Finite Group Databasebeta

← #44560309

Subgroups — #44560309

Order 32128.

410 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 137 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×#22548674C211yesmaximal, characteristic
2#30632025C214yesmaximal, complemented
3#30632020C212yesmaximal, complemented
4#30632015C212yesmaximal, complemented
5#30631727C212yesmaximal, complemented
6#30631696C212yesmaximal, complemented
7#30631651C211yesmaximal, characteristic
8C2×C22⋊Q8×C251C211yesmaximal, characteristic
9C2×#16860794C2212yes—
10#22548674—21no—
11#22548674—21no—
12#22548674C2212yes—
13#22548674C2212yes—
14C2×C4⋊Dic251C2211yescharacteristic
15C2×C4⋊Dic251C2211yescharacteristic
16C2×C22⋊Dic251—21no—
17C2×C4×Dic251—21no—
18#22548648C2214yescomplemented
19#22548646C2214yescomplemented
20#22548639C2214yescomplemented
21#22548639C2214yescomplemented
22C22⋊(C251⋊C8)C2212yes—
23C22⋊(C251⋊C8)C2212yes—
24C22×Q8×C251C2211yescharacteristic
25C23×C4×C251C2211yescharacteristic
26C22⋊Q8×C251—21no—
27C22⋊Q8×C251—21no—
28C22⋊Q8×C251C2212yes—
29C22⋊Q8×C251C2212yes—
30C2×C4⋊C4×C251—21no—
31C2×C4⋊C4×C251C2211yescharacteristic
32C2×C22⋊C4×C251—21no—
33C22×Dic251—21no—
34C22×Dic251—21no—
35#16860794—24no—
36C2×C251⋊C8C2314yes—
37C4⋊Dic251—21no—
38C4⋊Dic251—21no—
39C4⋊Dic251C2312yes—
40C4⋊Dic251C2312yes—
41C22⋊Dic251—22no—
42C22⋊Dic251—22no—
43C4×Dic251—22no—
44C4×Dic251—22no—
45C24×C251D411yescharacteristic
46C2×Q8×C251—21no—
47C2×Q8×C251—24no—
48C2×Q8×C251C2312yes—
49C22×C4×C251—24no—
50C22×C4×C251—21no—
51C22×C4×C251—21no—
52C22×C4×C251D411yescharacteristic
53C22×C4×C251D411yescharacteristic
54C22×C4×C251D411yescharacteristic
55C22×C4×C251C2311yescharacteristic
56C22×C4×C251C2311yescharacteristic
57C22×C4×C251C2311yescharacteristic
58C4⋊C4×C251—22no—
59C4⋊C4×C251—22no—
60C4⋊C4×C251—21no—
61C4⋊C4×C251C2312yes—
62C22⋊C4×C251—22no—
63C22⋊C4×C251—22no—
64C2×Dic251—22no—
65C2×Dic251—22no—
66C2×Dic251—42no—
67C2×Dic251—42no—
68C251⋊C8—24no—
69C23×C251—22no—
70C23×C251—22no—
71C23×C251—41no—
72C23×C251C2×D411yescharacteristic
73C23×C251C2×D411yescharacteristic
74C23×C251C2×D411yescharacteristic
75Q8×C251—24no—
76Q8×C251—42no—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251—22no—
80C2×C4×C251—22no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251—22no—
84C2×C4×C251—42no—
85C2×C4×C251—42no—
86C2×C4×C251C2×D411yescharacteristic
87C2×C4×C251C2×D411yescharacteristic
88C2×C4×C251C2×D411yescharacteristic
89C2×C4×C251#4114yes—
90C2×C4×C251C2411yescharacteristic
91C22×C251—21no—
92C22×C251—21no—
93C22×C251—21no—
94C22×C251—21no—
95C22×C251—21no—
96C22×C251—21no—
97C22×C251—21no—
98C22×C251—44no—
99C22×C251#12312yes—
100C22×C251#13712yes—
101C22×C251C22×D411yescharacteristic
102C4×C251—21no—
103C4×C251—21no—
104C4×C251—42no—
105C4×C251—42no—
106C4×C251#12312yes—
107C4×C251C2×#4112yes—
108Dic251—42no—
109Dic251—42no—
110C2×C251—22no—
111C2×C251—22no—
112C2×C251—41no—
113C2×C251C2×#12311yescharacteristic
114C2×C251C2×#13711yescharacteristic
