Finite Group Databasebeta

← #44558649

Subgroups — #44558649

Order 32128.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D251C211yesmaximal, characteristic
2C2×#22548691C212yesmaximal, complemented
3#30631559C212yesmaximal, complemented
4#30631558C214yesmaximal, complemented
5#30631547C211yesmaximal, characteristic, complemented
6#30631544C212yesmaximal, complemented
7#30631542C212yesmaximal
8C2×C22⋊Q8×C251C211yesmaximal, characteristic, complemented
9C24×D251C2211yescharacteristic
10C23×Dic251C2211yescharacteristic
11C22×C4×D251—28no—
12C22×C4×D251—24no—
13C22×C4×D251C2212yes—
14C22×C4×D251C2212yes—
15#22548691—28no—
16C2×C4⋊Dic251C2212yescomplemented
17C2×Dic251⋊C4C2214yescomplemented
18C2×C22⋊Dic251C2212yes—
19C2×(C2×D251)⋊C4C2214yes—
20C2×(C2×D251)⋊C4C2214yes—
21C251⋊#95C2214yescomplemented
22C251⋊#95C2212yescomplemented
23C22×Q8×C251C2211yescharacteristic, complemented
24C23×C4×C251C2211yescharacteristic
25C22⋊Q8×C251—24no—
26C2×C4⋊C4×C251C2211yescharacteristic, complemented
27C2×C4⋊C4×C251C2212yescomplemented
28C2×C22⋊C4×C251C2212yes—
29C23×D251—24no—
30C23×D251—24no—
31C23×D251—22no—
32C23×D251—22no—
33C23×D251Q814yescomplemented
34C23×D251C2312yes—
35C22×Dic251—24no—
36C22×Dic251—22no—
37C22×Dic251C2314yes—
38C22×Dic251C2312yes—
39C2×C4×D251—232no—
40C2×C4×D251—24no—
41C2×C4×D251—24no—
42C2×C4×D251—24no—
43C2×C4×D251—24no—
44C2×C4×D251—24no—
45C2×C4×D251D418yes—
46C4⋊Dic251—24no—
47Dic251⋊C4—28no—
48C22⋊Dic251—24no—
49(C2×D251)⋊C4—28no—
50(C2×D251)⋊C4—28no—
51C24×C251C2311yescharacteristic
52C2×Q8×C251—22no—
53C2×Q8×C251—24no—
54C22×C4×C251—24no—
55C22×C4×C251D414yes—
56C22×C4×C251C2312yes—
57C22×C4×C251C2312yes—
58C22×C4×C251C2311yescharacteristic
59C22×C4×C251C2311yescharacteristic
60C4⋊C4×C251—22no—
61C4⋊C4×C251—24no—
62C22⋊C4×C251—24no—
63C22×D251—28no—
64C22×D251—216no—
65C22×D251—28no—
66C22×D251—28no—
67C22×D251—22no—
68C22×D251—22no—
69C22×D251—22no—
70C22×D251—22no—
71C22×D251—216no—
72C22×D251C2×D414yes—
73C22×D251C2×Q814yes—
74C22×D251#4114yes—
75C2×Dic251—28no—
76C2×Dic251—22no—
77C2×Dic251—22no—
78C2×Dic251—24no—
79C2×Dic251—24no—
80C2×Dic251—24no—
81C2×Dic251C2×D414yes—
82C4×D251—216no—
83C4×D251—216no—
84C23×C251—22no—
85C23×C251—22no—
86C23×C251C2×D412yes—
87C23×C251C2×Q812yes—
88C23×C251#4112yes—
89C23×C251C2411yescharacteristic
90Q8×C251—44no—
91C2×C4×C251—22no—
92C2×C4×C251—28no—
93C2×C4×C251—22no—
94C2×C4×C251—22no—
95C2×C4×C251—24no—
96C2×C4×C251—24no—
97C2×C4×C251C2×D414yes—
98C2×C4×C251C2×D412yes—
99C2×C4×C251C2×D412yes—
100C22×C251—22no—
101C22×C251—24no—
102C22×C251—22no—
103C22×C251—24no—
104C22×C251C22⋊Q814yescomplemented
105C22×C251C22×D412yes—
106C22×C251C22×D411yescharacteristic
107C22×C251C22×Q811yescharacteristic
108C22×C251C2×#4112yes—
109C22×C251C2×#4111yescharacteristic
110C2×D251—232no—
111C2×D251—24no—
112C2×D251—24no—
113C2×D251—24no—
114C2×D251—24no—
115C2×D251—24no—
116C2×D251—24no—
117C2×D251C22⋊Q818yescomplemented
118C4×C251—22no—
119C4×C251—42no—
120C4×C251—42no—
121C4×C251#12014yes—
122Dic251—24no—
123Dic251—44no—
124C2×C251—24no—
125C2×C251C2×#12011yescharacteristic
126C2×C251C2×C22⋊Q812yes—
127C2×C251C2×C22⋊Q811yescharacteristic
128C2×C251#52511yescharacteristic
129C2×C251#52512yes—
130D251—28no—
131C251#284811yescharacteristic, complemented, Sylow
132#2848—2511nomaximal, Sylow
133C24×C4—2511no—
134C2×C22⋊Q8—2512no—
135C2×C22⋊Q8D25111yescharacteristic, complemented
