Finite Group Databasebeta

← #10819957

Subgroups — #10819957

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
1C2×#5086C211yesmaximal, characteristic
2#82549C211yesmaximal, characteristic, complemented
3#82548C211yesmaximal, characteristic
4#78948C211yesmaximal, characteristic, complemented
5#78947C211yesmaximal, characteristic
6C3×#1886C211yesmaximal, characteristic, complemented
7#71784C211yesmaximal, characteristic, complemented
8#70336C211yesmaximal, characteristic, complemented
9#70335C211yesmaximal, characteristic
10#70108C211yesmaximal, characteristic, complemented
11#70107C211yesmaximal, characteristic
12#68766C211yesmaximal, characteristic
13#68759C211yesmaximal, characteristic
14#68742C211yesmaximal, characteristic, complemented
15#68742C211yesmaximal, characteristic, complemented
16#23203—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C3×#519C2211yescharacteristic
19#5812C2211yescharacteristic
20C22×#216C2211yescharacteristic
21C22×#216C2211yescharacteristic
22C22×#215C2211yescharacteristic
23C22×#215C2211yescharacteristic
24#5521C2211yescharacteristic
25#5516C2211yescharacteristic
26C2×#875C2211yescharacteristic
27C3×#419C2211yescharacteristic
28C3×#419C2211yescharacteristic
29C2×C3×#101C2211yescharacteristic
30C2×#836C2211yescharacteristic
31#5225C2211yescharacteristic
32#5225C2211yescharacteristic
33C2×#824C2211yescharacteristic
34#5131C2211yescharacteristic
35C2×#823C2211yescharacteristic
36C2×#822C2211yescharacteristic
37C2×#822C2211yescharacteristic
38#5122C2211yescharacteristic
39#5122C2211yescharacteristic
40#5120C2211yescharacteristic
41#5086C418yescomplemented
42#5030—22no—
43C4×#216—22no—
44#5026—22no—
45C4×#215—22no—
46#5004C2211yescharacteristic
47#5001C2211yescharacteristic
48#4998C2211yescharacteristic
49#4998C2211yescharacteristic
50#4997C2211yescharacteristic
51#4997C2211yescharacteristic
52C2×#803C2211yescharacteristic
53#4817—22no—
54#4790—22no—
55#4789—22no—
56#4788—22no—
57C3×#339C2211yescharacteristic
58C3×#339C2211yescharacteristic
59#4523C2211yescharacteristic
60#4523C2211yescharacteristic
61C2×#497—31no—
62#2950—31no—
63#2950—31no—
64#2949—31no—
65#2949—31no—
66#1979—31no—
67#1979—31no—
68#1977—31no—
69#1977—31no—
70#1888—31no—
71#1886—31no—
72#1886D311yescharacteristic, complemented
73#1885—31no—
74#1773—31no—
75#1773—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C2×C3×M16C2311yescharacteristic
81C2×C3×M16C2311yescharacteristic
82C3×C4⋊Q8—41no—
83C3×C4⋊Q8C2×C414yescomplemented
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×#117—41no—
87C3×#117C2311yescharacteristic
88C3×#117C2311yescharacteristic
89C2×C3×C4⋊C4—41no—
90C2×C3×C42C2311yescharacteristic
91Q8×Dic3—22no—
92Q8×Dic3—22no—
93C2×#216—21no—
94C2×#216—21no—
95C2×#216—21no—
96C2×#216—21no—
97C2×#216—22no—
98C2×#216—22no—
99C2×#216C2×C414yescomplemented
100C2×#216C2×C414yescomplemented
101C2×#215—21no—
102C2×#215—21no—
103C2×#215—21no—
104C2×#215—21no—
105C2×#215—22no—
106C2×#215—22no—
107C2×#215C2×C414yes—
108C2×#215C2×C414yes—
109C2×(C2×D3)⋊C4—41no—
110#926—41no—
111#926C2311yescharacteristic
112#926C2311yescharacteristic
113C2×#208C2311yescharacteristic
114C2×#208C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#893—22no—
118#887—22no—
119D3×C22⋊C4—41no—
120#877—41no—
121#875—41no—
122#875C2×C414yes—
123C4×D12—22no—
124C4×Dic6—22no—
125C3×#103—22no—
126C3×#103—22no—
127C3×#101—41no—
128#836—41no—
129#835—22no—
130#835—22no—
131#824—41no—
132#823—41no—
133#822—22no—
134#822—22no—
135#821—22no—
136#816—22no—
137#815—21no—
138#815—21no—
139C8×Dic3—21no—
140C8×Dic3—21no—
141#810—22no—
142#809—22no—
143#808D412yes—
144#808D412yes—
145#807D412yes—
146#807D412yes—
147#803—21no—
148#803—21no—
149#803C2×C414yescomplemented
150C4×C3⋊C8—21no—
151C4×C3⋊C8—21no—
152C22×Q16—31no—
