Finite Group Databasebeta

← #82720

Subgroups — #82720

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×C3×#121C211yesmaximal, characteristic, complemented
2C2×#937C211yesmaximal, characteristic, complemented
3C2×#937C211yesmaximal, characteristic, complemented
4C2×#929C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#5222C211yesmaximal, characteristic, complemented
8#5221C211yesmaximal, characteristic, complemented
9#5213C211yesmaximal, characteristic, complemented
10#4994C211yesmaximal, characteristic, complemented
11#4990C211yesmaximal, characteristic, complemented
12#4961C211yesmaximal, characteristic, complemented
13#4957C211yesmaximal, characteristic, complemented
14#4955C211yesmaximal, characteristic, complemented
15#4952C211yesmaximal, characteristic, complemented
16#2678—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×C3×D4C2211yescharacteristic, complemented
21C23×C3×C4C2211yescharacteristic, complemented
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic
25C22×#67C2211yescharacteristic
26C3×#121—24no—
27C2×C3×C4⋊C4C2211yescharacteristic, complemented
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic, complemented
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic, complemented
34C2×#217C2211yescharacteristic, complemented
35C2×#217C2211yescharacteristic, complemented
36C2×#217C2211yescharacteristic, complemented
37#937—24no—
38#937—24no—
39#929—24no—
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×(C2×D3)⋊C4C2211yescharacteristic
48C2×C4⋊Dic3C2211yescharacteristic, 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×#121D311yescharacteristic, complemented
58C2×#120—31no—
59C2×#120—31no—
60#397—31no—
61#396—31no—
62#394—31no—
63#394—31no—
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
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—41no—
87C2×C3×D4—41no—
88C2×C3×D4—41no—
89C2×C3×D4—41no—
90C2×C3×D4—41no—
91C2×C3×D4D414yescomplemented
92C22×C3×C4—22no—
93C22×C3×C4—22no—
94C22×C3×C4—41no—
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C22×C3×C4C2311yescharacteristic
98C22×C3×C4C2311yescharacteristic
99C2×#67—22no—
100C2×#67—22no—
101C2×#67—22no—
102C2×#67—22no—
103C2×#67—22no—
104C2×#67—22no—
105C2×#67—22no—
106C2×#67—41no—
107C2×#67—41no—
108C2×#67—41no—
109C2×#67—41no—
110C2×#67—41no—
111C22×Dic3C2311yescharacteristic
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115C22×Dic3C2311yescharacteristic
116C22×Dic3C2311yescharacteristic
117C3×C4⋊C4—22no—
118C3×C22⋊C4—22no—
119C3×C22⋊C4—22no—
120#217—22no—
121#217—22no—
122#217—22no—
123#217—41no—
124#217—41no—
125#217—41no—
126#217—41no—
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—
134(C2×D3)⋊C4D414yescomplemented
135C4⋊Dic3—41no—
136Dic3⋊C4—22no—
137Dic3⋊C4—22no—
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×C4C2×D311yescharacteristic, complemented
148#123—64no—
149#121—24no—
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⋊C4—31no—
170C2×C22⋊C4—31no—
171C2×C22⋊C4C2×D311yescharacteristic
172C2×C22⋊C4C2×D311yescharacteristic, complemented
173C2×C42—31no—
174#95—31no—
175#95—31no—
176#95—31no—
177#95—31no—
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—44no—
235C2×C3×C4C2×D412yes—
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×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—21no—
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—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4—62no—
282C2×D4—62no—
283C2×D4—62no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4—121no—
291C2×D4#6714yescomplemented
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—41no—
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—22no—
311C22⋊C4—34no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C22⋊C4—62no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C22⋊C4—121no—
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—22no—
371C3×C4—22no—
372C3×C4—41no—
373C3×C4—41no—
374Dic3—22no—
375Dic3—41no—
376Dic3—41no—
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—
420C23C2×#6712yes—
421C23C2×#6712yes—
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—32no—
441C2×C4—44no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
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×C4D3×D412yes—
460C2×C4C2×#6712yes—
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
510C22#100411yescharacteristic
511C22#100411yescharacteristic
512C22#100411yescharacteristic
513C22D3×#4111yescharacteristic
514C22#100911yescharacteristic
515C22C22×#6711yescharacteristic
516C4—22no—
517C4—22no—
518C4—41no—
519C4—41no—
520C4—62no—
521C4—121no—
522C4—121no—
523C4—121no—
524C4—121no—
525C4—121no—
526C3#267811yescharacteristic, complemented, Sylow
527C2—42no—
528C2—42no—
529C2—81no—
530C2—122no—
531C2—122no—
532C2#560711yescharacteristic
533C2#561211yescharacteristic
534C2#561311yescharacteristic
535C2#562011yescharacteristic
536C2D4×#6711yescharacteristic
537C2#580411yescharacteristic
538C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2678
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#121C2semidirect
2. C2×#937C2semidirect
3. C2×#937C2semidirect
4. C2×#929C2semidirect
5. #5223C2semidirect
6. #5223C2semidirect
7. #5222C2semidirect
8. #5221C2semidirect
9. #5213C2semidirect
10. #4994C2semidirect
11. #4990C2semidirect
12. #4961C2semidirect
13. #4957C2semidirect
14. #4955C2semidirect
15. #4952C2semidirect
20. C22×C3×D4C22semidirect
21. C23×C3×C4C22semidirect
27. C2×C3×C4⋊C4C22semidirect
29. C2×C3×C22⋊C4C22semidirect
33. C2×#217C22semidirect
34. C2×#217C22semidirect
35. C2×#217C22semidirect
36. C2×#217C22semidirect
48. C2×C4⋊Dic3C22semidirect
49. C2×Dic3⋊C4C22semidirect
50. C2×Dic3⋊C4C22semidirect
51. C2×C4×Dic3C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
57. C2×#121D3semidirect
91. C2×C3×D4D4semidirect
134. (C2×D3)⋊C4D4semidirect
146. C22×D4C2×D3semidirect
147. C23×C4C2×D3semidirect
155. C2×C4⋊C4C2×D3semidirect
172. C2×C22⋊C4C2×D3semidirect
291. C2×D4#67semidirect
526. C3#2678semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×#9372
C2×#9292
#52232
#52222
#52212
#52132
#49942
#49902
#49612
#49572
#49552
#49522
#26783