Finite Group Databasebeta

← #44558713

Subgroups — #44558713

Order 32128.

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

normal subgroups only

This group has 644 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
1C2×#22548713C211yesmaximal, characteristic, complemented
2C2×#22548694C211yesmaximal, characteristic
3C22×C4×Dic251C211yesmaximal, characteristic
4#30631594C211yesmaximal, characteristic, complemented
5#30631592C211yesmaximal, characteristic, complemented
6#30631582C214yesmaximal, complemented
7#30631556C211yesmaximal, characteristic, complemented
8#30631545C214yesmaximal
9C2×C22⋊Q8×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C23×Dic251C2212yes—
12#22548713—24no—
13#22548694—24no—
14C2×C4⋊Dic251C2211yescharacteristic
15C2×Dic251⋊C4C2214yes—
16C2×Dic251⋊C4C2214yes—
17C2×Dic251⋊C4C2212yes—
18C2×C22⋊Dic251C2212yes—
19C2×C4×Dic251—24no—
20C2×C4×Dic251C2212yes—
21C2×C4×Dic251C2211yescharacteristic
22C2×C4×Dic251C2211yescharacteristic
23C251⋊#95C2214yes—
24C251⋊#95C2214yes—
25C22×Q8×C251C2211yescharacteristic
26C23×C4×C251C2211yescharacteristic
27C22⋊Q8×C251—24no—
28C2×C4⋊C4×C251C2211yescharacteristic
29C2×C4⋊C4×C251C2212yes—
30C2×C22⋊C4×C251C2212yes—
31C2×Dic502—22no—
32C2×Dic502—22no—
33C2×Dic502—22no—
34C22×Dic251—24no—
35C22×Dic251—24no—
36C22×Dic251Q818yescomplemented
37C22×Dic251C2312yes—
38C22×Dic251C2312yes—
39C22×Dic251C2312yes—
40C22×Dic251C2312yes—
41C4⋊Dic251—22no—
42Dic251⋊C4—28no—
43Dic251⋊C4—28no—
44Dic251⋊C4—24no—
45C22⋊Dic251—24no—
46C4×Dic251—24no—
47C4×Dic251—22no—
48C4×Dic251D414yes—
49C24×C251C2311yescharacteristic
50C2×Q8×C251—22no—
51C2×Q8×C251—22no—
52C2×Q8×C251—22no—
53C22×C4×C251—22no—
54C22×C4×C251—22no—
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—
63Dic502—44no—
64C2×Dic251—22no—
65C2×Dic251—22no—
66C2×Dic251—216no—
67C2×Dic251—22no—
68C2×Dic251—22no—
69C2×Dic251—22no—
70C2×Dic251—24no—
71C2×Dic251—22no—
72C2×Dic251C2×D414yes—
73C2×Dic251C2×Q814yes—
74C2×Dic251C2×Q814yes—
75C2×Dic251#4114yes—
76C23×C251—21no—
77C23×C251—21no—
78C23×C251—22no—
79C23×C251C2×D412yes—
80C23×C251C2×Q814yes—
81C23×C251C2411yescharacteristic
82Q8×C251—44no—
83C2×C4×C251—21no—
84C2×C4×C251—21no—
85C2×C4×C251—28no—
86C2×C4×C251—22no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—24no—
91C2×C4×C251C2×D412yes—
92C2×C4×C251C2×D412yes—
93C2×C4×C251C2×D412yes—
94C2×C4×C251#4112yes—
95C22×C251—22no—
96C22×C251—24no—
97C22×C251—22no—
98C22×C251—24no—
99C22×C251C4⋊Q814yescomplemented
100C22×C251C22×D411yescharacteristic
101C22×C251C22×D411yescharacteristic
102C22×C251C22×Q812yes—
103C22×C251C2×#4112yes—
104C22×C251C2×#4111yescharacteristic
105C4×C251—22no—
106C4×C251—42no—
107C4×C251—42no—
108C4×C251#12114yes—
109Dic251—24no—
110Dic251—42no—
111Dic251—42no—
112Dic251C22⋊Q818yescomplemented
113C2×C251—24no—
114C2×C251C2×#12111yescharacteristic
115C2×C251C2×C22⋊Q812yes—
116C2×C251C2×C4⋊Q811yescharacteristic
117C2×C251#53211yescharacteristic
118C2×C251#53312yes—
119C251#285711yescharacteristic, complemented, Sylow
