Finite Group Databasebeta

← #10804135

Subgroups — #10804135

Order 768.

1554 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5524C211yesmaximal, characteristic
2D3×#381C211yesmaximal, characteristic, complemented
3#74050C211yesmaximal, characteristic, complemented
4C3×#1820C211yesmaximal, characteristic, complemented
5#71594C211yesmaximal, characteristic, complemented
6#69537C211yesmaximal, characteristic, complemented
7#69537C211yesmaximal, characteristic, complemented
8#68970C214yesmaximal, complemented
9#68954C214yesmaximal, complemented
10#21748—31nomaximal, Sylow
11C22×#235C2211yescharacteristic
12C23×C4×D3C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14#5524—22no—
15#5524—22no—
16#5524—22no—
17#5524—22no—
18C2×#897C412yescomplemented
19C2×#895C412yescomplemented
20C2×#886C2212yes—
21C2×#882C2212yes—
22C2×D3×C22⋊C4C2212yes—
23C2×#877C2212yes—
24C2×C4×D12C414yescomplemented
25C3×#381C2211yescharacteristic
26C3×#381C2211yescharacteristic
27#5211C2211yescharacteristic
28#5211C2211yescharacteristic
29#4955C2212yes—
30C2×#806C2214yes—
31C2×#806C2214yes—
32#4943C2212yes—
33#4667—22no—
34#4666—22no—
35#4664—22no—
36D3×#95—22no—
37C3×#342C2214yes—
38#4526C2214yes—
39C2×#525—31no—
40#2280—31no—
41C2×#381—31no—
42#1899—34no—
43#1898—34no—
44#1820—31no—
45#1820D311yescharacteristic, complemented
46#1782—31no—
47#1782—31no—
48C24×D3C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C22×#67C2312yes—
51C23×Dic3C2311yescharacteristic
52C2×#235—22no—
53C2×#235—22no—
54C2×#235—44no—
55C22×D12C2×C414yes—
56C22×D12C2×C411yescharacteristic, complemented
57C22×C4×D3—22no—
58C22×C4×D3—44no—
59C22×C4×D3C2×C412yes—
60C22×C4×D3C2×C412yes—
61C22×C4×D3C2×C412yescomplemented
62C22×C4×D3D412yes—
63C22×C4×D3D412yes—
64C22×C4×D3D412yes—
65C22×C4×D3D412yes—
66C22×Dic6C2×C411yescharacteristic, complemented
67C3×#117—22no—
68C3×#117—22no—
69C2×C3×C4⋊C4C2×C412yescomplemented
70C2×C3×C22⋊C4C2312yes—
71C2×C3×C22⋊C4C2312yes—
72C2×C3×C22⋊C4C2312yes—
73C2×C3×C42C2×C412yescomplemented
74C2×#217C2312yes—
75C2×#217C2312yes—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—24no—
78C2×(C2×D3)⋊C4C2×C414yes—
79C2×(C2×D3)⋊C4C2×C414yescomplemented
80C2×C4⋊Dic3C2×C414yescomplemented
81C2×Dic3⋊C4—22no—
82C2×C4×Dic3—22no—
83#897—22no—
84#897—22no—
85#897—22no—
86#897—22no—
87#895—22no—
88#895—22no—
89#895—22no—
90#895—22no—
91#886—24no—
92#886—22no—
93#886—22no—
94#882—22no—
95#882—24no—
96#882—22no—
97D3×C22⋊C4—44no—
98#877—44no—
99C4×D12—24no—
100C4×D12—24no—
101C4×D12—24no—
102C4×D12—24no—
103C3×#95—22no—
104C3×#95—22no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107C3⋊#95—24no—
108#806—44no—
109#806—44no—
110C22×#41—31no—
111C24×C4—31no—
112#525—62no—
113#525—62no—
114#525—62no—
115#525—62no—
116C2×#123—32no—
117C2×C22⋊Q8—32no—
118C2×#121—32no—
119C2×#120—32no—
