Finite Group Databasebeta

← #41012568

Subgroups — #41012568

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#27213866C211yesmaximal, characteristic
2#27213496C211yesmaximal, characteristic, complemented
3#27213438C211yesmaximal, characteristic, complemented
4#27212893C211yesmaximal, characteristic, complemented
5#27212876C211yesmaximal, characteristic, complemented
6D4⋊(C211⋊C8)C211yesmaximal, characteristic
7#476×C211C211yesmaximal, characteristic
8#20269977C2211yescharacteristic
9#20269970—22no—
10#20269964—21no—
11#20269956—21no—
12C4×Dic422C2211yescharacteristic
13#20269942—21no—
14#20269936—21no—
15#20269910—21no—
16C4⋊(C211⋊C8)C2211yescharacteristic
17#20269901—22no—
18#20269900—21no—
19C4×C211⋊C8C2211yescharacteristic
20C4⋊Q8×C211C2211yescharacteristic
21C4×D4×C211C2211yescharacteristic
22#106×C211—21no—
23C4⋊C8×C211C2211yescharacteristic
24#102×C211—21no—
25C2×Dic422—21no—
26C2×Dic422D411yescharacteristic
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C4⋊Dic211—21no—
30C4⋊Dic211D412yes—
31C4⋊Dic211D411yescharacteristic
32Dic211⋊C4—22no—
33Dic211⋊C4—21no—
34C22⋊Dic211—22no—
35C22⋊Dic211—22no—
36C4×Dic211—22no—
37C4×Dic211—21no—
38C2×Q8×C211—21no—
39C2×D4×C211D411yescharacteristic
40C22×C4×C211—21no—
41C2×C8×C211—21no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211—21no—
44C4⋊C4×C211D411yescharacteristic
45C22⋊C4×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—
56C23×C211—21no—
57Q8×C211—42no—
58D4×C211—21no—
59D4×C211—21no—
60C2×C4×C211—21no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211C2×D411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C8×C211—41no—
68C22×C211—41no—
69C22×C211—41no—
70C22×C211—41no—
71C22×C211#12011yescharacteristic
72C4×C211—21no—
73C4×C211—41no—
74C4×C211—81no—
75C4×C211#13511yescharacteristic
76C4×C211#13511yescharacteristic
77C4×C211#13711yescharacteristic
78C4×C211#13711yescharacteristic
79Dic211—41no—
80Dic211—42no—
81Dic211—41no—
82C2×C211—81no—
83C2×C211#45211yescharacteristic
84C2×C211#45211yescharacteristic
85C2×C211#45411yescharacteristic
86C211#163811yescharacteristic, complemented, Sylow
87#1638—2111nomaximal, Sylow
88#556—2111no—
89#487—2111no—
90#477—2111no—
91#476D21111yescharacteristic
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×Q8—2111no—
102C4×D4C2×D21111yescharacteristic
103#117—4221no—
104#107—4221no—
105#106—21no—
106C4⋊C8—2111no—
107C4⋊C8C2×D21111yescharacteristic
108#103—4221no—
109#103—4222no—
110#102—21no—
111C4×C8—2111no—
112C2×Q8—21no—
113C2×Q8—4221no—
114C2×Q8—2111no—
115C2×D4C22⋊D21111yescharacteristic
116C22×C4—21no—
117C2×C8—21no—
118C2×C8—4221no—
119C2×C8—4221no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—4222no—
123C4⋊C4—4221no—
124C4⋊C4—4221no—
125C4⋊C4—2112no—
126C4⋊C4—2111no—
127C4⋊C4C22⋊D21111yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—4222no—
130C22⋊C4—4222no—
131C42—4222no—
132C42—4221no—
133C42C22×D21111yescharacteristic
134C23—21no—
135Q8—42no—
136Q8—8441no—
137Q8—8441no—
138Q8—4221no—
139Q8—4221no—
140D4—21no—
141D4—21no—
142C2×C4—21no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—4221no—
147C2×C4—4221no—
148C2×C4—4222no—
149C2×C4D4×D21111yescharacteristic
150C2×C4D4×D21111yescharacteristic
151C2×C4C2×C22⋊D21111yescharacteristic
152C8—41no—
153C8—8441no—
154C8—8441no—
155C22—41no—
156C22—41no—
157C22—41no—
158C22#2026994511yescharacteristic
159C4—21no—
160C4—41no—
161C4—81no—
162C4—8441no—
163C4—8441no—
164C4—8442no—
165C4#2026999711yescharacteristic
166C4#2026999811yescharacteristic
167C4#2027000611yescharacteristic
168C4#2027000811yescharacteristic
169C2—81no—
170C2#2721330511yescharacteristic
171C2#2721330611yescharacteristic
172C2#2721331711yescharacteristic
subgroupG/H
centerC22#20269945
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#476×C211C2
socleC22×C211#120
prime radicalC2×D4×C211D4
hypercenter#476D211
hypocenterC211#1638
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #27213496C2semidirect
3. #27213438C2semidirect
4. #27212893C2semidirect
5. #27212876C2semidirect
86. C211#1638semidirect

Maximal subgroups

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

subgroupindex
#272138662
#272134962
#272134382
#272128932
#272128762
D4⋊(C211⋊C8)2
#476×C2112
#1638211