Finite Group Databasebeta

← #10819927

Subgroups — #10819927

Order 768.

914 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5168C211yesmaximal, characteristic
2#83843C211yesmaximal, characteristic, complemented
3#78948C211yesmaximal, characteristic, complemented
4#78915C211yesmaximal, characteristic
5#78572C211yesmaximal, characteristic, complemented
6C3×#1977C211yesmaximal, characteristic, complemented
7#70708C211yesmaximal, characteristic, complemented
8#70702C211yesmaximal, characteristic, complemented
9#70343C211yesmaximal, characteristic, complemented
10#70325C211yesmaximal, characteristic, complemented
11#70108C211yesmaximal, characteristic, complemented
12#70104C211yesmaximal, characteristic, complemented
13#70074C211yesmaximal, characteristic
14#70069C211yesmaximal, characteristic, complemented
15#69889C211yesmaximal, characteristic, complemented
16#23199—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×#947C2211yescharacteristic
19#5812C2211yescharacteristic
20C22×#216C2211yescharacteristic
21#5798C2211yescharacteristic
22C2×#936C2211yescharacteristic
23C22×#213C2211yescharacteristic
24C22×#204C2211yescharacteristic
25#5521C2211yescharacteristic
26#5516C2211yescharacteristic
27C2×C3×#108C2211yescharacteristic
28C3×#419C2211yescharacteristic
29C3×#418C2211yescharacteristic
30C2×C3×#101C2211yescharacteristic
31C2×C3×#100C2211yescharacteristic
32C3×#405C2211yescharacteristic
33#5225C2211yescharacteristic
34#5215C2211yescharacteristic
35#5168C418yescomplemented
36#5164—22no—
37Dic3×SD16—22no—
38C2×#824C2211yescharacteristic
39#5131C2211yescharacteristic
40#5131C2211yescharacteristic
41C2×#823C2211yescharacteristic
42C2×#822C2211yescharacteristic
43#5122C2211yescharacteristic
44#5122C2211yescharacteristic
45#5120C2211yescharacteristic
46#5120C2211yescharacteristic
47#5106C2211yescharacteristic
48#5101C2211yescharacteristic
49C2×C8×Dic3C2211yescharacteristic
50#4998C2211yescharacteristic
51#4790—22no—
52#4789—22no—
53#4785—22no—
54#4756—22no—
55#4710—22no—
56C4×#204—22no—
57#4555C2211yescharacteristic
58#4555C2211yescharacteristic
59#4555C2211yescharacteristic
60#4532C2211yescharacteristic
61C2×#495—31no—
62#2950—31no—
63#2949—31no—
64#2949—31no—
65#2948—31no—
66#1979—31no—
67#1978—31no—
68#1977—31no—
69#1977D311yescharacteristic, complemented
70#1888—31no—
71#1888—31no—
72#1885—31no—
73#1885—31no—
74#1772—31no—
75#1772—31no—
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C2×C3×SD16—21no—
81C2×C3×SD16—21no—
82C2×C3×SD16—22no—
83C2×C3×SD16C2×C414yescomplemented
84C2×C3×M16C2311yescharacteristic
85C2×C3×M16C2311yescharacteristic
86C2×C3×M16C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×#117C2311yescharacteristic
89C2×#217—41no—
90C2×#217—41no—
91#947—41no—
92#947C2×C414yescomplemented
93Q8×Dic3—22no—
94Q8×Dic3—22no—
95C2×#216—21no—
96C2×#216—21no—
97C2×#216—22no—
98C2×#216C2×C414yescomplemented
99#936—41no—
100#936C2×C414yescomplemented
101D4×Dic3—22no—
102D4×Dic3—22no—
103C2×#213—21no—
104C2×#213—21no—
105C2×#213—22no—
106C2×#213C2×C414yes—
107C2×(C2×D3)⋊C4—41no—
108#926—41no—
