Finite Group Databasebeta

← #81563

Subgroups — #81563

Order 384.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5729C211yesmaximal, characteristic, complemented
2#5715C211yesmaximal, characteristic
3C3×#486C211yesmaximal, characteristic, complemented
4#5089C211yesmaximal, characteristic, complemented
5#5071C211yesmaximal, characteristic, complemented
6#4772C211yesmaximal, characteristic, complemented
7#4771C211yesmaximal, characteristic, complemented
8#4770C211yesmaximal, characteristic, complemented
9D3×#102C211yesmaximal, characteristic, complemented
10#4768C211yesmaximal, characteristic, complemented
11#4765C211yesmaximal, characteristic, complemented
12#4764C211yesmaximal, characteristic, complemented
13#4760C211yesmaximal, characteristic, complemented
14#4697C211yesmaximal, characteristic, complemented
15#4689C211yesmaximal, characteristic, complemented
16#3136—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic, complemented
20C3×#127C2211yescharacteristic
21#946—21no—
22#944—21no—
23#940C2211yescharacteristic
24#939—21no—
25#939—21no—
26#937—21no—
27#937—21no—
28#936C2211yescharacteristic
29C2×ΓL(1,𝔽25)C2211yescharacteristic
30C2×D3×C8C2211yescharacteristic
31#896—21no—
32#892C2211yescharacteristic
33D3×C4⋊C4C2211yescharacteristic
34#890C2211yescharacteristic, complemented
35#889—21no—
36#888C2211yescharacteristic, complemented
37#875C2211yescharacteristic
38#872C2211yescharacteristic
39#870C2211yescharacteristic, complemented
40C3×#102C2211yescharacteristic
41C3×#102C2211yescharacteristic
42C3×#102C2211yescharacteristic, complemented
43C3×#102C2211yescharacteristic, complemented
44C3×C4×C8C2211yescharacteristic, complemented
45#832C2211yescharacteristic
46#832C2211yescharacteristic
47#832C2211yescharacteristic, complemented
48#832C2211yescharacteristic, complemented
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#816C2211yescharacteristic, complemented
52#816C2211yescharacteristic, complemented
53#815C2211yescharacteristic, complemented
54C8×Dic3C2211yescharacteristic, complemented
55#810C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#809C2211yescharacteristic
59#803C2211yescharacteristic, complemented
60#544—31no—
61#535—31no—
62#490—31no—
63#490—31no—
64#489—31no—
65#489—31no—
66#488—31no—
67#488—31no—
68#486—31no—
69#486D311yescharacteristic, complemented
70#418—31no—
71#417—31no—
72#416—31no—
73C2×#102—31no—
74#405—31no—
75C23×D3—21no—
76C2×C3×Q8—21no—
77C2×C3×D4—21no—
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3—21no—
86#237—22no—
87#237—22no—
88D3×D4—22no—
89D3×D4—22no—
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3C2311yescharacteristic
93C2×Dic6—21no—
94C2×Dic6C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C42C2311yescharacteristic
101#217—21no—
102#217—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4D411yescharacteristic
107(C2×D3)⋊C4D411yescharacteristic
108C4⋊Dic3—21no—
109C4⋊Dic3C2311yescharacteristic
110C4⋊Dic3C2311yescharacteristic
111Dic3⋊C4—21no—
112Dic3⋊C4—21no—
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4D411yescharacteristic
117Dic3⋊C4D411yescharacteristic
118C4×Dic3—21no—
119C4×Dic3C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122ΓL(1,𝔽25)—22no—
123D3×C8—22no—
124C2×#41—31no—
125C22×D4—31no—
126C2×M16—31no—
127C22×C8—31no—
128C4⋊Q8—31no—
129C4⋊Q8—61no—
130C4⋊Q8C2×D311yescharacteristic, complemented
131#127—31no—
132#127C2×D311yescharacteristic
133#125—31no—
134#125—31no—
135#125—61no—
136#124—31no—
137#124—31no—
138C22⋊Q8—61no—
139C22⋊Q8—61no—
140#121—61no—
141#121—61no—
142#120—61no—
143#120—61no—
144#117—31no—
145#117—31no—
146C2×C4⋊C4—31no—
147#103—31no—
148#103—31no—
149#103—31no—
150#103—31no—
151#102—31no—
152#102—31no—
153#102—31no—
154#102—31no—
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102C2×D311yescharacteristic
160#102C2×D311yescharacteristic
161#102C2×D311yescharacteristic, complemented
162#102C2×D311yescharacteristic, complemented
163#97—31no—
164#97—31no—
165C4×C8—31no—
166C4×C8C2×D311yescharacteristic, complemented
167C23×C3—21no—
168C23×C3—21no—
169C22×D3—21no—
170C22×D3—42no—
171C22×D3—81no—
172C22×D3—81no—
173C22×D3C2×D411yescharacteristic
174C3×Q8—81no—
175C3×D4—21no—
