Finite Group Databasebeta

← #44558329

Subgroups — #44558329

Order 32128.

392 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 149 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#30631620C212yesmaximal
2#30631619C212yesmaximal
3#30631608C212yesmaximal
4#30631594C212yesmaximal
5#30631593C211yesmaximal, characteristic
6#30631585C211yesmaximal, characteristic
7#30631584C212yesmaximal
8#30631570C212yesmaximal
9#391×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2211yescharacteristic
13C2×C4⋊Dic251C2212yes—
14C2×C4⋊Dic251C2212yes—
15C2×C4⋊Dic251C2211yescharacteristic
16C2×Dic251⋊C4C2212yes—
17C2×Dic251⋊C4C2212yes—
18C2×Dic251⋊C4C2212yes—
19C2×C4×Dic251C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C22×Q8×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C42×C251C2211yescharacteristic
29#95×C251C2212yes—
30#95×C251C2212yes—
31C2×Dic502—22no—
32C2×Dic502—41no—
33C2×Dic502—41no—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C22×Dic251C2312yes—
38C4⋊Dic251—41no—
39C4⋊Dic251—42no—
40C4⋊Dic251—41no—
41C4⋊Dic251—42no—
42C4⋊Dic251Q814yescomplemented
43Dic251⋊C4—24no—
44Dic251⋊C4—42no—
45Dic251⋊C4—42no—
46C4×Dic251—42no—
47C2×Q8×C251—41no—
48C2×Q8×C251—41no—
49C2×Q8×C251Q814yescomplemented
50C22×C4×C251C2312yes—
51C22×C4×C251C2311yescharacteristic
52C22×C4×C251C2312yes—
53C22×C4×C251C2311yescharacteristic
54C22×C4×C251C2311yescharacteristic
55C4⋊C4×C251—41no—
56C42×C251—41no—
57Dic502—44no—
58C2×Dic251—22no—
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×Dic251C2×Q814yes—
70C23×C251C2411yescharacteristic
71Q8×C251—44no—
72C2×C4×C251—21no—
73C2×C4×C251—21no—
74C2×C4×C251—21no—
75C2×C4×C251—21no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—21no—
79C2×C4×C251—21no—
80C2×C4×C251—22no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251C2×Q812yes—
84C2×C4×C251C2×Q814yes—
85C2×C4×C251C2×Q812yes—
86C2×C4×C251#4112yes—
87C22×C251C22×Q811yescharacteristic
88C22×C251C22×Q811yescharacteristic
89C22×C251C2×#4111yescharacteristic
90C22×C251H(5,𝔽2)11yescharacteristic
91C22×C251H(5,𝔽2)11yescharacteristic
92C22×C251H(5,𝔽2)11yescharacteristic
93C22×C2512−1+411yescharacteristic
94C4×C251—22no—
95C4×C251—22no—
96C4×C251—24no—
97C4×C251—22no—
98C4×C251—42no—
99Dic251—24no—
100Dic251—42no—
101Dic251—42no—
102Dic251—42no—
103C2×C251#53811yescharacteristic
104C2×C251#54311yescharacteristic
105C2×C251#54411yescharacteristic
106C2×C251#55411yescharacteristic
107C2×C251#55711yescharacteristic
108C2×C251Q8211yescharacteristic
109C2×C251#56211yescharacteristic
110C251#280611yescharacteristic, complemented, Sylow
111#2806—2511nomaximal, Sylow
112#398—2512no—
113#398—2512no—
114#395—2512no—
115#391—2512no—
116#391—2511no—
117#391D25111yescharacteristic
118#389—2511no—
119#389—2512no—
120#387—2512no—
121C22×Q8—2511no—
122C22×Q8C2×D25111yescharacteristic
123C2×C4⋊C4—2512no—
124C2×C4⋊C4—2511no—
125C2×C4⋊C4—2512no—
126C2×C4⋊C4—2511no—
127C2×C4⋊C4—2512no—
128C2×C4⋊C4—2512no—
129C2×C4⋊C4—2512no—
130C2×C4⋊C4—2511no—
131C2×C4⋊C4C2×D25111yescharacteristic
132C2×C42—2512no—
133C2×C42C2×D25111yescharacteristic
134#95—2512no—
135#95—2512no—
136#95—2512no—
137#95—2512no—
138#95—2512no—
139#95—2512no—
140#95C2×D25112yes—
141#95C2×D25112yes—
142C2×Q8—41no—
143C2×Q8—41no—
144C2×Q8—10041no—
145C2×Q8—10041no—
146C2×Q8—5022no—
147C2×Q8Dic50214yescomplemented
148C22×C4—2512no—
149C22×C4—2512no—
150C22×C4—2512no—
151C22×C4—2512no—
152C22×C4C22×D25112yes—
153C22×C4C22×D25111yescharacteristic
154C22×C4C22×D25111yescharacteristic
155C22×C4C22×D25112yes—
156C22×C4C22×D25111yescharacteristic
157C4⋊C4—41no—
158C4⋊C4—10042no—
159C4⋊C4—10042no—
160C4⋊C4—10041no—
161C4⋊C4—10042no—
162C4⋊C4—10041no—
163C4⋊C4—10042no—
164C4⋊C4—5024no—
165C4⋊C4—2514no—
166C42—41no—
167C42—10042no—
168C23C23×D25111yescharacteristic
169Q8—44no—
170Q8—10044no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—5022no—
183C2×C4—5022no—
184C2×C4—5022no—
185C2×C4—5022no—
186C2×C4—5022no—
187C2×C4—5022no—
188C2×C4—5022no—
189C2×C4—5022no—
190C2×C4—5022no—
191C2×C4—5022no—
192C2×C4—5022no—
193C2×C4—2514no—
194C2×C4Q8×D25112yes—
195C2×C4C2×Dic50212yes—
196C2×C4C2×Dic50214yes—
197C2×C4#1686081112yes—
198C22C2×Q8×D25111yescharacteristic
199C22C22×Dic50211yescharacteristic
200C22C2×#1686081111yescharacteristic
201C22#2254876011yescharacteristic
202C22#2254876011yescharacteristic
203C22#2254876111yescharacteristic
204C22#2254876211yescharacteristic
205C4—22no—
206C4—22no—
207C4—24no—
208C4—22no—
209C4—42no—
210C4—10042no—
211C4—10042no—
212C4—10042no—
213C4—5024no—
214C2#3063227711yescharacteristic
215C2#3063231311yescharacteristic
216C2#3063231611yescharacteristic
217C2#3063239811yescharacteristic
218C2#3063241611yescharacteristic
219C2Q8×Dic50211yescharacteristic
220C2#3063243111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#391×C251C2
hypercenter#391D251
hypocenterC251#2806
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
42. C4⋊Dic251Q8semidirect
49. C2×Q8×C251Q8semidirect
110. C251#2806semidirect
147. C2×Q8Dic502semidirect

Maximal subgroups

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

subgroupindex
#306316202
#306316192
#306316082
#306315942
#306315932
#306315852
#306315842
#306315702
#391×C2512
#2806251