109#926C2311yescharacteristic
110#926C2311yescharacteristic
111#926C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C2×#208C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C2×#204—21no—
116C2×#204—21no—
117C2×#204—22no—
118C2×#204C2×C414yescomplemented
119#893—22no—
120#887—22no—
121D3×C22⋊C4—41no—
122#877—41no—
123C4×D12—22no—
124C4×Dic6—22no—
125C3×#108—22no—
126C3×#103—22no—
127C3×#102—22no—
128C3×#101—41no—
129C3×#100—41no—
130C3×#97—21no—
131C3×C4×C8—21no—
132#835—22no—
133#832—22no—
134#824—41no—
135#823—41no—
136#822—22no—
137#821—22no—
138#821—22no—
139#818D412yes—
140#817D412yes—
141#816—22no—
142#816—22no—
143#815—21no—
144#815—21no—
145#815—21no—
146C8×Dic3—21no—
147C8×Dic3—21no—
148C8×Dic3—21no—
149C8×Dic3C2×C414yescomplemented
150#808D412yes—
151#807D412yes—
152C22×Q16—31no—
153C22×SD16—31no—
154C22×SD16C2×D311yescharacteristic
155C22×D8—31no—
156C2×#124—31no—
157C2×#124—31no—
158#519—31no—
159#519—31no—
160#518—31no—
161#518—31no—
162#495—38no—
163#442—62no—
164#441—62no—
165#440—62no—
166#440—62no—
167C4×Q16—62no—
168C4×SD16—62no—
169C4×SD16—62no—
170C4×D8—62no—
171C2×#108—31no—
172C2×#108C2×D311yescharacteristic
173#427—31no—
174#427—31no—
175#419—31no—
176#419—31no—
177#419—31no—
178#419C2×D311yescharacteristic
179#418—31no—
180#418—31no—
181#418—31no—
182#418C2×D311yescharacteristic
183C2×#101—31no—
184C2×#101C2×D311yescharacteristic
185C2×#100—31no—
186C2×#100C2×D311yescharacteristic
187#405—31no—
188#405—31no—
189#405—31no—
190#405C2×D311yescharacteristic
191C2×C4×C8—31no—
192#339—31no—
193#339—31no—
194#339—31no—
195#339—31no—
196C24×C3—21no—
197C23×D3—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8C22×C412yes—
204C2×C3×D4—21no—
205C2×C3×D4—21no—
206C2×C3×D4—21no—
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4C22×C412yes—
210C22×C3×C4—21no—
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—42no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12—21no—
220C2×D12C22×C412yes—
221C2×C4×D3—42no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6C22×C412yes—
228C3×SD16—24no—
229C3×SD16—24no—
230C3×M16—21no—
231C3×M16—21no—
232C3×M16—21no—
233C3×M16—21no—
234C3×M16—21no—
235C3×M16—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8C22×C412yes—
240C2×C3×C8C2×D411yescharacteristic
241C2×C3×C8C2×D411yescharacteristic
242C2×C3×C8#4112yes—
243C3×C4⋊C4C2×D412yes—
244C3×C22⋊C4—21no—
245C3×C42—21no—
246#217—21no—
247#217—21no—
248#217—21no—
249#217—44no—
250#217—42no—
251#216—24no—
252#216—24no—
253#213—24no—
254#213—24no—
255(C2×D3)⋊C4—44no—
256(C2×D3)⋊C4—42no—
257C4⋊Dic3—42no—
258C4⋊Dic3C2×D412yes—
259C4⋊Dic3C2×D412yes—
260C4⋊Dic3C2×D412yes—
261Dic3⋊C4—42no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—42no—
267C4×Dic3—42no—
268C4×Dic3C22×C412yes—
269#208—21no—
270#208—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8C22×C412yes—
273C2×C3⋊C8C2×D411yescharacteristic
274C2×C3⋊C8C2×D411yescharacteristic
275C2×C3⋊C8#4112yes—
276#204—24no—
277#204—24no—
278C22×Q8—31no—
279C22×Q8C22×D311yescharacteristic
280C22×D4—31no—
281C22×D4C22×D311yescharacteristic
282C2×Q16—34no—
