Finite Group Databasebeta

← #77416

Subgroups — #77416

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×#895C211yesmaximal, characteristic, complemented
2C2×#895C211yesmaximal, characteristic, complemented
3C2×#893C211yesmaximal, characteristic, complemented
4C2×#882C211yesmaximal, characteristic, complemented
5C2×#882C211yesmaximal, characteristic, complemented
6C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
7C3×#396C211yesmaximal, characteristic, complemented
8#4991C211yesmaximal, characteristic, complemented
9#4990C211yesmaximal, characteristic, complemented
10#4989C211yesmaximal, characteristic, complemented
11#4960C211yesmaximal, characteristic, complemented
12#4957C211yesmaximal, characteristic, complemented
13#4956C211yesmaximal, characteristic, complemented
14#4672C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2424—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×D12C2211yescharacteristic
22C22×D12C2211yescharacteristic
23C22×D12C2211yescharacteristic
24C22×D12C2211yescharacteristic, complemented
25C22×C4×D3C2211yescharacteristic
26C22×C4×D3C2211yescharacteristic
27C22×C4×D3C2211yescharacteristic, complemented
28C2×C3×C4⋊C4C2211yescharacteristic, complemented
29C2×C3×C4⋊C4C2211yescharacteristic, complemented
30C2×C3×C4⋊C4C2211yescharacteristic, complemented
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
42C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C2×Dic3⋊C4C2211yescharacteristic, complemented
47C2×C4×Dic3C2211yescharacteristic, complemented
48#895—24no—
49#895—24no—
50#893—24no—
51#882—24no—
52#882—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#396D311yescharacteristic, complemented
65#394—31no—
66#394—31no—
67#392—31no—
68#390—31no—
69#388—31no—
70#386—31no—
71#386—31no—
72#385—31no—
73C24×C3C2311yescharacteristic
74C23×D3—22no—
75C23×D3—22no—
76C23×D3—22no—
77C23×D3—22no—
78C23×D3—22no—
79C23×D3—41no—
80C23×D3—41no—
81C23×D3C2311yescharacteristic
82C23×D3C2311yescharacteristic
83C23×D3C2311yescharacteristic
84C23×D3C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—22no—
94C2×#67—22no—
95C2×#67—41no—
96C2×#67—41no—
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C2×D12—22no—
102C2×D12—22no—
103C2×D12—22no—
104C2×D12—22no—
105C2×D12—22no—
106C2×D12—22no—
107C2×D12—22no—
108C2×D12—22no—
109C2×D12—22no—
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—41no—
113C2×D12—41no—
114C2×D12—41no—
115C2×D12D414yescomplemented
116C2×C4×D3—22no—
117C2×C4×D3—22no—
118C2×C4×D3—22no—
119C2×C4×D3—22no—
120C2×C4×D3—22no—
121C2×C4×D3—22no—
122C2×C4×D3—41no—
123C2×C4×D3—41no—
124C2×C4×D3—41no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C22⋊C4—22no—
129C3×C22⋊C4—22no—
130C3×C22⋊C4—22no—
131#217—22no—
132(C2×D3)⋊C4—22no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—22no—
136(C2×D3)⋊C4—22no—
137(C2×D3)⋊C4—22no—
138(C2×D3)⋊C4—22no—
139(C2×D3)⋊C4—22no—
140(C2×D3)⋊C4—41no—
141(C2×D3)⋊C4D414yescomplemented
142(C2×D3)⋊C4D414yescomplemented
143Dic3⋊C4—41no—
144C4×Dic3—22no—
145C25—31no—
146C22×D4—31no—
147C22×D4—31no—
148C22×D4—31no—
149C22×D4—31no—
150C22×D4—31no—
151C22×D4—31no—
152C22×D4—31no—
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⋊C4C2×D311yescharacteristic, complemented
163C2×C4⋊C4C2×D311yescharacteristic, complemented
164C2×C4⋊C4C2×D311yescharacteristic, complemented
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
180C2×C22⋊C4C2×D311yescharacteristic
