Finite Group Databasebeta

← #41012374

Subgroups — #41012374

Order 27008.

238 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 53 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#27213777C211yesmaximal, characteristic
2#27213304C212yesmaximal
3#27213066C211yesmaximal, characteristic
4#27212884C212yesmaximal
5#324×C211C211yesmaximal, characteristic
6C22×Dic422C2211yescharacteristic
7C2×Dic844—22no—
8#20269976—22no—
9#20269968—22no—
10#20269956—21no—
11#20269956C2212yes—
12#20269936—21no—
13C2×Dic211⋊C4—21no—
14#20269902—22no—
15C211⋊#95—21no—
16C211⋊#95C2211yescharacteristic
17C2×C4⋊C4×C211C2211yescharacteristic
18C22⋊C8×C211C2212yes—
19C2×Dic422—21no—
20C2×Dic422—22no—
21C2×Dic422—42no—
22C2×Dic422D412yes—
23C22×Dic211—21no—
24C22×Dic211—21no—
25Dic844—44no—
26C4⋊Dic211—21no—
27C4⋊Dic211—21no—
28C4⋊Dic211—21no—
29C4⋊Dic211D412yes—
30Dic211⋊C4—22no—
31Dic211⋊C4—22no—
32Dic211⋊C4—42no—
33C22⋊Dic211—22no—
34C4×Dic211—22no—
35C22×C4×C211D411yescharacteristic
36C22×C4×C211D411yescharacteristic
37C22×C4×C211C2311yescharacteristic
38C2×C8×C211—22no—
39C4⋊C4×C211—22no—
40Dic422—24no—
41Dic422—42no—
42Dic422—42no—
43Dic422—81no—
44C2×Dic211—22no—
45C2×Dic211—22no—
46C2×Dic211—22no—
47C2×Dic211—22no—
48C2×Dic211—41no—
49C2×Dic211—41no—
50C2×Dic211—41no—
51C2×Dic211—41no—
52C23×C211C2×D411yescharacteristic
53C2×C4×C211—21no—
54C2×C4×C211—21no—
55C2×C4×C211—21no—
56C2×C4×C211—41no—
57C2×C4×C211—41no—
58C2×C4×C211Q1612yescomplemented
59C2×C4×C211C2×D412yes—
60C8×C211—42no—
61C22×C211—21no—
62C22×C211—21no—
63C22×C211#12011yescharacteristic
64C22×C211C2×Q1611yescharacteristic
65C22×C211#13711yescharacteristic
66C4×C211—22no—
67C4×C211—22no—
68C4×C211—41no—
69Dic211—42no—
70Dic211—42no—
71Dic211—42no—
72Dic211—42no—
73C2×C211—21no—
74C2×C211—21no—
75C2×C211#45111yescharacteristic
76C2×C211#45611yescharacteristic
77C2×C211B(4,𝔽2)11yescharacteristic
78C211#160511yescharacteristic, complemented, Sylow
79#1605—2111nomaximal, Sylow
80#544—2111no—
81#451—2112no—
82#395—2111no—
83#328—2112no—
84#324D21111yescharacteristic
85C22×Q8—2111no—
86C2×Q16—4222no—
87C4⋊Q8—4222no—
88#125—4222no—
89C22⋊Q8—4221no—
90C22⋊Q8—2112no—
91#117—4221no—
92C2×C4⋊C4—4221no—
93C2×C4⋊C4C2×D21111yescharacteristic
94#103—4222no—
95C22⋊C8C2×D21112yes—
96#95—4221no—
97#95—2111no—
98C2×Q8—8442no—
99C2×Q8—4221no—
100C2×Q8—4222no—
101C2×Q8—2112no—
102C22×C4—4221no—
103C22×C4—4221no—
104C22×C4D84411yescharacteristic
105C22×C4D84411yescharacteristic
106C22×C4C22×D21111yescharacteristic
107Q16—8444no—
108C2×C8—22no—
109C4⋊C4—22no—
110C4⋊C4—8442no—
111C4⋊C4—4221no—
112C4⋊C4—4221no—
113C4⋊C4—4221no—
114C4⋊C4—4222no—
115C4⋊C4—4222no—
116C4⋊C4—2112no—
117C22⋊C4—4222no—
118C42—4222no—
119C23C2×D84411yescharacteristic
120Q8—8442no—
121Q8—8442no—
122Q8—16881no—
123Q8—4224no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—41no—
129C2×C4—8441no—
130C2×C4—8441no—
131C2×C4—8441no—
132C2×C4—8441no—
133C2×C4—4222no—
134C2×C4—4222no—
135C2×C4—4222no—
136C2×C4—4222no—
137C2×C4Dic84412yescomplemented
138C2×C4D4×D21112yes—
139C8—42no—
140C22—21no—
141C22—21no—
142C22#2026994711yescharacteristic
143C22C2×Dic84411yescharacteristic
144C22#2027001011yescharacteristic
145C4—22no—
146C4—22no—
147C4—41no—
148C4—8442no—
149C4—8442no—
150C4—8442no—
151C4—8442no—
152C2—21no—
153C2—21no—
154C2#2721330411yescharacteristic
155C2#2721333111yescharacteristic
156C2#2721333411yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC22×C4×C211C23
FrattiniC22×C4C22×D211
Fitting#324×C211C2
socleC22×C211#120
prime radicalC23×C211C2×D4
hypercenter#324D211
hypocenterC211#1605
LCSG ▷ C22×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
58. C2×C4×C211Q16semidirect
78. C211#1605semidirect
137. C2×C4Dic844semidirect

Maximal subgroups

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

subgroupindex
#272137772
#272133042
#272130662
#272128842
#324×C2112
#1605211