Finite Group Databasebeta

← #81543

Subgroups — #81543

Order 384.

410 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5628C211yesmaximal, characteristic, complemented
2#5525C211yesmaximal, characteristic
3C3×#485C211yesmaximal, characteristic, complemented
4#5046C211yesmaximal, characteristic, complemented
5#4996C211yesmaximal, characteristic, complemented
6#4886C211yesmaximal, characteristic, complemented
7#4885C211yesmaximal, characteristic, complemented
8#4866C211yesmaximal, characteristic, complemented
9#4864C211yesmaximal, characteristic, complemented
10#4812C211yesmaximal, characteristic, complemented
11#4807C211yesmaximal, characteristic, complemented
12#4806C211yesmaximal, characteristic, complemented
13#4806C211yesmaximal, characteristic, complemented
14#4731C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3200—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic, complemented
18C2×#235C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C3×C22⋊Q8C2211yescharacteristic
21C3×#117C2211yescharacteristic
22#946C2211yescharacteristic, complemented
23C2×#215C2211yescharacteristic
24C2×#215C2211yescharacteristic
25#924C2211yescharacteristic
26C2×#204C2211yescharacteristic
27C2×#204C2211yescharacteristic
28C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
29C2×D3×C8C2211yescharacteristic, complemented
30#897C2211yescharacteristic, complemented
31#895C2211yescharacteristic
32#895C2211yescharacteristic
33#895C2211yescharacteristic
34#895C2211yescharacteristic
35#894—21no—
36#892C2211yescharacteristic, complemented
37#888—21no—
38#885—21no—
39#883—21no—
40#882—21no—
41#882—21no—
42#875—21no—
43#874—21no—
44C3×#107C2211yescharacteristic, complemented
45C3×#106C2211yescharacteristic, complemented
46C3×#103C2211yescharacteristic, complemented
47C3×#103C2211yescharacteristic, complemented
48C3×C22⋊C8C2211yescharacteristic
49#830C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54#810C2211yescharacteristic, complemented
55#810C2211yescharacteristic, complemented
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#536—31no—
61#535—31no—
62#485—31no—
63#485D311yescharacteristic, complemented
64#482—31no—
65#482—31no—
66#468—31no—
67#468—31no—
68#465—31no—
69#465—31no—
70#450—31no—
71#450—31no—
72#448—31no—
73#448—31no—
74#408—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C2×#67D411yescharacteristic
79C2×#67D411yescharacteristic
80D3×Q8—41no—
81D3×Q8—41no—
82#235—41no—
83#235—41no—
84C2×D12—21no—
85C2×D12—21no—
86C2×D12—21no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12—41no—
90C2×D12—41no—
91C2×D12D411yescharacteristic
92C2×D12C2311yescharacteristic
93C2×D12C2311yescharacteristic
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C2×C4×D3C2311yescharacteristic
97C2×Dic6—21no—
98C2×Dic6D411yescharacteristic
99C2×Dic6C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C2×C3×C8C2311yescharacteristic
102C3×C4⋊C4—21no—
103C3×C4⋊C4C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C4⋊C4C2311yescharacteristic
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108C3×C42—21no—
109#217—21no—
110#215—41no—
111#215—41no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3C2311yescharacteristic
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123C4×Dic3—21no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126#204—41no—
127#204—41no—
128ΓL(1,𝔽25)—41no—
129D3×C8—41no—
130C2×#41—31no—
131C22×Q8—31no—
132C22×D4—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×M16—31no—
138C22×C8—31no—
139C4⋊Q8—61no—
140#127—61no—
141#124—61no—
142#124—61no—
143#123—61no—
144#123—61no—
145C22⋊Q8—31no—
146C22⋊Q8—31no—
147C22⋊Q8—31no—
148C22⋊Q8C2×D311yescharacteristic
149#121—31no—
150#121—31no—
151#121—31no—
152#121—31no—
153#120—61no—
154#120—61no—
155#117—31no—
156#117C2×D311yescharacteristic
157#107—31no—
158#107C2×D311yescharacteristic, complemented
159#106—31no—
160#106C2×D311yescharacteristic, complemented
161#103—31no—
162#103—31no—
163#103C2×D311yescharacteristic, complemented
164#103C2×D311yescharacteristic, complemented
165#102—31no—
166#102—31no—
167#102—31no—
168#102—31no—
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8C2×D311yescharacteristic
173C23×C3C2×D411yescharacteristic
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—42no—
177C22×D3—81no—
178C22×D3—81no—
179C22×D3C2×D411yescharacteristic
