Finite Group Databasebeta

← #10819953

Subgroups — #10819953

Order 768.

892 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2949C211yesmaximal, characteristic
2C2×#5167C211yesmaximal, characteristic
3#82549C211yesmaximal, characteristic, complemented
4#82543C211yesmaximal, characteristic
5#79169C211yesmaximal, characteristic, complemented
6#72052C211yesmaximal, characteristic
7#71660C211yesmaximal, characteristic, complemented
8#71649C211yesmaximal, characteristic
9#70660C211yesmaximal, characteristic, complemented
10#70345C211yesmaximal, characteristic, complemented
11#70336C211yesmaximal, characteristic, complemented
12#70325C211yesmaximal, characteristic
13#70315C211yesmaximal, characteristic
14#69904C211yesmaximal, characteristic, complemented
15#69890C211yesmaximal, characteristic, complemented
16#23203—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C3×#519C2211yescharacteristic
19C3×#518C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C2×#936C2211yescharacteristic
23C22×#214C2211yescharacteristic
24C22×Dic12C2211yescharacteristic
25#5516C2211yescharacteristic
26#5516C2211yescharacteristic
27C3×#440—22no—
28C3×C4×SD16—22no—
29C3×#427C2211yescharacteristic
30C3×#419C2211yescharacteristic
31C3×#418C2211yescharacteristic
32C3×#405C2211yescharacteristic
33C2×#836C2211yescharacteristic
34#5225C2211yescharacteristic
35C2×#833C2211yescharacteristic
36#5215C2211yescharacteristic
37#5167C418yescomplemented
38C2×#824C2211yescharacteristic
39C2×#824C2211yescharacteristic
40C2×#822C2211yescharacteristic
41C2×#819C2211yescharacteristic
42#5101C2211yescharacteristic
43C2×#815C2211yescharacteristic
44#5030—22no—
45C4×#216—22no—
46#5018—22no—
47C4×#214—22no—
48#5004C2211yescharacteristic
49#5004C2211yescharacteristic
50#5004C2211yescharacteristic
51#5004C2211yescharacteristic
52#4998C2211yescharacteristic
53#4997C2211yescharacteristic
54#4997C2211yescharacteristic
55#4873—22no—
56#4853—22no—
57#4555C2211yescharacteristic
58#4532C2211yescharacteristic
59#4532C2211yescharacteristic
60#4532C2211yescharacteristic
61C2×#497—31no—
62#2950—31no—
63#2950—31no—
64#2949—31no—
65#2949D311yescharacteristic
66#1979—31no—
67#1979—31no—
68#1977—31no—
69#1977—31no—
70#1888—31no—
71#1886—31no—
72#1886—31no—
73#1885—31no—
74#1773—31no—
75#1773—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C2×C3×SD16—21no—
81C2×C3×SD16—21no—
82C2×C3×SD16—22no—
83C2×C3×SD16C2×C414yes—
84C2×C3×M16C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×#117—41no—
91C3×#117C2311yescharacteristic
92C3×#117C2311yescharacteristic
93C3×#117C2311yescharacteristic
94C2×C3×C22⋊C4—41no—
95C2×#217—41no—
96#944—41no—
97#944C2×C414yescomplemented
98C2×#216—21no—
99C2×#216—21no—
100C2×#216—22no—
101C2×#216C2×C414yescomplemented
102#936—41no—
103#936C2×C414yes—
104C2×#214—21no—
105C2×#214—21no—
106C2×#214—22no—
107C2×#214C2×C414yes—
108#926C2311yescharacteristic
109C2×Dic3⋊C4—41no—
110C2×C4×Dic3C2311yescharacteristic
111C2×#208C2311yescharacteristic
112C2×#208C2311yescharacteristic
113C2×#208C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C2×Dic12—21no—
116C2×Dic12—21no—
117C2×Dic12—22no—
118C2×Dic12C2×C414yescomplemented
119#887—22no—
120#887—22no—
121#877—41no—
122#877—41no—
123C4×Dic6—22no—
124C4×Dic6—22no—
125C3×#107D412yes—
126C3×#106D412yes—
127C3×#103—22no—
128C3×#102—22no—
129C3×#97—21no—
130C3×C4×C8—21no—
131#836—41no—
132#835—22no—
133#833—41no—
134#832—22no—
135#824—41no—
136#824—41no—
137#822—22no—
138#819—22no—
139#815—21no—
140#815C2×C414yescomplemented
141C8×Dic3—21no—
142#810—22no—
143#810—22no—
144#810—22no—
145#810—22no—
146#808D412yes—
147#807D412yes—
148#803—21no—
149#803—21no—
150C4×C3⋊C8—21no—
151C4×C3⋊C8—21no—
152C22×Q16—31no—
153C22×Q16—31no—
154C22×SD16—31no—
155C22×SD16C2×D311yescharacteristic
156C2×C4⋊Q8—31no—
157C2×#124—31no—
158#519—31no—
159#519—31no—
160#519C2×D311yescharacteristic
