Finite Group Databasebeta

← #76756

Subgroups — #76756

Order 384.

922 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#939C211yesmaximal, characteristic, complemented
2C2×#938C211yesmaximal, characteristic, complemented
3C2×#937C211yesmaximal, characteristic, complemented
4C2×#937C211yesmaximal, characteristic, complemented
5C2×#67⋊C4C211yesmaximal, characteristic, complemented
6C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
7C3×#392C211yesmaximal, characteristic, complemented
8#5221C211yesmaximal, characteristic, complemented
9#5220C211yesmaximal, characteristic, complemented
10#4961C211yesmaximal, characteristic, complemented
11#4959C211yesmaximal, characteristic, complemented
12#4956C211yesmaximal, characteristic, complemented
13#4948C211yesmaximal, characteristic, complemented
14#4670C211yesmaximal, characteristic, complemented
15#4668C211yesmaximal, characteristic, complemented
16#2424—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×C3×D4C2211yescharacteristic, complemented
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic
25C23×Dic3C2211yescharacteristic, complemented
26C22×C4×D3C2211yescharacteristic
27C22×C4×D3C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic, complemented
30C2×C3×C22⋊C4C2211yescharacteristic, complemented
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic, complemented
35C2×#217C2211yescharacteristic, complemented
36C2×#217C2211yescharacteristic, complemented
37#939—24no—
38#938—24no—
39#937—24no—
40#937—24no—
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic
45C2×(C2×D3)⋊C4C2211yescharacteristic
46C2×(C2×D3)⋊C4C2211yescharacteristic
47C2×C4⋊Dic3C2211yescharacteristic, complemented
48C2×C4⋊Dic3C2211yescharacteristic, complemented
49C2×Dic3⋊C4C2211yescharacteristic, complemented
50C2×Dic3⋊C4C2211yescharacteristic, complemented
51C2×C4×Dic3C2211yescharacteristic, complemented
52#67⋊C4—24no—
53D3×C22⋊C4—24no—
54C3×#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C3⋊#95C2211yescharacteristic, complemented
57C3⋊#95C2211yescharacteristic, complemented
58C2×#121—31no—
59C2×#121—31no—
60C2×#120—31no—
61C2×#120—31no—
62C2×C4×D4—31no—
63C22×C22⋊C4—31no—
64#396—31no—
65#394—31no—
66#394—31no—
67#392D311yescharacteristic, complemented
68#390—31no—
69#388—31no—
70#386—31no—
71#386—31no—
72#385—31no—
73C24×C3C2311yescharacteristic
74C24×C3C2311yescharacteristic
75C24×C3C2311yescharacteristic
76C23×D3—22no—
77C23×D3—22no—
78C23×D3—22no—
79C23×D3—22no—
80C23×D3—22no—
81C23×D3—41no—
82C23×D3—41no—
83C23×D3C2311yescharacteristic
84C23×D3C2311yescharacteristic
85C2×C3×D4—22no—
86C2×C3×D4—22no—
87C2×C3×D4—22no—
88C2×C3×D4—22no—
89C2×C3×D4—22no—
90C2×C3×D4—22no—
91C2×C3×D4—41no—
92C2×C3×D4—41no—
93C2×C3×D4—41no—
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C22×C3×C4C2311yescharacteristic
98C2×#67—22no—
99C2×#67—22no—
100C2×#67—22no—
101C2×#67—22no—
102C2×#67—22no—
103C2×#67—22no—
104C2×#67—22no—
105C2×#67—22no—
106C2×#67—22no—
107C2×#67—41no—
108C2×#67—41no—
109C2×#67D414yescomplemented
110C22×Dic3—22no—
111C22×Dic3—22no—
112C22×Dic3—41no—
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115C22×Dic3C2311yescharacteristic
116C22×Dic3C2311yescharacteristic
117C22×Dic3C2311yescharacteristic
118C22×Dic3C2311yescharacteristic
119C2×C4×D3—22no—
120C2×C4×D3—22no—
121C2×C4×D3—22no—
122C2×C4×D3—22no—
123C2×C4×D3—41no—
124C2×C4×D3—41no—
125C3×C22⋊C4—22no—
126C3×C22⋊C4—22no—
127C3×C22⋊C4—41no—
128#217—22no—
129#217—22no—
130#217—22no—
131#217—22no—
132#217—22no—
133#217—22no—
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—22no—
136(C2×D3)⋊C4—22no—
137(C2×D3)⋊C4—22no—
