Finite Group Databasebeta

← #80982

Subgroups — #80982

Order 384.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5730C211yesmaximal, characteristic, complemented
2#5554C211yesmaximal, characteristic
3C3×#474C211yesmaximal, characteristic, complemented
4#5089C211yesmaximal, characteristic, complemented
5#5018C211yesmaximal, characteristic, complemented
6#4887C211yesmaximal, characteristic, complemented
7#4886C211yesmaximal, characteristic, complemented
8#4884C211yesmaximal, characteristic, complemented
9#4882C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4783C211yesmaximal, characteristic, complemented
13#4776C211yesmaximal, characteristic, complemented
14#4771C211yesmaximal, characteristic, complemented
15D3×#102C211yesmaximal, characteristic, complemented
16#3314—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic, complemented
20C3×C4×D4C2211yescharacteristic
21#946—21no—
22#946—21no—
23C2×#214C2211yescharacteristic
24#930—21no—
25C2×(C2×D3)⋊C4—21no—
26C2×Dic12C2211yescharacteristic, complemented
27C2×#204C2211yescharacteristic
28C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
29C2×D3×C8C2211yescharacteristic
30#897C2211yescharacteristic
31#897C2211yescharacteristic
32#897C2211yescharacteristic, complemented
33#895C2211yescharacteristic
34#893—21no—
35D3×C4⋊C4C2211yescharacteristic
36D3×C4⋊C4C2211yescharacteristic, complemented
37#890—21no—
38#886—21no—
39#882—21no—
40#875C2211yescharacteristic
41C4×D12C2211yescharacteristic
42C4×Dic6C2211yescharacteristic, complemented
43C3×#107C2211yescharacteristic, complemented
44C3×#107C2211yescharacteristic, complemented
45C3×C4⋊C8C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic
47C3×#102C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic, complemented
52#816C2211yescharacteristic, complemented
53#810C2211yescharacteristic, complemented
54#810C2211yescharacteristic, complemented
55#810C2211yescharacteristic, complemented
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#807C2211yescharacteristic, complemented
59#803C2211yescharacteristic, complemented
60#554—31no—
61#546—31no—
62#489—31no—
63#484—31no—
64#480—31no—
65#474D311yescharacteristic, complemented
66#468—31no—
67#467—31no—
68#464—31no—
69#452—31no—
70#450—31no—
71#440—31no—
72#435—31no—
73#417—31no—
74C2×#102—31no—
75C23×D3—21no—
76C2×C3×Q8—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C2×#67—21no—
80C2×#67—21no—
81C22×Dic3—21no—
82#237—22no—
83#237—41no—
84D3×D4—22no—
85D3×D4—41no—
86C2×D12—21no—
87C2×D12D411yescharacteristic
88C2×D12C2311yescharacteristic
89C2×C4×D3—21no—
90C2×C4×D3—21no—
91C2×C4×D3C2311yescharacteristic
92C2×C4×D3C2311yescharacteristic
93C2×Dic6C2311yescharacteristic
94C2×Dic6C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C22⋊C4—21no—
102C3×C42C2311yescharacteristic
103#214—22no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4D411yescharacteristic
111(C2×D3)⋊C4D411yescharacteristic
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3D411yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120C4×Dic3—21no—
121C2×C3⋊C8C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123Dic12—41no—
124#204—41no—
125ΓL(1,𝔽25)—41no—
126D3×C8—41no—
127C2×#41—31no—
128C22×D4—31no—
129C2×Q16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8C2×D311yescharacteristic, complemented
135#125—61no—
136#124—31no—
137#123—61no—
138C22⋊Q8—31no—
139C22⋊Q8—31no—
140C22⋊Q8—31no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143#121—31no—
144#121—61no—
145#120—61no—
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—61no—
149C4×D4C2×D311yescharacteristic
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C22⋊C4—61no—
153#107—31no—
154#107C2×D311yescharacteristic, complemented
155#107C2×D311yescharacteristic, complemented
156C4⋊C8C2×D311yescharacteristic, complemented
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3—21no—
171C22×D3—21no—
172C22×D3—42no—
173C22×D3—42no—
174C22×D3—81no—
175C22×D3C2×D411yescharacteristic
176C22×D3C2×D411yescharacteristic
177C3×Q8—81no—
178C3×D4—21no—
179C3×D4#4112yes—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—81no—
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2411yescharacteristic
