Finite Group Databasebeta

← #44560907

Subgroups — #44560907

Order 32128.

380 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 157 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#30632244C211yesmaximal, characteristic
2#30632093C214yesmaximal, complemented
3#30632090C212yesmaximal, complemented
4#30632083C212yesmaximal, complemented
5#30631727C214yesmaximal
6C4×#16860794C211yesmaximal, characteristic
7C2×C4⋊Q8×C251C211yesmaximal, characteristic
8C2×#16860794C2212yes—
9#22548715C2211yescharacteristic
10#22548715C2211yescharacteristic
11#22548707C2212yes—
12#22548694—21no—
13#22548694—21no—
14C4×Dic502—21no—
15C4×Dic502—21no—
16#22548674C2212yes—
17#22548648C2218yes—
18#22548646C2218yes—
19#22548626C2212yes—
20C4×C251⋊C8C2212yes—
21C22×Q8×C251—21no—
22C4⋊Q8×C251—21no—
23C4⋊Q8×C251—21no—
24C4⋊Q8×C251C2212yes—
25C4⋊Q8×C251C2212yes—
26C2×C4⋊C4×C251—21no—
27C2×C4⋊C4×C251C2212yes—
28C2×C42×C251C2211yescharacteristic
29C2×Dic502—21no—
30C2×Dic502—21no—
31#16860794Q818yescomplemented
32C2×C251⋊C8C2314yes—
33C4⋊Dic251—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251C2312yes—
36C4⋊Dic251C2312yes—
37Dic251⋊C4—22no—
38Dic251⋊C4—22no—
39C22⋊Dic251—22no—
40C4×Dic251—22no—
41C2×Q8×C251—22no—
42C2×Q8×C251—22no—
43C2×Q8×C251—44no—
44C22×C4×C251—22no—
45C22×C4×C251D411yescharacteristic
46C22×C4×C251D411yescharacteristic
47C22×C4×C251C2311yescharacteristic
48C4⋊C4×C251—22no—
49C4⋊C4×C251—22no—
50C4⋊C4×C251—22no—
51C4⋊C4×C251C2314yes—
52C42×C251D411yescharacteristic
53C42×C251D411yescharacteristic
54C42×C251C2311yescharacteristic
55C42×C251C2311yescharacteristic
56Dic502—42no—
57Dic502—42no—
58C2×Dic251—22no—
59C2×Dic251—22no—
60C251⋊C8C2×Q818yescomplemented
61C23×C251C2×D411yescharacteristic
62Q8×C251—44no—
63Q8×C251—44no—
64C2×C4×C251—22no—
65C2×C4×C251—22no—
66C2×C4×C251—24no—
67C2×C4×C251—44no—
68C2×C4×C251C2×D411yescharacteristic
69C2×C4×C251C2×D411yescharacteristic
70C2×C4×C251C2×D411yescharacteristic
71C2×C4×C251C2×D411yescharacteristic
72C2×C4×C251C2×D411yescharacteristic
73C2×C4×C251C2×Q814yes—
74C2×C4×C251C2411yescharacteristic
75C22×C251—21no—
76C22×C251—21no—
77C22×C251C4⋊Q812yes—
78C22×C251C22×D411yescharacteristic
79C4×C251—21no—
80C4×C251—21no—
81C4×C251—44no—
82C4×C251C4⋊Q812yes—
83C4×C251AGL(1,ℤ/8)12yes—
84C4×C251#13712yes—
85C4×C251C22×Q812yes—
86Dic251—42no—
87Dic251—42no—
88C2×C251—22no—
89C2×C251C2×C4⋊Q811yescharacteristic
90C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
91C2×C251C2×#13711yescharacteristic
92C251#316511yescharacteristic, complemented, Sylow
93#3165—2511nomaximal, Sylow
94#533—2511no—
95C2×C4⋊Q8D25111yescharacteristic
96#501—2514no—
97#500—2512no—
98#498—2512no—
99#428—2514no—
100C4×M16—2511no—
101C22×Q8—21no—
102C2×M16—2512no—
103C4⋊Q8—21no—
104C4⋊Q8—21no—
105C4⋊Q8—2511no—
106C4⋊Q8—2511no—
107C4⋊Q8C2×D25112yes—
108C4⋊Q8C2×D25112yes—
109#125—2512no—
110C22⋊Q8—5021no—
111C22⋊Q8—5021no—
112C4×Q8—5021no—
113C4×Q8—5021no—
114#117—2512no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C2×D25112yes—
117C2×C42C2×D25111yescharacteristic
118#107—2518no—
119#106—2518no—
120#97—2512no—
121C4×C8—2512no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—44no—
125C2×Q8—5021no—
126C2×Q8—5021no—
127C22×C4—22no—
128C22×C4C22⋊D25111yescharacteristic
129C22×C4C22⋊D25111yescharacteristic
130C22×C4C22×D25111yescharacteristic
131M16—2518no—
132C2×C8—2514no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—5022no—
137C4⋊C4—5022no—
138C4⋊C4—5021no—
139C4⋊C4—5021no—
140C4⋊C4—2512no—
141C4⋊C4—2512no—
142C4⋊C4C22×D25114yes—
143C22⋊C4—5022no—
144C42—5022no—
145C42C22⋊D25111yescharacteristic
146C42C22⋊D25111yescharacteristic
147C42C22×D25111yescharacteristic
148C42C22×D25111yescharacteristic
149C23C2×C22⋊D25111yescharacteristic
150Q8—44no—
151Q8—44no—
152Q8—10042no—
153Q8—10042no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—44no—
158C2×C4—5022no—
159C2×C4—5022no—
160C2×C4C2×C22⋊D25111yescharacteristic
161C2×C4C2×C22⋊D25111yescharacteristic
162C2×C4C2×C22⋊D25111yescharacteristic
163C2×C4C2×C22⋊D25111yescharacteristic
164C2×C4C2×C22⋊D25111yescharacteristic
165C2×C4Q8×D25114yes—
166C2×C4C23×D25111yescharacteristic
167C8—2518no—
168C22—21no—
169C22—21no—
170C22#2254871312yes—
171C22C22×C22⋊D25111yescharacteristic
172C4—21no—
173C4—21no—
174C4—44no—
175C4—10042no—
176C4—10042no—
177C4#2254871312yes—
178C4#2254874112yes—
179C4#2254874612yes—
180C4C2×Q8×D25112yes—
181C2—22no—
182C2C2×#2254871311yescharacteristic
183C2C2×#2254874111yescharacteristic
184C2C2×#2254874611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×C4⋊Q8×C251C2
socleC22×C251C22×D4
prime radicalC23×C251C2×D4
hypercenterC2×C4⋊Q8D251
hypocenterC251#3165
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #30632093C2semidirect
3. #30632090C2semidirect
4. #30632083C2semidirect
31. #16860794Q8semidirect
60. C251⋊C8C2×Q8semidirect
92. C251#3165semidirect

Maximal subgroups

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

subgroupindex
#306322442
#306320932
#306320902
#306320832
#306317272
C4×#168607942
C2×C4⋊Q8×C2512
#3165251