Finite Group Databasebeta

← #76771

Subgroups — #76771

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×#939C211yesmaximal, characteristic, complemented
2C2×#937C211yesmaximal, characteristic, complemented
3C2×#937C211yesmaximal, characteristic, complemented
4C2×#935C211yesmaximal, characteristic, complemented
5C3×#392C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#5222C211yesmaximal, characteristic, complemented
8#4959C211yesmaximal, characteristic, complemented
9#4958C211yesmaximal, characteristic, complemented
10#4957C211yesmaximal, characteristic, complemented
11#4957C211yesmaximal, characteristic, complemented
12#4955C211yesmaximal, characteristic, complemented
13#4944C211yesmaximal, characteristic, complemented
14#4676C211yesmaximal, characteristic, complemented
15#4675C211yesmaximal, characteristic, complemented
16#2678—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
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic, complemented
28C2×C3×C22⋊C4C2211yescharacteristic, complemented
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic, complemented
33C2×#217C2211yescharacteristic, complemented
34C2×#217C2211yescharacteristic, complemented
35#939—24no—
36#937—24no—
37#937—24no—
38#935—24no—
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
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×Dic3⋊C4C2211yescharacteristic, complemented
48C2×Dic3⋊C4C2211yescharacteristic, complemented
49C2×Dic3⋊C4C2211yescharacteristic, complemented
50C2×Dic3⋊C4C2211yescharacteristic, complemented
51C2×C4×Dic3C2211yescharacteristic, complemented
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#394—31no—
64#392—31no—
65#392D311yescharacteristic, complemented
66#390—31no—
67#390—31no—
68#388—31no—
69#386—31no—
70#379—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C24×C3C2311yescharacteristic
74C23×D3—22no—
75C23×D3—22no—
76C23×D3—22no—
77C23×D3—41no—
78C23×D3—41no—
79C23×D3—41no—
80C23×D3—41no—
81C23×D3C2311yescharacteristic
82C23×D3C2311yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—22no—
86C2×C3×D4—22no—
87C2×C3×D4—41no—
88C2×C3×D4—41no—
89C2×C3×D4—41no—
90C2×C3×D4—41no—
91C2×C3×D4—41no—
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C2×#67—22no—
97C2×#67—22no—
98C2×#67—22no—
99C2×#67—22no—
100C2×#67—22no—
101C2×#67—22no—
102C2×#67—41no—
103C2×#67—41no—
104C2×#67—41no—
105C2×#67—41no—
106C2×#67—41no—
107C2×#67D414yescomplemented
108C22×Dic3—22no—
109C22×Dic3—22no—
110C22×Dic3—41no—
111C22×Dic3C2311yescharacteristic
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115C22×Dic3C2311yescharacteristic
116C22×Dic3C2311yescharacteristic
117C3×C22⋊C4—41no—
118C3×C22⋊C4—41no—
119C3×C22⋊C4—41no—
120#217—22no—
121#217—22no—
122#217—22no—
123#217—22no—
124#217—41no—
125#217D414yescomplemented
126(C2×D3)⋊C4—22no—
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—22no—
129(C2×D3)⋊C4—22no—
130(C2×D3)⋊C4—22no—
131(C2×D3)⋊C4—41no—
132(C2×D3)⋊C4—41no—
133(C2×D3)⋊C4—41no—
134Dic3⋊C4—22no—
135Dic3⋊C4—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×D4C2×D311yescharacteristic
145C22×D4C2×D311yescharacteristic
146C22×D4C2×D311yescharacteristic, complemented
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⋊C4—31no—
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⋊C4—31no—
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×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3C2×D412yes—
198C23×C3C2×D412yes—
199C23×C3C2411yescharacteristic
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—44no—
213C22×D3—44no—
214C22×D3—44no—
215C22×D3—44no—
216C22×D3—44no—
217C22×D3—44no—
218C22×D3—44no—
219C22×D3C2×D412yes—
220C22×D3#4112yes—
221C3×D4—44no—
222C3×D4—44no—
223C3×D4—44no—
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—
236#67—44no—
237#67—44no—
238#67—44no—
239#67—44no—
240C2×Dic3—21no—
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×Dic3—21no—
255C2×Dic3—44no—
256C2×Dic3C2×D412yes—
257C2×Dic3C2×D412yes—
258C2×Dic3C2×D412yes—
259C24—31no—