136#385—2514no—
137#385—2512no—
138#380—2511no—
139#380—2512no—
140#379—2512no—
141C25—2511no—
142C22×Q8C2×D25111yescharacteristic, complemented
143C23×C4—5028no—
144C23×C4—5024no—
145C23×C4—2511no—
146C23×C4—2512no—
147C23×C4—2512no—
148C23×C4C2×D25111yescharacteristic
149C22⋊Q8—24no—
150C22⋊Q8—5028no—
151C2×C4⋊C4—2512no—
152C2×C4⋊C4—2514no—
153C2×C4⋊C4C2×D25111yescharacteristic, complemented
154C2×C4⋊C4C2×D25112yescomplemented
155C2×C22⋊C4—2512no—
156C2×C22⋊C4—2514no—
157C2×C22⋊C4—2514no—
158C2×C22⋊C4C2×D25112yes—
159#95—2512no—
160#95—2514no—
161C24—5024no—
162C24—5024no—
163C24—5022no—
164C24—5022no—
165C24—2514no—
166C24—2512no—
167C24C22×D25111yescharacteristic
168C2×Q8—24no—
169C2×Q8—22no—
170C22×C4—24no—
171C22×C4—50232no—
172C22×C4—5024no—
173C22×C4—5024no—
174C22×C4—5022no—
175C22×C4—5024no—
176C22×C4—5024no—
177C22×C4—5024no—
178C22×C4—5024no—
179C22×C4—2514no—
180C22×C4—2512no—
181C22×C4—2518no—
182C22×C4C22⋊D25114yes—
183C22×C4C22×D25112yes—
184C22×C4C22×D25112yes—
185C22×C4C22×D25111yescharacteristic
186C22×C4C22×D25111yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—5028no—
190C4⋊C4—5024no—
191C22⋊C4—24no—
192C22⋊C4—5028no—
193C22⋊C4—5028no—
194C22⋊C4—5024no—
195C23—22no—
196C23—22no—
197C23—5028no—
198C23—5028no—
199C23—50216no—
200C23—50216no—
201C23—5022no—
202C23—5022no—
203C23—5022no—
204C23—5022no—
205C23—5028no—
206C23—2514no—
207C23—2514no—
208C23—2514no—
209C23C2×C22⋊D25112yes—
210C23Q8×D25112yes—
211C23#1686081112yes—
212C23C23×D25111yescharacteristic
213Q8—44no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—24no—
220C2×C4—50216no—
221C2×C4—5024no—
222C2×C4—5024no—
223C2×C4—5024no—
224C2×C4—5028no—
225C2×C4—5022no—
226C2×C4—5022no—
227C2×C4—50216no—
228C2×C4—2514no—
229C2×C4D4×D25114yes—
230C2×C4C2×C22⋊D25112yes—
231C2×C4C2×C22⋊D25112yes—
232C22—22no—
233C22—24no—
234C22—22no—
235C22—24no—
236C22—50232no—
237C22—5024no—
238C22—5024no—
239C22—5024no—
240C22—5024no—
241C22—5024no—
242C22—5024no—
243C22—2518no—
244C22#2254869114yescomplemented
245C22C2×D4×D25112yes—
246C22C22×C22⋊D25111yescharacteristic
247C22C2×Q8×D25111yescharacteristic
248C22#41×D25112yes—
249C22C2×#1686081111yescharacteristic
250C4—22no—
251C4—42no—
252C4—42no—
253C4—10044no—
254C4—5024no—
255C4#2254868314yes—
256C2—24no—
257C2—5028no—
258C2C2×#2254868311yescharacteristic
259C2C22⋊Q8×D25112yes—
260C2C2×#2254869111yescharacteristic
261C2#3063219411yescharacteristic
262C2#3063220012yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×C22⋊Q8×C251C2
prime radicalC24×D251C22
hypercenterC2×C22⋊Q8D251
hypocenterC251#2848
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#22548691C2semidirect
3. #30631559C2semidirect
4. #30631558C2semidirect
5. #30631547C2semidirect
6. #30631544C2semidirect
8. C2×C22⋊Q8×C251C2semidirect
16. C2×C4⋊Dic251C22semidirect
17. C2×Dic251⋊C4C22semidirect
21. C251⋊#95C22semidirect
22. C251⋊#95C22semidirect
23. C22×Q8×C251C22semidirect
26. C2×C4⋊C4×C251C22semidirect
27. C2×C4⋊C4×C251C22semidirect
33. C23×D251Q8semidirect
104. C22×C251C22⋊Q8semidirect
117. C2×D251C22⋊Q8semidirect
131. C251#2848semidirect
135. C2×C22⋊Q8D251semidirect
142. C22×Q8C2×D251semidirect
153. C2×C4⋊C4C2×D251semidirect
154. C2×C4⋊C4C2×D251semidirect
244. C22#22548691semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D2512
C2×#225486912
#306315592
#306315582
#306315472
#306315442
#306315422
C2×C22⋊Q8×C2512
#2848251