153C22×Q16—31no—
154C22×SD16—31no—
155C22×SD16—31no—
156C2×C4⋊Q8C2×D311yescharacteristic
157C2×#124—31no—
158#519—31no—
159#519—31no—
160#519C2×D311yescharacteristic
161#518—31no—
162#497—38no—
163#441—62no—
164#441—62no—
165#440—62no—
166#440—62no—
167C4×Q16—62no—
168C4×Q16—62no—
169C4×SD16—62no—
170C4×SD16—62no—
171C2×#108—31no—
172C2×#108—31no—
173#427—31no—
174#427—31no—
175#419—31no—
176#419—31no—
177#419—31no—
178#419—31no—
179#419C2×D311yescharacteristic
180#419C2×D311yescharacteristic
181#418—31no—
182#418—31no—
183C2×#101—31no—
184C2×#101—31no—
185C2×#101C2×D311yescharacteristic
186C2×#100—31no—
187#405—31no—
188#405—31no—
189#405—31no—
190#405—31no—
191C2×#97—31no—
192#339—31no—
193#339—31no—
194#339C2×D311yescharacteristic
195#339C2×D311yescharacteristic
196C23×D3—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—21no—
207C2×C3×Q8C22×C412yes—
208C2×C3×Q8C22×C412yes—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12C22×C412yes—
220C2×C4×D3—42no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6C22×C412yes—
227C3×M16—21no—
228C3×M16—21no—
229C3×M16—21no—
230C3×M16—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C3×C4⋊C4—44no—
234C3×C4⋊C4—42no—
235C3×C4⋊C4C2×D412yes—
236C3×C4⋊C4C2×D412yes—
237C3×C22⋊C4—21no—
238C3×C22⋊C4—21no—
239C3×C42—21no—
240C3×C42—21no—
241C3×C42—21no—
242C3×C42—42no—
243C3×C42C22×C412yes—
244#217—21no—
245#217—21no—
246#216—24no—
247#216—24no—
248#216—24no—
249#216—24no—
250#215—24no—
251#215—24no—
252#215—24no—
253#215—24no—
254(C2×D3)⋊C4—44no—
255(C2×D3)⋊C4—42no—
256C4⋊Dic3—42no—
257C4⋊Dic3C2×D412yes—
258C4⋊Dic3C2×D412yes—
259Dic3⋊C4—42no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—42no—
263C4×Dic3—42no—
264#208—21no—
265#208—21no—
266#208—21no—
267#208—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8C22×C412yes—
271C2×C3⋊C8C22×C412yes—
272C2×C3⋊C8C2×D411yescharacteristic
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8#4112yes—
277C2×C3⋊C8#4112yes—
278C22×Q8—31no—
279C22×Q8C22×D311yescharacteristic
280C22×Q8C22×D311yescharacteristic
281C22×D4—31no—
282C2×Q16—34no—
283C2×Q16—34no—
284C2×Q16—62no—
285C2×Q16—62no—
286C2×Q16—61no—
287C2×Q16—61no—
288C2×Q16—61no—
289C2×Q16—61no—
290C2×SD16—34no—
291C2×SD16—34no—
292C2×SD16—62no—
293C2×SD16—62no—
294C2×SD16—61no—
295C2×SD16—61no—
296C2×SD16—61no—
297C2×SD16—61no—
298C2×M16—31no—
299C2×M16—31no—
300C2×M16C22×D311yescharacteristic
301C2×M16C22×D311yescharacteristic
302C22×C8—31no—
303C22×C8—31no—
304C4⋊Q8—41no—
305C4⋊Q8C4×D314yescomplemented
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—62no—
315C4×D4—62no—
316#117—31no—
317#117—31no—
318#117—41no—
319#117—121no—
320#117—121no—
321#117C22×D311yescharacteristic
322#117C22×D311yescharacteristic
323C2×C4⋊C4—41no—
324C2×C22⋊C4—121no—
325C2×C22⋊C4—121no—
326C2×C42C22×D311yescharacteristic
327#108—62no—
328#108—62no—
329#107—32no—
330#107—32no—
331#106—32no—
332#106—32no—
333#103—22no—
334#103—22no—
335#103—62no—
336#103—62no—
337#103—62no—
338#103—62no—
339#102—62no—
340#102—62no—
341#101—41no—
342#101—121no—
343#101—121no—
344#100—121no—
345#97—34no—
346#97—61no—
347#97—61no—
348#97—61no—
349#97—61no—
350C4×C8—61no—
351C4×C8—61no—
352C4×C8—61no—
353C4×C8—61no—
354C23×C3H(5,𝔽2)11yescharacteristic
355C22×D3—22no—
356C22×D3—41no—
357C22×D3—41no—
358C22×D3—82no—
359C3×Q8—22no—
360C3×Q8—22no—
361C3×Q8—22no—
362C3×Q8—22no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—41no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—22no—
372C2×C3×C4—42no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4C23×C411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388D12—22no—
389D12—22no—
390D12—41no—
391D12—41no—
392C4×D3—162no—
393Dic6—22no—
394Dic6—22no—
395Dic6—41no—
396Dic6—41no—
397C3×C8—41no—
