Finite Group Databasebeta

← #44558653

Subgroups — #44558653

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×#22548695C212yesmaximal, complemented
3C2×#22548694C211yesmaximal, characteristic, complemented
4#30631562C212yesmaximal, complemented
5#30631561C214yesmaximal, complemented
6#30631544C212yesmaximal, complemented
7#30631542C212yesmaximal
8#380×C251C211yesmaximal, characteristic, complemented
9C24×D251C2211yescharacteristic
10C22×Dic502C2211yescharacteristic, complemented
11C23×Dic251C2211yescharacteristic
12C22×C4×D251—28no—
13C22×C4×D251—24no—
14C22×C4×D251C2212yes—
15C22×C4×D251C2212yes—
16#22548695—28no—
17#22548694—24no—
18C2×C4⋊Dic251C2211yescharacteristic, complemented
19C2×C4⋊Dic251C2214yescomplemented
20C2×Dic251⋊C4C2212yescomplemented
21C2×C22⋊Dic251C2212yes—
22C2×(C2×D251)⋊C4C2214yes—
23C2×(C2×D251)⋊C4C2214yes—
24C251⋊#95C2214yescomplemented
25C23×C4×C251C2211yescharacteristic
26C2×C4⋊C4×C251C2212yescomplemented
27C2×C22⋊C4×C251C2212yes—
28#95×C251C2212yescomplemented
29C23×D251—24no—
30C23×D251—24no—
31C23×D251—22no—
32C23×D251—22no—
33C23×D251Q814yescomplemented
34C23×D251C2312yes—
35C2×Dic502—22no—
36C2×Dic502—24no—
37C22×Dic251—24no—
38C22×Dic251—22no—
39C22×Dic251C2312yes—
40C22×Dic251C2312yes—
41C22×Dic251C2312yes—
42C2×C4×D251—232no—
43C2×C4×D251—24no—
44C2×C4×D251—24no—
45C2×C4×D251—24no—
46C2×C4×D251—24no—
47C2×C4×D251—24no—
48C2×C4×D251D418yes—
49C4⋊Dic251—28no—
50C4⋊Dic251—22no—
51Dic251⋊C4—24no—
52C22⋊Dic251—24no—
53(C2×D251)⋊C4—28no—
54(C2×D251)⋊C4—28no—
55C24×C251C2311yescharacteristic
56C22×C4×C251—24no—
57C22×C4×C251D414yes—
58C22×C4×C251C2314yes—
59C22×C4×C251C2311yescharacteristic
60C22×C4×C251C2311yescharacteristic
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—
75Dic502—44no—
76C2×Dic251—28no—
77C2×Dic251—22no—
78C2×Dic251—22no—
79C2×Dic251—24no—
80C2×Dic251—22no—
81C2×Dic251—22no—
82C2×Dic251—24no—
83C2×Dic251C2×D414yes—
84C4×D251—216no—
85C4×D251—216no—
86C23×C251—22no—
87C23×C251—22no—
88C23×C251C2×D412yes—
89C23×C251C2×Q812yes—
90C23×C251#4112yes—
91C23×C251C2411yescharacteristic
92C2×C4×C251—22no—
93C2×C4×C251—28no—
94C2×C4×C251—24no—
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—44no—
120C4×C251#12014yes—
121Dic251—24no—
122Dic251—42no—
123Dic251—42no—
124C2×C251—24no—
125C2×C251C2×#12011yescharacteristic
126C2×C251C2×C22⋊Q812yes—
127C2×C251C2×C22⋊Q811yescharacteristic
128C2×C251#52512yes—
129C2×C251#52511yescharacteristic
130D251—28no—
131C251#284811yescharacteristic, complemented, Sylow
132#2848—2511nomaximal, Sylow
133C24×C4—2511no—
134C2×C22⋊Q8—2511no—
135C2×C22⋊Q8—2512no—
136#385—2512no—
137#385—2514no—
138#380—2512no—
139#380D25111yescharacteristic, complemented
140#379—2512no—
141C25—2511no—
142C22×Q8—2511no—
143C23×C4—5028no—
144C23×C4—5024no—
145C23×C4—2511no—
146C23×C4—2512no—
147C23×C4—2512no—
148C23×C4C2×D25111yescharacteristic
149C22⋊Q8—5028no—
