Finite Group Databasebeta

← #44548707

Subgroups — #44548707

Order 32128.

236 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 69 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×#22548680C211yesmaximal, characteristic
2#30631741C212yesmaximal
3#30631413C211yesmaximal, characteristic
4#30631412C212yesmaximal
5#324×C251C211yesmaximal, characteristic
6C22×Dic502C411yescharacteristic
7#22548680—22no—
8#22548680—22no—
9C2×Dic251⋊C4C411yescharacteristic
10C2×Dic251⋊C4C2211yescharacteristic
11C2×C4×Dic251C411yescharacteristic
12C2×C4×Dic251C411yescharacteristic
13C2×C4×Dic251C2211yescharacteristic
14Dic251⋊C8—22no—
15C22⋊(C251⋊C8)C2212yes—
16#22548625—22no—
17C2×C4⋊C4×C251C2211yescharacteristic
18C22⋊C8×C251C2212yes—
19C2×Dic502—22no—
20C2×Dic502—22no—
21C2×Dic502—22no—
22C22×Dic251C2×C411yescharacteristic
23C22×Dic251C2×C411yescharacteristic
24C22×Dic251C2×C411yescharacteristic
25C22×Dic251C2×C411yescharacteristic
26C2×C251⋊C8—22no—
27Dic251⋊C4—22no—
28Dic251⋊C4—22no—
29C4×Dic251—21no—
30C4×Dic251—21no—
31C4×Dic251—21no—
32C4×Dic251—21no—
33C4×Dic251D412yes—
34C4×Dic251D412yes—
35C22×C4×C251C2×C411yescharacteristic
36C22×C4×C251C2×C411yescharacteristic
37C22×C4×C251C2311yescharacteristic
38C2×C8×C251—22no—
39C4⋊C4×C251—22no—
40Dic502—28no—
41C2×Dic251—21no—
42C2×Dic251—21no—
43C2×Dic251—21no—
44C2×Dic251—21no—
45C2×Dic251—22no—
46C2×Dic251—21no—
47C2×Dic251—41no—
48C2×Dic251C22⋊C411yescharacteristic
49C2×Dic251C22⋊C412yes—
50C2×Dic251C22⋊C411yescharacteristic
51C2×Dic251M1612yes—
52C251⋊C8—42no—
53C23×C251C22×C411yescharacteristic
54C2×C4×C251—21no—
55C2×C4×C251—21no—
56C2×C4×C251—21no—
57C2×C4×C251—41no—
58C2×C4×C251—41no—
59C2×C4×C251M1612yes—
60C2×C4×C251C2×D412yes—
61C8×C251—42no—
62C22×C251—21no—
63C22×C251—21no—
64C22×C251C2×C22⋊C411yescharacteristic
65C22×C251C2×M1611yescharacteristic
66C22×C251C2×M1611yescharacteristic
67C4×C251—22no—
68C4×C251—22no—
69C4×C251—41no—
70Dic251—22no—
71Dic251—22no—
72Dic251—21no—
73Dic251—41no—
74Dic251#10112yes—
75C2×C251—21no—
76C2×C251—21no—
77C2×C251#40711yescharacteristic
78C2×C251#41011yescharacteristic
79C2×C251C2×#10111yescharacteristic
80C251#146711yescharacteristic, complemented, Sylow
81#1467—2511nomaximal, Sylow
82C2×C4×Q8—2511no—
83#432—2512no—
84#324—2511no—
85#324—2512no—
86#324D25111yescharacteristic
87C22×Q8—2511no—
88C4×Q8—5022no—
89C4×Q8—5022no—
90C2×C4⋊C4—2511no—
91C2×C4⋊C4—2511no—
92C2×C4⋊C4C2×D25111yescharacteristic
93C2×C42—2511no—
94C2×C42—2511no—
95C2×C42—2511no—
96C4⋊C8—5022no—
97C22⋊C8—2512no—
98C22⋊C8C2×D25112yes—
99#97—5022no—
100C2×Q8—5022no—
101C2×Q8—5022no—
102C2×Q8—5022no—
103C22×C4—2511no—
104C22×C4—2511no—
105C22×C4—2511no—
106C22×C4—2511no—
107C22×C4C4×D25111yescharacteristic
108C22×C4C4×D25111yescharacteristic
109C22×C4C22×D25111yescharacteristic
110C2×C8—22no—
111C2×C8—5022no—
112C4⋊C4—22no—
113C4⋊C4—5022no—
114C4⋊C4—5022no—
115C42—5021no—
116C42—5021no—
117C42—5021no—
118C42—5021no—
119C42—2512no—
120C42—2512no—
121C23C2×C4×D25111yescharacteristic
122Q8—5028no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—10041no—
129C2×C4—5021no—
130C2×C4—5021no—
131C2×C4—5021no—
132C2×C4—5021no—
133C2×C4—5021no—
134C2×C4—5022no—
135C2×C4—2511no—
136C2×C4—2512no—
137C2×C4—2512no—
138C2×C4—2511no—
139C2×C4#1686079312yes—
140C2×C4D4×D25112yes—
141C8—42no—
142C8—10042no—
143C22—21no—
144C22—21no—
145C22C22⋊C4×D25111yescharacteristic
146C22M16×D25111yescharacteristic
147C22C2×#1686079311yescharacteristic
148C4—22no—
149C4—22no—
150C4—41no—
151C4—10041no—
152C4—5022no—
153C4—5022no—
154C4—5021no—
155C4—2512no—
156C2—21no—
157C2—21no—
158C2#3063164911yescharacteristic
159C2#3063165911yescharacteristic
160C2#101×D25111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutator, prime radicalC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#324×C251C2
socleC22×C251C2×C22⋊C4
cosocleC22×C4×C251C23
hypercenter#324D251
hypocenterC251#1467
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
80. C251#1467semidirect

Maximal subgroups

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

subgroupindex
C2×#225486802
#306317412
#306314132
#306314122
#324×C2512
#1467251