Finite Group Databasebeta

← #41012559

Subgroups — #41012559

Order 27008.

188 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 57 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#27213875C211yesmaximal, characteristic
2#27213497C211yesmaximal, characteristic
3#27213442C211yesmaximal, characteristic
4#27213441C211yesmaximal, characteristic
5#27212893C211yesmaximal, characteristic
6#27212878C211yesmaximal, characteristic
7Q8⋊C8×C211C211yesmaximal, characteristic
8#20269977—21no—
9#20269977C2211yescharacteristic
10#20269977C2211yescharacteristic
11#20269968—22no—
12#20269968—21no—
13Q8×Dic211—21no—
14C4×Dic422—21no—
15C4×Dic422C2211yescharacteristic
16#20269911—21no—
17#20269910—21no—
18C4⋊(C211⋊C8)C2211yescharacteristic
19#20269902—21no—
20#20269902—22no—
21#20269901—21no—
22C4×Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4×C8×C211C2211yescharacteristic
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422D411yescharacteristic
28C2×C211⋊C8—21no—
29C4⋊Dic211—22no—
30C4⋊Dic211—22no—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211D412yes—
35C4⋊Dic211D411yescharacteristic
36Dic211⋊C4—22no—
37Dic211⋊C4—21no—
38C4×Dic211—22no—
39C4×Dic211—21no—
40C2×Q8×C211D411yescharacteristic
41C2×C8×C211—21no—
42C2×C8×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211D411yescharacteristic
45C42×C211—21no—
46C42×C211C2311yescharacteristic
47Dic422—21no—
48Dic422—21no—
49Dic422—41no—
50Dic422—41no—
51Dic422—42no—
52C2×Dic211—21no—
53C2×Dic211—21no—
54C2×Dic211—22no—
55C2×Dic211—41no—
56C211⋊C8—41no—
57Q8×C211—21no—
58Q8×C211Q1612yes—
59C2×C4×C211—21no—
60C2×C4×C211—21no—
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211C2×D411yescharacteristic
63C2×C4×C211C2×D411yescharacteristic
64C8×C211—41no—
65C8×C211—41no—
66C22×C211#12011yescharacteristic
67C4×C211—22no—
68C4×C211—21no—
69C4×C211—41no—
70C4×C211C2×Q1611yescharacteristic
71C4×C211#13511yescharacteristic
72C4×C211#13711yescharacteristic
73C4×C211#13711yescharacteristic
74Dic211—41no—
75Dic211—42no—
76Dic211—41no—
77Dic211—81no—
78C2×C211#45111yescharacteristic
79C2×C211#45211yescharacteristic
80C2×C211#45411yescharacteristic
81C211#163611yescharacteristic, complemented, Sylow
82#1636—2111nomaximal, Sylow
83#557—2111no—
84#487—2111no—
85#477—2111no—
86#477—2111no—
87#332—2111no—
88Q8⋊C8—2111no—
89Q8⋊C8D21111yescharacteristic
90C4⋊Q8—4221no—
91C4⋊Q8—2111no—
92C4⋊Q8—2111no—
93#125—4222no—
94#125—4221no—
95C4×Q8—4221no—
96C4×Q8—4221no—
97C4×Q8—2111no—
98C4×Q8C2×D21111yescharacteristic
99#107—4221no—
100#107—4221no—
101C4⋊C8—2111no—
102C4⋊C8C2×D21111yescharacteristic
103#103—4221no—
104#103—4221no—
105#103—4222no—
106C4×C8C2×D21111yescharacteristic
107C2×Q8—4221no—
108C2×Q8—4221no—
109C2×Q8—2111no—
110C2×Q8D84411yescharacteristic
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—4221no—
114C4⋊C4—21no—
115C4⋊C4—4222no—
116C4⋊C4—4222no—
117C4⋊C4—4222no—
118C4⋊C4—4221no—
119C4⋊C4—4221no—
120C4⋊C4—4221no—
121C4⋊C4—4221no—
122C4⋊C4—2112no—
123C4⋊C4—2111no—
124C4⋊C4D84411yescharacteristic
125C42—21no—
126C42—4222no—
127C42—4221no—
128C42C22×D21111yescharacteristic
129Q8—21no—
130Q8—8442no—
131Q8—8441no—
132Q8—8441no—
133Q8—4221no—
134Q8—4221no—
135Q8Dic84412yes—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—8441no—
139C2×C4—4221no—
140C2×C4—4221no—
141C2×C4—4222no—
142C2×C4D4×D21111yescharacteristic
143C2×C4D4×D21111yescharacteristic
144C2×C4C2×D84411yescharacteristic
145C8—41no—
146C8—41no—
147C8—8441no—
148C22#2026994711yescharacteristic
149C4—21no—
150C4—22no—
151C4—41no—
152C4—8441no—
153C4—8442no—
154C4—8441no—
155C4—16881no—
156C4C2×Dic84411yescharacteristic
157C4#2026999811yescharacteristic
158C4#2027000911yescharacteristic
159C4#2027001011yescharacteristic
160C2#2721330411yescharacteristic
161C2#2721331111yescharacteristic
162C2#2721332011yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingQ8⋊C8×C211C2
socle, prime radicalC22×C211#120
hypercenterQ8⋊C8D211
hypocenterC211#1636
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
81. C211#1636semidirect

Maximal subgroups

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

subgroupindex
#272138752
#272134972
#272134422
#272134412
#272128932
#272128782
Q8⋊C8×C2112
#1636211