283C2×Q16—62no—
284C2×Q16—61no—
285C2×Q16—61no—
286C2×SD16—21no—
287C2×SD16—21no—
288C2×SD16—22no—
289C2×SD16—34no—
290C2×SD16—61no—
291C2×SD16—61no—
292C2×SD16—62no—
293C2×SD16C4×D314yescomplemented
294C2×D8—34no—
295C2×D8—62no—
296C2×D8—61no—
297C2×D8—61no—
298C2×M16—31no—
299C2×M16C22×D311yescharacteristic
300C2×M16C22×D311yescharacteristic
301C2×M16C22×D311yescharacteristic
302C22×C8—31no—
303C22×C8C22×D311yescharacteristic
304#124—34no—
305#124—34no—
306#124—121no—
307#124—121no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×Q8—62no—
311C4×Q8—62no—
312C4×D4—62no—
313C4×D4—62no—
314C4×D4—62no—
315C4×D4—62no—
316#117—31no—
317#117—31no—
318#117—31no—
319#117—121no—
320#117—121no—
321#117C22×D311yescharacteristic
322C2×C22⋊C4—121no—
323C2×C22⋊C4—121no—
324C2×C22⋊C4—121no—
325C2×C22⋊C4—121no—
326C2×C42—31no—
327#108—22no—
328#108—62no—
329#107—32no—
330#107—32no—
331#106—32no—
332#106—32no—
333#103—22no—
334#103—62no—
335#103—62no—
336#103—62no—
337#102—22no—
338#102—62no—
339#102—62no—
340#102—62no—
341#101—41no—
342#101—121no—
343#100—41no—
344#100—121no—
345#97—21no—
346#97—61no—
347#97—61no—
348#97—61no—
349C4×C8—21no—
350C4×C8—34no—
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
359C22×D3—22no—
360C22×D3—41no—
361C22×D3—41no—
362C22×D3—82no—
363C3×Q8—22no—
364C3×Q8—22no—
365C3×Q8—41no—
366C3×Q8—41no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4—41no—
370C3×D4—41no—
371C2×C3×C4—22no—
372C2×C3×C4—22no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4C23×C411yescharacteristic
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—42no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—162no—
390D12—22no—
391D12—22no—
392D12—41no—
393D12—41no—
394C4×D3—162no—
395Dic6—22no—
396Dic6—22no—
397Dic6—41no—
398Dic6—41no—
399C3×C8—41no—
400C3×C8—41no—
401C3×C8—41no—
402C3×C8C4×D412yes—
403C3×C8C4×D412yes—
404C3⋊C8—41no—
405C3⋊C8C4×D412yes—
406C3⋊C8C4×D412yes—
407C24—21no—
408C24—61no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—32no—
415C2×Q8—61no—
416C2×Q8—61no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8C2×C4×D312yes—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—32no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4C2×C4×D312yes—
433C22×C4—21no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—122no—
437C22×C4—122no—
438C22×C4C23×D311yescharacteristic
439Q16—64no—
440Q16—64no—
441SD16—24no—
442SD16—24no—
443SD16—64no—
444SD16—64no—
445D8—64no—
446D8—64no—
447M16—21no—
448M16—21no—
449M16—21no—
450M16—21no—
451M16—21no—
452M16—21no—
453M16—61no—
454M16—61no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—32no—
459C2×C8—31no—
460C2×C8—31no—
461C2×C8—32no—
462C2×C8—61no—
463C2×C8C2×C4×D312yes—
464C2×C8D3×D411yescharacteristic
465C2×C8D3×D411yescharacteristic
466C2×C8#23712yes—
467C4⋊C4—32no—
468C4⋊C4—32no—
469C4⋊C4—32no—
470C4⋊C4—122no—
471C4⋊C4—122no—
472C4⋊C4D3×D412yes—
473C22⋊C4—21no—
474C22⋊C4—61no—
475C22⋊C4—61no—
476C22⋊C4—61no—
477C22⋊C4—124no—
478C22⋊C4—122no—
479C22⋊C4—124no—
480C22⋊C4—122no—
481C42—21no—
482C42—32no—
483C42—61no—
484C42—61no—
