Finite Group Databasebeta

← #49133071

Subgroups — #49133071

Order 36224.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827118C211yesmaximal, characteristic, complemented
2D4×D1132C211yesmaximal, characteristic, complemented
3#31826813C211yesmaximal, characteristic, complemented
4#31826780C212yesmaximal, complemented
5#31826702C212yesmaximal, complemented
6#31826661C211yesmaximal, characteristic, complemented
7#31826600C212yesmaximal, complemented
8C4×#17985377C211yesmaximal, characteristic, complemented
9#31826530C211yesmaximal, characteristic, complemented
10#102×D283C212yesmaximal, complemented
11#474×C283C211yesmaximal, characteristic, complemented
12C2×#17985393—21no—
13C22×D1132—21no—
14C2×D4×D283C2212yes—
15C2×D2264C2212yescomplemented
16C2×#17985377C2211yescharacteristic
17C2×C8×D283C2212yescomplemented
18#22583607C2211yescharacteristic
19#22583595—22no—
20#22583586—21no—
21#22583585C2212yes—
22#22583585C2212yescomplemented
23#22583584—21no—
24#22583580—22no—
25C4×D1132C2211yescharacteristic
26C4×D1132C2211yescharacteristic, complemented
27C4⋊C4×D283C2212yescomplemented
28#22583543C2211yescharacteristic, complemented
29#22583532C2212yescomplemented
30#22583531C2211yescharacteristic
31#22583530C2211yescharacteristic, complemented
32#22583530C2212yescomplemented
33#22583530C2212yescomplemented
34(C2×D283)⋊C8C2212yescomplemented
35C4×C283⋊C8C2211yescharacteristic, complemented
36C4⋊Q8×C283C2211yescharacteristic, complemented
37C4×D4×C283C2211yescharacteristic
38#107×C283C2212yescomplemented
39C4⋊C8×C283C2211yescharacteristic, complemented
40#102×C283C2212yes—
41C23×D283—22no—
42#17985393—44no—
43C2×D1132—22no—
44C2×D1132—22no—
45C2×D1132—21no—
46C2×D1132—21no—
47C2×D1132—44no—
48C2×D1132D411yescharacteristic
49C2×D1132C2311yescharacteristic
50C2×D1132C2312yes—
51C2×C22⋊D283—22no—
52D4×D283—24no—
53D4×D283—42no—
54C2×C4×D283—22no—
55C2×C4×D283—21no—
56C2×C4×D283C2312yes—
57D2264—42no—
58#17985377—22no—
59C2×C283⋊C8C2311yescharacteristic
60C2×C283⋊C8C2311yescharacteristic
61C8×D283—42no—
62C4⋊Dic283D411yescharacteristic
63C4⋊Dic283C2311yescharacteristic
64Dic283⋊C4—22no—
65(C2×D283)⋊C4—22no—
66(C2×D283)⋊C4—22no—
67(C2×D283)⋊C4—21no—
68(C2×D283)⋊C4D412yes—
69C2×Q8×C283—21no—
70C2×D4×C283C2311yescharacteristic
71C22×C4×C283—21no—
72C2×C8×C283C2312yes—
73C4⋊C4×C283—21no—
74C4⋊C4×C283C2312yes—
75C4⋊C4×C283C2311yescharacteristic
76C22⋊C4×C283—21no—
77C42×C283C2311yescharacteristic
78C22×D283—21no—
79C22×D283—22no—
80C22×D283—44no—
81C22×D283—44no—
82C22×D283—82no—
83C22×D283C2×D412yes—
84D1132—21no—
85D1132—22no—
86D1132—21no—
87D1132—22no—
88D1132—44no—
89D1132—44no—
90D1132—42no—
91D1132—81no—
92D1132D814yescomplemented
93C22⋊D283—44no—
94C2×Dic283—21no—
95C2×Dic283C2×D412yes—
96C4×D283—22no—
97C4×D283—22no—
98C4×D283—44no—
99C4×D283—42no—
100C283⋊C8—21no—
101C283⋊C8—21no—
102C23×C283—21no—
103Q8×C283—42no—
104D4×C283—21no—
105D4×C283#4112yescomplemented
106C2×C4×C283—21no—
107C2×C4×C283—22no—
108C2×C4×C283—42no—
109C2×C4×C283C2×D411yescharacteristic
110C2×C4×C283C2×D411yescharacteristic
111C2×C4×C283C2411yescharacteristic
112C8×C283—22no—
113C22×C283—22no—
114C22×C283—41no—
115C22×C283—41no—
116C22×C283C22×D411yescharacteristic
117C2×D283—22no—
118C2×D283—22no—
119C2×D283—42no—
120C2×D283—41no—
121C2×D283—41no—
122C2×D283—42no—
123C2×D283—42no—
124C2×D283—41no—
125C2×D283—88no—
126C2×D283C2×D814yes—
127C4×C283—21no—
128C4×C283—41no—
129C4×C283—42no—
130C4×C283C2×D812yes—
131C4×C283C2×#4111yescharacteristic
132C4×C283H(5,𝔽2)11yescharacteristic
133Dic283—22no—
134Dic283—41no—
135C2×C283—42no—
136C2×C283#54611yescharacteristic
137C2×C283C22×D811yescharacteristic
138C2×C283#57711yescharacteristic
139D283—24no—
140D283—81no—
141D283—82no—
142C283#329411yescharacteristic, complemented, Sylow
143#3294—2831nomaximal, Sylow
144#547—2831no—
145D42—2831no—
146#486—2831no—
147#480—2832no—
