Finite Group Databasebeta

← #44558489

Subgroups — #44558489

Order 32128.

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

normal subgroups only

This group has 508 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#30631616C212yesmaximal, complemented
2#30631605C212yesmaximal, complemented
3#30631604C211yesmaximal, characteristic, complemented
4#30631604C212yesmaximal, complemented
5#30631597C212yesmaximal, complemented
6#30631597C212yesmaximal, complemented
7#30631588C211yesmaximal, characteristic, complemented
8#30631576C212yesmaximal, complemented
9#400×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C22×D1004C2212yes—
12C22×D1004C2211yescharacteristic
13C22×D1004C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×(C2×D251)⋊C4C2212yes—
16C2×(C2×D251)⋊C4C2212yes—
17C2×(C2×D251)⋊C4C2212yes—
18C2×(C2×D251)⋊C4C2212yes—
19C2×(C2×D251)⋊C4C2212yes—
20C2×(C2×D251)⋊C4C2212yes—
21C2×(C2×D251)⋊C4C2212yes—
22C2×(C2×D251)⋊C4C2212yes—
23C2×C4×Dic251C2211yescharacteristic
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C2×C4⋊C4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28#95×C251C2212yes—
29#95×C251C2212yes—
30#95×C251C2211yescharacteristic
31C23×D251C2312yes—
32C23×D251C2312yes—
33C2×D1004—22no—
34C2×D1004—41no—
35C2×D1004—41no—
36C2×D1004—41no—
37C2×D1004—42no—
38C2×D1004—42no—
39C2×D1004—42no—
40C2×D1004—42no—
41C2×D1004—42no—
42C2×D1004—42no—
43C2×D1004—42no—
44C22×Dic251C2312yes—
45C22×Dic251C2312yes—
46Dic251⋊C4—42no—
47(C2×D251)⋊C4—24no—
48(C2×D251)⋊C4—42no—
49(C2×D251)⋊C4—42no—
50(C2×D251)⋊C4—42no—
51(C2×D251)⋊C4—42no—
52(C2×D251)⋊C4—42no—
53(C2×D251)⋊C4—42no—
54(C2×D251)⋊C4—42no—
55C4×Dic251—41no—
56C22×C4×C251C2312yes—
57C22×C4×C251C2312yes—
58C22×C4×C251C2311yescharacteristic
59C22×C4×C251C2311yescharacteristic
60C22×C4×C251C2311yescharacteristic
61C4⋊C4×C251—41no—
62C4⋊C4×C251—41no—
63C22×D251—22no—
64C22×D251—22no—
65C22×D251—22no—
66C22×D251—22no—
67C22×D251—22no—
68C22×D251—22no—
69C22×D251—22no—
70C22×D251—22no—
71C22×D251—22no—
72C22×D251—22no—
73C22×D251—22no—
74C22×D251—22no—
75C22×D251—22no—
76C22×D251—22no—
77D1004—44no—
78D1004—44no—
79D1004—48no—
80D1004—48no—
81C2×Dic251—22no—
82C2×Dic251—22no—
83C2×Dic251—22no—
84C2×Dic251—22no—
85C2×Dic251—22no—
86C2×Dic251—22no—
87C23×C251C2411yescharacteristic
88C2×C4×C251—21no—
89C2×C4×C251—21no—
90C2×C4×C251—21no—
91C2×C4×C251—21no—
92C2×C4×C251—22no—
93C2×C4×C251—22no—
94C2×C4×C251—22no—
95C2×C4×C251—22no—
96C2×C4×C251—22no—
97C2×C4×C251—22no—
98C2×C4×C251—22no—
99C2×C4×C251—22no—
100C2×C4×C251#4112yes—
101C22×C251C2×#4111yescharacteristic
102C22×C251H(5,𝔽2)11yescharacteristic
103C22×C251H(5,𝔽2)11yescharacteristic
104C22×C251H(5,𝔽2)12yes—
105C22×C251H(5,𝔽2)12yes—
106C2×D251—42no—
107C2×D251—42no—
108C2×D251—42no—
109C2×D251—42no—
110C2×D251—42no—
111C2×D251—42no—
112C2×D251—42no—
113C2×D251—42no—
114C2×D251—42no—
115C2×D251—42no—
116C2×D251—42no—
117C2×D251—42no—
118C2×D251—42no—
119C2×D251—42no—
120C4×C251—22no—
121C4×C251—41no—
122C4×C251—41no—
123C4×C251—42no—
124C4×C251—42no—
125Dic251—42no—
126Dic251—42no—
127C2×C251#53811yescharacteristic
128C2×C251#54012yes—
129C2×C251#56011yescharacteristic