180C22×D3C2×D411yescharacteristic
181C3×Q8—21no—
182C3×Q8—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2411yescharacteristic
191#67—41no—
192#67—41no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197D12—21no—
198D12—21no—
199D12—21no—
200D12—21no—
201D12—41no—
202D12—42no—
203D12—81no—
204D12—81no—
205D12—81no—
206D12—81no—
207C4×D3—21no—
208C4×D3—21no—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—42no—
212Dic6—21no—
213Dic6—21no—
214Dic6—41no—
215Dic6—42no—
216C3×C8—21no—
217C3×C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220C24—61no—
221#41—121no—
222#41—121no—
223C2×Q8—31no—
224C2×Q8—31no—
225C2×Q8—61no—
226C2×Q8—121no—
227C2×Q8—121no—
228C2×Q8C22×D311yescharacteristic
229C2×D4—31no—
230C2×D4—31no—
231C2×D4—31no—
232C2×D4—31no—
233C2×D4—31no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—121no—
240C2×D4—121no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—61no—
244C22×C4C22×D311yescharacteristic
245SD16—121no—
246SD16—121no—
247SD16—121no—
248SD16—121no—
249M16—121no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8—121no—
253C2×C8C22×D311yescharacteristic
254C2×C8C22×D311yescharacteristic
255C4⋊C4—21no—
256C4⋊C4—31no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4C22×D311yescharacteristic
261C4⋊C4C22×D311yescharacteristic
262C4⋊C4C22×D311yescharacteristic
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C42—21no—
274C42—61no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3C22×D411yescharacteristic
279C2×D3—21no—
280C2×D3—21no—
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—21no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—84no—
292C3×C4—21no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4—41no—
297C3×C4H(5,𝔽2)11yescharacteristic
298C3×C4H(5,𝔽2)11yescharacteristic
299Dic3—21no—
300Dic3—21no—
301Dic3—41no—
302Dic3—41no—
303C23—31no—
304C23—31no—
305C23—61no—
306C23—61no—
307C23—122no—
308C23—241no—
309C23—241no—
310C23D3×D411yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—61no—
314Q8—61no—
315Q8—121no—
316Q8—122no—
317D4—61no—
318D4—61no—
319D4—61no—
320D4—61no—
321D4—121no—
322D4—122no—
323D4—121no—
324D4—121no—
325D4—241no—
326D4—241no—
327D4—241no—
328D4—241no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—122no—
344C2×C4D3×D411yescharacteristic
345C2×C4C23×D311yescharacteristic
346C8—21no—
347C8—21no—
348C8—61no—
349C8—61no—
350C2×C3—41no—
351C2×C3#53411yescharacteristic
352C2×C3#57711yescharacteristic
353C2×C3#57711yescharacteristic
354D3—41no—
355D3—41no—
356D3—81no—
357D3—81no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—61no—
362C22—61no—
363C22—61no—
364C22—61no—
365C22—61no—
366C22—61no—
367C22—121no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22—121no—
373C22—244no—
374C22C2×D3×D411yescharacteristic
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—61no—
381C4—61no—
382C4—121no—
383C4—121no—
384C4#100911yescharacteristic
385C4#100911yescharacteristic
386C3#320011yescharacteristic, complemented, Sylow
387C2—41no—
388C2—121no—
389C2—121no—
390C2—241no—
391C2—241no—
392C2#565311yescharacteristic
393C2#577011yescharacteristic
394C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#485C2
socleC22×C3C22×D4
prime radical#5525C2
hypercenter#485D3
hypocenterC3#3200
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #5628C2semidirect
3. C3×#485C2semidirect
4. #5046C2semidirect
5. #4996C2semidirect
6. #4886C2semidirect
7. #4885C2semidirect
8. #4866C2semidirect
9. #4864C2semidirect
10. #4812C2semidirect
11. #4807C2semidirect
12. #4806C2semidirect
13. #4806C2semidirect
14. #4731C2semidirect
15. #4727C2semidirect
17. C2×D3×Q8C22semidirect
22. #946C22semidirect
28. C2×ΓL(1,𝔽25)C22semidirect
29. C2×D3×C8C22semidirect
30. #897C22semidirect
36. #892C22semidirect
44. C3×#107C22semidirect
45. C3×#106C22semidirect
46. C3×#103C22semidirect
47. C3×#103C22semidirect
52. (C2×D3)⋊C8C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. #810C22semidirect
55. #810C22semidirect
58. #808C22semidirect
59. #807C22semidirect
63. #485D3semidirect
158. #107C2×D3semidirect
160. #106C2×D3semidirect
163. #103C2×D3semidirect
164. #103C2×D3semidirect
386. C3#3200semidirect

Maximal subgroups

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

subgroupindex
#56282
#55252
C3×#4852
#50462
#49962
#48862
#48852
#48662
#48642
#48122
#48072
#48062
#47312
#47272
#32003