148#474D28311yescharacteristic, complemented
149#466—2832no—
150#459—2831no—
151#447—2832no—
152C4×D8—2831no—
153C8×D4—2831no—
154C2×#102—2832no—
155C2×#41—5661no—
156C22×D4—2832no—
157C22×D4—5661no—
158C2×D8—2831no—
159C2×D8—2832no—
160C22×C8—2832no—
161C4⋊Q8C2×D28311yescharacteristic, complemented
162#127—2831no—
163#123—5662no—
164C22⋊Q8—5661no—
165#121—2832no—
166#121—2832no—
167#121—5661no—
168#120—5662no—
169C4×D4—2831no—
170C4×D4—2831no—
171C4×D4C2×D28311yescharacteristic
172C2×C4⋊C4—2832no—
173#107—2831no—
174#107C2×D28312yescomplemented
175C4⋊C8C2×D28311yescharacteristic, complemented
176#102—2831no—
177#102—2832no—
178#102—2832no—
179#102—2831no—
180#102—2832no—
181#102C2×D28312yes—
182C22⋊C8—2832no—
183C4×C8—2831no—
184C24—5662no—
185#41—11324no—
186C2×Q8—21no—
187C2×D4—2831no—
188C2×D4—2832no—
189C2×D4—2831no—
190C2×D4—5662no—
191C2×D4—5662no—
192C2×D4—5662no—
193C2×D4—5664no—
194C2×D4—5661no—
195C2×D4—5661no—
196C2×D4—11324no—
197C2×D4—11322no—
198C2×D4C22×D28311yescharacteristic
199C22×C4—21no—
200C22×C4—2832no—
201C22×C4—5662no—
202C22×C4—5661no—
203D8—5662no—
204D8—11322no—
205C2×C8—2831no—
206C2×C8—2831no—
207C2×C8—11322no—
208C2×C8C22×D28312yes—
209C4⋊C4—21no—
210C4⋊C4—2831no—
211C4⋊C4—2831no—
212C4⋊C4—5662no—
213C4⋊C4C22×D28312yes—
214C4⋊C4C22×D28311yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—2832no—
217C22⋊C4—5662no—
218C22⋊C4—5662no—
219C22⋊C4—5661no—
220C42C22×D28311yescharacteristic
221C23—21no—
222C23—2832no—
223C23—5661no—
224C23—5662no—
225C23—11324no—
226C23—11324no—
227C23—22642no—
228Q8—42no—
229D4—21no—
230D4—2834no—
231D4—5661no—
232D4—5662no—
233D4—5661no—
234D4—5662no—
235D4—11324no—
236D4—11324no—
237D4—11324no—
238D4—11322no—
239D4—22641no—
240D4#1798539312yescomplemented
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—2832no—
245C2×C4—5662no—
246C2×C4—5662no—
247C2×C4—5661no—
248C2×C4—11324no—
249C2×C4—11322no—
250C2×C4D4×D28311yescharacteristic
251C2×C4D4×D28311yescharacteristic
252C2×C4C23×D28311yescharacteristic
253C8—22no—
254C8—5661no—
255C8—5661no—
256C22—22no—
257C22—41no—
258C22—41no—
259C22—2834no—
260C22—5662no—
261C22—5662no—
262C22—11321no—
263C22—11322no—
264C22—11322no—
265C22—11321no—
266C22—11322no—
267C22—11321no—
268C22—22648no—
269C22C2×D4×D28311yescharacteristic
270C4—21no—
271C4—41no—
272C4—42no—
273C4—5662no—
274C4—11321no—
275C4D8×D28312yes—
276C4C2×#1798539311yescharacteristic
277C4#2258365611yescharacteristic
278C2—42no—
279C2—5664no—
280C2—22642no—
281C2—22641no—
282C2#3182710611yescharacteristic
283C2C2×D8×D28311yescharacteristic
284C2#3182727411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#474×C283C2
socleC22×C283C22×D4
hypercenter#474D283
hypocenterC283#3294
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #31827118C2semidirect
2. D4×D1132C2semidirect
3. #31826813C2semidirect
4. #31826780C2semidirect
5. #31826702C2semidirect
6. #31826661C2semidirect
7. #31826600C2semidirect
8. C4×#17985377C2semidirect
9. #31826530C2semidirect
10. #102×D283C2semidirect
11. #474×C283C2semidirect
15. C2×D2264C22semidirect
17. C2×C8×D283C22semidirect
22. #22583585C22semidirect
26. C4×D1132C22semidirect
27. C4⋊C4×D283C22semidirect
28. #22583543C22semidirect
29. #22583532C22semidirect
31. #22583530C22semidirect
32. #22583530C22semidirect
33. #22583530C22semidirect
34. (C2×D283)⋊C8C22semidirect
35. C4×C283⋊C8C22semidirect
36. C4⋊Q8×C283C22semidirect
38. #107×C283C22semidirect
39. C4⋊C8×C283C22semidirect
92. D1132D8semidirect
105. D4×C283#41semidirect
142. C283#3294semidirect
148. #474D283semidirect
161. C4⋊Q8C2×D283semidirect
174. #107C2×D283semidirect
175. C4⋊C8C2×D283semidirect
240. D4#17985393semidirect

Maximal subgroups

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

subgroupindex
#318271182
D4×D11322
#318268132
#318267802
#318267022
#318266612
#318266002
C4×#179853772
#318265302
#102×D2832
#474×C2832
#3294283