130C2×C251#56012yes—
131C2×C251H(3,𝔽4)11yescharacteristic
132D251—82no—
133D251—82no—
134C251#282811yescharacteristic, complemented, Sylow
135#2828—2511nomaximal, Sylow
136#400D25111yescharacteristic, complemented
137#397—2512no—
138#394—2512no—
139#394—2511no—
140#394—2512no—
141#392—2512no—
142#392—2512no—
143#390—2511no—
144#388—2512no—
145C22×D4—2512no—
146C22×D4—2511no—
147C22×D4—2512no—
148C22×D4—2511no—
149C2×C4⋊C4—2512no—
150C2×C4⋊C4C2×D25111yescharacteristic
151C2×C4⋊C4C2×D25111yescharacteristic
152C2×C22⋊C4—2512no—
153C2×C22⋊C4—2512no—
154C2×C22⋊C4—2512no—
155C2×C22⋊C4—2512no—
156C2×C22⋊C4—2512no—
157C2×C22⋊C4—2512no—
158C2×C22⋊C4—2512no—
159C2×C22⋊C4—2512no—
160C2×C42—2511no—
161#95—2512no—
162#95—2511no—
163#95C2×D25112yes—
164#95C2×D25112yes—
165#95C2×D25111yescharacteristic
166C24—2512no—
167C24—2512no—
168C2×D4—10041no—
169C2×D4—10041no—
170C2×D4—10041no—
171C2×D4—10042no—
172C2×D4—10042no—
173C2×D4—10042no—
174C2×D4—10042no—
175C2×D4—10042no—
176C2×D4—10042no—
177C2×D4—10042no—
178C2×D4—5022no—
179C22×C4—2512no—
180C22×C4—2512no—
181C22×C4C22×D25112yes—
182C22×C4C22×D25111yescharacteristic
183C22×C4C22×D25112yes—
184C22×C4C22×D25111yescharacteristic
185C22×C4C22×D25111yescharacteristic
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—10042no—
189C22⋊C4—10042no—
190C22⋊C4—10042no—
191C22⋊C4—10042no—
192C22⋊C4—10042no—
193C22⋊C4—10042no—
194C22⋊C4—10042no—
195C22⋊C4—10042no—
196C22⋊C4—5024no—
197C42—10041no—
198C23—5022no—
199C23—5022no—
200C23—5022no—
201C23—5022no—
202C23—5022no—
203C23—5022no—
204C23—5022no—
205C23—5022no—
206C23—5022no—
207C23—5022no—
208C23—5022no—
209C23—5022no—
210C23—5022no—
211C23—5022no—
212C23C23×D25111yescharacteristic
213D4—10044no—
214D4—10044no—
215D4—10048no—
216D4—10048no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—5022no—
230C2×C4—5022no—
231C2×C4—5022no—
232C2×C4—5022no—
233C2×C4—5022no—
234C2×C4—5022no—
235C2×C4#1686081112yes—
236C22—10042no—
237C22—10042no—
238C22—10042no—
239C22—10042no—
240C22—10042no—
241C22—10042no—
242C22—10042no—
243C22—10042no—
244C22—10042no—
245C22—10042no—
246C22—10042no—
247C22—10042no—
248C22—10042no—
249C22—10042no—
250C22C2×#1686081111yescharacteristic
251C22#2254876011yescharacteristic
252C22#2254876111yescharacteristic
253C22#2254876112yes—
254C22#2254876112yes—
255C4—22no—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—42no—
260C4—10042no—
261C4—10042no—
262C2—20082no—
263C2—20082no—
264C2#3063227711yescharacteristic
265C2#3063229312yes—
266C2#3063242711yescharacteristic
267C2#3063242812yes—
268C2#3063242911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#400×C251C2
hypercenter#400D251
hypocenterC251#2828
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. #30631616C2semidirect
2. #30631605C2semidirect
3. #30631604C2semidirect
4. #30631604C2semidirect
5. #30631597C2semidirect
6. #30631597C2semidirect
7. #30631588C2semidirect
8. #30631576C2semidirect
9. #400×C251C2semidirect
134. C251#2828semidirect
136. #400D251semidirect

Maximal subgroups

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

subgroupindex
#306316162
#306316052
#306316042
#306315972
#306315882
#306315762
#400×C2512
#2828251