120C2×C4×D4—34no—
121C2×#117—32no—
122C2×#117C2×D311yescharacteristic
123C22×C22⋊C4—32no—
124C2×#99—34no—
125C2×#99—34no—
126#381—31no—
127#381—31no—
128#381—32no—
129#381—62no—
130#381C2×D311yescharacteristic
131#381C2×D311yescharacteristic
132#380—62no—
133#379—32no—
134C4×C22⋊C4—62no—
135C2×#95—62no—
136#342—34no—
137#342C2×D314yes—
138C24×C3C2411yescharacteristic
139C23×D3—24no—
140C23×D3—44no—
141C23×D3C22×C412yes—
142C23×D3C22×C412yes—
143C23×D3C2×D411yescharacteristic
144C23×D3C2×D412yes—
145C23×D3C2×D411yescharacteristic
146C22×C3×C4—24no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4C22×C411yescharacteristic
150C22×C3×C4C22×C411yescharacteristic
151C22×C3×C4C22×C414yes—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C2×#67—22no—
155C2×#67—22no—
156C2×#67—24no—
157C2×#67—24no—
158C22×Dic3—24no—
159C22×Dic3—22no—
160C22×Dic3—22no—
161C22×Dic3C22×C412yes—
162C22×Dic3C22×C412yes—
163C22×Dic3C2×D412yes—
164C22×Dic3C2×D411yescharacteristic
165C22×Dic3C2×D411yescharacteristic
166#235—48no—
167#235—48no—
168C2×D12—24no—
169C2×D12—24no—
170C2×D12—22no—
171C2×D12—42no—
172C2×D12—48no—
173C2×C4×D3—24no—
174C2×C4×D3—44no—
175C2×C4×D3—44no—
176C2×C4×D3—44no—
177C2×C4×D3—44no—
178C2×C4×D3—44no—
179C2×C4×D3—44no—
180C2×C4×D3—42no—
181C2×C4×D3—42no—
182C2×C4×D3—44no—
183C2×C4×D3—48no—
184C2×C4×D3C22⋊C412yes—
185C2×C4×D3C22⋊C412yes—
186C2×C4×D3C22⋊C412yes—
187C2×C4×D3C22⋊C412yes—
188C2×C4×D3C22⋊C412yes—
189C2×C4×D3C22⋊C412yes—
190C2×C4×D3C22⋊C412yes—
191C2×C4×D3C22⋊C412yes—
192C2×Dic6—22no—
193C2×Dic6—42no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—22no—
198C3×C22⋊C4—24no—
199C3×C22⋊C4—24no—
200C3×C42—24no—
201#217—24no—
202#217—24no—
203(C2×D3)⋊C4—24no—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4—24no—
207(C2×D3)⋊C4—44no—
208(C2×D3)⋊C4—48no—
209C4⋊Dic3—24no—
210C4⋊Dic3—24no—
211Dic3⋊C4—44no—
212C4×Dic3—44no—
213C25—31no—
214C2×#41—62no—
215C2×#41—62no—
216C2×#41—124no—
217C22×Q8—31no—
218C22×D4—32no—
219C22×D4—34no—
220C22×D4—31no—
221C23×C4—31no—
222C23×C4—32no—
223C23×C4—32no—
224C23×C4—32no—
225C23×C4—32no—
226C23×C4—32no—
227C23×C4—32no—
228C23×C4—32no—
229C23×C4—62no—
230C23×C4—124no—
231C23×C4C22×D311yescharacteristic
232#123—64no—
233#123—62no—
234#123—62no—
235C22⋊Q8—62no—
236C22⋊Q8—62no—
237C22⋊Q8—62no—
238C22⋊Q8—62no—
239#121—62no—
240#121—62no—
241#121—62no—
242#121—62no—
243#120—62no—
244#120—64no—
245#120—62no—
246C4×D4—64no—
247C4×D4—64no—
248C4×D4—64no—
249C4×D4—64no—
250#117—22no—
251#117—22no—
252#117—124no—
253C2×C4⋊C4—34no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4C4×D312yescomplemented
256C2×C22⋊C4—34no—
257C2×C22⋊C4—34no—
258C2×C22⋊C4—32no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—64no—
262C2×C22⋊C4—124no—
263C2×C22⋊C4C22×D312yes—
264C2×C22⋊C4C22×D312yes—
265C2×C22⋊C4C22×D312yes—
