Finite Group Databasebeta

← #41011740

Subgroups — #41011740

Order 27008.

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×#20269942C211yesmaximal, characteristic
2#27213200C212yesmaximal
3#27212872C211yesmaximal, characteristic
4#27212871C212yesmaximal
5#324×C211C211yesmaximal, characteristic
6C22×Dic422C411yescharacteristic
7#20269942—22no—
8#20269942—22no—
9C2×Dic211⋊C4C411yescharacteristic
10C2×Dic211⋊C4C2211yescharacteristic
11C2×C4×Dic211C411yescharacteristic
12C2×C4×Dic211C411yescharacteristic
13C2×C4×Dic211C2211yescharacteristic
14Dic211⋊C8—22no—
15C22⋊(C211⋊C8)C2212yes—
16#20269887—22no—
17C2×C4⋊C4×C211C2211yescharacteristic
18C22⋊C8×C211C2212yes—
19C2×Dic422—22no—
20C2×Dic422—22no—
21C2×Dic422—22no—
22C22×Dic211C2×C411yescharacteristic
23C22×Dic211C2×C411yescharacteristic
24C22×Dic211C2×C411yescharacteristic
25C22×Dic211C2×C411yescharacteristic
26C2×C211⋊C8—22no—
27Dic211⋊C4—22no—
28Dic211⋊C4—22no—
29C4×Dic211—21no—
30C4×Dic211—21no—
31C4×Dic211—21no—
32C4×Dic211—21no—
33C4×Dic211D412yes—
34C4×Dic211D412yes—
35C22×C4×C211C2×C411yescharacteristic
36C22×C4×C211C2×C411yescharacteristic
37C22×C4×C211C2311yescharacteristic
38C2×C8×C211—22no—
39C4⋊C4×C211—22no—
40Dic422—28no—
41C2×Dic211—21no—
42C2×Dic211—21no—
43C2×Dic211—21no—
44C2×Dic211—21no—
45C2×Dic211—22no—
46C2×Dic211—21no—
47C2×Dic211—41no—
48C2×Dic211C22⋊C411yescharacteristic
49C2×Dic211C22⋊C412yes—
50C2×Dic211C22⋊C411yescharacteristic
51C2×Dic211M1612yes—
52C211⋊C8—42no—
53C23×C211C22×C411yescharacteristic
54C2×C4×C211—21no—
55C2×C4×C211—21no—
56C2×C4×C211—21no—
57C2×C4×C211—41no—
58C2×C4×C211—41no—
59C2×C4×C211M1612yes—
60C2×C4×C211C2×D412yes—
61C8×C211—42no—
62C22×C211—21no—
63C22×C211—21no—
64C22×C211C2×C22⋊C411yescharacteristic
65C22×C211C2×M1611yescharacteristic
66C22×C211C2×M1611yescharacteristic
67C4×C211—22no—
68C4×C211—22no—
69C4×C211—41no—
70Dic211—22no—
71Dic211—22no—
72Dic211—21no—
73Dic211—41no—
74Dic211#10112yes—
75C2×C211—21no—
76C2×C211—21no—
77C2×C211#40711yescharacteristic
78C2×C211#41011yescharacteristic
79C2×C211C2×#10111yescharacteristic
80C211#146711yescharacteristic, complemented, Sylow
81#1467—2111nomaximal, Sylow
82C2×C4×Q8—2111no—
83#432—2112no—
84#324—2111no—
85#324—2112no—
86#324D21111yescharacteristic
87C22×Q8—2111no—
88C4×Q8—4222no—
89C4×Q8—4222no—
90C2×C4⋊C4—2111no—
91C2×C4⋊C4—2111no—
92C2×C4⋊C4C2×D21111yescharacteristic
93C2×C42—2111no—
94C2×C42—2111no—
95C2×C42—2111no—
96C4⋊C8—4222no—
97C22⋊C8—2112no—
98C22⋊C8C2×D21112yes—
99#97—4222no—
100C2×Q8—4222no—
101C2×Q8—4222no—
102C2×Q8—4222no—
103C22×C4—2111no—
104C22×C4—2111no—
105C22×C4—2111no—
106C22×C4—2111no—
107C22×C4C4×D21111yescharacteristic
108C22×C4C4×D21111yescharacteristic
109C22×C4C22×D21111yescharacteristic
110C2×C8—22no—
111C2×C8—4222no—
112C4⋊C4—22no—
113C4⋊C4—4222no—
114C4⋊C4—4222no—
115C42—4221no—
116C42—4221no—
117C42—4221no—
118C42—4221no—
119C42—2112no—
120C42—2112no—
121C23C2×C4×D21111yescharacteristic
122Q8—4228no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—8441no—
129C2×C4—4221no—
130C2×C4—4221no—
131C2×C4—4221no—
132C2×C4—4221no—
133C2×C4—4221no—
134C2×C4—4222no—
135C2×C4—2111no—
136C2×C4—2112no—
137C2×C4—2112no—
138C2×C4—2111no—
139C2×C4#1680994512yes—
140C2×C4D4×D21112yes—
141C8—42no—
142C8—8442no—
143C22—21no—
144C22—21no—
145C22C22⋊C4×D21111yescharacteristic
146C22M16×D21111yescharacteristic
147C22C2×#1680994511yescharacteristic
148C4—22no—
149C4—22no—
150C4—41no—
151C4—8441no—
152C4—4222no—
153C4—4222no—
154C4—4221no—
155C4—2112no—
156C2—21no—
157C2—21no—
158C2#2721310811yescharacteristic
159C2#2721311811yescharacteristic
160C2#101×D21111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutator, prime radicalC23×C211C22×C4
FrattiniC22×C4C22×D211
Fitting#324×C211C2
socleC22×C211C2×C22⋊C4
cosocleC22×C4×C211C23
hypercenter#324D211
hypocenterC211#1467
LCSG ▷ C23×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
80. C211#1467semidirect

Maximal subgroups

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

subgroupindex
C2×#202699422
#272132002
#272128722
#272128712
#324×C2112
#1467211