Finite Group Databasebeta

← #41012495

Subgroups — #41012495

Order 27008.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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#27213871C211yesmaximal, characteristic, complemented
2#27213494C211yesmaximal, characteristic
3#27213347C211yesmaximal, characteristic
4#27213347C211yesmaximal, characteristic
5#27212876C211yesmaximal, characteristic, complemented
6#27212876C211yesmaximal, characteristic, complemented
7#332×C211C211yesmaximal, characteristic, complemented
8C2×#16809951—21no—
9C2×#16809951—21no—
10#20269975—21no—
11#20269974C2211yescharacteristic
12#20269968—21no—
13#20269967—22no—
14Q8×Dic211—21no—
15C4×Dic422C2211yescharacteristic
16C4×Dic422C2211yescharacteristic
17#20269942—21no—
18#20269899—21no—
19#20269899—21no—
20#20269897—22no—
21C4×C211⋊C8C2211yescharacteristic
22C4⋊Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4⋊C8×C211C2211yescharacteristic
25C2×Dic422D411yescharacteristic
26C2×Dic422D411yescharacteristic
27C2×D844—21no—
28C2×D844—21no—
29C2×D844—21no—
30#16809951—42no—
31#16809951—42no—
32C2×C211⋊C8—21no—
33C4⋊Dic211—22no—
34C4⋊Dic211D411yescharacteristic
35C4⋊Dic211D411yescharacteristic
36Dic211⋊C4—22no—
37Dic211⋊C4—22no—
38Dic211⋊C4—21no—
39Dic211⋊C4—21no—
40C4×Dic211—22no—
41C4×Dic211—21no—
42C4×Dic211—21no—
43C2×Q8×C211—21no—
44C2×C8×C211—21no—
45C2×C8×C211—21no—
46C4⋊C4×C211—21no—
47C4⋊C4×C211D412yes—
48C42×C211C2311yescharacteristic
49C22×D211—41no—
50Dic422—21no—
51Dic422—21no—
52Dic422—21no—
53Dic422SD1612yescomplemented
54D844—42no—
55D844—41no—
56D844—41no—
57D844—41no—
58D844—41no—
59C2×Dic211—21no—
60C2×Dic211—21no—
61C2×Dic211—21no—
62C2×Dic211—21no—
63C211⋊C8—41no—
64Q8×C211—41no—
65Q8×C211—41no—
66C2×C4×C211—22no—
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C8×C211—41no—
71C8×C211—41no—
72C22×C211#12011yescharacteristic
73C2×D211—81no—
74C2×D211—81no—
75C2×D211—81no—
76C4×C211—21no—
77C4×C211—42no—
78C4×C211C2×SD1611yescharacteristic
79C4×C211#13511yescharacteristic
80C4×C211AGL(1,ℤ/8)11yescharacteristic
81C4×C211#13711yescharacteristic
82Dic211—22no—
83Dic211—41no—
84Dic211—41no—
85Dic211—41no—
86C2×C211#44811yescharacteristic
87C2×C211#44911yescharacteristic
88C2×C211#45411yescharacteristic
89D211—161no—
90C211#162611yescharacteristic, complemented, Sylow
91#1626—2111nomaximal, Sylow
92#557—2111no—
93#486—2111no—
94#460—2111no—
95#460—2111no—
96#332D21111yescharacteristic, complemented
97Q8⋊C8—2111no—
98Q8⋊C8—2111no—
99C2×SD16—4221no—
100C2×SD16—4221no—
101C4⋊Q8—4221no—
102C4⋊Q8C2×D21111yescharacteristic
103#127—2111no—
104#125—4221no—
105#125—4222no—
106C4×Q8—4221no—
107C4×Q8—4221no—
108C4×Q8—2111no—
109C4×Q8—2111no—
110C4⋊C8C2×D21111yescharacteristic
111C4⋊C8C2×D21111yescharacteristic
112#102—4221no—
113#102—4221no—
114#102—4222no—
115C4×C8—2111no—
116C2×Q8—21no—
117C2×Q8—2111no—
118C2×Q8—2111no—
119C2×D4—4221no—
120C2×D4—4221no—
121C2×D4—4221no—
122SD16—8442no—
123SD16—8442no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—4221no—
127C4⋊C4—21no—
128C4⋊C4—4222no—
129C4⋊C4—4222no—
130C4⋊C4—4222no—
131C4⋊C4—4221no—
132C4⋊C4—4221no—
133C4⋊C4—2111no—
134C4⋊C4—2111no—
135C4⋊C4D84412yes—
136C42—4222no—
137C42—4221no—
138C42—4221no—
139C42C22×D21111yescharacteristic
140C23—8441no—
141Q8—41no—
142Q8—41no—
143Q8—4221no—
144Q8—4221no—
145Q8—4221no—
146Q8—2112no—
147D4—8442no—
148D4—8441no—
149D4—8441no—
150D4—8441no—
151D4—8441no—
152C2×C4—22no—
153C2×C4—4221no—
154C2×C4—4221no—
155C2×C4—4221no—
156C2×C4—4221no—
157C2×C4D4×D21111yescharacteristic
158C2×C4D4×D21111yescharacteristic
159C2×C4C2×D84411yescharacteristic
160C8—41no—
161C8—41no—
162C8—8441no—
163C22—16881no—
164C22—16881no—
165C22—16881no—
166C22#2026994711yescharacteristic
167C4—21no—
168C4—42no—
169C4—8441no—
170C4—8441no—
171C4—8441no—
172C4—4222no—
173C4SD16×D21111yescharacteristic
174C4#2026999811yescharacteristic
175C4#2027000111yescharacteristic
176C4#2027000911yescharacteristic
177C2—33761no—
178C2#2721328611yescharacteristic
179C2#2721329211yescharacteristic
180C2#2721332111yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
Fitting#332×C211C2
socleC22×C211#120
prime radical#20269974C22
hypercenter#332D211
hypocenterC211#1626
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #27213871C2semidirect
5. #27212876C2semidirect
6. #27212876C2semidirect
7. #332×C211C2semidirect
53. Dic422SD16semidirect
90. C211#1626semidirect
96. #332D211semidirect

Maximal subgroups

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

subgroupindex
#272138712
#272134942
#272133472
#272128762
#332×C2112
#1626211