485C42—61no—
486C42—61no—
487C42—122no—
488C42—122no—
489C22×C3—22no—
490C22×C3—42no—
491C22×C3—82no—
492C22×C3—82no—
493C22×C3—161no—
494C22×C3#51811yescharacteristic
495C22×C3D4211yescharacteristic
496C22×C3#55911yescharacteristic
497C2×D3—42no—
498C2×D3—82no—
499C2×D3—82no—
500C2×D3—161no—
501C3×C4—42no—
502C3×C4—42no—
503C3×C4C2×C4×D411yescharacteristic
504C3×C4C2×C4×D411yescharacteristic
505C3×C4#55011yescharacteristic
506C3×C4#55011yescharacteristic
507Dic3—42no—
508Dic3—42no—
509Dic3—42no—
510Dic3—42no—
511Dic3—42no—
512C23—22no—
513C23—41no—
514C23—41no—
515C23—62no—
516C23—82no—
517C23—121no—
518C23—121no—
519C23—242no—
520C23#100911yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525Q8—62no—
526Q8—62no—
527Q8—121no—
528Q8—121no—
529D4—22no—
530D4—22no—
531D4—41no—
532D4—41no—
533D4—62no—
534D4—62no—
535D4—121no—
536D4—121no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—62no—
546C2×C4—122no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—482no—
550C2×C4—482no—
551C2×C4C22×C4×D311yescharacteristic
552C2×C4C2×D3×D411yescharacteristic
553C2×C4C2×D3×D411yescharacteristic
554C2×C4C2×#23711yescharacteristic
555C2×C4#100411yescharacteristic
556C2×C4D3×#4111yescharacteristic
557C8—32no—
558C8—32no—
559C8—41no—
560C8—41no—
561C8—41no—
562C8—121no—
563C8#67⋊C412yes—
564C8#67⋊C412yes—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—82no—
568C2×C3#236611yescharacteristic
569C2×C3#329012yes—
570D3—82no—
571C22—22no—
572C22—42no—
573C22—82no—
574C22—82no—
575C22—122no—
576C22—161no—
577C22—242no—
578C22—242no—
579C22—481no—
580C22#554511yescharacteristic
581C22#560911yescharacteristic
582C22#568111yescharacteristic
583C4—42no—
584C4—42no—
585C4—122no—
586C4—122no—
587C4—122no—
588C4—122no—
589C4—122no—
590C4C2×#67⋊C411yescharacteristic
591C4C4×D3×D411yescharacteristic
592C4#559911yescharacteristic
593C4#571811yescharacteristic
594C3#2319911yescharacteristic, complemented, Sylow
595C2—21no—
596C2—21no—
597C2—82no—
598C2—242no—
599C2#7630611yescharacteristic
600C2#8093412yes—
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1977C2
socleC22×C3D42
prime radicalC22×#213C22
cosocleC22×C3×C4C24
hypercenter#1977D3
hypocenterC3#23199
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1977

Complements

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

NH ≅ G/Ntype
2. #83843C2semidirect
3. #78948C2semidirect
5. #78572C2semidirect
6. C3×#1977C2semidirect
7. #70708C2semidirect
8. #70702C2semidirect
9. #70343C2semidirect
10. #70325C2semidirect
11. #70108C2semidirect
12. #70104C2semidirect
14. #70069C2semidirect
15. #69889C2semidirect
35. #5168C4semidirect
69. #1977D3semidirect
83. C2×C3×SD16C2×C4semidirect
92. #947C2×C4semidirect
98. C2×#216C2×C4semidirect
100. #936C2×C4semidirect
118. C2×#204C2×C4semidirect
149. C8×Dic3C2×C4semidirect
293. C2×SD16C4×D3semidirect
594. C3#23199semidirect

Maximal subgroups

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

subgroupindex
C2×#51682
#838432
#789482
#789152
#785722
C3×#19772
#707082
#707022
#703432
#703252
#701082
#701042
#700742
#700692
#698892
#231993