Finite Group Databasebeta

← #49124415

Subgroups — #49124415

Order 36224.

576 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#22583584C211yesmaximal, characteristic
2#31826365C212yesmaximal, complemented
3#31826355C212yesmaximal, complemented
4#31826350C212yesmaximal, complemented
5#31826343C212yesmaximal, complemented
6#31826341C212yesmaximal, complemented
7#31826336C212yesmaximal, complemented
8#31826328C211yesmaximal, characteristic, complemented
9#380×C283C211yesmaximal, characteristic, complemented
10C22×D1132C2211yescharacteristic
11C22×C22⋊D283C2212yes—
12C22×C4×D283C2212yes—
13#22583584C418yescomplemented
14C2×C4⋊Dic283C2211yescharacteristic
15C2×Dic283⋊C4C2212yes—
16C2×C22⋊Dic283C2212yes—
17C2×(C2×D283)⋊C4C2212yes—
18C2×(C2×D283)⋊C4C2212yes—
19C2×(C2×D283)⋊C4C2212yes—
20C2×(C2×D283)⋊C4C2212yes—
21C2×C4×Dic283C2212yes—
22C283⋊#95C2211yescharacteristic
23C283⋊#95C2212yes—
24C283⋊#95C2211yescharacteristic
25C283⋊#95C2212yes—
26C283⋊#95C2212yes—
27C23×C4×C283C2211yescharacteristic
28C2×C4⋊C4×C283C2212yes—
29C2×C22⋊C4×C283C2211yescharacteristic
30C2×C22⋊C4×C283C2211yescharacteristic
31#95×C283C2212yes—
32C23×D283C2312yes—
33C2×D1132—42no—
34C2×D1132C2×C414yes—
35C2×C22⋊D283—42no—
36C2×C22⋊D283—42no—
37C2×C22⋊D283C2×C418yescomplemented
38C22×Dic283C2312yes—
39C22×Dic283C2312yes—
40C22×Dic283C2312yes—
41C2×C4×D283—42no—
42C2×C4×D283—42no—
43C2×C4×D283—42no—
44C4⋊Dic283C2×C414yescomplemented
45Dic283⋊C4—42no—
46C22⋊Dic283—42no—
47(C2×D283)⋊C4—42no—
48(C2×D283)⋊C4—42no—
49(C2×D283)⋊C4—42no—
50(C2×D283)⋊C4C2×C418yescomplemented
51C4×Dic283—42no—
52C24×C283C2311yescharacteristic
53C22×C4×C283—42no—
54C22×C4×C283C2×C414yescomplemented
55C22×C4×C283C2312yes—
56C22×C4×C283C2312yes—
57C22×C4×C283C2311yescharacteristic
58C22×C4×C283C2311yescharacteristic
59C4⋊C4×C283—42no—
60C22⋊C4×C283—41no—
61C22⋊C4×C283—41no—
62C22×D283—22no—
63C22×D283—22no—
64C22×D283—22no—
65C22×D283—22no—
66C22×D283—22no—
67C22×D283—22no—
68C22×D283C22×C414yes—
69D1132—44no—
70C22⋊D283—48no—
71C2×Dic283—22no—
72C2×Dic283—22no—
73C2×Dic283—22no—
74C2×Dic283—22no—
75C2×Dic283—22no—
76C2×Dic283—22no—
77C2×Dic283—22no—
78C2×Dic283—22no—
79C2×Dic283C22×C414yes—
80C4×D283—48no—
81C23×C283—21no—
82C23×C283—22no—
83C23×C283—22no—
84C23×C283—21no—
85C23×C283C22×C412yes—
86C23×C283C2411yescharacteristic
87C2×C4×C283—22no—
88C2×C4×C283—22no—
89C2×C4×C283—22no—
90C2×C4×C283—22no—
91C2×C4×C283—22no—
92C2×C4×C283—22no—
93C2×C4×C283—22no—
94C2×C4×C283—22no—
95C2×C4×C283—44no—
96C2×C4×C283C22×C412yes—
97C2×C4×C283C22×C412yes—
98C22×C283—22no—
99C22×C283—22no—
100C22×C283—22no—
101C22×C283—42no—
102C22×C283—42no—
103C22×C283C23×C411yescharacteristic
104C22×C283H(5,𝔽2)12yes—
105C22×C283H(5,𝔽2)11yescharacteristic
106C22×C283H(5,𝔽2)12yes—
107C22×C2832−1+411yescharacteristic
108C2×D283—24no—
109C2×D283—24no—
110C2×D283—24no—
111C2×D283—42no—
112C2×D283—42no—
113C2×D283—42no—
114C2×D283—42no—
115C4×C283—22no—
116C4×C283—22no—
117C4×C283—42no—
118C4×C283—42no—
119Dic283—24no—
120Dic283—42no—
121Dic283—42no—
122C2×C283—42no—
123C2×C283#51811yescharacteristic
124C2×C283#51811yescharacteristic
125C2×C283#52011yescharacteristic
126C2×C283#56012yes—
127C2×C283#56212yes—
128D283—44no—
129C283#237711yescharacteristic, complemented, Sylow
