Finite Group Databasebeta

← #44558537

Subgroups — #44558537

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#30631625C212yesmaximal, complemented
2#30631616C212yesmaximal, complemented
3#30631604C211yesmaximal, characteristic
4#30631604C212yesmaximal
5#30631600C212yesmaximal, complemented
6#30631598C212yesmaximal, complemented
7#30631592C211yesmaximal, characteristic, complemented
8#30631576C212yesmaximal, complemented
9#395×C251C211yesmaximal, characteristic, complemented
10C22×Dic502C2211yescharacteristic
11C22×D1004C2211yescharacteristic
12C2×Dic251⋊C4C2212yes—
13C2×(C2×D251)⋊C4C2212yes—
14C2×(C2×D251)⋊C4C2212yes—
15C2×(C2×D251)⋊C4C2212yes—
16C2×(C2×D251)⋊C4C2212yes—
17C2×(C2×D251)⋊C4C2212yes—
18C2×(C2×D251)⋊C4C2212yes—
19C2×C4×Dic251C2211yescharacteristic
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C22×Q8×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2212yes—
29#95×C251C2212yes—
30#95×C251C2211yescharacteristic
31C23×D251C2312yes—
32C2×Dic502—22no—
33C2×Dic502—41no—
34C2×Dic502—41no—
35C2×D1004—41no—
36C2×D1004—42no—
37C22×Dic251C2312yes—
38C22×Dic251C2312yes—
39C22×Dic251C2312yes—
40Dic251⋊C4—42no—
41(C2×D251)⋊C4—24no—
42(C2×D251)⋊C4—42no—
43(C2×D251)⋊C4—42no—
44(C2×D251)⋊C4—42no—
45(C2×D251)⋊C4—42no—
46(C2×D251)⋊C4—42no—
47C4×Dic251—41no—
48C2×Q8×C251—41no—
49C2×Q8×C251—42no—
50C22×C4×C251C2311yescharacteristic
51C22×C4×C251C2312yes—
52C22×C4×C251C2312yes—
53C22×C4×C251C2311yescharacteristic
54C22×C4×C251C2311yescharacteristic
55C4⋊C4×C251—41no—
56C4⋊C4×C251—42no—
57C22×D251—22no—
58C22×D251—22no—
59C22×D251—22no—
60C22×D251—22no—
61C22×D251—22no—
62C22×D251—22no—
63C22×D251—22no—
64Dic502—44no—
65D1004—44no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251—22no—
69C2×Dic251—22no—
70C2×Dic251—22no—
71C2×Dic251—22no—
72C2×Dic251—22no—
73C2×Dic251—22no—
74C2×Dic251—22no—
75C23×C251C2411yescharacteristic
76Q8×C251—44no—
77C2×C4×C251—21no—
78C2×C4×C251—21no—
79C2×C4×C251—21no—
80C2×C4×C251—22no—
81C2×C4×C251—22no—
82C2×C4×C251—21no—
83C2×C4×C251—22no—
84C2×C4×C251—22no—
85C2×C4×C251—22no—
86C2×C4×C251—22no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251#4112yes—
90C22×C251C2×#4111yescharacteristic
91C22×C251H(5,𝔽2)11yescharacteristic
92C22×C251H(5,𝔽2)12yes—
93C22×C251H(5,𝔽2)11yescharacteristic
94C22×C2512−1+412yes—
95C2×D251—42no—
96C2×D251—42no—
97C2×D251—42no—
98C2×D251—42no—
99C2×D251—42no—
100C2×D251—42no—
101C2×D251—42no—
102C4×C251—22no—
103C4×C251—41no—
104C4×C251—42no—
105C4×C251—41no—
106C4×C251—42no—
107Dic251—42no—
108Dic251—42no—
109Dic251—42no—
110C2×C251#53811yescharacteristic
111C2×C251#54212yes—
112C2×C251H(3,𝔽4)11yescharacteristic
113C2×C251#56212yes—
114C2×C251#56311yescharacteristic
115D251—82no—
116C251#283211yescharacteristic, complemented, Sylow
117#2832—2511nomaximal, Sylow
118#400—2512no—
119#397—2512no—
120#395D25111yescharacteristic, complemented
121#394—2512no—
122#394—2511no—
123#393—2512no—
124#393—2512no—
125#391—2511no—
126#388—2512no—
127C22×Q8—2511no—
128C22×Q8C2×D25111yescharacteristic
129C22×D4—2511no—
130C2×C4⋊C4—2512no—
131C2×C4⋊C4C2×D25111yescharacteristic
132C2×C4⋊C4C2×D25112yes—
133C2×C22⋊C4—2512no—
134C2×C22⋊C4—2512no—
135C2×C22⋊C4—2512no—
136C2×C22⋊C4—2512no—
137C2×C22⋊C4—2512no—
138C2×C22⋊C4—2512no—
139C2×C42—2511no—
140#95—2512no—
141#95—2512no—
142#95—2512no—
143#95—2512no—
144#95—2512no—
145#95—2511no—
146#95C2×D25112yes—
147#95C2×D25111yescharacteristic
148C24—2512no—
149C2×Q8—41no—
150C2×Q8—42no—
151C2×Q8—10041no—
152C2×Q8—10041no—
153C2×Q8—5022no—
154C2×D4—10041no—
155C2×D4—10042no—
156C22×C4—2512no—
157C22×C4—2512no—
158C22×C4—2512no—
159C22×C4C22×D25111yescharacteristic
160C22×C4C22×D25112yes—
161C22×C4C22×D25111yescharacteristic
162C22×C4C22×D25112yes—
163C22×C4C22×D25111yescharacteristic
164C4⋊C4—41no—
165C4⋊C4—42no—
166C4⋊C4—10042no—
167C22⋊C4—10042no—
168C22⋊C4—10042no—
169C22⋊C4—10042no—
170C22⋊C4—10042no—
171C22⋊C4—10042no—
172C22⋊C4—5024no—
173C42—10041no—
174C23—5022no—
175C23—5022no—
176C23—5022no—
177C23—5022no—
178C23—5022no—
179C23—5022no—
180C23—5022no—
181C23C23×D25111yescharacteristic
182Q8—44no—
183Q8—10044no—
184D4—10044no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
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#1686081112yes—
207C22—10042no—
208C22—10042no—
209C22—10042no—
210C22—10042no—
211C22—10042no—
212C22—10042no—
213C22—10042no—
214C22C2×#1686081111yescharacteristic
215C22#2254876011yescharacteristic
216C22#2254876012yes—
217C22#2254876111yescharacteristic
218C22#2254876312yes—
219C4—22no—
220C4—41no—
221C4—42no—
222C4—41no—
223C4—42no—
224C4—10042no—
225C4—10042no—
226C4—10042no—
227C2—20082no—
228C2#3063227711yescharacteristic
229C2#3063231112yes—
230C2#3063243011yescharacteristic
231C2#3063243312yes—
232C2#3063243611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#395×C251C2
prime radicalC22×D1004C22
hypercenter#395D251
hypocenterC251#2832
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. #30631625C2semidirect
2. #30631616C2semidirect
5. #30631600C2semidirect
6. #30631598C2semidirect
7. #30631592C2semidirect
8. #30631576C2semidirect
9. #395×C251C2semidirect
116. C251#2832semidirect
120. #395D251semidirect

Maximal subgroups

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

subgroupindex
#306316252
#306316162
#306316042
#306316002
#306315982
#306315922
#306315762
#395×C2512
#2832251