Finite Group Databasebeta

← #80671

Subgroups — #80671

Order 384.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5639C211yesmaximal, characteristic, complemented
2#5612C211yesmaximal, characteristic
3C3×#465C211yesmaximal, characteristic, complemented
4#5169C211yesmaximal, characteristic, complemented
5#5166C211yesmaximal, characteristic, complemented
6#5113C211yesmaximal, characteristic, complemented
7C8×#67C211yesmaximal, characteristic, complemented
8#5047C211yesmaximal, characteristic, complemented
9#5035C211yesmaximal, characteristic, complemented
10#4871C211yesmaximal, characteristic, complemented
11#4862C211yesmaximal, characteristic, complemented
12#4813C211yesmaximal, characteristic, complemented
13#4788C211yesmaximal, characteristic, complemented
14D3×#102C211yesmaximal, characteristic, complemented
15#4768C211yesmaximal, characteristic, complemented
16#3296—31nomaximal, Sylow
17C2×D3×D4C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C22×C3×C8C2211yescharacteristic, complemented
21C3×C22⋊Q8C2211yescharacteristic, complemented
22C3×#121C2211yescharacteristic
23#946C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic, complemented
25C2×#216C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic
27#938—21no—
28#937—21no—
29#936C2211yescharacteristic
30C4×#67C2211yescharacteristic
31#926C2211yescharacteristic, complemented
32C2×D3×C8C2211yescharacteristic
33#897C2211yescharacteristic
34#893C2211yescharacteristic
35#892—21no—
36D3×C4⋊C4C2211yescharacteristic
37#890C2211yescharacteristic, complemented
38#889—21no—
39#883—21no—
40#883—21no—
41D3×C22⋊C4—21no—
42#879—21no—
43C3×#106C2211yescharacteristic, complemented
44C3×#103C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#830C2211yescharacteristic, complemented
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#816C2211yescharacteristic, complemented
53#816C2211yescharacteristic, complemented
54Dic3⋊C8C2211yescharacteristic, complemented
55C8×Dic3C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#552—31no—
61#546—31no—
62#488—31no—
63#483—31no—
64#482—31no—
65#478—31no—
66#467—31no—
67#465D311yescharacteristic, complemented
68#464—31no—
69#452—31no—
70#450—31no—
71C4×SD16—31no—
72C8×D4—31no—
73#416—31no—
74C2×#102—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67—21no—
82C2×#67D411yescharacteristic
83D3×D4—22no—
84D3×D4—41no—
85#235—22no—
86#235—41no—
87C2×D12C2311yescharacteristic
88C2×C4×D3—21no—
89C2×C4×D3—21no—
90C2×C4×D3—21no—
91C2×C4×D3C2311yescharacteristic
92C2×Dic6C2311yescharacteristic
93C3×SD16—41no—
94C2×C3×C8—41no—
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C22⋊C4—21no—
101C3×C22⋊C4—21no—
102#217—21no—
103#217—21no—
104#217—21no—
105#217D411yescharacteristic
106#216—41no—
107#215—22no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4D411yescharacteristic
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4D411yescharacteristic
121C4×Dic3—21no—
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D3×C8—41no—
127C2×#41—31no—
128C22×D4—31no—
129C2×Q16—31no—
130C2×SD16—31no—
131C2×SD16C2×D311yescharacteristic
132C22×C8—31no—
133C22×C8C2×D311yescharacteristic, complemented
134#126—61no—
135#125—31no—
136#125—61no—
137#124—31no—
138#123—61no—
139#123—61no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8C2×D311yescharacteristic, complemented
143#121—61no—
144#121C2×D311yescharacteristic
145#120—61no—
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—31no—
149#117—31no—
150#117—61no—
151C2×C4⋊C4—31no—
152C2×C22⋊C4—61no—
153#107—31no—
154#106—31no—
155#106C2×D311yescharacteristic, complemented
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103C2×D311yescharacteristic, complemented
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3—21no—
173C22×D3—42no—
174C22×D3—42no—
175C22×D3—81no—
176C22×D3C2×D411yescharacteristic
177C3×Q8—21no—
178C3×Q8—21no—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—81no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189#67—22no—