115C2×C251#57511yescharacteristic
116C251#309511yescharacteristic, complemented, Sylow
117#3095—2511nomaximal, Sylow
118C2×C22⋊Q8D25111yescharacteristic
119C2×#117—2511no—
120#485—2514no—
121#484—2512no—
122#483—2512no—
123#428—2512no—
124#419—2512no—
125#407—2511no—
126C22×Q8C2×D25111yescharacteristic
127C23×C4C2×D25111yescharacteristic
128C2×M16—2512no—
129C22⋊Q8—21no—
130C22⋊Q8—21no—
131C22⋊Q8C2×D25112yes—
132C22⋊Q8C2×D25112yes—
133#117—5021no—
134#117—5021no—
135#117—2512no—
136#117—2512no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—2511no—
139C2×C4⋊C4—2511no—
140C2×C4⋊C4C2×D25111yescharacteristic
141C2×C22⋊C4—21no—
142C2×C22⋊C4—5021no—
143C2×C42—5021no—
144#107—2514no—
145#106—2514no—
146#103—2514no—
147#103—2514no—
148C22⋊C8—2512no—
149C22⋊C8—2512no—
150C24C22⋊D25111yescharacteristic
151C2×Q8—21no—
152C2×Q8—24no—
153C2×Q8C22×D25112yes—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—5021no—
158C22×C4—5021no—
159C22×C4C22⋊D25111yescharacteristic
160C22×C4C22⋊D25111yescharacteristic
161C22×C4C22⋊D25111yescharacteristic
162C22×C4C22×D25111yescharacteristic
163C22×C4C22×D25111yescharacteristic
164C22×C4C22×D25111yescharacteristic
165M16—5024no—
166C2×C8—2514no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—5021no—
171C4⋊C4—5021no—
172C4⋊C4—2512no—
173C4⋊C4—2512no—
174C4⋊C4C22×D25112yes—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—5022no—
178C22⋊C4—5022no—
179C42—5022no—
180C42—5022no—
181C23—22no—
182C23—22no—
183C23—41no—
184C23C2×C22⋊D25111yescharacteristic
185C23C2×C22⋊D25111yescharacteristic
186C23C2×C22⋊D25111yescharacteristic
187Q8—24no—
188Q8—42no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—10042no—
199C2×C4—10042no—
200C2×C4—5022no—
201C2×C4—5022no—
202C2×C4C2×C22⋊D25111yescharacteristic
203C2×C4C2×C22⋊D25111yescharacteristic
204C2×C4C2×C22⋊D25111yescharacteristic
205C2×C4#1686080914yes—
206C2×C4C23×D25111yescharacteristic
207C8—5024no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—44no—
216C22#2254869612yes—
217C22#2254874612yes—
218C22C22×C22⋊D25111yescharacteristic
219C4—21no—
220C4—21no—
221C4—42no—
222C4—42no—
223C4—10042no—
224C4—10042no—
225C4#2254869612yes—
226C4C2×#1686080912yes—
227C2—22no—
228C2—22no—
229C2—41no—
230C2C2×#2254869611yescharacteristic
231C2C2×#2254874611yescharacteristic
232C2#3063248511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×C22⋊Q8×C251C2
socleC22×C251C22×D4
prime radicalC24×C251D4
hypercenterC2×C22⋊Q8D251
hypocenterC251#3095
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #30632025C2semidirect
3. #30632020C2semidirect
4. #30632015C2semidirect
5. #30631727C2semidirect
6. #30631696C2semidirect
18. #22548648C22semidirect
19. #22548646C22semidirect
20. #22548639C22semidirect
21. #22548639C22semidirect
116. C251#3095semidirect

Maximal subgroups

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

subgroupindex
C2×#225486742
#306320252
#306320202
#306320152
#306317272
#306316962
#306316512
C2×C22⋊Q8×C2512
#3095251