161#518C2×D311yescharacteristic
162#497—38no—
163#441—62no—
164#441—62no—
165#440—22no—
166#440—62no—
167C4×Q16—62no—
168C4×Q16—62no—
169C4×SD16—22no—
170C4×SD16—62no—
171C2×#108—31no—
172C2×#108—31no—
173#427—31no—
174#427C2×D311yescharacteristic
175#419—31no—
176#419—31no—
177#419—31no—
178#419—31no—
179#419—31no—
180#419C2×D311yescharacteristic
181#418—31no—
182#418C2×D311yescharacteristic
183C2×#101—31no—
184C2×#101—31no—
185C2×#101—31no—
186C2×#100—31no—
187#405—31no—
188#405—31no—
189#405—31no—
190#405C2×D311yescharacteristic
191C2×#97—31no—
192#339—31no—
193#339—31no—
194#339—31no—
195#339—31no—
196C24×C3—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8C22×C412yes—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4C22×C412yes—
209C22×C3×C4—21no—
210C22×C3×C4—42no—
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6C22×C412yes—
226C2×Dic6C22×C412yes—
227C3×SD16—24no—
228C3×SD16—24no—
229C3×M16—21no—
230C3×M16—21no—
231C2×C3×C8—21no—
232C2×C3×C8C22×C412yes—
233C2×C3×C8C2×D411yescharacteristic
234C2×C3×C8C2×D411yescharacteristic
235C2×C3×C8#4112yes—
236C3×C4⋊C4—42no—
237C3×C4⋊C4C2×D412yes—
238C3×C4⋊C4C2×D412yes—
239C3×C4⋊C4C2×D412yes—
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—21no—
243C3×C22⋊C4—42no—
244C3×C42—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—42no—
248#217—21no—
249#217—44no—
250#216—24no—
251#216—24no—
252#214—24no—
253#214—24no—
254C4⋊Dic3C2×D412yes—
255Dic3⋊C4—44no—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—42no—
261C4×Dic3—42no—
262C4×Dic3C22×C412yes—
263#208—21no—
264#208—21no—
265#208—21no—
266#208—21no—
267#208—21no—
268#208—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8C22×C412yes—
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8#4112yes—
276Dic12—24no—
277Dic12—24no—
278C22×Q8—31no—
279C22×Q8—31no—
280C22×Q8C22×D311yescharacteristic
281C22×D4C22×D311yescharacteristic
282C2×Q16—34no—
283C2×Q16—34no—
284C2×Q16—62no—
285C2×Q16—61no—
286C2×Q16—61no—
287C2×Q16—62no—
288C2×Q16—61no—
289C2×Q16—61no—
290C2×SD16—22no—
291C2×SD16—21no—
292C2×SD16—21no—
293C2×SD16—34no—
294C2×SD16—62no—
295C2×SD16—61no—
296C2×SD16—61no—
297C2×SD16C4×D314yes—
298C2×M16—31no—
299C2×M16—31no—
300C2×M16—31no—
301C2×M16C22×D311yescharacteristic
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304C4⋊Q8—34no—
305C4⋊Q8—121no—
306#124—34no—
307#124—121no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—62no—
311C4×Q8—62no—
312C4×Q8—62no—
313C4×Q8—62no—
314C4×D4—22no—
315C4×D4—22no—
316#117—31no—
317#117—41no—
318#117—121no—
319#117—121no—
320#117C22×D311yescharacteristic
321#117C22×D311yescharacteristic
322#117C22×D311yescharacteristic
323C2×C4⋊C4—121no—
324C2×C22⋊C4—41no—
325C2×C22⋊C4—121no—
326C2×C42—31no—
327#108—62no—
328#108—62no—
329#107—32no—
330#107#6712yes—
331#106—32no—
332#106#6712yes—
333#103—22no—
334#103—62no—
335#103—62no—
336#103—62no—
337#103—62no—
338#103—62no—
339#102—22no—
340#102—62no—
341#101—121no—
342#101—121no—
343#101—121no—
344#100—121no—
345#97—21no—
346#97—34no—
347#97—61no—
348#97—61no—
349#97—61no—
350C4×C8—21no—
351C4×C8—61no—
352C4×C8—61no—
353C4×C8—61no—
354C23×C3—22no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3—82no—
358C23×C3H(5,𝔽2)11yescharacteristic
359C3×Q8—22no—
360C3×Q8—22no—
361C3×Q8—41no—
362C3×Q8—41no—
363C3×D4—22no—
364C3×D4—22no—
365C3×D4—41no—
366C3×D4—41no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—162no—
374C2×C3×C4C23×C411yescharacteristic
375C2×C3×C4C22×D411yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4H(5,𝔽2)11yescharacteristic
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—42no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389Dic6—22no—
390Dic6—22no—
391Dic6—22no—
392Dic6—22no—
393Dic6—41no—
394Dic6—41no—
395Dic6—41no—
396Dic6—41no—
397C3×C8—41no—
398C3×C8C4×D412yes—
399C3×C8C4×D412yes—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C3⋊C8C4×D412yes—