260C24—31no—
261C24—62no—
262C24—62no—
263C24—62no—
264C24—121no—
265C24—121no—
266C24—121no—
267C24—121no—
268C24C22×D311yescharacteristic
269C24C22×D311yescharacteristic
270C24C22×D311yescharacteristic
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—34no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—62no—
282C2×D4—62no—
283C2×D4—62no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—62no—
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—31no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4—121no—
301C22×C4C22×D311yescharacteristic
302C22×C4C22×D311yescharacteristic
303C22×C4C22×D311yescharacteristic
304C22×C4C22×D311yescharacteristic
305C4⋊C4—62no—
306C4⋊C4—62no—
307C4⋊C4—62no—
308C4⋊C4—121no—
309C22⋊C4—34no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—62no—
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—22no—
331C22×C3—22no—
332C22×C3—22no—
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×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3C22×D411yescharacteristic
349C22×C3C22×D411yescharacteristic
350C22×C3C2×#4111yescharacteristic
351C22×C3H(5,𝔽2)11yescharacteristic
352C22×C3H(5,𝔽2)11yescharacteristic
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—416no—
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—41no—
371C3×C4—41no—
372C3×C4—41no—
373C3×C4—41no—
374Dic3—22no—
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—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—32no—
400C23—32no—
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—124no—
414C23—124no—
415C23—124no—
416C23—124no—
417C23—124no—
418C23—124no—
419C23—124no—
420C23D3×D412yes—
421C23D3×D412yes—
422C23C23×D311yescharacteristic
423D4—44no—
424D4—44no—
425D4—44no—
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—32no—
443C2×C4—32no—
444C2×C4—32no—
445C2×C4—61no—
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—61no—
460C2×C4—124no—
461C2×C3—42no—
462C2×C3—42no—
463C2×C3—81no—
464C2×C3#53411yescharacteristic
465C2×C3#53511yescharacteristic
466C2×C3#53811yescharacteristic
467C2×C3D4211yescharacteristic
468C2×C3#54611yescharacteristic
469C2×C3#54611yescharacteristic
470C2×C3#56011yescharacteristic
471D3—42no—
472D3—42no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—62no—
495C22—62no—
496C22—62no—
497C22—62no—
498C22—62no—
499C22—62no—
500C22—1216no—
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
513C22#100411yescharacteristic
514C22#100411yescharacteristic
515C22D3×#4111yescharacteristic
516C4—41no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—62no—
521C4—62no—
522C4—62no—
523C4—121no—
524C4—121no—
525C4—121no—
526C3#267811yescharacteristic, complemented, Sylow
527C2—42no—
528C2—42no—
529C2—81no—
530C2—122no—
531C2—122no—
532C2#558911yescharacteristic
533C2#559011yescharacteristic
534C2#559211yescharacteristic
535C2#561011yescharacteristic
536C2#561011yescharacteristic
537C2#561311yescharacteristic
538C2#572011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2678
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×#937C2semidirect
3. C2×#937C2semidirect
4. C2×#935C2semidirect
5. C3×#392C2semidirect
6. #5223C2semidirect
7. #5222C2semidirect
8. #4959C2semidirect
9. #4958C2semidirect
10. #4957C2semidirect
11. #4957C2semidirect
12. #4955C2semidirect
13. #4944C2semidirect
14. #4676C2semidirect
15. #4675C2semidirect
20. C22×C3×D4C22semidirect
25. C23×Dic3C22semidirect
27. C2×C3×C22⋊C4C22semidirect
28. C2×C3×C22⋊C4C22semidirect
32. C2×#217C22semidirect
33. C2×#217C22semidirect
34. C2×#217C22semidirect
47. C2×Dic3⋊C4C22semidirect
48. C2×Dic3⋊C4C22semidirect
49. C2×Dic3⋊C4C22semidirect
50. C2×Dic3⋊C4C22semidirect
51. C2×C4×Dic3C22semidirect
52. C3×#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
65. #392D3semidirect
107. C2×#67D4semidirect
125. #217D4semidirect
146. C22×D4C2×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×#9392
C2×#9372
C2×#9352
C3×#3922
#52232
#52222
#49592
#49582
#49572
#49552
#49442
#46762
#46752
#26783