398C3×C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C3⋊C8C4×D412yes—
402C3⋊C8C4×D412yes—
403C3⋊C8C4×D412yes—
404C3⋊C8C4×D412yes—
405C24—61no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8C2×C4×D312yes—
423C2×Q8C2×C4×D312yes—
424C2×D4—32no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—61no—
434C22×C4—122no—
435C22×C4C23×D311yescharacteristic
436Q16—64no—
437Q16—64no—
438Q16—64no—
439Q16—64no—
440SD16—64no—
441SD16—64no—
442SD16—64no—
443SD16—64no—
444M16—21no—
445M16—21no—
446M16—21no—
447M16—21no—
448M16—61no—
449M16—61no—
450M16—61no—
451M16—61no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—32no—
455C2×C8—32no—
456C2×C8—31no—
457C2×C8—31no—
458C2×C8—31no—
459C2×C8—31no—
460C2×C8—32no—
461C2×C8—32no—
462C2×C8—61no—
463C2×C8—61no—
464C4⋊C4—32no—
465C4⋊C4—32no—
466C4⋊C4—44no—
467C4⋊C4—42no—
468C4⋊C4—122no—
469C4⋊C4—122no—
470C4⋊C4D3×D412yes—
471C4⋊C4D3×D412yes—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—61no—
475C22⋊C4—61no—
476C22⋊C4—124no—
477C22⋊C4—122no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—42no—
482C42—61no—
483C42—61no—
484C42—122no—
485C42—122no—
486C42C2×C4×D312yes—
487C22×C3—22no—
488C22×C3#51811yescharacteristic
489C22×C3D4211yescharacteristic
490C22×C3#55911yescharacteristic
491C2×D3—42no—
492C2×D3—82no—
493C2×D3—82no—
494C2×D3—161no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4—42no—
498C3×C4—42no—
499C3×C4—42no—
500C3×C4C2×C4×D411yescharacteristic
501C3×C4C2×C4×D411yescharacteristic
502C3×C4#55011yescharacteristic
503C3×C4#55011yescharacteristic
504Dic3—42no—
505Dic3—42no—
506Dic3—42no—
507C23—62no—
508C23—121no—
509C23—121no—
510C23—242no—
511C23#100411yescharacteristic
512Q8—22no—
513Q8—22no—
514Q8—22no—
515Q8—22no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—41no—
520Q8—62no—
521Q8—62no—
522Q8—121no—
523Q8—121no—
524D4—62no—
525D4—62no—
526D4—121no—
527D4—121no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—482no—
544C2×C4C22×C4×D311yescharacteristic
545C2×C4C2×D3×D411yescharacteristic
546C2×C4C2×D3×D411yescharacteristic
547C2×C4#100411yescharacteristic
548C2×C4D3×#4111yescharacteristic
549C2×C4D3×#4111yescharacteristic
550C8—32no—
551C8—32no—
552C8—32no—
553C8—32no—
554C8—41no—
555C8—41no—
556C8—121no—
557C8—121no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3#236611yescharacteristic
561C2×C3#328912yes—
562D3—82no—
563C22—22no—
564C22—122no—
565C22—242no—
566C22—242no—
567C22—481no—
568C22#548611yescharacteristic
569C22#559011yescharacteristic
570C22#566411yescharacteristic
571C4—42no—
572C4—42no—
573C4—42no—
574C4—42no—
575C4—42no—
576C4—122no—
577C4—122no—
578C4—122no—
579C4C4×D3×D411yescharacteristic
580C4C4×D3×D411yescharacteristic
581C4#559911yescharacteristic
582C4#559911yescharacteristic
583C3#2320311yescharacteristic, complemented, Sylow
584C2—21no—
585C2—21no—
586C2—242no—
587C2#7467911yescharacteristic
588C2#8030112yes—
subgroupG/H
centerC22#5590
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1886C2
socleC22×C3D42
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1886D3
hypocenterC3#23203
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1886

Complements

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

NH ≅ G/Ntype
2. #82549C2semidirect
4. #78948C2semidirect
6. C3×#1886C2semidirect
7. #71784C2semidirect
8. #70336C2semidirect
10. #70108C2semidirect
14. #68742C2semidirect
15. #68742C2semidirect
41. #5086C4semidirect
72. #1886D3semidirect
83. C3×C4⋊Q8C2×C4semidirect
99. C2×#216C2×C4semidirect
100. C2×#216C2×C4semidirect
149. #803C2×C4semidirect
305. C4⋊Q8C4×D3semidirect
583. C3#23203semidirect

Maximal subgroups

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

subgroupindex
C2×#50862
#825492
#825482
#789482
#789472
C3×#18862
#717842
#703362
#703352
#701082
#701072
#687662
#687592
#687422
#232033