Finite Group Databasebeta

← #41012569

Subgroups — #41012569

Order 27008.

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

normal subgroups only

This group has 194 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#27213869C211yesmaximal, characteristic
2#27213496C211yesmaximal, characteristic, complemented
3#27213433C211yesmaximal, characteristic
4#27212891C211yesmaximal, characteristic, complemented
5Q8⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
6#27212873C211yesmaximal, characteristic
7#477×C211C211yesmaximal, characteristic, complemented
8#20269977C2211yescharacteristic
9#20269972—22no—
10#20269965—21no—
11#20269953—21no—
12C4×Dic422—21no—
13C4×D844C2211yescharacteristic
14#20269935—21no—
15#20269908—21no—
16C4⋊(C211⋊C8)C2211yescharacteristic
17#20269901—22no—
18#20269897—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×D844D411yescharacteristic
27C2×C4×D211—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211D412yes—
32C4⋊Dic211D411yescharacteristic
33Dic211⋊C4—22no—
34(C2×D211)⋊C4—22no—
35(C2×D211)⋊C4—22no—
36(C2×D211)⋊C4—21no—
37C4×Dic211—22no—
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
47C22×D211—21no—
48Dic422—41no—
49Dic422—41no—
50D844—21no—
51D844—21no—
52C2×Dic211—21no—
53C2×Dic211—22no—
54C4×D211—41no—
55C4×D211—41no—
56C211⋊C8—41no—
57C211⋊C8—41no—
58Q8×C211—21no—
59Q8×C211—21no—
60Q8×C211—42no—
61C2×C4×C211—21no—
62C2×C4×C211—21no—
63C2×C4×C211—41no—
64C2×C4×C211C2×D411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C8×C211—41no—
68C22×C211#12011yescharacteristic
69C2×D211—41no—
70C2×D211—41no—
71C2×D211—41no—
72C4×C211—21no—
73C4×C211—41no—
74C4×C211—41no—
75C4×C211—81no—
76C4×C211#13511yescharacteristic
77C4×C211#13511yescharacteristic
78C4×C211#13711yescharacteristic
79C4×C211#13711yescharacteristic
80Dic211—41no—
81Dic211—42no—
82C2×C211#45211yescharacteristic
83C2×C211#45211yescharacteristic
84C2×C211#45411yescharacteristic
85D211—81no—
86C211#163811yescharacteristic, complemented, Sylow
87#1638—2111nomaximal, Sylow
88#556—2111no—
89#487—2111no—
90#477D21111yescharacteristic, complemented
91#476—2111no—
92#332—2111no—
93Q8⋊C8—2111no—
94D4⋊C8—2111no—
95C4⋊Q8—2111no—
96C4⋊Q8C2×D21111yescharacteristic
97#126—4222no—
98#124—4221no—
99C22⋊Q8—4221no—
100C4×Q8—4221no—
101C4×Q8C2×D21111yescharacteristic
102C4×D4—2111no—
103#117—4221no—
104#107—21no—
105#106—4221no—
106C4⋊C8—2111no—
107C4⋊C8C2×D21111yescharacteristic
108#103—21no—
109#103—4222no—
110#102—4221no—
111C4×C8—2111no—
112C2×Q8—21no—
113C2×Q8—4221no—
114C2×Q8C22⋊D21111yescharacteristic
115C2×D4—2111no—
116C22×C4—4221no—
117C2×C8—21no—
118C2×C8—4221no—
119C2×C8—4221no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—4222no—
124C4⋊C4—4221no—
125C4⋊C4—2112no—
126C4⋊C4—2111no—
127C4⋊C4C22⋊D21111yescharacteristic
128C22⋊C4—4222no—
129C22⋊C4—4222no—
130C22⋊C4—4221no—
131C42—21no—
132C42—4222no—
133C42C22×D21111yescharacteristic
134C23—4221no—
135Q8—21no—
136Q8—21no—
137Q8—42no—
138Q8—8441no—
139Q8—8441no—
140D4—4221no—
141D4—4221no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—8441no—
146C2×C4—8441no—
147C2×C4—4221no—
148C2×C4—4222no—
149C2×C4D4×D21111yescharacteristic
150C2×C4D4×D21111yescharacteristic
151C2×C4C2×C22⋊D21111yescharacteristic
152C8—41no—
153C8—8441no—
154C8—8441no—
155C22—8441no—
156C22—8441no—
157C22—8441no—
158C22#2026994511yescharacteristic
159C4—21no—
160C4—41no—
161C4—41no—
162C4—81no—
163C4—8441no—
164C4—8442no—
165C4#2026999711yescharacteristic
166C4#2027000011yescharacteristic
167C4#2027000811yescharacteristic
168C4#2027000911yescharacteristic
169C2—16881no—
170C2#2721330611yescharacteristic
171C2#2721330711yescharacteristic
172C2#2721331711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#477×C211C2
socleC22×C211#120
prime radicalC2×D844D4
hypercenter#477D211
hypocenterC211#1638
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #27213496C2semidirect
4. #27212891C2semidirect
5. Q8⋊(C211⋊C8)C2semidirect
7. #477×C211C2semidirect
86. C211#1638semidirect
90. #477D211semidirect

Maximal subgroups

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

subgroupindex
#272138692
#272134962
#272134332
#272128912
Q8⋊(C211⋊C8)2
#272128732
#477×C2112
#1638211