404C3⋊C8C4×D412yes—
405C24—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—32no—
412C2×Q8—32no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8C2×C4×D312yes—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4C2×C4×D312yes—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—61no—
433C22×C4—61no—
434C22×C4—61no—
435C22×C4C23×D311yescharacteristic
436Q16—64no—
437Q16—64no—
438Q16—64no—
439Q16—64no—
440SD16—24no—
441SD16—24no—
442SD16—64no—
443SD16—64no—
444M16—21no—
445M16—21no—
446M16—61no—
447M16—61no—
448M16—61no—
449M16—61no—
450M16—61no—
451M16—61no—
452C2×C8—21no—
453C2×C8—32no—
454C2×C8—31no—
455C2×C8—31no—
456C2×C8—32no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8C2×C4×D312yes—
461C2×C8#23512yes—
462C2×C8C2×#6711yescharacteristic
463C2×C8C2×#6711yescharacteristic
464C4⋊C4—32no—
465C4⋊C4—42no—
466C4⋊C4—124no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4D3×D412yes—
470C4⋊C4C2×#6712yes—
471C4⋊C4C2×#6712yes—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—21no—
475C22⋊C4—42no—
476C22⋊C4—61no—
477C22⋊C4—124no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—32no—
482C42—42no—
483C42—61no—
484C42—61no—
485C42—122no—
486C42—122no—
487C22×C3—22no—
488C22×C3—42no—
489C22×C3—82no—
490C22×C3—82no—
491C22×C3—161no—
492C22×C3#51811yescharacteristic
493C22×C3D4211yescharacteristic
494C22×C3#55911yescharacteristic
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4C2×C4×D411yescharacteristic
500C3×C4C2×C4×D411yescharacteristic
501C3×C4#55011yescharacteristic
502C3×C4#55011yescharacteristic
503Dic3—42no—
504Dic3—42no—
505Dic3—42no—
506Dic3—42no—
507C23—22no—
508C23—41no—
509C23—41no—
510C23—82no—
511C23#100411yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516Q8—62no—
517Q8—62no—
518Q8—62no—
519Q8—62no—
520Q8—121no—
521Q8—121no—
522Q8—121no—
523Q8—121no—
524D4—22no—
525D4—22no—
526D4—41no—
527D4—41no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—122no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—162no—
544C2×C4C22×C4×D311yescharacteristic
545C2×C4C2×#23511yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4D3×#4111yescharacteristic
548C2×C4#100911yescharacteristic
549C2×C4C22×#6711yescharacteristic
550C8—32no—
551C8—32no—
552C8—41no—
553C8—121no—
554C8—121no—
555C8—121no—
556C8C4×#6712yes—
557C8C4×#6712yes—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—82no—
561C2×C3#236611yescharacteristic
562C2×C3#328912yes—
563C22—22no—
564C22—42no—
565C22—82no—
566C22—82no—
567C22—161no—
568C22#554511yescharacteristic
569C22#555911yescharacteristic
570C22D4×#6711yescharacteristic
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—122no—
576C4—122no—
577C4—122no—
578C4—122no—
579C4C4×D3×D411yescharacteristic
580C4#555311yescharacteristic
581C4C2×C4×#6711yescharacteristic
582C4#583411yescharacteristic
583C3#2320311yescharacteristic, complemented, Sylow
584C2—21no—
585C2—21no—
586C2—82no—
587C2#8234311yescharacteristic
588C2#8401312yes—
subgroupG/H
centerC22D4×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2949C2
socleC22×C3D42
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#2949D3
hypocenterC3#23203
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2949

Complements

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

NH ≅ G/Ntype
3. #82549C2semidirect
5. #79169C2semidirect
7. #71660C2semidirect
9. #70660C2semidirect
10. #70345C2semidirect
11. #70336C2semidirect
14. #69904C2semidirect
15. #69890C2semidirect
37. #5167C4semidirect
97. #944C2×C4semidirect
101. C2×#216C2×C4semidirect
118. C2×Dic12C2×C4semidirect
140. #815C2×C4semidirect
583. C3#23203semidirect

Maximal subgroups

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

subgroupindex
C3×#29492
C2×#51672
#825492
#825432
#791692
#720522
#716602
#716492
#706602
#703452
#703362
#703252
#703152
#699042
#698902
#232033