Finite Group Databasebeta

← #44558668

Subgroups — #44558668

Order 32128.

696 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 177 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×#22548704C211yesmaximal, characteristic, complemented
2C2×#22548689C211yesmaximal, characteristic
3C22×C4×Dic251C211yesmaximal, characteristic, complemented
4#30631594C211yesmaximal, characteristic, complemented
5#30631588C211yesmaximal, characteristic, complemented
6#30631577C214yesmaximal, complemented
7#30631564C214yesmaximal, complemented
8#30631552C211yesmaximal, characteristic, complemented
9C2×C22⋊Q8×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C22×C22⋊D251C2212yes—
12C23×Dic251C2212yes—
13#22548704—24no—
14#22548689—24no—
15C2×C4⋊Dic251C2211yescharacteristic
16C2×(C2×D251)⋊C4C2214yes—
17C2×(C2×D251)⋊C4C2214yes—
18C2×(C2×D251)⋊C4C2212yes—
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—
31C23×D251C2312yes—
32C2×D1004—22no—
33C2×D1004—22no—
34C2×D1004—22no—
35C2×C22⋊D251—24no—
36C2×C22⋊D251—28no—
37C22×Dic251—24no—
38C22×Dic251—24no—
39C22×Dic251—24no—
40C22×Dic251C2312yes—
41C22×Dic251C2312yes—
42C22×Dic251C2312yes—
43C4⋊Dic251—22no—
44(C2×D251)⋊C4—28no—
45(C2×D251)⋊C4—28no—
46(C2×D251)⋊C4—24no—
47C4×Dic251—24no—
48C4×Dic251—22no—
49C4×Dic251D414yescomplemented
50C24×C251C2311yescharacteristic
51C2×Q8×C251—22no—
52C2×Q8×C251—22no—
53C2×Q8×C251—22no—
54C22×C4×C251—22no—
55C22×C4×C251—22no—
56C22×C4×C251D414yes—
57C22×C4×C251C2312yes—
58C22×C4×C251C2312yes—
59C22×C4×C251C2311yescharacteristic
60C22×C4×C251C2311yescharacteristic
61C4⋊C4×C251—22no—
62C4⋊C4×C251—24no—
63C22⋊C4×C251—24no—
64C22×D251—22no—
65C22×D251—22no—
66C22×D251—24no—
67C22×D251—24no—
68C22×D251—22no—
69D1004—44no—
70C22⋊D251—48no—
71C2×Dic251—216no—
72C2×Dic251—22no—
73C2×Dic251—22no—
74C2×Dic251—22no—
75C2×Dic251—22no—
76C2×Dic251—22no—
77C2×Dic251—24no—
78C2×Dic251C2×D414yes—
79C2×Dic251#4114yes—
80C23×C251—21no—
81C23×C251—21no—
82C23×C251—22no—
83C23×C251C2×D412yes—
84C23×C251#4114yes—
85C23×C251C2411yescharacteristic
86Q8×C251—44no—
87C2×C4×C251—21no—
88C2×C4×C251—21no—
89C2×C4×C251—28no—
90C2×C4×C251—22no—
91C2×C4×C251—22no—
92C2×C4×C251—22no—
93C2×C4×C251—22no—
94C2×C4×C251—24no—
95C2×C4×C251C2×D412yes—
96C2×C4×C251C2×D412yes—
97C2×C4×C251C2×D412yes—
98C2×C4×C251#4112yes—
99C22×C251—22no—
100C22×C251—24no—
101C22×C251—24no—
102C22×C251—22no—
103C22×C251#12414yescomplemented
104C22×C251C22×D411yescharacteristic
105C22×C251C22×D411yescharacteristic
106C22×C251C2×#4112yes—
107C22×C251C2×#4111yescharacteristic
108C22×C251C2×#4112yes—
109C2×D251—42no—
110C2×D251—44no—
111C2×D251—42no—
112C2×D251—44no—
113C2×D251—42no—
114C4×C251—22no—
115C4×C251—42no—
116C4×C251—42no—
117C4×C251#12114yes—
118Dic251—24no—
119Dic251—24no—
120Dic251—42no—
121C2×C251—24no—
122C2×C251C2×#12111yescharacteristic
123C2×C251#52512yes—
124C2×C251C2×#12411yescharacteristic
125C2×C251#52912yes—
126C2×C251#53211yescharacteristic
127D251—82no—
128C251#285011yescharacteristic, complemented, Sylow
129#2850—2511nomaximal, Sylow
130C2×#124—2511no—
131C2×C22⋊Q8D25111yescharacteristic, complemented
132C2×#121—2511no—
133C22×C42—2511no—
