Finite Group Databasebeta

← #77137

Subgroups — #77137

Order 384.

802 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#938C211yesmaximal, characteristic, complemented
2C2×#937C211yesmaximal, characteristic, complemented
3C2×#894C211yesmaximal, characteristic, complemented
4C2×#882C211yesmaximal, characteristic, complemented
5C3×#394C211yesmaximal, characteristic, complemented
6#4994C211yesmaximal, characteristic, complemented
7#4960C211yesmaximal, characteristic, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4959C211yesmaximal, characteristic, complemented
10#4957C211yesmaximal, characteristic, complemented
11#4957C211yesmaximal, characteristic, complemented
12#4955C211yesmaximal, characteristic, complemented
13#4952C211yesmaximal, characteristic, complemented
14#4670C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2678—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic, complemented
23C22×D12C2211yescharacteristic
24C22×D12C2211yescharacteristic
25C22×C4×D3C2211yescharacteristic, complemented
26C2×C3×C4⋊C4C2211yescharacteristic, complemented
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic, complemented
30C2×C3×C22⋊C4C2211yescharacteristic, complemented
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic, complemented
34#938—24no—
35#937—24no—
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
46C2×C4⋊Dic3C2211yescharacteristic, complemented
47C2×Dic3⋊C4C2211yescharacteristic, complemented
48C2×Dic3⋊C4C2211yescharacteristic, complemented
49C2×C4×Dic3C2211yescharacteristic, complemented
50#894—24no—
51#882—24no—
52C3×#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C2×#123—31no—
57C2×#121—31no—
58C2×#120—31no—
59C2×#120—31no—
60#397—31no—
61#396—31no—
62#394—31no—
63#394D311yescharacteristic, complemented
64#392—31no—
65#392—31no—
66#390—31no—
67#390—31no—
68#388—31no—
69#386—31no—
70#379—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C23×D3—22no—
74C23×D3—22no—
75C23×D3—22no—
76C23×D3—41no—
77C23×D3—41no—
78C23×D3—41no—
79C23×D3—41no—
80C23×D3C2311yescharacteristic
81C23×D3C2311yescharacteristic
82C23×D3C2311yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—41no—
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—
97C2×#67—41no—
98C2×#67—41no—
99C2×#67—41no—
100C2×#67—41no—
101C2×#67—41no—
102C2×#67D414yescomplemented
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108C2×D12—22no—
109C2×D12—22no—
110C2×D12—22no—
111C2×D12—22no—
112C2×D12—41no—
113C2×D12—41no—
114C2×C4×D3—22no—
115C2×C4×D3—22no—
116C2×C4×D3—41no—
117C3×C4⋊C4—22no—
118C3×C22⋊C4—22no—
119C3×C22⋊C4—41no—
120C3×C22⋊C4—41no—
121C3×C22⋊C4—41no—
122#217—22no—
123#217—22no—
124#217—22no—
125(C2×D3)⋊C4—22no—
126(C2×D3)⋊C4—22no—
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—22no—
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—41no—
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—41no—
133(C2×D3)⋊C4—41no—
134(C2×D3)⋊C4D414yescomplemented
135C4⋊Dic3—22no—
136Dic3⋊C4—22no—
137Dic3⋊C4—41no—
138C4×Dic3—41no—
139C25—31no—
140C22×D4—31no—
141C22×D4—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—31no—
145C22×D4—31no—
146C22×D4C2×D311yescharacteristic
147C23×C4—31no—
148#123—64no—
149#121—64no—
150#120—64no—
151#120—64no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4C2×D311yescharacteristic, complemented
156C2×C22⋊C4—31no—
157C2×C22⋊C4—31no—
158C2×C22⋊C4—31no—
159C2×C22⋊C4—31no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4—31no—
162C2×C22⋊C4—31no—
163C2×C22⋊C4—31no—
164C2×C22⋊C4—31no—
165C2×C22⋊C4—31no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—31no—
169C2×C22⋊C4C2×D311yescharacteristic
170C2×C22⋊C4C2×D311yescharacteristic
171C2×C22⋊C4C2×D311yescharacteristic, complemented
172C2×C22⋊C4C2×D311yescharacteristic, complemented
173C2×C42—31no—
174#95—31no—
175#95—31no—
176#95—31no—
177#95C2×D311yescharacteristic, complemented
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3C2×D412yes—
192C23×C3C2411yescharacteristic
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—21no—
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—44no—
212C22×D3—44no—
213C22×D3—44no—
214C22×D3—44no—
215C22×D3—44no—
216C22×D3—44no—
217C22×D3—44no—
218C22×D3C2×D412yes—
219C22×D3C2×D412yes—
220C22×D3#4112yes—
221C3×D4—44no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
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×C4C2×D412yes—
237#67—44no—
238#67—44no—
239#67—44no—
240#67—44no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3C2×D412yes—
255C2×Dic3C2×D412yes—
256D12—44no—
257D12—44no—
258C4×D3—44no—
259C24—31no—
260C24—31no—
261C24—31no—
262C24—62no—
263C24—62no—
264C24—62no—
265C24—121no—
266C24—121no—
267C24—121no—
268C24—121no—
269C24C22×D311yescharacteristic
270C24C22×D311yescharacteristic
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—34no—
274C2×D4—41no—
275C2×D4—62no—
276C2×D4—62no—