138(C2×D3)⋊C4D414yescomplemented
139(C2×D3)⋊C4D414yescomplemented
140C4⋊Dic3—22no—
141C4⋊Dic3—41no—
142Dic3⋊C4—22no—
143Dic3⋊C4—22no—
144C4×Dic3—22no—
145C25—31no—
146C22×D4—31no—
147C22×D4—31no—
148C22×D4—31no—
149C22×D4—31no—
150C22×D4C2×D311yescharacteristic
151C22×D4C2×D311yescharacteristic
152C22×D4C2×D311yescharacteristic, complemented
153C23×C4—31no—
154C23×C4—31no—
155C23×C4—31no—
156#121—64no—
157#121—64no—
158#120—64no—
159#120—64no—
160C4×D4—64no—
161C2×C4⋊C4—31no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—31no—
165C2×C22⋊C4—31no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—31no—
169C2×C22⋊C4—31no—
170C2×C22⋊C4—31no—
171C2×C22⋊C4—31no—
172C2×C22⋊C4—31no—
173C2×C22⋊C4—31no—
174C2×C22⋊C4—31no—
175C2×C22⋊C4—31no—
176C2×C22⋊C4—31no—
177C2×C22⋊C4—64no—
178C2×C22⋊C4C2×D311yescharacteristic
179C2×C22⋊C4C2×D311yescharacteristic, complemented
180C2×C22⋊C4C2×D311yescharacteristic, complemented
181C2×C42—31no—
182#95—31no—
183#95—31no—
184#95—31no—
185#95C2×D311yescharacteristic, complemented
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—21no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—21no—
205C23×C3C2×D412yes—
206C23×C3#4112yes—
207C23×C3C2411yescharacteristic
208C22×D3—21no—
209C22×D3—21no—
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—28no—
213C22×D3—21no—
214C22×D3—21no—
215C22×D3—21no—
216C22×D3—21no—
217C22×D3—21no—
218C22×D3—21no—
219C22×D3—44no—
220C22×D3—44no—
221C22×D3—44no—
222C22×D3—44no—
223C22×D3—44no—
224C22×D3—44no—
225C22×D3C2×D412yes—
226C22×D3C2×D412yes—
227C22×D3C2×D412yes—
228C22×D3#4112yes—
229C3×D4—44no—
230C3×D4—44no—
231C3×D4—44no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4C2×D412yes—
243C2×C3×C4C2×D412yes—
244#67—28no—
245#67—44no—
246#67—44no—
247#67—44no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—21no—
260C2×Dic3—21no—
261C2×Dic3—21no—
262C2×Dic3—21no—
263C2×Dic3—44no—
264C2×Dic3C2×D412yes—
265C2×Dic3C2×D412yes—
266C2×Dic3C2×D412yes—
267C4×D3—44no—
268C4×D3—44no—
269C24—31no—
270C24—31no—
271C24—62no—
272C24—62no—
273C24—62no—
274C24—62no—
275C24—62no—
276C24—121no—
277C24—121no—
278C24C22×D311yescharacteristic
279C24C22×D311yescharacteristic
280C24C22×D311yescharacteristic
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—34no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—62no—
296C2×D4—62no—
297C2×D4—62no—
298C2×D4—62no—
299C2×D4—62no—
300C2×D4—121no—
301C2×D4—121no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—62no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—62no—
313C22×C4—62no—
314C22×C4—121no—
315C22×C4—121no—
316C22×C4—121no—
317C22×C4C22×D311yescharacteristic
318C22×C4C22×D311yescharacteristic
319C22×C4C22×D311yescharacteristic
320C22×C4C22×D311yescharacteristic
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—121no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—34no—
328C22⋊C4—34no—
329C22⋊C4—41no—
330C22⋊C4—62no—
331C22⋊C4—62no—
332C22⋊C4—62no—
333C22⋊C4—62no—
334C22⋊C4—62no—
335C22⋊C4—62no—
336C22⋊C4—62no—
337C22⋊C4—62no—
338C22⋊C4—62no—
339C22⋊C4—62no—
340C42—62no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—22no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3C22×D411yescharacteristic
363C22×C3C22×D411yescharacteristic
364C22×C3C22×D411yescharacteristic
365C22×C3C2×#4111yescharacteristic
366C22×C3C2×#4111yescharacteristic
367C22×C3H(5,𝔽2)11yescharacteristic
368C22×C3H(5,𝔽2)11yescharacteristic
369C2×D3—22no—
370C2×D3—22no—
371C2×D3—22no—
372C2×D3—22no—
373C2×D3—22no—
374C2×D3—22no—
375C2×D3—22no—
376C2×D3—22no—
377C2×D3—22no—
378C2×D3—416no—
379C2×D3—41no—
380C2×D3—41no—