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#4112yes—
193C23×C3C2411yescharacteristic
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—28no—
199C22×D3—21no—
200C22×D3—21no—
201C22×D3—21no—
202C22×D3—21no—
203C22×D3—21no—
204C22×D3—21no—
205C22×D3—21no—
206C22×D3—21no—
207C22×D3—21no—
208C22×D3—21no—
209C22×D3—21no—
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—21no—
213C22×D3—21no—
214C22×D3—21no—
215C22×D3—21no—
216C22×D3—21no—
217C22×D3—21no—
218C22×D3—44no—
219C22×D3—44no—
220C22×D3—44no—
221C22×D3—44no—
222C22×D3—44no—
223C22×D3—44no—
224C22×D3C2×D412yes—
225C22×D3C2×D412yes—
226C22×D3C2×D412yes—
227C22×D3C2×D412yes—
228C22×D3#4112yes—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
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×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4C2×D412yes—
245C2×C3×C4C2×D412yes—
246C2×C3×C4C2×D412yes—
247#67—44no—
248#67—44no—
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×Dic3C2×D412yes—
260C2×Dic3C2×D412yes—
261D12—28no—
262D12—44no—
263D12—44no—
264D12—44no—
265D12—44no—
266C4×D3—44no—
267C4×D3—44no—
268C4×D3—44no—
269C24—31no—
270C24—31no—
271C24—31no—
272C24—31no—
273C24—62no—
274C24—62no—
275C24—62no—
276C24—62no—
277C24—62no—
278C24—121no—
279C24—121no—
280C24C22×D311yescharacteristic
281C2×D4—34no—
282C2×D4—62no—
283C2×D4—62no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—62no—
287C2×D4—62no—
288C2×D4—62no—
289C2×D4—62no—
290C2×D4—62no—
291C2×D4—62no—
292C2×D4—62no—
293C2×D4—62no—
294C2×D4—62no—
295C2×D4—62no—
296C2×D4—62no—
297C2×D4—121no—
298C2×D4—121no—
299C2×D4—121no—
300C2×D4—121no—
301C2×D4—121no—
302C22×C4—31no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—31no—
306C22×C4—62no—
307C22×C4—62no—
308C22×C4—62no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—121no—
313C22×C4—121no—
314C22×C4—121no—
315C22×C4C22×D311yescharacteristic
316C22×C4C22×D311yescharacteristic
317C22×C4C22×D311yescharacteristic
318C22×C4C22×D311yescharacteristic
319C22×C4C22×D311yescharacteristic
320C22×C4C22×D311yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—121no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—34no—
329C22⋊C4—34no—
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—121no—
340C42—62no—
341C22×C3—22no—
342C22×C3—22no—
343C22×C3—22no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3C22×D411yescharacteristic
349C22×C3C22×D411yescharacteristic
350C22×C3C22×D411yescharacteristic
351C22×C3C2×#4111yescharacteristic
352C22×C3C2×#4111yescharacteristic
353C22×C3H(5,𝔽2)11yescharacteristic
354C22×C3H(5,𝔽2)11yescharacteristic
355C2×D3—22no—
356C2×D3—22no—
357C2×D3—22no—
358C2×D3—22no—
359C2×D3—22no—
360C2×D3—22no—
361C2×D3—22no—
362C2×D3—22no—
363C2×D3—22no—
364C2×D3—22no—
365C2×D3—22no—
366C2×D3—22no—
367C2×D3—416no—
368C2×D3—41no—
369C2×D3—41no—
370C2×D3—41no—
371C2×D3—41no—
372C2×D3—41no—
373C2×D3—41no—
374C2×D3—41no—
375C2×D3—41no—
376C2×D3—41no—
377C2×D3—41no—
378C2×D3—41no—
379C2×D3—41no—
380C2×D3—41no—
381C2×D3—41no—
382C2×D3—41no—
383C2×D3#12114yescomplemented
384C3×C4—22no—
385C3×C4—22no—
386C3×C4—22no—
387C3×C4—41no—
388C3×C4—41no—
389C3×C4—41no—
390Dic3—22no—
391Dic3—22no—
392Dic3—41no—
393Dic3—41no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—32no—
401C23—32no—
402C23—32no—
403C23—32no—
404C23—32no—
405C23—68no—
406C23—61no—
407C23—61no—
408C23—61no—
409C23—61no—
410C23—61no—
411C23—61no—