266C2×C42—62no—
267C2×C42C4×D312yescomplemented
268#99—124no—
269#99—124no—
270#95—22no—
271#95—22no—
272#95—64no—
273#95—62no—
274#95—62no—
275C23×C3—42no—
276C23×C3C23×C411yescharacteristic
277C23×C3C22×D411yescharacteristic
278C23×C3C22×D411yescharacteristic
279C23×C3H(5,𝔽2)14yes—
280C22×D3—22no—
281C22×D3—24no—
282C22×D3—24no—
283C22×D3—22no—
284C22×D3—24no—
285C22×D3—24no—
286C22×D3—24no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—416no—
294C22×D3—44no—
295C22×D3C2×C22⋊C412yes—
296C22×D3C2×C22⋊C412yes—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—44no—
303C2×C3×C4—44no—
304C2×C3×C4—44no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4C2×C22⋊C412yes—
308C2×C3×C4C2×C22⋊C412yes—
309#67—44no—
310#67—44no—
311C2×Dic3—24no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—24no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—44no—
319C2×Dic3—44no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3C2×C22⋊C412yes—
323C2×Dic3C2×C22⋊C412yes—
324D12—44no—
325D12—48no—
326D12—48no—
327C4×D3—42no—
328C4×D3—42no—
329C4×D3—42no—
330C4×D3—42no—
331C4×D3—42no—
332C4×D3—42no—
333C4×D3—42no—
334C4×D3—42no—
335C4×D3—48no—
336C4×D3—88no—
337Dic6—44no—
338C24—31no—
339C24—32no—
340C24—32no—
341C24—31no—
342C24—32no—
343C24—64no—
344C24—124no—
345C24C23×D311yescharacteristic
346#41—128no—
347#41—128no—
348C2×Q8—62no—
349C2×Q8—122no—
350C2×D4—64no—
351C2×D4—64no—
352C2×D4—62no—
353C2×D4—62no—
354C2×D4—64no—
355C2×D4—64no—
356C2×D4—62no—
357C2×D4—122no—
358C2×D4—128no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—32no—
363C22×C4—32no—
364C22×C4—31no—
365C22×C4—31no—
366C22×C4—32no—
367C22×C4—32no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—32no—
371C22×C4—32no—
372C22×C4—32no—
373C22×C4—32no—
374C22×C4—32no—
375C22×C4—64no—
376C22×C4—64no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—124no—
380C22×C4—124no—
381C22×C4—124no—
382C22×C4—124no—
383C22×C4—124no—
384C22×C4—124no—
385C22×C4—122no—
386C22×C4—122no—
387C22×C4—124no—
388C22×C4—128no—
389C22×C4C2×C4×D311yescharacteristic
390C22×C4C2×C4×D311yescharacteristic
391C22×C4C2×C4×D314yes—
392C22×C4D3×D412yes—
393C22×C4D3×D412yes—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—64no—
397C4⋊C4—64no—
398C4⋊C4—124no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—64no—
404C22⋊C4—64no—
405C22⋊C4—64no—
406C22⋊C4—64no—
407C22⋊C4—64no—
408C22⋊C4—64no—
409C22⋊C4—124no—
410C22⋊C4—128no—
411C42—24no—
412C42—124no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—22no—
416C22×C3—44no—
417C22×C3C22×C22⋊C411yescharacteristic
418C22×C3#51812yes—
419C22×C3#53412yes—
420C22×C3#53412yes—
421C2×D3—28no—
422C2×D3—22no—
423C2×D3—22no—
424C2×D3—22no—
425C2×D3—22no—
426C2×D3—22no—
427C2×D3—48no—
428C2×D3—48no—
429C2×D3—44no—
430C2×D3—42no—
431C2×D3—44no—
432C2×D3—82no—
433C2×D3—82no—