190#67—42no—
191#67—42no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198D12—21no—
199D12#4112yes—
200C4×D3—21no—
201C4×D3—21no—
202C4×D3—42no—
203C4×D3—42no—
204C4×D3—81no—
205Dic6—21no—
206Dic6—21no—
207C3×C8—21no—
208C3×C8—21no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211C24—61no—
212#41—62no—
213#41—121no—
214C2×Q8—31no—
215C2×Q8C22×D311yescharacteristic
216C2×D4—21no—
217C2×D4—31no—
218C2×D4—31no—
219C2×D4—61no—
220C2×D4—62no—
221C2×D4—61no—
222C2×D4—121no—
223C2×D4C22×D311yescharacteristic
224C22×C4—31no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C22×D311yescharacteristic
229Q16—121no—
230SD16—41no—
231SD16—62no—
232C2×C8—31no—
233C2×C8—31no—
234C2×C8—41no—
235C2×C8—121no—
236C2×C8C22×D311yescharacteristic
237C2×C8C22×D311yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4C22×D311yescharacteristic
249C4⋊C4C22×D311yescharacteristic
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—31no—
253C22⋊C4—31no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C42—31no—
261C42—61no—
262C42—61no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3—21no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3C22×D411yescharacteristic
270C2×D3—21no—
271C2×D3—21no—
272C2×D3—22no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—84no—
276C2×D3#13512yes—
277C3×C4—21no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4C2×#4111yescharacteristic
282C3×C4H(5,𝔽2)11yescharacteristic
283Dic3—21no—
284Dic3—21no—
285Dic3—21no—
286Dic3—41no—
287Dic3—41no—
288Dic3—41no—
289C23—21no—
290C23—31no—
291C23—61no—
292C23—122no—
293C23—122no—
294C23—241no—
295C23D3×D411yescharacteristic
296Q8—21no—
297Q8—21no—
298Q8—61no—
299Q8—61no—
300D4—21no—
301D4—21no—
302D4—32no—
303D4—62no—
304D4—61no—
305D4—81no—
306D4—122no—
307D4—122no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—122no—
322C2×C4—122no—
323C2×C4—241no—
324C2×C4D3×D411yescharacteristic
325C2×C4C23×D311yescharacteristic
326C8—21no—
327C8—21no—
328C8—61no—
329C8—61no—
330C2×C3—41no—
331C2×C3—81no—
332C2×C3#54611yescharacteristic
333C2×C3C2×#13511yescharacteristic
334C2×C3#57711yescharacteristic
335D3—22no—
336D3—42no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—32no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—61no—
345C22—61no—
346C22—62no—
347C22—121no—
348C22—121no—
349C22—244no—
350C22C2×D3×D411yescharacteristic
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—61no—
356C4—61no—
357C4—61no—
358C4—121no—
359C4—121no—
360C4—121no—
361C4#100411yescharacteristic
362C4D3×#4111yescharacteristic
363C3#329611yescharacteristic, complemented, Sylow
364C2—41no—
365C2—62no—
366C2—81no—
367C2—122no—
368C2#561011yescharacteristic
369C2#576211yescharacteristic
370C2D3×#13511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
prime radical#5612C2
hypercenter#465D3
hypocenterC3#3296
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #5639C2semidirect
3. C3×#465C2semidirect
4. #5169C2semidirect
5. #5166C2semidirect
6. #5113C2semidirect
7. C8×#67C2semidirect
8. #5047C2semidirect
9. #5035C2semidirect
10. #4871C2semidirect
11. #4862C2semidirect
12. #4813C2semidirect
13. #4788C2semidirect
14. D3×#102C2semidirect
15. #4768C2semidirect
20. C22×C3×C8C22semidirect
21. C3×C22⋊Q8C22semidirect
24. Q8×Dic3C22semidirect
25. C2×#216C22semidirect
31. #926C22semidirect
37. #890C22semidirect
43. C3×#106C22semidirect
44. C3×#103C22semidirect
48. #830C22semidirect
52. #816C22semidirect
53. #816C22semidirect
54. Dic3⋊C8C22semidirect
55. C8×Dic3C22semidirect
56. #810C22semidirect
58. #808C22semidirect
59. #807C22semidirect
67. #465D3semidirect
133. C22×C8C2×D3semidirect
142. C22⋊Q8C2×D3semidirect
155. #106C2×D3semidirect
160. #103C2×D3semidirect
363. C3#3296semidirect

Maximal subgroups

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

subgroupindex
#56392
#56122
C3×#4652
#51692
#51662
#51132
C8×#672
#50472
#50352
#48712
#48622
#48132
#47882
D3×#1022
#47682
#32963