187#67—42no—
188#67—42no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—42no—
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195D12—22no—
196D12—21no—
197D12—21no—
198D12—81no—
199C4×D3—21no—
200C4×D3—21no—
201C4×D3—21no—
202C4×D3—21no—
203C4×D3—81no—
204C4×D3—81no—
205Dic6—21no—
206Dic6—21no—
207Dic6—21no—
208Dic6—21no—
209C3×C8—21no—
210C3×C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213C24—61no—
214#41—62no—
215#41—121no—
216C2×Q8—21no—
217C2×Q8—31no—
218C2×Q8—31no—
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—62no—
224C2×D4—61no—
225C2×D4—121no—
226C2×D4C22×D311yescharacteristic
227C22×C4—21no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4—61no—
233Q16—121no—
234SD16—62no—
235SD16—121no—
236M16—121no—
237C2×C8—31no—
238C2×C8—31no—
239C2×C8—121no—
240C2×C8C22×D311yescharacteristic
241C2×C8C22×D311yescharacteristic
242C4⋊C4—21no—
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4C22×D311yescharacteristic
252C4⋊C4C22×D311yescharacteristic
253C4⋊C4C22×D311yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—31no—
256C22⋊C4—31no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C42—61no—
264C42C22×D311yescharacteristic
265C22×C3—22no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3C22×D411yescharacteristic
269C2×D3—21no—
270C2×D3—21no—
271C2×D3—21no—
272C2×D3—21no—
273C2×D3—21no—
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—41no—
277C2×D3—84no—
278C2×D3#13712yes—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4C2×#4111yescharacteristic
285C3×C4H(5,𝔽2)11yescharacteristic
286Dic3—21no—
287Dic3—21no—
288Dic3—41no—
289Dic3—41no—
290Dic3—41no—
291C23—21no—
292C23—31no—
293C23—31no—
294C23—61no—
295C23—122no—
296C23—122no—
297C23—241no—
298Q8—61no—
299Q8—61no—
300Q8—61no—
301Q8—61no—
302Q8—81no—
303D4—21no—
304D4—62no—
305D4—61no—
306D4—61no—
307D4—122no—
308D4—122no—
309D4—241no—
310D4#23912yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—31no—
315C2×C4—31no—
316C2×C4—61no—
317C2×C4—61no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—81no—
324C2×C4—122no—
325C2×C4—241no—
326C2×C4—241no—
327C2×C4D3×D411yescharacteristic
328C2×C4D3×D411yescharacteristic
329C2×C4C23×D311yescharacteristic
330C8—21no—
331C8—21no—
332C8—61no—
333C8—61no—
334C2×C3—42no—
335C2×C3#54611yescharacteristic
336C2×C3C2×#13711yescharacteristic
337C2×C3#57611yescharacteristic
338D3—22no—
339D3—41no—
340D3—81no—
341C22—22no—
342C22—32no—
343C22—41no—
344C22—41no—
345C22—61no—
346C22—61no—
347C22—61no—
348C22—61no—
349C22—61no—
350C22—121no—
351C22—121no—
352C22—121no—
353C22—244no—
354C22C2×D3×D411yescharacteristic
355C4—21no—
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—61no—
361C4—61no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4C2×#23911yescharacteristic
366C4#100911yescharacteristic
367C3#331411yescharacteristic, complemented, Sylow
368C2—42no—
369C2—62no—
370C2—121no—
371C2—241no—
372C2#563011yescharacteristic
373C2#575711yescharacteristic
374C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#474C2
socleC22×C3C22×D4
prime radical#5554C2
hypercenter#474D3
hypocenterC3#3314
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #5730C2semidirect
3. C3×#474C2semidirect
4. #5089C2semidirect
5. #5018C2semidirect
6. #4887C2semidirect
7. #4886C2semidirect
8. #4884C2semidirect
9. #4882C2semidirect
10. #4845C2semidirect
11. #4836C2semidirect
12. #4783C2semidirect
13. #4776C2semidirect
14. #4771C2semidirect
15. D3×#102C2semidirect
19. C3×C4⋊Q8C22semidirect
26. C2×Dic12C22semidirect
28. C2×ΓL(1,𝔽25)C22semidirect
32. #897C22semidirect
36. D3×C4⋊C4C22semidirect
42. C4×Dic6C22semidirect
43. C3×#107C22semidirect
44. C3×#107C22semidirect
45. C3×C4⋊C8C22semidirect
51. (C2×D3)⋊C8C22semidirect
52. #816C22semidirect
53. #810C22semidirect
54. #810C22semidirect
55. #810C22semidirect
58. #807C22semidirect
59. #803C22semidirect
65. #474D3semidirect
134. C4⋊Q8C2×D3semidirect
154. #107C2×D3semidirect
155. #107C2×D3semidirect
156. C4⋊C8C2×D3semidirect
367. C3#3314semidirect

Maximal subgroups

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

subgroupindex
#57302
#55542
C3×#4742
#50892
#50182
#48872
#48862
#48842
#48822
#48452
#48362
#47832
#47762
#47712
D3×#1022
#33143