Finite Group Databasebeta

← #41012669

Subgroups — #41012669

Order 27008.

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

normal subgroups only

This group has 200 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#27213542C211yesmaximal, characteristic, complemented
3#27213427C211yesmaximal, characteristic
4#27213357C211yesmaximal, characteristic, complemented
5#27212876C211yesmaximal, characteristic, complemented
6#27212873C211yesmaximal, characteristic
7#332×C211C211yesmaximal, characteristic, complemented
8C2×Dic844—21no—
9#20269977C2211yescharacteristic
10#20269972—22no—
11#20269966—21no—
12#20269957—21no—
13Q8×Dic211—21no—
14C4×Dic422C2211yescharacteristic
15C4×D844C2211yescharacteristic
16#20269935—21no—
17#20269911—21no—
18#20269908—22no—
19#20269902—21no—
20#20269899—21no—
21C4×C211⋊C8C2211yescharacteristic
22C4⋊Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4⋊C8×C211C2211yescharacteristic
25C2×Dic422—21no—
26C2×Dic422D411yescharacteristic
27C2×D844D411yescharacteristic
28C2×C4×D211—21no—
29Dic844—42no—
30C2×C211⋊C8—21no—
31C4⋊Dic211—22no—
32C4⋊Dic211—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211D411yescharacteristic
35C4⋊Dic211D411yescharacteristic
36Dic211⋊C4—21no—
37(C2×D211)⋊C4—22no—
38(C2×D211)⋊C4—22no—
39(C2×D211)⋊C4—21no—
40C4×Dic211—22no—
41C4×Dic211—21no—
42C2×Q8×C211—21no—
43C2×C8×C211—21no—
44C2×C8×C211—21no—
45C4⋊C4×C211—21no—
46C4⋊C4×C211D412yes—
47C42×C211C2311yescharacteristic
48C22×D211—21no—
49Dic422—21no—
50Dic422—21no—
51Dic422—41no—
52Dic422—41no—
53D844—21no—
54D844—21no—
55C2×Dic211—21no—
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211—41no—
59C4×D211—41no—
60C4×D211—41no—
61C211⋊C8—22no—
62Q8×C211—41no—
63Q8×C211—41no—
64C2×C4×C211—22no—
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C8×C211—41no—
69C8×C211—41no—
70C22×C211#12011yescharacteristic
71C2×D211—41no—
72C2×D211—41no—
73C2×D211—41no—
74C4×C211—21no—
75C4×C211—42no—
76C4×C211#13511yescharacteristic
77C4×C211#13511yescharacteristic
78C4×C211AGL(1,ℤ/8)11yescharacteristic
79C4×C211#13711yescharacteristic
80Dic211—41no—
81Dic211—41no—
82Dic211—41no—
83Dic211—81no—
84C2×C211#44911yescharacteristic
85C2×C211#45211yescharacteristic
86C2×C211#45611yescharacteristic
87D211—81no—
88C211#165211yescharacteristic, complemented, Sylow
89#1652—2111nomaximal, Sylow
90#556—2111no—
91#498—2111no—
92#474—2111no—
93#462—2111no—
94#332D21111yescharacteristic, complemented
95Q8⋊C8—2111no—
96D4⋊C8—2111no—
97C2×Q16—4221no—
98C4⋊Q8—2111no—
99C4⋊Q8C2×D21111yescharacteristic
100#126—4222no—
101#124—4221no—
102C22⋊Q8—4221no—
103C4×Q8—4221no—
104C4×Q8—2111no—
105C4×D4—2111no—
106#117—4221no—
107#107—4221no—
108#106—4222no—
109C4⋊C8C2×D21111yescharacteristic
110C4⋊C8C2×D21111yescharacteristic
111#103—4221no—
112#102—4221no—
113C4×C8—2111no—
114C2×Q8—21no—
115C2×Q8—4221no—
116C2×Q8—2111no—
117C2×D4—2111no—
118C22×C4—4221no—
119Q16—8442no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—4221no—
123C4⋊C4—21no—
124C4⋊C4—4222no—
125C4⋊C4—4221no—
126C4⋊C4—4221no—
127C4⋊C4—4221no—
128C4⋊C4—2111no—
129C4⋊C4—2111no—
130C4⋊C4D84412yes—
131C22⋊C4—4222no—
132C22⋊C4—4222no—
133C22⋊C4—4221no—
134C42—4222no—
135C42—4221no—
136C42C22×D21111yescharacteristic
137C23—4221no—
138Q8—41no—
139Q8—41no—
140Q8—8441no—
141Q8—8441no—
142Q8—4221no—
143Q8—4221no—
144D4—4221no—
145D4—4221no—
146C2×C4—22no—
147C2×C4—8441no—
148C2×C4—8441no—
149C2×C4—8441no—
150C2×C4—4221no—
151C2×C4—4221no—
152C2×C4—4221no—
153C2×C4D4×D21111yescharacteristic
154C2×C4D4×D21111yescharacteristic
155C2×C4C2×D84411yescharacteristic
156C8—41no—
157C8—41no—
158C8—4222no—
159C22—8441no—
160C22—8441no—
161C22—8441no—
162C22#2026994711yescharacteristic
163C4—21no—
164C4—42no—
165C4—8441no—
166C4—8441no—
167C4—8441no—
168C4—16881no—
169C4#2026999611yescharacteristic
170C4#2027000011yescharacteristic
171C4#2027000411yescharacteristic
172C4#2027000611yescharacteristic
173C2—16881no—
174C2#2721329411yescharacteristic
175C2#2721330911yescharacteristic
176C2#2721333111yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#332×C211C2
socleC22×C211#120
prime radicalC2×D844D4
hypercenter#332D211
hypocenterC211#1652
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27213542C2semidirect
4. #27213357C2semidirect
5. #27212876C2semidirect
7. #332×C211C2semidirect
88. C211#1652semidirect
94. #332D211semidirect

Maximal subgroups

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

subgroupindex
#272138702
#272135422
#272134272
#272133572
#272128762
#272128732
#332×C2112
#1652211