Finite Group Databasebeta

← #41012537

Subgroups — #41012537

Order 27008.

206 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×Dic422C211yesmaximal, characteristic
2#27213525C211yesmaximal, characteristic
3#27213438C211yesmaximal, characteristic
4#27212894C211yesmaximal, characteristic
5Q8⋊(C211⋊C8)C211yesmaximal, characteristic
6#27212876C211yesmaximal, characteristic
7#477×C211C211yesmaximal, characteristic
8C2×#16809952—22no—
9#20269977—21no—
10#20269977C2211yescharacteristic
11#20269976—22no—
12#20269976—21no—
13Q8×Dic211—21no—
14C4×Dic422—21no—
15C4×Dic422C2211yescharacteristic
16#20269910—21no—
17C4⋊(C211⋊C8)C2211yescharacteristic
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—22no—
26C2×Dic422D411yescharacteristic
27C2×Dic422D412yes—
28#16809952—44no—
29C2×C211⋊C8—21no—
30C2×C211⋊C8—21no—
31C4⋊Dic211—22no—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211D411yescharacteristic
35Dic211⋊C4—22no—
36Dic211⋊C4—21no—
37C4×Dic211—22no—
38C4×Dic211—21no—
39C2×Q8×C211—21no—
40C2×Q8×C211D411yescharacteristic
41C2×C8×C211—21no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211D411yescharacteristic
46C42×C211—21no—
47C42×C211C2311yescharacteristic
48Dic422—21no—
49Dic422—42no—
50Dic422—82no—
51Dic422Q1612yescomplemented
52C2×Dic211—21no—
53C2×Dic211—22no—
54C2×Dic211—21no—
55C211⋊C8—22no—
56C211⋊C8—41no—
57Q8×C211—21no—
58Q8×C211—42no—
59Q8×C211Q1612yescomplemented
60C2×C4×C211—21no—
61C2×C4×C211—21no—
62C2×C4×C211—41no—
63C2×C4×C211C2×D411yescharacteristic
64C2×C4×C211C2×D411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C8×C211—41no—
67C22×C211#12011yescharacteristic
68C4×C211—22no—
69C4×C211—21no—
70C4×C211—41no—
71C4×C211—81no—
72C4×C211C2×Q1611yescharacteristic
73C4×C211C2×Q1611yescharacteristic
74C4×C211#13711yescharacteristic
75C4×C211#13711yescharacteristic
76Dic211—22no—
77Dic211—42no—
78Dic211—41no—
79C2×C211#45111yescharacteristic
80C2×C211#45111yescharacteristic
81C2×C211#45311yescharacteristic
82C211#163311yescharacteristic, complemented, Sylow
83#1633—2111nomaximal, Sylow
84Q82—2111no—
85#494—2111no—
86#477—2111no—
87#477D21111yescharacteristic
88#333—2111no—
89Q8⋊C8—2111no—
90Q8⋊C8—2111no—
91C2×Q16—4222no—
92C4⋊Q8—4221no—
93C4⋊Q8—4222no—
94C4⋊Q8—4221no—
95C4⋊Q8—2111no—
96C4⋊Q8C2×D21111yescharacteristic
97C4×Q8—4221no—
98C4×Q8—4221no—
99C4×Q8—2111no—
100C4×Q8C2×D21111yescharacteristic
101#107—21no—
102#107—4221no—
103C4⋊C8—2111no—
104C4⋊C8C2×D21111yescharacteristic
105#103—21no—
106#103—4221no—
107C4×C8—2111no—
108C2×Q8—21no—
109C2×Q8—4222no—
110C2×Q8—2112no—
111C2×Q8—2111no—
112C2×Q8C22⋊D21111yescharacteristic
113Q16—8444no—
114C2×C8—21no—
115C2×C8—4221no—
116C2×C8—4221no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—4222no—
121C4⋊C4—4222no—
122C4⋊C4—4221no—
123C4⋊C4—4221no—
124C4⋊C4—4221no—
125C4⋊C4—2111no—
126C4⋊C4C22⋊D21111yescharacteristic
127C42—21no—
128C42—4222no—
129C42—4221no—
130C42C22×D21111yescharacteristic
131Q8—21no—
132Q8—42no—
133Q8—8442no—
134Q8—16882no—
135Q8—4221no—
136Q8—2112no—
137Q8#1680995212yescomplemented
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—41no—
141C2×C4—4222no—
142C2×C4—4221no—
143C2×C4—4221no—
144C2×C4D4×D21111yescharacteristic
145C2×C4D4×D21111yescharacteristic
146C2×C4C2×C22⋊D21111yescharacteristic
147C8—41no—
148C8—8441no—
149C8—4222no—
150C22#2026994511yescharacteristic
151C4—21no—
152C4—22no—
153C4—41no—
154C4—81no—
155C4—8442no—
156C4—8441no—
157C4—4222no—
158C4Q16×D21111yescharacteristic
159C4C2×#1680995211yescharacteristic
160C4#2027000711yescharacteristic
161C4#2027000911yescharacteristic
162C2#2721330011yescharacteristic
163C2#2721330111yescharacteristic
164C2#2721331211yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#477×C211C2
socle, prime radicalC22×C211#120
hypercenter#477D211
hypocenterC211#1633
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
51. Dic422Q16semidirect
59. Q8×C211Q16semidirect
82. C211#1633semidirect
137. Q8#16809952semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic4222
#272135252
#272134382
#272128942
Q8⋊(C211⋊C8)2
#272128762
#477×C2112
#1633211