Finite Group Databasebeta

← #44558066

Subgroups — #44558066

Order 32128.

472 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 137 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#30631626C211yesmaximal, characteristic, complemented
2#30631610C212yesmaximal, complemented
3#30631607C211yesmaximal, characteristic, complemented
4#30631599C212yesmaximal, complemented
5#30631571C212yesmaximal, complemented
6#30631567C212yesmaximal
7#30631559C212yesmaximal, complemented
8#30631545C212yesmaximal, complemented
9#390×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C23×Dic251C2212yes—
12C2×C4⋊Dic251C2211yescharacteristic
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×C22⋊Dic251C2212yes—
17C2×C22⋊Dic251C2212yes—
18C2×C22⋊Dic251C2212yes—
19C251⋊#95C2211yescharacteristic
20C251⋊#95C2212yes—
21C251⋊#95C2211yescharacteristic
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C22×D4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C22⋊C4×C251C2212yes—
29C2×C22⋊C4×C251C2212yes—
30C2×C42×C251C2211yescharacteristic
31C2×Dic502—42no—
32C2×Dic502—41no—
33C22×Dic251—24no—
34C22×Dic251—42no—
35C22×Dic251—42no—
36C22×Dic251C2312yes—
37C22×Dic251C2312yes—
38C22×Dic251C2312yes—
39C22×Dic251C2312yes—
40C4⋊Dic251—41no—
41Dic251⋊C4—24no—
42Dic251⋊C4—42no—
43Dic251⋊C4—42no—
44C22⋊Dic251—24no—
45C22⋊Dic251—42no—
46C22⋊Dic251—42no—
47C24×C251C2312yes—
48C2×D4×C251—41no—
49C2×D4×C251—41no—
50C2×D4×C251Q814yescomplemented
51C22×C4×C251C2311yescharacteristic
52C22×C4×C251C2312yes—
53C22×C4×C251C2311yescharacteristic
54C22×C4×C251C2311yescharacteristic
55C4⋊C4×C251—41no—
56C22⋊C4×C251—42no—
57C22⋊C4×C251—42no—
58C42×C251—41no—
59Dic502—44no—
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—
71C2×Dic251—48no—
72C2×Dic251#4114yes—
73C23×C251—22no—
74C23×C251—22no—
75C23×C251—22no—
76C23×C251—22no—
77C23×C251—22no—
78C23×C251—22no—
79C23×C251C2×Q814yes—
80C23×C251C2411yescharacteristic
81D4×C251—44no—
82C2×C4×C251—21no—
83C2×C4×C251—21no—
84C2×C4×C251—22no—
85C2×C4×C251—22no—
86C2×C4×C251—22no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—21no—
90C2×C4×C251—21no—
91C2×C4×C251C2×Q812yes—
92C22×C251—24no—
93C22×C251—24no—
94C22×C251—24no—
95C22×C251—42no—
96C22×C251—42no—
97C22×C251—42no—
98C22×C251—42no—
99C22×C251C22×Q811yescharacteristic
100C22×C251C2×#4112yes—
101C22×C251H(5,𝔽2)11yescharacteristic
102C22×C251H(5,𝔽2)12yes—
103C22×C251H(5,𝔽2)11yescharacteristic
104C4×C251—22no—
105C4×C251—42no—
106C4×C251—41no—
107C4×C251—41no—
108Dic251—24no—
109Dic251—42no—
110Dic251—42no—
111Dic251—42no—
112C2×C251—44no—
113C2×C251#53812yes—
114C2×C251#54311yescharacteristic
115C2×C251#55412yes—
116C2×C251#55511yescharacteristic
117C2×C251#56011yescharacteristic
118C251#278411yescharacteristic, complemented, Sylow
119#2784—2511nomaximal, Sylow
120#400—2511no—
121#396—2512no—
122#395—2511no—
123#393—2512no—
124#390D25111yescharacteristic
125#387—2512no—
126#386—2512no—
127#385—2512no—
128#380—2512no—
129C22×Q8—2511no—
130C22×D4C2×D25111yescharacteristic
131C23×C4—2512no—
132C2×C4⋊C4—2512no—
133C2×C4⋊C4—2512no—
134C2×C4⋊C4—2511no—
135C2×C4⋊C4—2512no—
136C2×C4⋊C4C2×D25111yescharacteristic
137C2×C22⋊C4—2512no—
138C2×C22⋊C4—2512no—
139C2×C22⋊C4—2512no—
140C2×C22⋊C4C2×D25112yes—
141C2×C22⋊C4C2×D25112yes—
142C2×C42C2×D25111yescharacteristic
143#95—2511no—
144#95—2512no—
145#95—2511no—
146#95—2512no—
147#95—2512no—
148#95—2512no—
149#95—2512no—
150C24C22×D25112yes—
151C2×Q8—10042no—
152C2×Q8—10041no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4Dic50214yescomplemented
156C22×C4—10042no—
157C22×C4—10042no—
158C22×C4—5024no—
159C22×C4—2512no—
160C22×C4—2512no—
161C22×C4—2512no—
162C22×C4—2512no—
163C22×C4C22×D25111yescharacteristic
164C22×C4C22×D25112yes—
165C22×C4C22×D25111yescharacteristic
166C22×C4C22×D25111yescharacteristic
167C4⋊C4—41no—
168C4⋊C4—10042no—
169C4⋊C4—10042no—
170C4⋊C4—10041no—
171C4⋊C4—5024no—
172C22⋊C4—42no—
173C22⋊C4—42no—
174C22⋊C4—10042no—
175C22⋊C4—10042no—
176C22⋊C4—5024no—
177C42—41no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23C2×Dic50214yes—
185C23C23×D25111yescharacteristic
186Q8—10044no—
187D4—44no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—10048no—
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—5022no—
207C2×C4—5022no—
208C2×C4—5022no—
209C2×C4—2514no—
210C2×C4C2×Dic50212yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—42no—
215C22—42no—
216C22—42no—
217C22—42no—
218C22C22×Dic50211yescharacteristic
219C22#41×D25112yes—
220C22#2254876011yescharacteristic
221C22#2254876012yes—
222C22#2254876011yescharacteristic
223C4—22no—
224C4—42no—
225C4—41no—
226C4—41no—
227C4—10042no—
228C4—10042no—
229C4—10042no—
230C4—5024no—
231C2—44no—
232C2#3063226912yes—
233C2#3063231311yescharacteristic
234C2#3063239612yes—
235C2#3063239911yescharacteristic
236C2#3063242411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#390×C251C2
prime radicalC22×D4×C251C22
hypercenter#390D251
hypocenterC251#2784
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #30631626C2semidirect
2. #30631610C2semidirect
3. #30631607C2semidirect
4. #30631599C2semidirect
5. #30631571C2semidirect
7. #30631559C2semidirect
8. #30631545C2semidirect
50. C2×D4×C251Q8semidirect
118. C251#2784semidirect
155. C2×D4Dic502semidirect

Maximal subgroups

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

subgroupindex
#306316262
#306316102
#306316072
#306315992
#306315712
#306315672
#306315592
#306315452
#390×C2512
#2784251