—
584C4—81no—
585C4—2921no—
586C4—2921no—
587C4—2921no—
588C4—2921no—
589C4#1798923611yescharacteristic
590C4#1798924711yescharacteristic
591C4#1798925011yescharacteristic
592C4Q8×C22⋊D7311yescharacteristic
593C2—21no—
594C2—21no—
595C2—82no—
596C2#2261352611yescharacteristic
597C2#2261964711yescharacteristic
598C2#2261965511yescharacteristic
subgroupG/H
centerC22C2×#14556562
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#478×C73C2
socleC22×C73C2×#121
prime radicalC22×D4×C73C23
hypercenterC2×#478D73
hypocenterC73#25493
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #22620480C2semidirect
3. #22618874C2semidirect
4. #22611442C2semidirect
5. #22611437C2semidirect
8. #22610925C2semidirect
9. #22610714C2semidirect
10. #22610709C2semidirect
11. #22610707C2semidirect
108. #14556508Q8semidirect
315. C73#25493semidirect

Maximal subgroups

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

subgroupindex
#226204802
#226197282
#226188742
#226114422
#226114372
#226112892
#226109252
#226107142
#226107092
#226107072
#226105902
C4×#145565082
C2×#478×C732
#2549373