Finite Group Databasebeta

← #49133068

Subgroups — #49133068

Order 36224.

494 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×D1132C211yesmaximal, characteristic, complemented
2#31826813C211yesmaximal, characteristic, complemented
3#31826778C212yesmaximal, complemented
4#31826742C211yesmaximal, characteristic, complemented
5#31826699C212yesmaximal, complemented
6#31826660C211yesmaximal, characteristic, complemented
7#31826598C212yesmaximal, complemented
8C4×#17985377C211yesmaximal, characteristic, complemented
9D4×C283⋊C8C211yesmaximal, characteristic, complemented
10C2×#22583530C212yesmaximal, complemented
11#547×C283C211yesmaximal, characteristic, complemented
12C22×D1132C2212yes—
13C2×D4×D283—21no—
14C2×#17985377C2211yescharacteristic
15C2×#17985377C2212yescomplemented
16C22×C283⋊C8C2212yescomplemented
17#22583607C2211yescharacteristic
18#22583585—21no—
19#22583584C2212yes—
20#22583580—22no—
21C4×D1132C2211yescharacteristic
22#22583543C2212yescomplemented
23#22583543C2211yescharacteristic, complemented
24C4⋊(C283⋊C8)C2211yescharacteristic, complemented
25#22583531C2211yescharacteristic, complemented
26#22583530C2211yescharacteristic
27#22583530C2212yes—
28#22583530C2212yescomplemented
29#22583530C2212yescomplemented
30#22583530C2212yescomplemented
31C22⋊(C283⋊C8)C2212yescomplemented
32C4×C283⋊C8C2211yescharacteristic, complemented
33C2×#41×C283—21no—
34C4⋊Q8×C283C2211yescharacteristic, complemented
35#123×C283—22no—
36C22⋊Q8×C283—21no—
37#121×C283C2212yescomplemented
38C4×D4×C283C2211yescharacteristic
39C4×D4×C283C2211yescharacteristic, complemented
40C2×C4⋊C4×C283C2212yescomplemented
41C23×D283—22no—
42C2×D1132—24no—
43C2×D1132—22no—
44C2×D1132—21no—
45C2×D1132—21no—
46C2×D1132—42no—
47C2×D1132C2312yes—
48C2×D1132C2311yescharacteristic
49C2×C22⋊D283—22no—
50D4×D283—44no—
51C2×C4×D283—21no—
52#17985377—22no—
53#17985377—42no—
54C2×C283⋊C8—42no—
55C2×C283⋊C8C2311yescharacteristic
56C2×C283⋊C8C2312yes—
57C2×C283⋊C8C2311yescharacteristic
58C4⋊Dic283C2311yescharacteristic
59(C2×D283)⋊C4—21no—
60(C2×D283)⋊C4—22no—
61#41×C283—44no—
62C2×Q8×C283—21no—
63C2×D4×C283—22no—
64C2×D4×C283D411yescharacteristic
65C2×D4×C283C2311yescharacteristic
66C22×C4×C283—22no—
67C22×C4×C283—21no—
68C22×C4×C283C2312yes—
69C4⋊C4×C283—22no—
70C4⋊C4×C283—21no—
71C4⋊C4×C283D411yescharacteristic
72C4⋊C4×C283C2311yescharacteristic
73C4⋊C4×C283C2312yes—
74C22⋊C4×C283—21no—
75C22⋊C4×C283—22no—
76C22⋊C4×C283D412yes—
77C42×C283C2311yescharacteristic
78C22×D283—21no—
79C22×D283—22no—
80C22×D283—44no—
81C22×D283—44no—
82C22×D283—82no—
83D1132—22no—
84D1132—21no—
85D1132—22no—
86D1132—44no—
87D1132—42no—
88D1132—81no—
89D1132#4112yescomplemented
90C22⋊D283—44no—
91C2×Dic283—21no—
92C4×D283—42no—
93C283⋊C8—21no—
94C283⋊C8—21no—
95C283⋊C8—22no—
96C23×C283—21no—
97C23×C283C2×D412yes—
98Q8×C283—42no—
99D4×C283—21no—
100D4×C283—21no—
101D4×C283—44no—
102D4×C283D814yescomplemented
103C2×C4×C283—22no—
104C2×C4×C283—22no—
105C2×C4×C283—21no—
106C2×C4×C283—22no—
107C2×C4×C283—44no—
108C2×C4×C283—42no—
109C2×C4×C283C2×D411yescharacteristic
110C2×C4×C283C2×D412yes—
111C2×C4×C283C2×D411yescharacteristic
112C2×C4×C283C2411yescharacteristic
113C22×C283—22no—
114C22×C283—22no—
115C22×C283—41no—
116C22×C283—41no—
117C22×C283—41no—
118C22×C283C2×D814yes—
119C22×C283C22×D411yescharacteristic
120C2×D283—22no—
121C2×D283—41no—
122C2×D283—41no—
123C2×D283—42no—
124C2×D283—42no—
125C2×D283—42no—
126C2×D283—88no—
127C4×C283—22no—
128C4×C283—21no—
129C4×C283—41no—
130C4×C283—42no—
131C4×C283C2×D812yes—
132C4×C283C2×#4111yescharacteristic
133C4×C283H(5,𝔽2)11yescharacteristic
134Dic283—41no—
135C2×C283—24no—
136C2×C283—81no—
137C2×C283#54611yescharacteristic
138C2×C283C22×D811yescharacteristic
139C2×C283#57711yescharacteristic
140D283—42no—
141D283—82no—
142C283#329411yescharacteristic, complemented, Sylow
143#3294—2831nomaximal, Sylow
144#547D28311yescharacteristic, complemented