134#391—2511no—
135#390—2511no—
136#388—2514no—
137#386—2514no—
138#382—2511no—
139C22×Q8C2×D25111yescharacteristic
140C22×D4—2512no—
141C22×D4—2511no—
142C23×C4—2512no—
143C23×C4C2×D25111yescharacteristic
144#124—5024no—
145C22⋊Q8—24no—
146#121—5024no—
147C2×C4⋊C4—2511no—
148C2×C4⋊C4C2×D25111yescharacteristic
149C2×C4⋊C4C2×D25112yes—
150C2×C22⋊C4—2512no—
151C2×C22⋊C4—2514no—
152C2×C22⋊C4—2514no—
153C2×C22⋊C4C2×D25112yes—
154C2×C42—5024no—
155C2×C42—2512no—
156C2×C42—2511no—
157C2×C42—2511no—
158#95—2514no—
159#95—2514no—
160C24—2512no—
161C24C22×D25111yescharacteristic
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×D4—5024no—
166C2×D4—5028no—
167C2×D4—5022no—
168C2×D4—5022no—
169C2×D4—5022no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—5024no—
173C22×C4—5024no—
174C22×C4—5024no—
175C22×C4—2512no—
176C22×C4—2512no—
177C22×C4—2512no—
178C22×C4C22⋊D25114yes—
179C22×C4C22×D25112yes—
180C22×C4C22×D25112yes—
181C22×C4C22×D25111yescharacteristic
182C22×C4C22×D25111yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—5022no—
186C22⋊C4—24no—
187C22⋊C4—5028no—
188C22⋊C4—5028no—
189C22⋊C4—5024no—
190C42—5024no—
191C42—5022no—
192C42—2514no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—5022no—
197C23—5022no—
198C23—5024no—
199C23—5022no—
200C23—5024no—
201C23C2×C22⋊D25112yes—
202C23#1686081114yes—
203C23C23×D25111yescharacteristic
204Q8—44no—
205D4—10048no—
206D4—10044no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—28no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—22no—
215C2×C4—5022no—
216C2×C4—5024no—
217C2×C4—50216no—
218C2×C4—5022no—
219C2×C4—5022no—
220C2×C4—5022no—
221C2×C4—5022no—
222C2×C4—2514no—
223C2×C4—2514no—
224C2×C4D4×D25112yes—
225C2×C4C2×C22⋊D25112yes—
226C2×C4C2×C22⋊D25112yes—
227C2×C4#1686080912yes—
228C22—24no—
229C22—24no—
230C22—22no—
231C22—22no—
232C22—10044no—
233C22—10042no—
234C22—10044no—
235C22—10042no—
236C22—10042no—
237C22#2254870414yescomplemented
238C22C2×D4×D25111yescharacteristic
239C22C22×C22⋊D25111yescharacteristic
240C22#41×D25112yes—
241C22C2×#1686080911yescharacteristic
242C22C2×#1686081112yes—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—10042no—
247C4—5024no—
248C4—5024no—
249C4#2254868714yes—
250C2—24no—
251C2—20082no—
252C2C2×#2254868711yescharacteristic
253C2#3063220012yes—
254C2C2×#2254870411yescharacteristic
255C2#3063222312yes—
256C2#3063223311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×C22⋊Q8×C251C2
prime radicalC2×#22548689C2
hypercenterC2×C22⋊Q8D251
hypocenterC251#2850
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×#22548704C2semidirect
3. C22×C4×Dic251C2semidirect
4. #30631594C2semidirect
5. #30631588C2semidirect
6. #30631577C2semidirect
7. #30631564C2semidirect
8. #30631552C2semidirect
9. C2×C22⋊Q8×C251C2semidirect
49. C4×Dic251D4semidirect
103. C22×C251#124semidirect
128. C251#2850semidirect
131. C2×C22⋊Q8D251semidirect
237. C22#22548704semidirect

Maximal subgroups

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

subgroupindex
C2×#225487042
C2×#225486892
C22×C4×Dic2512
#306315942
#306315882
#306315772
#306315642
#306315522
C2×C22⋊Q8×C2512
#2850251