150C22⋊Q8—5024no—
151C2×C4⋊C4—2511no—
152C2×C4⋊C4—2512no—
153C2×C4⋊C4—2514no—
154C2×C4⋊C4C2×D25112yescomplemented
155C2×C22⋊C4—2512no—
156C2×C22⋊C4—2514no—
157C2×C22⋊C4—2514no—
158C2×C22⋊C4C2×D25112yes—
159#95—2514no—
160#95C2×D25112yescomplemented
161C24—5024no—
162C24—5024no—
163C24—5022no—
164C24—5022no—
165C24—2514no—
166C24—2512no—
167C24C22×D25111yescharacteristic
168C2×Q8—5024no—
169C2×Q8—5022no—
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—2512no—
180C22×C4—2512no—
181C22×C4—2512no—
182C22×C4—2518no—
183C22×C4D100414yes—
184C22×C4C22×D25114yes—
185C22×C4C22×D25111yescharacteristic
186C22×C4C22×D25111yescharacteristic
187C4⋊C4—24no—
188C4⋊C4—5028no—
189C4⋊C4—5024no—
190C4⋊C4—5022no—
191C22⋊C4—24no—
192C22⋊C4—5028no—
193C22⋊C4—5028no—
194C22⋊C4—5024no—
195C23—22no—
196C23—22no—
197C23—5028no—
198C23—50216no—
199C23—5028no—
200C23—50216no—
201C23—5022no—
202C23—5022no—
203C23—5022no—
204C23—5022no—
205C23—5028no—
206C23—2514no—
207C23—2514no—
208C23—2514no—
209C23C2×D100412yes—
210C23Q8×D25112yes—
211C23#1686080912yes—
212C23C23×D25111yescharacteristic
213Q8—10044no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—50216no—
220C2×C4—5024no—
221C2×C4—5022no—
222C2×C4—5022no—
223C2×C4—5024no—
224C2×C4—5028no—
225C2×C4—5022no—
226C2×C4—5022no—
227C2×C4—50216no—
228C2×C4—2514no—
229C2×C4D4×D25114yes—
230C2×C4C2×D100412yes—
231C2×C4C2×D100412yes—
232C22—24no—
233C22—22no—
234C22—22no—
235C22—24no—
236C22—50232no—
237C22—5024no—
238C22—5024no—
239C22—5024no—
240C22—5024no—
241C22—5024no—
242C22—5024no—
243C22—2518no—
244C22#2254869514yescomplemented
245C22C2×D4×D25112yes—
246C22C22×D100411yescharacteristic
247C22C2×Q8×D25111yescharacteristic
248C22#41×D25112yes—
249C22C2×#1686080911yescharacteristic
250C4—22no—
251C4—44no—
252C4—10042no—
253C4—10042no—
254C4—5024no—
255C4#2254868514yes—
256C2—24no—
257C2—5028no—
258C2C2×#2254868511yescharacteristic
259C2C22⋊Q8×D25112yes—
260C2C2×#2254869511yescharacteristic
261C2#3063219712yes—
262C2#3063219911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#380×C251C2
prime radicalC24×D251C22
hypercenter#380D251
hypocenterC251#2848
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#22548695C2semidirect
3. C2×#22548694C2semidirect
4. #30631562C2semidirect
5. #30631561C2semidirect
6. #30631544C2semidirect
8. #380×C251C2semidirect
10. C22×Dic502C22semidirect
18. C2×C4⋊Dic251C22semidirect
19. C2×C4⋊Dic251C22semidirect
20. C2×Dic251⋊C4C22semidirect
24. C251⋊#95C22semidirect
26. C2×C4⋊C4×C251C22semidirect
28. #95×C251C22semidirect
33. C23×D251Q8semidirect
104. C22×C251C22⋊Q8semidirect
117. C2×D251C22⋊Q8semidirect
131. C251#2848semidirect
139. #380D251semidirect
154. C2×C4⋊C4C2×D251semidirect
160. #95C2×D251semidirect
244. C22#22548695semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D2512
C2×#225486952
C2×#225486942
#306315622
#306315612
#306315442
#306315422
#380×C2512
#2848251