381C2×D3—41no—
382C2×D3—41no—
383C2×D3#12114yescomplemented
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—41no—
387C3×C4—41no—
388Dic3—22no—
389Dic3—22no—
390Dic3—22no—
391Dic3—41no—
392Dic3—41no—
393Dic3—41no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—32no—
414C23—32no—
415C23—32no—
416C23—32no—
417C23—68no—
418C23—61no—
419C23—61no—
420C23—61no—
421C23—61no—
422C23—61no—
423C23—61no—
424C23—61no—
425C23—61no—
426C23—61no—
427C23—61no—
428C23—124no—
429C23—124no—
430C23—124no—
431C23—124no—
432C23—124no—
433C23—124no—
434C23D3×D412yes—
435C23#23712yes—
436C23C23×D311yescharacteristic
437D4—44no—
438D4—44no—
439D4—44no—
440D4—68no—
441D4—124no—
442D4—124no—
443D4—124no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—32no—
455C2×C4—32no—
456C2×C4—32no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—61no—
463C2×C4—61no—
464C2×C4—61no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—124no—
473C2×C4—124no—
474C2×C4—124no—
475C2×C4D3×D412yes—
476C2×C4D3×D412yes—
477C2×C3—42no—
478C2×C3—42no—
479C2×C3—81no—
480C2×C3C2×#12111yescharacteristic
481C2×C3#52511yescharacteristic
482C2×C3#53811yescharacteristic
483C2×C3D4211yescharacteristic
484C2×C3D4211yescharacteristic
485C2×C3#54611yescharacteristic
486C2×C3#54611yescharacteristic
487D3—24no—
488D3—42no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—34no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—62no—
512C22—62no—
513C22—62no—
514C22—62no—
515C22—62no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—62no—
520C22—1216no—
521C22—121no—
522C22—121no—
523C22—121no—
524C22—121no—
525C22C2×D3×D411yescharacteristic
526C22C2×D3×D411yescharacteristic
527C22C2×D3×D411yescharacteristic
528C22C2×#23711yescharacteristic
529C22#100411yescharacteristic
530C22#100411yescharacteristic
531C22D3×#4111yescharacteristic
532C4—22no—
533C4—22no—
534C4—41no—
535C4—41no—
536C4—62no—
537C4—62no—
538C4—62no—
539C4—121no—
540C4—121no—
541C4—121no—
542C3#242411yescharacteristic, complemented, Sylow
543C2—42no—
544C2—42no—
545C2—64no—
546C2—81no—
547C2—122no—
548C2#559011yescharacteristic
549C2#559111yescharacteristic
550C2#559111yescharacteristic
551C2D3×#12111yescharacteristic
552C2#560611yescharacteristic
553C2#561311yescharacteristic
554C2#571711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2424
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#939C2semidirect
2. C2×#938C2semidirect
3. C2×#937C2semidirect
4. C2×#937C2semidirect
5. C2×#67⋊C4C2semidirect
6. C2×D3×C22⋊C4C2semidirect
7. C3×#392C2semidirect
8. #5221C2semidirect
9. #5220C2semidirect
10. #4961C2semidirect
11. #4959C2semidirect
12. #4956C2semidirect
13. #4948C2semidirect
14. #4670C2semidirect
15. #4668C2semidirect
20. C22×C3×D4C22semidirect
25. C23×Dic3C22semidirect
29. C2×C3×C22⋊C4C22semidirect
30. C2×C3×C22⋊C4C22semidirect
34. C2×#217C22semidirect
35. C2×#217C22semidirect
36. C2×#217C22semidirect
47. C2×C4⋊Dic3C22semidirect
48. C2×C4⋊Dic3C22semidirect
49. C2×Dic3⋊C4C22semidirect
50. C2×Dic3⋊C4C22semidirect
51. C2×C4×Dic3C22semidirect
54. C3×#95C22semidirect
55. C3⋊#95C22semidirect
56. C3⋊#95C22semidirect
57. C3⋊#95C22semidirect
67. #392D3semidirect
109. C2×#67D4semidirect
138. (C2×D3)⋊C4D4semidirect
139. (C2×D3)⋊C4D4semidirect
152. C22×D4C2×D3semidirect
179. C2×C22⋊C4C2×D3semidirect
180. C2×C22⋊C4C2×D3semidirect
185. #95C2×D3semidirect
383. C2×D3#121semidirect
542. C3#2424semidirect

Maximal subgroups

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

subgroupindex
C2×#9392
C2×#9382
C2×#9372
C2×#67⋊C42
C2×D3×C22⋊C42
C3×#3922
#52212
#52202
#49612
#49592
#49562
#49482
#46702
#46682
#24243