Finite Group Databasebeta

← #44558536

Subgroups — #44558536

Order 32128.

416 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 117 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#30631629C212yesmaximal, complemented
2#30631612C212yesmaximal, complemented
3#30631607C211yesmaximal, characteristic, complemented
4#30631601C212yesmaximal
5#30631599C212yesmaximal, complemented
6#30631599C212yesmaximal, complemented
7#30631592C211yesmaximal, characteristic, complemented
8#30631580C212yesmaximal, complemented
9#394×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C22×Dic502C2211yescharacteristic
12C2×C4⋊Dic251C2212yes—
13C2×Dic251⋊C4C2212yes—
14C2×C22⋊Dic251C2212yes—
15C2×C22⋊Dic251C2212yes—
16C2×C22⋊Dic251C2212yes—
17C2×C22⋊Dic251C2212yes—
18C2×C4×Dic251C2211yescharacteristic
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C22×D4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C22⋊C4×C251C2212yes—
29C2×C22⋊C4×C251C2212yes—
30#95×C251C2211yescharacteristic
31C2×Dic502—22no—
32C2×Dic502—41no—
33C2×Dic502—41no—
34C2×Dic502—42no—
35C2×Dic502—41no—
36C22×Dic251C2312yes—
37C22×Dic251C2312yes—
38C22×Dic251C2312yes—
39C22×Dic251C2312yes—
40C4⋊Dic251—42no—
41Dic251⋊C4—42no—
42C22⋊Dic251—42no—
43C22⋊Dic251—42no—
44C22⋊Dic251—42no—
45C22⋊Dic251—42no—
46C4×Dic251—41no—
47C24×C251C2312yes—
48C2×D4×C251—41no—
49C2×D4×C251—42no—
50C22×C4×C251C2311yescharacteristic
51C22×C4×C251C2312yes—
52C22×C4×C251C2311yescharacteristic
53C22×C4×C251C2311yescharacteristic
54C4⋊C4×C251—41no—
55C22⋊C4×C251—24no—
56C22⋊C4×C251—42no—
57Dic502—44no—
58Dic502—44no—
59C2×Dic251—22no—
60C2×Dic251—22no—
61C2×Dic251—22no—
62C2×Dic251—22no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251—22no—
69C2×Dic251—22no—
70C2×Dic251—22no—
71C23×C251—22no—
72C23×C251—22no—
73C23×C251—22no—
74C23×C251—22no—
75C23×C251—22no—
76C23×C251—22no—
77C23×C251—22no—
78C23×C251C2411yescharacteristic
79D4×C251—44no—
80C2×C4×C251—21no—
81C2×C4×C251—21no—
82C2×C4×C251—22no—
83C2×C4×C251—22no—
84C2×C4×C251—22no—
85C2×C4×C251—21no—
86C2×C4×C251—21no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251#4112yes—
90C22×C251—42no—
91C22×C251—42no—
92C22×C251—42no—
93C22×C251—42no—
94C22×C251—42no—
95C22×C251—42no—
96C22×C251—42no—
97C22×C251C2×#4111yescharacteristic
98C22×C251H(5,𝔽2)11yescharacteristic
99C22×C251H(5,𝔽2)12yes—
100C22×C251H(5,𝔽2)11yescharacteristic
101C22×C2512−1+412yes—
102C4×C251—22no—
103C4×C251—42no—
104C4×C251—41no—
105C4×C251—41no—
106Dic251—42no—
107Dic251—42no—
108Dic251—42no—
109Dic251—42no—
110C2×C251—82no—
111C2×C251#53811yescharacteristic
112C2×C251#54212yes—
113C2×C251H(3,𝔽4)11yescharacteristic
114C2×C251#56212yes—
115C2×C251#56311yescharacteristic
116C251#283211yescharacteristic, complemented, Sylow
117#2832—2511nomaximal, Sylow
118#400—2512no—
119#397—2512no—
120#395—2511no—
121#394—2512no—
122#394D25111yescharacteristic
123#393—2512no—
124#393—2512no—
125#391—2511no—
126#388—2512no—
127C22×Q8—2511no—
128C22×Q8—2511no—
129C22×D4C2×D25111yescharacteristic
130C2×C4⋊C4—2512no—
131C2×C4⋊C4—2512no—
132C2×C4⋊C4C2×D25111yescharacteristic
133C2×C22⋊C4—2512no—
134C2×C22⋊C4—2512no—
135C2×C22⋊C4—2512no—
136C2×C22⋊C4—2512no—
137C2×C22⋊C4C2×D25112yes—
138C2×C22⋊C4C2×D25112yes—
139C2×C42—2511no—
140#95—2512no—
141#95—2512no—
142#95—2512no—
143#95—2512no—
144#95—2512no—
145#95—2512no—
146#95—2511no—
147#95C2×D25111yescharacteristic
148C24C22×D25112yes—
149C2×Q8—10041no—
150C2×Q8—10041no—
151C2×Q8—10042no—
152C2×Q8—10041no—
153C2×Q8—5022no—
154C2×D4—41no—
155C2×D4—42no—
156C22×C4—2512no—
157C22×C4—2512no—
158C22×C4—2512no—
159C22×C4—2512no—
160C22×C4C22×D25112yes—
161C22×C4C22×D25111yescharacteristic
162C22×C4C22×D25111yescharacteristic
163C22×C4C22×D25111yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—10042no—
166C4⋊C4—10042no—
167C22⋊C4—24no—
168C22⋊C4—42no—
169C22⋊C4—10042no—
170C22⋊C4—10042no—
171C22⋊C4—10042no—
172C22⋊C4—10042no—
173C42—10041no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23C23×D25111yescharacteristic
182Q8—10044no—
183Q8—10044no—
184D4—44no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—5022no—
195C2×C4—5022no—
196C2×C4—5022no—
197C2×C4—5022no—
198C2×C4—5022no—
199C2×C4—5022no—
200C2×C4—5022no—
201C2×C4—5022no—
202C2×C4—5022no—
203C2×C4—5022no—
204C2×C4—5022no—
205C2×C4—5022no—
206C2×C4#1686080912yes—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22C2×#1686080911yescharacteristic
215C22#2254876011yescharacteristic
216C22#2254876012yes—
217C22#2254876111yescharacteristic
218C22#2254876212yes—
219C4—22no—
220C4—42no—
221C4—41no—
222C4—41no—
223C4—10042no—
224C4—10042no—
225C4—10042no—
226C4—10042no—
227C2—82no—
228C2#3063227411yescharacteristic
229C2#3063230412yes—
230C2#3063243011yescharacteristic
231C2#3063243412yes—
232C2#3063243711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#394×C251C2
prime radicalC22×D4×C251C22
hypercenter#394D251
hypocenterC251#2832
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #30631629C2semidirect
2. #30631612C2semidirect
3. #30631607C2semidirect
5. #30631599C2semidirect
6. #30631599C2semidirect
7. #30631592C2semidirect
8. #30631580C2semidirect
116. C251#2832semidirect

Maximal subgroups

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

subgroupindex
#306316292
#306316122
#306316072
#306316012
#306315992
#306315922
#306315802
#394×C2512
#2832251