176C3×D4—21no—
177C3×D4—21no—
178C3×D4—21no—
179C3×D4—81no—
180C3×D4—81no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2411yescharacteristic
186#67—42no—
187#67—42no—
188#67—81no—
189#67—81no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—42no—
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197D12—21no—
198D12—21no—
199C4×D3—81no—
200C4×D3#4112yes—
201C4×D3#4112yes—
202Dic6—21no—
203Dic6—21no—
204Dic6—81no—
205C3×C8—21no—
206C3×C8—21no—
207C3⋊C8—21no—
208C3⋊C8—21no—
209C24—61no—
210#41—62no—
211#41—62no—
212C2×Q8—21no—
213C2×Q8—31no—
214C2×Q8—61no—
215C2×D4—21no—
216C2×D4—21no—
217C2×D4—31no—
218C2×D4—62no—
219C2×D4—62no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4C22×D311yescharacteristic
225C2×D4C22×D311yescharacteristic
226C22×C4—31no—
227C22×C4—61no—
228C22×C4—61no—
229M16—62no—
230C2×C8—31no—
231C2×C8—31no—
232C2×C8—62no—
233C2×C8C22×D311yescharacteristic
234C2×C8C22×D311yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—31no—
239C4⋊C4—31no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4C22×D311yescharacteristic
247C4⋊C4C22×D311yescharacteristic
248C22⋊C4—31no—
249C22⋊C4—31no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C42—31no—
256C42—61no—
257C42C22×D311yescharacteristic
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3C22×D411yescharacteristic
265C2×D3—21no—
266C2×D3—21no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—84no—
271C2×D3#12412yes—
272C3×C4—21no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4C2×#4111yescharacteristic
277C3×C4C2×#4111yescharacteristic
278Dic3—21no—
279Dic3—21no—
280Dic3—41no—
281Dic3—41no—
282Dic3—41no—
283Dic3#12412yes—
284C23—21no—
285C23—21no—
286C23—31no—
287C23—61no—
288C23—122no—
289C23—241no—
290C23—241no—
291Q8—61no—
292Q8—61no—
293Q8—81no—
294Q8—241no—
295D4—21no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—61no—
300D4—61no—
301D4—81no—
302D4—81no—
303D4—122no—
304D4—122no—
305D4—241no—
306D4—241no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—31no—
310C2×C4—32no—
311C2×C4—32no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—61no—
317C2×C4—122no—
318C2×C4—241no—
319C2×C4D3×D411yescharacteristic
320C2×C4D3×D411yescharacteristic
321C2×C4C23×D311yescharacteristic
322C8—21no—
323C8—21no—
324C8—61no—
325C8—61no—
326C2×C3—81no—
327C2×C3—81no—
328C2×C3C2×#12411yescharacteristic
329C2×C3#57611yescharacteristic
330C2×C3#57711yescharacteristic
331D3—22no—
332D3—81no—
333C22—32no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—61no—
341C22—61no—
342C22—121no—
343C22—121no—
344C22—121no—
345C22—244no—
346C22C2×D3×D411yescharacteristic
347C4—21no—
348C4—21no—
349C4—32no—
350C4—41no—
351C4—41no—
352C4—61no—
353C4—61no—
354C4—121no—
355C4—121no—
356C4—121no—
357C4D3×#4111yescharacteristic
358C4D3×#4111yescharacteristic
359C3#313611yescharacteristic, complemented, Sylow
360C2—62no—
361C2—81no—
362C2—81no—
363C2—241no—
364C2D3×#12411yescharacteristic
365C2#575711yescharacteristic
366C2#576211yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#486C2
socleC22×C3C22×D4
prime radical#5715C2
hypercenter#486D3
hypocenterC3#3136
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #5729C2semidirect
3. C3×#486C2semidirect
4. #5089C2semidirect
5. #5071C2semidirect
6. #4772C2semidirect
7. #4771C2semidirect
8. #4770C2semidirect
9. D3×#102C2semidirect
10. #4768C2semidirect
11. #4765C2semidirect
12. #4764C2semidirect
13. #4760C2semidirect
14. #4697C2semidirect
15. #4689C2semidirect
19. C3×C4⋊Q8C22semidirect
34. #890C22semidirect
36. #888C22semidirect
39. #870C22semidirect
42. C3×#102C22semidirect
43. C3×#102C22semidirect
44. C3×C4×C8C22semidirect
47. #832C22semidirect
48. #832C22semidirect
51. #816C22semidirect
52. #816C22semidirect
53. #815C22semidirect
54. C8×Dic3C22semidirect
55. #810C22semidirect
56. #810C22semidirect
59. #803C22semidirect
69. #486D3semidirect
130. C4⋊Q8C2×D3semidirect
161. #102C2×D3semidirect
162. #102C2×D3semidirect
166. C4×C8C2×D3semidirect
359. C3#3136semidirect

Maximal subgroups

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

subgroupindex
#57292
#57152
C3×#4862
#50892
#50712
#47722
#47712
#47702
D3×#1022
#47682
#47652
#47642
#47602
#46972
#46892
#31363