434C2×D3#41014yes—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—44no—
438C3×C4—82no—
439C3×C4—82no—
440Dic3—22no—
441Dic3—22no—
442Dic3—42no—
443Dic3—82no—
444Dic3—82no—
445C23—32no—
446C23—32no—
447C23—42no—
448C23—64no—
449C23—64no—
450C23—64no—
451C23—62no—
452C23—62no—
453C23—64no—
454C23—64no—
455C23—62no—
456C23—62no—
457C23—62no—
458C23—122no—
459C23—122no—
460C23—122no—
461C23—124no—
462C23—1216no—
463C23C22×C4×D311yescharacteristic
464C23C2×D3×D411yescharacteristic
465C23C2×D3×D411yescharacteristic
466C23#100914yes—
467Q8—124no—
468D4—124no—
469D4—128no—
470D4—128no—
471D4—124no—
472D4—124no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—32no—
479C2×C4—32no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—64no—
488C2×C4—64no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—122no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—122no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—128no—
501C2×C4—124no—
502C2×C4—124no—
503C2×C4—122no—
504C2×C4—122no—
505C2×C4—248no—
506C2×C4D3×C22⋊C412yes—
507C2×C4D3×C22⋊C412yes—
508C2×C3—22no—
509C2×C3—22no—
510C2×C3—22no—
511C2×C3#230811yescharacteristic
512C2×C3C2×#41012yes—
513D3—24no—
514D3—42no—
515D3—82no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—34no—
520C22—44no—
521C22—68no—
522C22—62no—
523C22—62no—
524C22—62no—
525C22—62no—
526C22—62no—
527C22—128no—
528C22—128no—
529C22—124no—
530C22—124no—
531C22—122no—
532C22—242no—
533C22—242no—
534C22C2×D3×C22⋊C411yescharacteristic
535C22#552112yes—
536C22#565312yes—
537C22#565312yes—
538C4—22no—
539C4—22no—
540C4—44no—
541C4—62no—
542C4—62no—
543C4—82no—
544C4—82no—
545C4—122no—
546C4—242no—
547C4—242no—
548C3#2174811yescharacteristic, complemented, Sylow
549C2—22no—
550C2—22no—
551C2—22no—
552C2—64no—
553C2—122no—
554C2—242no—
555C2#7411911yescharacteristic
556C2D3×#41012yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#1820C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5524C2
cosocleC24×C3C24
hypercenter#1820D3
hypocenterC3#21748
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
2. D3×#381C2semidirect
3. #74050C2semidirect
4. C3×#1820C2semidirect
5. #71594C2semidirect
6. #69537C2semidirect
7. #69537C2semidirect
8. #68970C2semidirect
9. #68954C2semidirect
18. C2×#897C4semidirect
19. C2×#895C4semidirect
24. C2×C4×D12C4semidirect
45. #1820D3semidirect
56. C22×D12C2×C4semidirect
61. C22×C4×D3C2×C4semidirect
66. C22×Dic6C2×C4semidirect
69. C2×C3×C4⋊C4C2×C4semidirect
73. C2×C3×C42C2×C4semidirect
79. C2×(C2×D3)⋊C4C2×C4semidirect
80. C2×C4⋊Dic3C2×C4semidirect
255. C2×C4⋊C4C4×D3semidirect
267. C2×C42C4×D3semidirect
548. C3#21748semidirect

Maximal subgroups

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

subgroupindex
C2×#55242
D3×#3812
#740502
C3×#18202
#715942
#695372
#689702
#689542
#217483