Finite Group Databasebeta

← #41012555

Subgroups — #41012555

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#27213496C211yesmaximal, characteristic
3#27213438C211yesmaximal, characteristic
4#27212891C211yesmaximal, characteristic
5Q8⋊(C211⋊C8)C211yesmaximal, characteristic
6#27212876C211yesmaximal, characteristic
7#477×C211C211yesmaximal, characteristic
8#20269977—21no—
9#20269977C2211yescharacteristic
10#20269968—22no—
11#20269968—21no—
12Q8×Dic211—21no—
13C4×Dic422—21no—
14C4×Dic422C2211yescharacteristic
15#20269910—21no—
16C4⋊(C211⋊C8)C2211yescharacteristic
17#20269901—22no—
18#20269900—21no—
19C4×C211⋊C8C2211yescharacteristic
20C4⋊Q8×C211C2211yescharacteristic
21C4×Q8×C211C2211yescharacteristic
22#107×C211—21no—
23C4⋊C8×C211C2211yescharacteristic
24#103×C211—21no—
25C2×Dic422—21no—
26C2×Dic422D411yescharacteristic
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C4⋊Dic211—22no—
30C4⋊Dic211—22no—
31C4⋊Dic211—21no—
32C4⋊Dic211D412yes—
33C4⋊Dic211D411yescharacteristic
34Dic211⋊C4—22no—
35Dic211⋊C4—21no—
36C4×Dic211—22no—
37C4×Dic211—21no—
38C2×Q8×C211—21no—
39C2×Q8×C211D411yescharacteristic
40C2×C8×C211—21no—
41C4⋊C4×C211—21no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211D411yescharacteristic
45C42×C211—21no—
46C42×C211C2311yescharacteristic
47Dic422—21no—
48Dic422—21no—
49Dic422—41no—
50Dic422—41no—
51C2×Dic211—21no—
52C2×Dic211—21no—
53C2×Dic211—22no—
54C211⋊C8—41no—
55C211⋊C8—41no—
56Q8×C211—21no—
57Q8×C211—42no—
58Q8×C211Q1612yes—
59C2×C4×C211—21no—
60C2×C4×C211—21no—
61C2×C4×C211—41no—
62C2×C4×C211C2×D411yescharacteristic
63C2×C4×C211C2×D411yescharacteristic
64C2×C4×C211C2×D411yescharacteristic
65C8×C211—41no—
66C22×C211#12011yescharacteristic
67C4×C211—22no—
68C4×C211—21no—
69C4×C211—41no—
70C4×C211—81no—
71C4×C211C2×Q1611yescharacteristic
72C4×C211#13511yescharacteristic
73C4×C211#13711yescharacteristic
74C4×C211#13711yescharacteristic
75Dic211—41no—
76Dic211—42no—
77Dic211—41no—
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#477D21111yescharacteristic
87#332—2111no—
88Q8⋊C8—2111no—
89Q8⋊C8—2111no—
90C4⋊Q8—4221no—
91C4⋊Q8—2111no—
92C4⋊Q8C2×D21111yescharacteristic
93#125—4222no—
94#125—4221no—
95C4×Q8—4221no—
96C4×Q8—4221no—
97C4×Q8—2111no—
98C4×Q8C2×D21111yescharacteristic
99#107—21no—
100#107—4221no—
101C4⋊C8—2111no—
102C4⋊C8C2×D21111yescharacteristic
103#103—21no—
104#103—4221no—
105#103—4222no—
106C4×C8—2111no—
107C2×Q8—21no—
108C2×Q8—4221no—
109C2×Q8—2111no—
110C2×Q8C22⋊D21111yescharacteristic
111C2×C8—21no—
112C2×C8—4221no—
113C2×C8—4221no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—4222no—
118C4⋊C4—4222no—
119C4⋊C4—4222no—
120C4⋊C4—4221no—
121C4⋊C4—4221no—
122C4⋊C4—2112no—
123C4⋊C4—2111no—
124C4⋊C4C22⋊D21111yescharacteristic
125C42—21no—
126C42—4222no—
127C42—4221no—
128C42C22×D21111yescharacteristic
129Q8—21no—
130Q8—42no—
131Q8—8441no—
132Q8—8441no—
133Q8—4221no—
134Q8—4221no—
135Q8#1680995212yes—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—4221no—
140C2×C4—4221no—
141C2×C4—4222no—
142C2×C4D4×D21111yescharacteristic
143C2×C4D4×D21111yescharacteristic
144C2×C4C2×C22⋊D21111yescharacteristic
145C8—41no—
146C8—8441no—
147C8—8441no—
148C22#2026994511yescharacteristic
149C4—21no—
150C4—22no—
151C4—41no—
152C4—81no—
153C4—8441no—
154C4—8442no—
155C4—8441no—
156C4C2×#1680995211yescharacteristic
157C4#2027000011yescharacteristic
158C4#2027000711yescharacteristic
159C4#2027000911yescharacteristic
160C2#2721330111yescharacteristic
161C2#2721330711yescharacteristic
162C2#2721331711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#477×C211C2
socle, prime radicalC22×C211#120
hypercenter#477D211
hypocenterC211#1636
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #477

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
#272134962
#272134382
#272128912
Q8⋊(C211⋊C8)2
#272128762
#477×C2112
#1636211