Finite Group Databasebeta

← #41012518

Subgroups — #41012518

Order 27008.

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

normal subgroups only

This group has 218 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#27213881C211yesmaximal, characteristic
2#27213520C211yesmaximal, characteristic
3#27213438C211yesmaximal, characteristic, complemented
4#27212889C211yesmaximal, characteristic
5Q8⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6#27212876C211yesmaximal, characteristic, complemented
7#477×C211C211yesmaximal, characteristic, complemented
8C2×#16809949—22no—
9#20269974C2211yescharacteristic
10#20269966—21no—
11#20269965—21no—
12#20269962—22no—
13C4×Dic422C2211yescharacteristic
14C4×D844—21no—
15#20269939—21no—
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×Dic422D411yescharacteristic
26C2×D844—21no—
27C2×D844—21no—
28C2×D844D412yes—
29C2×C4×D211—22no—
30#16809949—44no—
31C2×C211⋊C8—21no—
32C2×C211⋊C8—21no—
33C4⋊Dic211D411yescharacteristic
34Dic211⋊C4—21no—
35(C2×D211)⋊C4—22no—
36(C2×D211)⋊C4—22no—
37(C2×D211)⋊C4—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
48C22×D211—22no—
49Dic422—21no—
50Dic422—21no—
51D844—42no—
52D844—41no—
53D844—41no—
54D844—41no—
55D844—41no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C4×D211—82no—
59C211⋊C8—22no—
60C211⋊C8—41no—
61Q8×C211—21no—
62Q8×C211—21no—
63Q8×C211—42no—
64C2×C4×C211—21no—
65C2×C4×C211—21no—
66C2×C4×C211—41no—
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C8×C211—41no—
71C22×C211#12011yescharacteristic
72C2×D211—42no—
73C2×D211—42no—
74C2×D211—42no—
75C4×C211—21no—
76C4×C211—41no—
77C4×C211—41no—
78C4×C211—81no—
79C4×C211#13511yescharacteristic
80C4×C211#13511yescharacteristic
81C4×C211#13711yescharacteristic
82C4×C211#13711yescharacteristic
83Dic211—41no—
84Dic211—41no—
85C2×C211#45211yescharacteristic
86C2×C211#45211yescharacteristic
87C2×C211#45311yescharacteristic
88D211—82no—
89C211#163011yescharacteristic, complemented, Sylow
90#1630—2111nomaximal, Sylow
91#559—2111no—
92#492—2111no—
93#477—2111no—
94#477D21111yescharacteristic, complemented
95#331—2111no—
96Q8⋊C8—2111no—
97Q8⋊C8—2111no—
98C2×SD16—4222no—
99C4⋊Q8C2×D21111yescharacteristic
100#127—2111no—
101#124—4221no—
102#124—4221no—
103#123—4222no—
104C4×Q8—2111no—
105C4×Q8C2×D21111yescharacteristic
106C4×D4—4221no—
107C4×D4—4221no—
108#107—21no—
109#107—4221no—
110C4⋊C8—2111no—
111C4⋊C8C2×D21111yescharacteristic
112#103—21no—
113#103—4221no—
114C4×C8—2111no—
115C2×Q8—21no—
116C2×Q8—2111no—
117C2×Q8C22⋊D21111yescharacteristic
118C2×D4—4221no—
119C2×D4—4221no—
120C2×D4—2112no—
121C22×C4—4222no—
122SD16—8444no—
123C2×C8—21no—
124C2×C8—4221no—
125C2×C8—4221no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—4221no—
130C4⋊C4—2111no—
131C4⋊C4C22⋊D21111yescharacteristic
132C22⋊C4—4222no—
133C22⋊C4—4222no—
134C22⋊C4—4222no—
135C42—21no—
136C42—4221no—
137C42C22×D21111yescharacteristic
138C23—4222no—
139Q8—21no—
140Q8—21no—
141Q8—42no—
142Q8—4221no—
143Q8—4221no—
144D4—8442no—
145D4—8441no—
146D4—8441no—
147D4—8441no—
148D4—8441no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—16882no—
153C2×C4—4221no—
154C2×C4—4221no—
155C2×C4D4×D21111yescharacteristic
156C2×C4D4×D21111yescharacteristic
157C2×C4C2×C22⋊D21111yescharacteristic
158C8—41no—
159C8—8441no—
160C8—4222no—
161C22—8442no—
162C22—8442no—
163C22—8442no—
164C22#2026994511yescharacteristic
165C4—21no—
166C4—41no—
167C4—41no—
168C4—81no—
169C4—8441no—
170C4—8441no—
171C4#2026999711yescharacteristic
172C4#2027000011yescharacteristic
173C4#2027000811yescharacteristic
174C4#2027000911yescharacteristic
175C2—16882no—
176C2#2721330611yescharacteristic
177C2#2721330711yescharacteristic
178C2#2721331211yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#477×C211C2
socleC22×C211#120
prime radical#20269974C22
hypercenter#477D211
hypocenterC211#1630
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
3. #27213438C2semidirect
5. Q8⋊(C211⋊C8)C2semidirect
6. #27212876C2semidirect
7. #477×C211C2semidirect
89. C211#1630semidirect
94. #477D211semidirect

Maximal subgroups

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

subgroupindex
#272138812
#272135202
#272134382
#272128892
Q8⋊(C211⋊C8)2
#272128762
#477×C2112
#1630211