Finite Group Databasebeta

← #41012567

Subgroups — #41012567

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#27213870C211yesmaximal, characteristic
2#27213438C211yesmaximal, characteristic, complemented
3#27213433C211yesmaximal, characteristic
4#27212892C211yesmaximal, characteristic, complemented
5#27212878C211yesmaximal, characteristic, complemented
6#27212875C211yesmaximal, characteristic
7#487×C211C211yesmaximal, characteristic, complemented
8#20269972—22no—
9#20269966—21no—
10#20269957—21no—
11Q8×Dic211—21no—
12C4×Dic422C2211yescharacteristic
13C4×D844C2211yescharacteristic
14#20269935—21no—
15#20269910—21no—
16#20269908—21no—
17C4⋊(C211⋊C8)C2211yescharacteristic
18C4⋊(C211⋊C8)C2211yescharacteristic
19#20269900—21no—
20#20269897—21no—
21C4⋊Q8×C211C2211yescharacteristic
22C4⋊Q8×C211C2211yescharacteristic
23#103×C211—22no—
24C4×C8×C211C2211yescharacteristic
25C2×Dic422D411yescharacteristic
26C2×D844D411yescharacteristic
27C2×C4×D211—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8—21no—
30C4⋊Dic211—22no—
31C4⋊Dic211D411yescharacteristic
32C4⋊Dic211D411yescharacteristic
33Dic211⋊C4—21no—
34(C2×D211)⋊C4—22no—
35(C2×D211)⋊C4—22no—
36(C2×D211)⋊C4—21no—
37C4×Dic211—22no—
38C4×Dic211—21no—
39C2×Q8×C211—21no—
40C2×Q8×C211—21no—
41C2×C8×C211—21no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211D412yes—
46C42×C211C2311yescharacteristic
47C22×D211—21no—
48Dic422—21no—
49Dic422—21no—
50D844—21no—
51D844—21no—
52C2×Dic211—21no—
53C2×Dic211—21no—
54C2×Dic211—21no—
55C4×D211—41no—
56C4×D211—41no—
57C211⋊C8—41no—
58C211⋊C8—41no—
59Q8×C211—42no—
60Q8×C211—41no—
61Q8×C211—41no—
62C2×C4×C211—22no—
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—42no—
74C4×C211—81no—
75C4×C211#13511yescharacteristic
76C4×C211#13511yescharacteristic
77C4×C211#13711yescharacteristic
78C4×C211#13711yescharacteristic
79Dic211—41no—
80Dic211—41no—
81Dic211—41no—
82C2×C211#45211yescharacteristic
83C2×C211#45211yescharacteristic
84C2×C211#45411yescharacteristic
85D211—81no—
86C211#163811yescharacteristic, complemented, Sylow
87#1638—2111nomaximal, Sylow
88#556—2111no—
89#487D21111yescharacteristic, complemented
90#477—2111no—
91#476—2111no—
92#332—2111no—
93Q8⋊C8—2111no—
94D4⋊C8—2111no—
95C4⋊Q8C2×D21111yescharacteristic
96C4⋊Q8C2×D21111yescharacteristic
97#126—4222no—
98#124—4221no—
99C22⋊Q8—4221no—
100C4×Q8—4221no—
101C4×Q8—2111no—
102C4×D4—2111no—
103#117—4221no—
104#107—4221no—
105#106—4221no—
106C4⋊C8—2111no—
107C4⋊C8—2111no—
108#103—22no—
109#103—4221no—
110#102—4221no—
111C4×C8C2×D21111yescharacteristic
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—2111no—
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—2111no—
126C4⋊C4—2111no—
127C4⋊C4C22⋊D21112yes—
128C22⋊C4—4222no—
129C22⋊C4—4222no—
130C22⋊C4—4221no—
131C42—4222no—
132C42—4221no—
133C42C22×D21111yescharacteristic
134C23—4221no—
135Q8—42no—
136Q8—41no—
137Q8—41no—
138Q8—4221no—
139Q8—4221no—
140D4—4221no—
141D4—4221no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—8441no—
145C2×C4—8441no—
146C2×C4—4221no—
147C2×C4—4221no—
148C2×C4—4221no—
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—42no—
161C4—81no—
162C4—8441no—
163C4—8441no—
164C4—8441no—
165C4#2026999811yescharacteristic
166C4#2027000011yescharacteristic
167C4#2027000611yescharacteristic
168C4#2027000911yescharacteristic
169C2—16881no—
170C2#2721330511yescharacteristic
171C2#2721330711yescharacteristic
172C2#2721331811yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#487×C211C2
socleC22×C211#120
prime radicalC2×D844D4
hypercenter#487D211
hypocenterC211#1638
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #27213438C2semidirect
4. #27212892C2semidirect
5. #27212878C2semidirect
7. #487×C211C2semidirect
86. C211#1638semidirect
89. #487D211semidirect

Maximal subgroups

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

subgroupindex
#272138702
#272134382
#272134332
#272128922
#272128782
#272128752
#487×C2112
#1638211