120#2857—2511nomaximal, Sylow
121C2×C4⋊Q8—2511no—
122C2×C22⋊Q8—2511no—
123C2×C22⋊Q8D25111yescharacteristic
124C22×C42—2511no—
125#391—2511no—
126#391—2511no—
127#389—2514no—
128#384—2511no—
129#380—2514no—
130C22×Q8—2511no—
131C22×Q8C2×D25111yescharacteristic
132C23×C4—2512no—
133C23×C4C2×D25111yescharacteristic
134C4⋊Q8—5024no—
135C22⋊Q8—24no—
136C22⋊Q8—5024no—
137C2×C4⋊C4—2511no—
138C2×C4⋊C4—2512no—
139C2×C4⋊C4—2514no—
140C2×C4⋊C4—2514no—
141C2×C4⋊C4C2×D25111yescharacteristic
142C2×C4⋊C4C2×D25112yes—
143C2×C22⋊C4—2512no—
144C2×C22⋊C4C2×D25112yes—
145C2×C42—5024no—
146C2×C42—2512no—
147C2×C42—2511no—
148C2×C42—2511no—
149#95—2514no—
150#95—2514no—
151C24C22×D25111yescharacteristic
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—5022no—
156C2×Q8—5022no—
157C2×Q8—5022no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—5024no—
161C22×C4—5024no—
162C22×C4—2512no—
163C22×C4—2512no—
164C22×C4—2518no—
165C22×C4—2512no—
166C22×C4—2512no—
167C22×C4C22⋊D25114yes—
168C22×C4C22×D25112yes—
169C22×C4C22×D25112yes—
170C22×C4C22×D25111yescharacteristic
171C22×C4C22×D25111yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—24no—
174C4⋊C4—5028no—
175C4⋊C4—5028no—
176C4⋊C4—5024no—
177C4⋊C4—5022no—
178C22⋊C4—24no—
179C22⋊C4—5024no—
180C42—5024no—
181C42—5022no—
182C42—2514no—
183C23—21no—
184C23—22no—
185C23—21no—
186C23C2×C22⋊D25112yes—
187C23Q8×D25114yes—
188C23C23×D25111yescharacteristic
189Q8—44no—
190Q8—10044no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—28no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—5022no—
200C2×C4—5022no—
201C2×C4—5022no—
202C2×C4—5024no—
203C2×C4—5022no—
204C2×C4—5022no—
205C2×C4—5022no—
206C2×C4—50216no—
207C2×C4—2514no—
208C2×C4—2514no—
209C2×C4—2514no—
210C2×C4—2514no—
211C2×C4D4×D25112yes—
212C2×C4C2×C22⋊D25112yes—
213C2×C4C2×C22⋊D25112yes—
214C2×C4#1686080912yes—
215C22—22no—
216C22—24no—
217C22—22no—
218C22—24no—
219C22#2254871314yescomplemented
220C22C2×D4×D25111yescharacteristic
221C22C22×C22⋊D25111yescharacteristic
222C22C2×Q8×D25112yes—
223C22#41×D25112yes—
224C22C2×#1686080911yescharacteristic
225C4—22no—
226C4—42no—
227C4—42no—
228C4—10042no—
229C4—10042no—
230C4—5024no—
231C4—2518no—
232C4#2254868714yes—
233C2—24no—
234C2C2×#2254868711yescharacteristic
235C2C22⋊Q8×D25112yes—
236C2C2×#2254871311yescharacteristic
237C2#3063223311yescharacteristic
238C2#3063223912yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×C22⋊Q8×C251C2
prime radicalC24×C251C23
hypercenterC2×C22⋊Q8D251
hypocenterC251#2857
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#22548713C2semidirect
4. #30631594C2semidirect
5. #30631592C2semidirect
6. #30631582C2semidirect
7. #30631556C2semidirect
36. C22×Dic251Q8semidirect
99. C22×C251C4⋊Q8semidirect
112. Dic251C22⋊Q8semidirect
119. C251#2857semidirect
219. C22#22548713semidirect

Maximal subgroups

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

subgroupindex
C2×#225487132
C2×#225486942
C22×C4×Dic2512
#306315942
#306315922
#306315822
#306315562
#306315452
C2×C22⋊Q8×C2512
#2857251