277C2×D4—62no—
278C2×D4—62no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4—62no—
282C2×D4—62no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4—121no—
292C22×C4—31no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—62no—
298C22×C4—62no—
299C22×C4—121no—
300C22×C4C22×D311yescharacteristic
301C22×C4C22×D311yescharacteristic
302C22×C4C22×D311yescharacteristic
303C22×C4C22×D311yescharacteristic
304C22×C4C22×D311yescharacteristic
305C4⋊C4—22no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C4⋊C4—121no—
309C22⋊C4—22no—
310C22⋊C4—34no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C22⋊C4—62no—
319C22⋊C4—62no—
320C22⋊C4—62no—
321C22⋊C4—62no—
322C22⋊C4—121no—
323C22⋊C4—121no—
324C22⋊C4—121no—
325C22⋊C4—121no—
326C42—121no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3C22×D411yescharacteristic
342C22×C3C22×D411yescharacteristic
343C22×C3C2×#4111yescharacteristic
344C22×C3H(5,𝔽2)11yescharacteristic
345C22×C3H(5,𝔽2)11yescharacteristic
346C22×C3H(5,𝔽2)11yescharacteristic
347C22×C3H(5,𝔽2)11yescharacteristic
348C2×D3—22no—
349C2×D3—22no—
350C2×D3—22no—
351C2×D3—22no—
352C2×D3—22no—
353C2×D3—22no—
354C2×D3—22no—
355C2×D3—22no—
356C2×D3—22no—
357C2×D3—416no—
358C2×D3—41no—
359C2×D3—41no—
360C2×D3—41no—
361C2×D3—41no—
362C2×D3—41no—
363C2×D3—41no—
364C2×D3—41no—
365C2×D3—41no—
366C2×D3—41no—
367C2×D3—41no—
368C2×D3—41no—
369C2×D3—41no—
370C3×C4—22no—
371C3×C4—41no—
372C3×C4—41no—
373C3×C4—41no—
374C3×C4—41no—
375Dic3—22no—
376Dic3—22no—
377Dic3—41no—
378Dic3—41no—
379Dic3—41no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—32no—
394C23—32no—
395C23—32no—
396C23—61no—
397C23—61no—
398C23—61no—
399C23—61no—
400C23—61no—
401C23—61no—
402C23—61no—
403C23—61no—
404C23—61no—
405C23—61no—
406C23—61no—
407C23—61no—
408C23—61no—
409C23—61no—
410C23—61no—
411C23—61no—
412C23—61no—
413C23—61no—
414C23—124no—
415C23—124no—
416C23—124no—
417C23—124no—
418C23—124no—
419C23—124no—
420C23—124no—
421C23D3×D412yes—
422C23C23×D311yescharacteristic
423D4—44no—
424D4—124no—
425D4—124no—
426D4—124no—
427D4—124no—
428D4—124no—
429D4—124no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—32no—
445C2×C4—32no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—61no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—61no—
459C2×C4—124no—
460C2×C4D3×D412yes—
461C2×C3—42no—
462C2×C3—81no—
463C2×C3#53411yescharacteristic
464C2×C3#53511yescharacteristic
465C2×C3#53811yescharacteristic
466C2×C3D4211yescharacteristic
467C2×C3#54611yescharacteristic
468C2×C3#54611yescharacteristic
469C2×C3#56011yescharacteristic
470D3—42no—
471D3—42no—
472D3—42no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—62no—
488C22—62no—
489C22—62no—
490C22—62no—
491C22—62no—
492C22—62no—
493C22—62no—
494C22—62no—
495C22—62no—
496C22—1216no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22—121no—
501C22—121no—
502C22—121no—
503C22—121no—
504C22—121no—
505C22—121no—
506C22—121no—
507C22—121no—
508C22—121no—
509C22C2×D3×D411yescharacteristic
510C22C2×D3×D411yescharacteristic
511C22#100411yescharacteristic
512C22#100411yescharacteristic
513C22D3×#4111yescharacteristic
514C22#100911yescharacteristic
515C22#100911yescharacteristic
516C4—22no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—62no—
522C4—62no—
523C4—121no—
524C4—121no—
525C4—121no—
526C3#267811yescharacteristic, complemented, Sylow
527C2—42no—
528C2—81no—
529C2—122no—
530C2—122no—
531C2—122no—
532C2#559011yescharacteristic
533C2#560711yescharacteristic
534C2#561011yescharacteristic
535C2#562011yescharacteristic
536C2#565311yescharacteristic
537C2#565711yescharacteristic
538C2#567611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2678
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#938C2semidirect
2. C2×#937C2semidirect
3. C2×#894C2semidirect
4. C2×#882C2semidirect
5. C3×#394C2semidirect
6. #4994C2semidirect
7. #4960C2semidirect
8. #4960C2semidirect
9. #4959C2semidirect
10. #4957C2semidirect
11. #4957C2semidirect
12. #4955C2semidirect
13. #4952C2semidirect
14. #4670C2semidirect
15. #4665C2semidirect
22. C22×#67C22semidirect
25. C22×C4×D3C22semidirect
26. C2×C3×C4⋊C4C22semidirect
29. C2×C3×C22⋊C4C22semidirect
30. C2×C3×C22⋊C4C22semidirect
33. C2×#217C22semidirect
44. C2×(C2×D3)⋊C4C22semidirect
45. C2×(C2×D3)⋊C4C22semidirect
46. C2×C4⋊Dic3C22semidirect
47. C2×Dic3⋊C4C22semidirect
48. C2×Dic3⋊C4C22semidirect
49. C2×C4×Dic3C22semidirect
52. C3×#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
63. #394D3semidirect
102. C2×#67D4semidirect
134. (C2×D3)⋊C4D4semidirect
155. C2×C4⋊C4C2×D3semidirect
171. C2×C22⋊C4C2×D3semidirect
172. C2×C22⋊C4C2×D3semidirect
177. #95C2×D3semidirect
526. C3#2678semidirect

Maximal subgroups

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

subgroupindex
C2×#9382
C2×#9372
C2×#8942
C2×#8822
C3×#3942
#49942
#49602
#49592
#49572
#49552
#49522
#46702
#46652
#26783