Finite Group Databasebeta

← #44558464

Subgroups — #44558464

Order 32128.

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

normal subgroups only

This group has 464 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#30631628C212yesmaximal, complemented
2#30631623C211yesmaximal, characteristic, complemented
3#30631615C212yesmaximal, complemented
4#30631610C212yesmaximal, complemented
5#30631601C212yesmaximal, complemented
6#30631601C212yesmaximal, complemented
7#30631567C212yesmaximal, complemented
8#30631559C211yesmaximal, characteristic, complemented
9#392×C251C211yesmaximal, characteristic
10C23×Dic251C2211yescharacteristic
11C2×C4⋊Dic251C2211yescharacteristic
12C2×Dic251⋊C4C2212yes—
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×C22⋊Dic251C2211yescharacteristic
16C2×C22⋊Dic251C2212yes—
17C2×C22⋊Dic251C2212yes—
18C2×C22⋊Dic251C2212yes—
19C2×C22⋊Dic251C2212yes—
20C2×C22⋊Dic251C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C22×D4×C251C2212yes—
27C22×D4×C251C2211yescharacteristic
28C2×C22⋊C4×C251C2211yescharacteristic
29C2×C22⋊C4×C251C2212yes—
30#95×C251C2211yescharacteristic
31C22×Dic251—41no—
32C22×Dic251—42no—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C4⋊Dic251—41no—
38Dic251⋊C4—42no—
39Dic251⋊C4—42no—
40Dic251⋊C4—42no—
41C22⋊Dic251—22no—
42C22⋊Dic251—42no—
43C22⋊Dic251—42no—
44C22⋊Dic251—42no—
45C22⋊Dic251—42no—
46C22⋊Dic251—42no—
47C24×C251C2312yes—
48C24×C251C2311yescharacteristic
49C2×D4×C251—24no—
50C2×D4×C251—42no—
51C2×D4×C251—41no—
52C2×D4×C251—42no—
53C2×D4×C251—42no—
54C22×C4×C251C2312yes—
55C22×C4×C251C2311yescharacteristic
56C22×C4×C251C2311yescharacteristic
57C22⋊C4×C251—42no—
58C22⋊C4×C251—41no—
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—
71C2×Dic251—44no—
72C23×C251—21no—
73C23×C251—22no—
74C23×C251—22no—
75C23×C251—21no—
76C23×C251—22no—
77C23×C251—22no—
78C23×C251—22no—
79C23×C251—22no—
80C23×C251—22no—
81C23×C251—22no—
82C23×C251—22no—
83C23×C251#4112yes—
84C23×C251C2411yescharacteristic
85D4×C251—48no—
86D4×C251—44no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—21no—
90C2×C4×C251—22no—
91C2×C4×C251—21no—
92C2×C4×C251—22no—
93C2×C4×C251—22no—
94C22×C251—24no—
95C22×C251—22no—
96C22×C251—41no—
97C22×C251—41no—
98C22×C251—42no—
99C22×C251—42no—
100C22×C251—42no—
101C22×C251—42no—
102C22×C251—42no—
103C22×C251—42no—
104C22×C251—42no—
105C22×C251—42no—
106C22×C251C2×#4111yescharacteristic
107C22×C251H(5,𝔽2)12yes—
108C22×C251H(5,𝔽2)12yes—
109C22×C251H(5,𝔽2)11yescharacteristic
110C22×C2512−1+411yescharacteristic
111C4×C251—42no—
112C4×C251—41no—
113C4×C251—41no—
114Dic251—42no—
115Dic251—42no—
116Dic251—42no—
117Dic251—42no—
118C2×C251—42no—
119C2×C251—82no—
120C2×C251#53812yes—
121C2×C251#53911yescharacteristic
122C2×C251#56011yescharacteristic
123C2×C251#56011yescharacteristic
124C2×C251#56212yes—
125C251#282611yescharacteristic, complemented, Sylow
126#2826—2511nomaximal, Sylow
127#400—2512no—
128#399—2511no—
129#397—2512no—
130#396—2512no—
131#394—2512no—
132#394—2512no—
133#392D25111yescharacteristic
134#386—2512no—
135#385—2511no—
136C22×D4C2×D25112yes—
137C22×D4C2×D25111yescharacteristic
138C23×C4—2511no—
139C2×C4⋊C4—2512no—
140C2×C4⋊C4—2512no—
141C2×C4⋊C4—2512no—
142C2×C4⋊C4—2511no—
143C2×C22⋊C4—2512no—
144C2×C22⋊C4—2512no—
145C2×C22⋊C4—2512no—
146C2×C22⋊C4—2512no—
147C2×C22⋊C4—2512no—
148C2×C22⋊C4—2511no—
149C2×C22⋊C4C2×D25111yescharacteristic
150C2×C22⋊C4C2×D25112yes—
151#95—2512no—
152#95—2512no—
153#95—2512no—
154#95—2512no—
155#95—2511no—
156#95C2×D25111yescharacteristic
157C24C22×D25112yes—
158C24C22×D25111yescharacteristic
159C2×D4—24no—
160C2×D4—42no—
161C2×D4—41no—
162C2×D4—42no—
163C2×D4—42no—
164C22×C4—10041no—
165C22×C4—10042no—
166C22×C4—2512no—
167C22×C4—2512no—
168C22×C4—2512no—
169C22×C4—2512no—
170C22×C4C22×D25112yes—
171C22×C4C22×D25111yescharacteristic
172C22×C4C22×D25111yescharacteristic
173C4⋊C4—10042no—
174C4⋊C4—10042no—
175C4⋊C4—10042no—
176C4⋊C4—10041no—
177C22⋊C4—41no—
178C22⋊C4—42no—
179C22⋊C4—10042no—
180C22⋊C4—10042no—
181C22⋊C4—10042no—
182C22⋊C4—10042no—
183C22⋊C4—10042no—
184C22⋊C4—5022no—
185C23—21no—
186C23—22no—
187C23—22no—
188C23—21no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23#1686080912yes—
197C23C23×D25111yescharacteristic
198D4—48no—
199D4—44no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—10044no—
208C2×C4—5022no—
209C2×C4—5022no—
210C2×C4—5022no—
211C2×C4—5022no—
212C2×C4—5022no—
213C2×C4—5022no—
214C2×C4—5022no—
215C2×C4—5022no—
216C2×C4—5022no—
217C2×C4—5022no—
218C2×C4—5022no—
219C2×C4—5022no—
220C22—24no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—42no—
232C22C2×#1686080911yescharacteristic
233C22#2254876012yes—
234C22#2254876012yes—
235C22#2254876011yescharacteristic
236C22#2254876211yescharacteristic
237C4—42no—
238C4—41no—
239C4—41no—
240C4—10042no—
241C4—10042no—
242C4—10042no—
243C4—10042no—
244C2—42no—
245C2—82no—
246C2#3063227212yes—
247C2#3063228211yescharacteristic
248C2#3063242411yescharacteristic
249C2#3063242611yescharacteristic
250C2#3063243212yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting, prime radical#392×C251C2
hypercenter#392D251
hypocenterC251#2826
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #30631628C2semidirect
2. #30631623C2semidirect
3. #30631615C2semidirect
4. #30631610C2semidirect
5. #30631601C2semidirect
6. #30631601C2semidirect
7. #30631567C2semidirect
8. #30631559C2semidirect
125. C251#2826semidirect

Maximal subgroups

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

subgroupindex
#306316282
#306316232
#306316152
#306316102
#306316012
#306315672
#306315592
#392×C2512
#2826251