Finite Group Databasebeta

← #44558508

Subgroups — #44558508

Order 32128.

408 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#30631617C211yesmaximal, characteristic, complemented
2#30631615C212yesmaximal, complemented
3#30631615C212yesmaximal, complemented
4#30631610C212yesmaximal, complemented
5#30631607C211yesmaximal, characteristic, complemented
6#30631601C212yesmaximal
7#30631599C212yesmaximal, complemented
8#30631571C212yesmaximal, complemented
9#390×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C2×C4⋊Dic251C2211yescharacteristic
12C2×Dic251⋊C4C2212yes—
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×C22⋊Dic251C2212yes—
16C2×C22⋊Dic251C2212yes—
17C2×C22⋊Dic251C2212yes—
18C2×C22⋊Dic251C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2211yescharacteristic
25C251⋊#95C2211yescharacteristic
26C22×D4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C22⋊C4×C251C2212yes—
29C2×C22⋊C4×C251C2212yes—
30C2×C42×C251C2211yescharacteristic
31C2×Dic502—42no—
32C2×Dic502—41no—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C4⋊Dic251—41no—
38Dic251⋊C4—24no—
39Dic251⋊C4—42no—
40Dic251⋊C4—42no—
41C22⋊Dic251—42no—
42C22⋊Dic251—42no—
43C22⋊Dic251—42no—
44C22⋊Dic251—42no—
45C24×C251C2312yes—
46C2×D4×C251—22no—
47C2×D4×C251—41no—
48C2×D4×C251—41no—
49C22×C4×C251C2311yescharacteristic
50C22×C4×C251C2312yes—
51C22×C4×C251C2311yescharacteristic
52C22×C4×C251C2311yescharacteristic
53C4⋊C4×C251—41no—
54C22⋊C4×C251—42no—
55C22⋊C4×C251—42no—
56C42×C251—41no—
57Dic502—44no—
58C2×Dic251—22no—
59C2×Dic251—22no—
60C2×Dic251—22no—
61C2×Dic251—22no—
62C2×Dic251—22no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251—22no—
69C2×Dic251—22no—
70C23×C251—22no—
71C23×C251—22no—
72C23×C251—22no—
73C23×C251—22no—
74C23×C251—22no—
75C23×C251—22no—
76C23×C251—22no—
77C23×C251C2411yescharacteristic
78D4×C251—44no—
79C2×C4×C251—21no—
80C2×C4×C251—21no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251—22no—
84C2×C4×C251—22no—
85C2×C4×C251—21no—
86C2×C4×C251—21no—
87C2×C4×C251—22no—
88C2×C4×C251#4112yes—
89C22×C251—42no—
90C22×C251—42no—
91C22×C251—42no—
92C22×C251—42no—
93C22×C251—42no—
94C22×C251—42no—
95C22×C251—42no—
96C22×C251C2×#4111yescharacteristic
97C22×C251H(5,𝔽2)11yescharacteristic
98C22×C251H(5,𝔽2)11yescharacteristic
99C22×C251H(5,𝔽2)12yes—
100C22×C2512−1+412yes—
101C4×C251—22no—
102C4×C251—41no—
103C4×C251—41no—
104C4×C251—42no—
105Dic251—42no—
106Dic251—42no—
107Dic251—42no—
108Dic251—42no—
109C2×C251—82no—
110C2×C251#53811yescharacteristic
111C2×C251#54212yes—
112C2×C251#56011yescharacteristic
113C2×C251#56212yes—
114C2×C251#56311yescharacteristic
115C251#283011yescharacteristic, complemented, Sylow
116#2830—2511nomaximal, Sylow
117#398—2511no—
118#397—2512no—
119#397—2512no—
120#396—2512no—
121#395—2511no—
122#394—2512no—
123#393—2512no—
124#390D25111yescharacteristic
125#387—2512no—
126C22×Q8—2511no—
127C22×D4C2×D25111yescharacteristic
128C2×C4⋊C4—2512no—
129C2×C4⋊C4—2512no—
130C2×C4⋊C4—2511no—
131C2×C4⋊C4—2512no—
132C2×C4⋊C4C2×D25111yescharacteristic
133C2×C22⋊C4—2512no—
134C2×C22⋊C4—2512no—
135C2×C22⋊C4—2512no—
136C2×C22⋊C4—2512no—
137C2×C22⋊C4C2×D25112yes—
138C2×C22⋊C4C2×D25112yes—
139C2×C42C2×D25111yescharacteristic
140#95—2512no—
141#95—2512no—
142#95—2512no—
143#95—2512no—
144#95—2512no—
145#95—2511no—
146#95—2511no—
147C24C22×D25112yes—
148C2×Q8—10042no—
149C2×Q8—10041no—
150C2×D4—22no—
151C2×D4—41no—
152C2×D4—41no—
153C22×C4—2512no—
154C22×C4—2512no—
155C22×C4—2512no—
156C22×C4—2512no—
157C22×C4C22×D25112yes—
158C22×C4C22×D25111yescharacteristic
159C22×C4C22×D25111yescharacteristic
160C22×C4C22×D25111yescharacteristic
161C4⋊C4—41no—
162C4⋊C4—10042no—
163C4⋊C4—10042no—
164C4⋊C4—10041no—
165C4⋊C4—5024no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—10042no—
169C22⋊C4—10042no—
170C22⋊C4—10042no—
171C22⋊C4—10042no—
172C42—41no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23C23×D25111yescharacteristic
181Q8—10044no—
182D4—44no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—5022no—
193C2×C4—5022no—
194C2×C4—5022no—
195C2×C4—5022no—
196C2×C4—5022no—
197C2×C4—5022no—
198C2×C4—5022no—
199C2×C4—5022no—
200C2×C4—5022no—
201C2×C4—5022no—
202C2×C4—5022no—
203C2×C4—5022no—
204C2×C4#1686081012yes—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22C2×#1686081011yescharacteristic
213C22#2254876011yescharacteristic
214C22#2254876011yescharacteristic
215C22#2254876012yes—
216C22#2254876212yes—
217C4—22no—
218C4—41no—
219C4—41no—
220C4—42no—
221C4—10042no—
222C4—10042no—
223C4—10042no—
224C4—10042no—
225C2—82no—
226C2#3063227511yescharacteristic
227C2#3063230212yes—
228C2#3063242411yescharacteristic
229C2#3063243212yes—
230C2#3063243711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#390×C251C2
prime radicalC22×D4×C251C22
hypercenter#390D251
hypocenterC251#2830
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #30631617C2semidirect
2. #30631615C2semidirect
3. #30631615C2semidirect
4. #30631610C2semidirect
5. #30631607C2semidirect
7. #30631599C2semidirect
8. #30631571C2semidirect
115. C251#2830semidirect

Maximal subgroups

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

subgroupindex
#306316172
#306316152
#306316102
#306316072
#306316012
#306315992
#306315712
#390×C2512
#2830251