130#2377—2831nomaximal, Sylow
131C2×#121—2831no—
132#390—2832no—
133#388—2832no—
134#387—2832no—
135#386—2832no—
136#386—2832no—
137#385—2832no—
138#381—2831no—
139#380D28311yescharacteristic, complemented
140C22×D4—2832no—
141C22×D4—2831no—
142C23×C4—2832no—
143C23×C4C2×D28311yescharacteristic
144#121—2838no—
145C2×C4⋊C4—2831no—
146C2×C4⋊C4—2832no—
147C2×C4⋊C4C2×D28312yes—
148C2×C22⋊C4—2832no—
149C2×C22⋊C4—2832no—
150C2×C22⋊C4—2832no—
151C2×C22⋊C4—2832no—
152C2×C22⋊C4—2832no—
153C2×C22⋊C4C2×D28311yescharacteristic
154C2×C22⋊C4C2×D28311yescharacteristic
155C2×C42—2832no—
156#95—2831no—
157#95—2832no—
158#95—2831no—
159#95—2832no—
160#95—2832no—
161#95C2×D28312yes—
162C24—2832no—
163C24C22×D28311yescharacteristic
164C2×D4—2838no—
165C2×D4—2834no—
166C2×D4—11322no—
167C2×D4—11322no—
168C2×D4—11322no—
169C22×C4—42no—
170C22×C4—2832no—
171C22×C4—2832no—
172C22×C4—2832no—
173C22×C4—11322no—
174C22×C4—11322no—
175C22×C4—11322no—
176C22×C4C4×D28314yescomplemented
177C22×C4C22×D28312yes—
178C22×C4C22×D28312yes—
179C22×C4C22×D28311yescharacteristic
180C22×C4C22×D28311yescharacteristic
181C4⋊C4—42no—
182C4⋊C4—2834no—
183C4⋊C4—11322no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—2838no—
187C22⋊C4—11322no—
188C22⋊C4—11322no—
189C22⋊C4—11322no—
190C22⋊C4—11322no—
191C42—11322no—
192C23—21no—
193C23—22no—
194C23—22no—
195C23—21no—
196C23—2834no—
197C23—5662no—
198C23—5662no—
199C23—5662no—
200C23—5662no—
201C23—5662no—
202C23—5662no—
203C23C2×C4×D28312yes—
204C23C23×D28311yescharacteristic
205D4—11324no—
206D4—11328no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—2834no—
217C2×C4—5662no—
218C2×C4—5662no—
219C2×C4—5662no—
220C2×C4—5662no—
221C2×C4—5662no—
222C2×C4—5662no—
223C2×C4—5662no—
224C2×C4—5662no—
225C2×C4—11328no—
226C2×C4C2×C4×D28312yes—
227C2×C4C2×C4×D28312yes—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—42no—
232C22—42no—
233C22—5664no—
234C22—5664no—
235C22—5664no—
236C22—11322no—
237C22—11322no—
238C22—11322no—
239C22—11322no—
240C22C22×C4×D28311yescharacteristic
241C22#2258365511yescharacteristic
242C22#2258365512yes—
243C22#2258365612yes—
244C22#2258365811yescharacteristic
245C4—22no—
246C4—22no—
247C4—42no—
248C4—42no—
249C4—5664no—
250C4—11322no—
251C4—11322no—
252C2—42no—
253C2—11324no—
254C2#3182693811yescharacteristic
255C2#3182693911yescharacteristic
256C2#3182694611yescharacteristic
257C2#3182720312yes—
258C2#3182721112yes—
subgroupG/H
center, FrattiniC23C23×D283
commutatorC22×C283C23×C4
Fitting#380×C283C2
socle, cosocleC23×C283C24
prime radicalC2×#22583584C2
hypercenter#380D283
hypocenterC283#2377
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #31826365C2semidirect
3. #31826355C2semidirect
4. #31826350C2semidirect
5. #31826343C2semidirect
6. #31826341C2semidirect
7. #31826336C2semidirect
8. #31826328C2semidirect
9. #380×C283C2semidirect
13. #22583584C4semidirect
37. C2×C22⋊D283C2×C4semidirect
44. C4⋊Dic283C2×C4semidirect
50. (C2×D283)⋊C4C2×C4semidirect
54. C22×C4×C283C2×C4semidirect
129. C283#2377semidirect
139. #380D283semidirect
176. C22×C4C4×D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835842
#318263652
#318263552
#318263502
#318263432
#318263412
#318263362
#318263282
#380×C2832
#2377283