412C23—61no—
413C23—61no—
414C23—61no—
415C23—61no—
416C23—61no—
417C23—61no—
418C23—61no—
419C23—61no—
420C23—61no—
421C23—61no—
422C23—61no—
423C23—61no—
424C23—61no—
425C23—61no—
426C23—61no—
427C23—61no—
428C23—61no—
429C23—124no—
430C23—124no—
431C23—124no—
432C23—124no—
433C23—124no—
434C23—124no—
435C23#23912yes—
436C23C23×D311yescharacteristic
437D4—68no—
438D4—124no—
439D4—124no—
440D4—124no—
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—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—32no—
460C2×C4—32no—
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—124no—
472C2×C4—124no—
473C2×C4—124no—
474C2×C4D3×D412yes—
475C2×C4D3×D412yes—
476C2×C4D3×D412yes—
477C2×C3—42no—
478C2×C3C2×#12111yescharacteristic
479C2×C3#52511yescharacteristic
480C2×C3#53811yescharacteristic
481C2×C3D4211yescharacteristic
482C2×C3D4211yescharacteristic
483C2×C3#54611yescharacteristic
484C2×C3#54611yescharacteristic
485D3—24no—
486D3—42no—
487D3—42no—
488D3—81no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—34no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—62no—
498C22—62no—
499C22—62no—
500C22—62no—
501C22—62no—
502C22—62no—
503C22—62no—
504C22—62no—
505C22—62no—
506C22—62no—
507C22—62no—
508C22—62no—
509C22—1216no—
510C22—121no—
511C22—121no—
512C22—121no—
513C22—121no—
514C22—121no—
515C22—121no—
516C22—121no—
517C22—121no—
518C22—121no—
519C22—121no—
520C22—121no—
521C22—121no—
522C22—121no—
523C22—121no—
524C22—121no—
525C22C2×D3×D411yescharacteristic
526C22C2×D3×D411yescharacteristic
527C22C2×D3×D411yescharacteristic
528C22C2×#23911yescharacteristic
529C22D3×#4111yescharacteristic
530C22#100911yescharacteristic
531C22#100911yescharacteristic
532C4—22no—
533C4—22no—
534C4—22no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—62no—
539C4—62no—
540C4—121no—
541C4—121no—
542C3#242411yescharacteristic, complemented, Sylow
543C2—42no—
544C2—64no—
545C2—122no—
546C2—122no—
547C2—241no—
548C2D3×#12111yescharacteristic
549C2#560911yescharacteristic
550C2#560911yescharacteristic
551C2#562711yescharacteristic
552C2#563011yescharacteristic
553C2#563011yescharacteristic
554C2#570411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#396C2
hypercenter#396D3
hypocenterC3#2424
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#895C2semidirect
2. C2×#895C2semidirect
3. C2×#893C2semidirect
4. C2×#882C2semidirect
5. C2×#882C2semidirect
6. C2×D3×C22⋊C4C2semidirect
7. C3×#396C2semidirect
8. #4991C2semidirect
9. #4990C2semidirect
10. #4989C2semidirect
11. #4960C2semidirect
12. #4957C2semidirect
13. #4956C2semidirect
14. #4672C2semidirect
15. #4665C2semidirect
24. C22×D12C22semidirect
27. C22×C4×D3C22semidirect
28. C2×C3×C4⋊C4C22semidirect
29. C2×C3×C4⋊C4C22semidirect
30. C2×C3×C4⋊C4C22semidirect
41. C2×(C2×D3)⋊C4C22semidirect
42. C2×(C2×D3)⋊C4C22semidirect
43. C2×(C2×D3)⋊C4C22semidirect
44. C2×(C2×D3)⋊C4C22semidirect
45. C2×(C2×D3)⋊C4C22semidirect
46. C2×Dic3⋊C4C22semidirect
47. C2×C4×Dic3C22semidirect
54. C3×#95C22semidirect
55. C3⋊#95C22semidirect
56. C3⋊#95C22semidirect
57. C3⋊#95C22semidirect
64. #396D3semidirect
115. C2×D12D4semidirect
141. (C2×D3)⋊C4D4semidirect
142. (C2×D3)⋊C4D4semidirect
162. C2×C4⋊C4C2×D3semidirect
163. C2×C4⋊C4C2×D3semidirect
164. C2×C4⋊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×#8952
C2×#8932
C2×#8822
C2×D3×C22⋊C42
C3×#3962
#49912
#49902
#49892
#49602
#49572
#49562
#46722
#46652
#24243