145D42—2831no—
146#486—2831no—
147#480—2832no—
148#474—2831no—
149#466—2832no—
150#459—2831no—
151#447—2832no—
152C4×D8—2831no—
153C8×D4—2831no—
154C2×#102—2832no—
155C2×#41—21no—
156C22×D4—2832no—
157C22×D4—5661no—
158C2×D8—2831no—
159C2×D8—2832no—
160C22×C8—2832no—
161C4⋊Q8C2×D28311yescharacteristic, complemented
162#127—2831no—
163#123—22no—
164C22⋊Q8—21no—
165#121—2832no—
166#121—5661no—
167#121C2×D28312yescomplemented
168#120—5662no—
169C4×D4—2831no—
170C4×D4C2×D28311yescharacteristic
171C4×D4C2×D28311yescharacteristic, complemented
172C2×C4⋊C4C2×D28312yescomplemented
173#107—2831no—
174#107—2832no—
175C4⋊C8—2831no—
176#102—2831no—
177#102—2831no—
178#102—2832no—
179#102—2832no—
180#102—2832no—
181#102—2832no—
182C22⋊C8—2832no—
183C4×C8—2831no—
184C24—5662no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—2831no—
189C2×D4—2832no—
190C2×D4—5664no—
191C2×D4—5662no—
192C2×D4—5662no—
193C2×D4—5661no—
194C2×D4—5661no—
195C2×D4—11322no—
196C2×D4—11324no—
197C2×D4C22⋊D28311yescharacteristic
198C2×D4C22×D28311yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—5661no—
202C22×C4C22×D28312yes—
203D8—5662no—
204D8—11322no—
205C2×C8—2831no—
206C2×C8—2831no—
207C2×C8—2832no—
208C2×C8—11322no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—2831no—
212C4⋊C4C22⋊D28311yescharacteristic
213C4⋊C4C22×D28311yescharacteristic
214C4⋊C4C22×D28312yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—5661no—
218C22⋊C4—5662no—
219C22⋊C4C22⋊D28312yes—
220C42C22×D28311yescharacteristic
221C23—21no—
222C23—5661no—
223C23—5662no—
224C23—11324no—
225C23—11324no—
226C23—22642no—
227C23C2×C22⋊D28312yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—2832no—
233D4—5662no—
234D4—5661no—
235D4—5662no—
236D4—11324no—
237D4—11324no—
238D4—11322no—
239D4—22641no—
240D4#1798537714yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—5661no—
248C2×C4—11322no—
249C2×C4C2×C22⋊D28311yescharacteristic
250C2×C4C2×C22⋊D28312yes—
251C2×C4C2×C22⋊D28311yescharacteristic
252C2×C4C23×D28311yescharacteristic
253C8—5661no—
254C8—5661no—
255C8—5662no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—5662no—
262C22—11321no—
263C22—11321no—
264C22—11322no—
265C22—11322no—
266C22—11322no—
267C22—22648no—
268C22C2×#1798537714yes—
269C22C22×C22⋊D28311yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—11321no—
275C4C2×#1798537712yes—
276C4#41×D28311yescharacteristic
277C4#2258365611yescharacteristic
278C2—24no—
279C2—81no—
280C2—11322no—
281C2—22642no—
282C2#3182710311yescharacteristic
283C2C22×#1798537711yescharacteristic
284C2#3182727311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#547×C283C2
socleC22×C283C22×D4
hypercenter#547D283
hypocenterC283#3294
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D1132C2semidirect
2. #31826813C2semidirect
3. #31826778C2semidirect
4. #31826742C2semidirect
5. #31826699C2semidirect
6. #31826660C2semidirect
7. #31826598C2semidirect
8. C4×#17985377C2semidirect
9. D4×C283⋊C8C2semidirect
10. C2×#22583530C2semidirect
11. #547×C283C2semidirect
15. C2×#17985377C22semidirect
16. C22×C283⋊C8C22semidirect
22. #22583543C22semidirect
23. #22583543C22semidirect
24. C4⋊(C283⋊C8)C22semidirect
25. #22583531C22semidirect
28. #22583530C22semidirect
29. #22583530C22semidirect
30. #22583530C22semidirect
31. C22⋊(C283⋊C8)C22semidirect
32. C4×C283⋊C8C22semidirect
34. C4⋊Q8×C283C22semidirect
37. #121×C283C22semidirect
39. C4×D4×C283C22semidirect
40. C2×C4⋊C4×C283C22semidirect
89. D1132#41semidirect
102. D4×C283D8semidirect
142. C283#3294semidirect
144. #547D283semidirect
161. C4⋊Q8C2×D283semidirect
167. #121C2×D283semidirect
171. C4×D4C2×D283semidirect
172. C2×C4⋊C4C2×D283semidirect
240. D4#17985377semidirect

Maximal subgroups

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

subgroupindex
D4×D11322
#318268132
#318267782
#318267422
#318266992
#318266602
#318265982
C4×#179853772
D4×C283⋊C82
C2×#225835302
#547×C2832
#3294283