Finite Group Databasebeta

← #41012671

Subgroups — #41012671

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#27213866C211yesmaximal, characteristic
2#27213544C211yesmaximal, characteristic, complemented
3#27213426C211yesmaximal, characteristic
4#27213357C211yesmaximal, characteristic, complemented
5#27212891C211yesmaximal, characteristic, complemented
6#27212878C211yesmaximal, characteristic, complemented
7D4⋊C8×C211C211yesmaximal, characteristic
8C2×Dic844—21no—
9#20269977C2211yescharacteristic
10#20269977C2211yescharacteristic
11#20269970—22no—
12#20269964—21no—
13#20269956—21no—
14C4×Dic422C2211yescharacteristic
15#20269942—21no—
16#20269936—21no—
17#20269910—21no—
18#20269909—22no—
19C4⋊(C211⋊C8)C2211yescharacteristic
20#20269902—21no—
21#20269898—21no—
22C4×D4×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4×C8×C211C2211yescharacteristic
25C2×Dic422—21no—
26C2×Dic422—21no—
27C2×Dic422D411yescharacteristic
28Dic844—42no—
29C2×C211⋊C8—21no—
30C4⋊Dic211—21no—
31C4⋊Dic211—21no—
32C4⋊Dic211—21no—
33C4⋊Dic211D411yescharacteristic
34C4⋊Dic211D412yes—
35Dic211⋊C4—22no—
36Dic211⋊C4—21no—
37C22⋊Dic211—22no—
38C22⋊Dic211—22no—
39C4×Dic211—22no—
40C4×Dic211—21no—
41C2×D4×C211D411yescharacteristic
42C22×C4×C211—21no—
43C2×C8×C211—21no—
44C2×C8×C211—21no—
45C4⋊C4×C211D411yescharacteristic
46C22⋊C4×C211—21no—
47C42×C211C2311yescharacteristic
48Dic422—21no—
49Dic422—21no—
50Dic422—41no—
51Dic422—41no—
52Dic422—41no—
53Dic422—41no—
54C2×Dic211—21no—
55C2×Dic211—21no—
56C2×Dic211—22no—
57C2×Dic211—41no—
58C211⋊C8—41no—
59C23×C211—21no—
60D4×C211—21no—
61D4×C211—21no—
62C2×C4×C211—21no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C8×C211—22no—
69C8×C211—41no—
70C22×C211—41no—
71C22×C211—41no—
72C22×C211—41no—
73C22×C211#12011yescharacteristic
74C4×C211—21no—
75C4×C211—41no—
76C4×C211#13511yescharacteristic
77C4×C211#13511yescharacteristic
78C4×C211AGL(1,ℤ/8)11yescharacteristic
79C4×C211#13711yescharacteristic
80Dic211—41no—
81Dic211—42no—
82Dic211—41no—
83Dic211—81no—
84C2×C211—81no—
85C2×C211#44911yescharacteristic
86C2×C211#45211yescharacteristic
87C2×C211#45611yescharacteristic
88C211#165211yescharacteristic, complemented, Sylow
89#1652—2111nomaximal, Sylow
90#556—2111no—
91#498—2111no—
92#474—2111no—
93#462—2111no—
94#332—2111no—
95Q8⋊C8—2111no—
96D4⋊C8D21111yescharacteristic
97C2×Q16—4221no—
98C4⋊Q8—2111no—
99C4⋊Q8—2111no—
100#126—4222no—
101#124—4221no—
102C22⋊Q8—4221no—
103C4×Q8—4221no—
104C4×Q8—2111no—
105C4×D4C2×D21111yescharacteristic
106#117—4221no—
107#107—4221no—
108#106—4222no—
109C4⋊C8—2111no—
110C4⋊C8C2×D21111yescharacteristic
111#103—4221no—
112#102—4221no—
113C4×C8C2×D21111yescharacteristic
114C2×Q8—4221no—
115C2×Q8—4221no—
116C2×Q8—2111no—
117C2×D4D84411yescharacteristic
118C22×C4—21no—
119Q16—8442no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—4221no—
123C4⋊C4—4222no—
124C4⋊C4—4221no—
125C4⋊C4—4221no—
126C4⋊C4—4221no—
127C4⋊C4—4221no—
128C4⋊C4—2111no—
129C4⋊C4—2112no—
130C4⋊C4D84411yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—4222no—
133C22⋊C4—4222no—
134C42—4222no—
135C42—4221no—
136C42C22×D21111yescharacteristic
137C23—21no—
138Q8—8441no—
139Q8—8441no—
140Q8—8441no—
141Q8—8441no—
142Q8—4221no—
143Q8—4221no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—8441no—
150C2×C4—4221no—
151C2×C4—4221no—
152C2×C4—4222no—
153C2×C4D4×D21111yescharacteristic
154C2×C4D4×D21111yescharacteristic
155C2×C4C2×D84411yescharacteristic
156C8—22no—
157C8—41no—
158C8—8441no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#2026994711yescharacteristic
163C4—21no—
164C4—41no—
165C4—8441no—
166C4—8441no—
167C4—8442no—
168C4—16881no—
169C4#2026999811yescharacteristic
170C4#2026999911yescharacteristic
171C4#2027000111yescharacteristic
172C4#2027001011yescharacteristic
173C2—81no—
174C2#2721329211yescharacteristic
175C2#2721331011yescharacteristic
176C2#2721333011yescharacteristic
subgroupG/H
centerC22#20269947
commutator, cosocleC42×C211C23
FrattiniC42C22×D211
FittingD4⋊C8×C211C2
socleC22×C211#120
prime radicalC2×D4×C211D4
hypercenterD4⋊C8D211
hypocenterC211#1652
LCSG ▷ C42×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #27213544C2semidirect
4. #27213357C2semidirect
5. #27212891C2semidirect
6. #27212878C2semidirect
88. C211#1652semidirect

Maximal subgroups

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

subgroupindex
#272138662
#272135442
#272134262
#272133572
#272128912
#272128782
D4⋊C8×C2112
#1652211