Finite Group Databasebeta

← #41012587

Subgroups — #41012587

Order 27008.

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

normal subgroups only

This group has 212 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#27213881C211yesmaximal, characteristic
2#27213493C211yesmaximal, characteristic
3#27213431C211yesmaximal, characteristic, complemented
4#27213357C211yesmaximal, characteristic, complemented
5#27212889C211yesmaximal, characteristic
6#27212878C211yesmaximal, characteristic, complemented
7Q8⋊C8×C211C211yesmaximal, characteristic, complemented
8C2×Dic844—21no—
9#20269977C2211yescharacteristic
10#20269974C2211yescharacteristic
11#20269966—21no—
12#20269965—21no—
13#20269962—22no—
14C4×Dic422C2211yescharacteristic
15C4×D844—21no—
16#20269939—21no—
17#20269908—21no—
18C4⋊(C211⋊C8)C2211yescharacteristic
19#20269902—21no—
20#20269901—21no—
21#20269899—22no—
22C4×Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4×C8×C211C2211yescharacteristic
25C2×Dic422—21no—
26C2×Dic422D411yescharacteristic
27C2×D844—21no—
28C2×D844—21no—
29C2×D844D412yes—
30C2×C4×D211—22no—
31Dic844—42no—
32C2×C211⋊C8—21no—
33C4⋊Dic211—21no—
34C4⋊Dic211—21no—
35C4⋊Dic211D411yescharacteristic
36Dic211⋊C4—21no—
37(C2×D211)⋊C4—22no—
38(C2×D211)⋊C4—22no—
39(C2×D211)⋊C4—22no—
40C4×Dic211—21no—
41C2×Q8×C211D411yescharacteristic
42C2×C8×C211—21no—
43C2×C8×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211D411yescharacteristic
46C42×C211—21no—
47C42×C211C2311yescharacteristic
48C22×D211—22no—
49Dic422—21no—
50Dic422—21no—
51Dic422—41no—
52Dic422—41no—
53D844—42no—
54D844—41no—
55D844—41no—
56D844—41no—
57D844—41no—
58C2×Dic211—21no—
59C2×Dic211—21no—
60C2×Dic211—41no—
61C4×D211—82no—
62C211⋊C8—41no—
63Q8×C211—21no—
64Q8×C211—21no—
65C2×C4×C211—21no—
66C2×C4×C211—21no—
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C2×C4×C211C2×D411yescharacteristic
70C8×C211—41no—
71C8×C211—41no—
72C22×C211#12011yescharacteristic
73C2×D211—42no—
74C2×D211—42no—
75C2×D211—42no—
76C4×C211—21no—
77C4×C211—41no—
78C4×C211—41no—
79C4×C211#13511yescharacteristic
80C4×C211#13511yescharacteristic
81C4×C211AGL(1,ℤ/8)11yescharacteristic
82C4×C211#13711yescharacteristic
83Dic211—41no—
84Dic211—41no—
85Dic211—81no—
86C2×C211#44911yescharacteristic
87C2×C211#45211yescharacteristic
88C2×C211#45511yescharacteristic
89D211—82no—
90C211#164011yescharacteristic, complemented, Sylow
91#1640—2111nomaximal, Sylow
92#559—2111no—
93#486—2111no—
94#475—2111no—
95#462—2111no—
96#331—2111no—
97Q8⋊C8—2111no—
98Q8⋊C8D21111yescharacteristic, complemented
99C2×Q16—4221no—
100C4⋊Q8—2111no—
101#127—2111no—
102#124—4221no—
103#124—4221no—
104#123—4222no—
105C4×Q8—2111no—
106C4×Q8C2×D21111yescharacteristic
107C4×D4—4221no—
108C4×D4—4221no—
109#106—4221no—
110C4⋊C8—2111no—
111C4⋊C8C2×D21111yescharacteristic
112#103—4221no—
113#103—4221no—
114#102—4222no—
115C4×C8C2×D21111yescharacteristic
116C2×Q8—4221no—
117C2×Q8—2111no—
118C2×Q8D84411yescharacteristic
119C2×D4—4221no—
120C2×D4—4221no—
121C2×D4—2112no—
122C22×C4—4222no—
123Q16—8442no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—4221no—
127C4⋊C4—21no—
128C4⋊C4—4221no—
129C4⋊C4—4221no—
130C4⋊C4—4221no—
131C4⋊C4—2111no—
132C4⋊C4D84411yescharacteristic
133C22⋊C4—4222no—
134C22⋊C4—4222no—
135C22⋊C4—4222no—
136C42—21no—
137C42—4221no—
138C42C22×D21111yescharacteristic
139C23—4222no—
140Q8—21no—
141Q8—21no—
142Q8—8441no—
143Q8—8441no—
144Q8—4221no—
145Q8—4221no—
146D4—8442no—
147D4—8441no—
148D4—8441no—
149D4—8441no—
150D4—8441no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—8441no—
154C2×C4—16882no—
155C2×C4—4221no—
156C2×C4—4221no—
157C2×C4D4×D21111yescharacteristic
158C2×C4D4×D21111yescharacteristic
159C2×C4C2×D84411yescharacteristic
160C8—41no—
161C8—41no—
162C8—8441no—
163C22—8442no—
164C22—8442no—
165C22—8442no—
166C22#2026994711yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—8441no—
171C4—8441no—
172C4—16881no—
173C4#2026999911yescharacteristic
174C4#2027000011yescharacteristic
175C4#2027000411yescharacteristic
176C4#2027001011yescharacteristic
177C2—16882no—
178C2#2721329411yescharacteristic
179C2#2721331011yescharacteristic
180C2#2721332511yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingQ8⋊C8×C211C2
socleC22×C211#120
prime radical#20269974C22
hypercenterQ8⋊C8D211
hypocenterC211#1640
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
3. #27213431C2semidirect
4. #27213357C2semidirect
6. #27212878C2semidirect
7. Q8⋊C8×C211C2semidirect
90. C211#1640semidirect
98. Q8⋊C8D211semidirect

Maximal subgroups

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

subgroupindex
#272138812
#272134932
#272134312
#272133572
#272128892
#272128782
Q8⋊C8×C2112
#1640211