Finite Group Databasebeta

← #41014342

Subgroups — #41014342

Order 27008.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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#27213703C211yesmaximal, characteristic
2C4×C4⋊Dic211C211yesmaximal, characteristic
3#27212907C214yesmaximal
4C4×M16×C211C211yesmaximal, characteristic
5#20269977C411yescharacteristic, complemented
6#20269977C411yescharacteristic, complemented
7#20269969C412yescomplemented
8#20269956—21no—
9#20269956—21no—
10C4×Dic422—21no—
11C4×Dic422—21no—
12#20269936C2212yes—
13C2×C4⋊Dic211—21no—
14C2×C4×Dic211C2212yes—
15C211⋊#95—21no—
16C2×M16×C211C2212yes—
17C2×C42×C211C2211yescharacteristic
18#97×C211—21no—
19C4×C8×C211—21no—
20C2×Dic422—21no—
21C2×Dic422—21no—
22C22×Dic211—22no—
23C4⋊Dic211—24no—
24C4⋊Dic211—21no—
25C4⋊Dic211—21no—
26C4⋊Dic211C2×C412yes—
27C4⋊Dic211C2×C412yes—
28Dic211⋊C4—22no—
29Dic211⋊C4—22no—
30C22⋊Dic211—22no—
31C4×Dic211—22no—
32C4×Dic211—24no—
33C22×C4×C211D411yescharacteristic
34C22×C4×C211D411yescharacteristic
35C22×C4×C211C2311yescharacteristic
36M16×C211—22no—
37M16×C211Q814yescomplemented
38C2×C8×C211—22no—
39C42×C211C2×C411yescharacteristic
40C42×C211C2×C411yescharacteristic
41C42×C211D411yescharacteristic
42C42×C211D411yescharacteristic
43Dic422—42no—
44Dic422—42no—
45C2×Dic211—24no—
46C2×Dic211—22no—
47C2×Dic211—22no—
48C2×Dic211—42no—
49C2×Dic211—42no—
50C23×C211C2×D411yescharacteristic
51C2×C4×C211—22no—
52C2×C4×C211—41no—
53C2×C4×C211C22⋊C411yescharacteristic
54C2×C4×C211C22⋊C411yescharacteristic
55C2×C4×C211C22⋊C411yescharacteristic
56C2×C4×C211C22⋊C411yescharacteristic
57C2×C4×C211C22×C411yescharacteristic
58C2×C4×C211C2×D411yescharacteristic
59C2×C4×C211C2×Q812yes—
60C2×C4×C211#4112yes—
61C8×C211—24no—
62C22×C211—21no—
63C22×C211—21no—
64C22×C211C2×C22⋊C411yescharacteristic
65C22×C211#12411yescharacteristic
66C22×C211C4⋊Q811yescharacteristic
67C4×C211—21no—
68C4×C211—21no—
69C4×C211—21no—
70C4×C211#10412yes—
71C4×C211C4×Q812yes—
72C4×C211C22⋊Q812yes—
73Dic211—44no—
74Dic211—42no—
75Dic211—42no—
76C2×C211—21no—
77C2×C211—21no—
78C2×C211#39111yescharacteristic
79C2×C211C2×#10411yescharacteristic
80C2×C211#42111yescharacteristic
81C211#198611yescharacteristic, complemented, Sylow
82#1986—2111nomaximal, Sylow
83#533—2111no—
84C4×M16D21111yescharacteristic
85C4×C4⋊C4—2111no—
86#339—2114no—
87C2×M16C2×D21112yes—
88C4⋊Q8—2111no—
89C4⋊Q8—2111no—
90#125—2112no—
91C22⋊Q8—4221no—
92C22⋊Q8—4221no—
93C4×Q8—4221no—
94C4×Q8—4221no—
95#117—2112no—
96C2×C4⋊C4—4221no—
97C2×C42—2112no—
98C2×C42C2×D21111yescharacteristic
99#97—21no—
100C4×C8—21no—
101#95—4221no—
102C2×Q8—4221no—
103C2×Q8—4221no—
104C22×C4—4222no—
105C22×C4D84411yescharacteristic
106C22×C4D84411yescharacteristic
107C22×C4C22×D21111yescharacteristic
108M16—22no—
109M16Dic42214yescomplemented
110C2×C8—22no—
111C4⋊C4—4224no—
112C4⋊C4—4222no—
113C4⋊C4—4222no—
114C4⋊C4—4221no—
115C4⋊C4—4221no—
116C4⋊C4—2112no—
117C4⋊C4—2112no—
118C22⋊C4—4222no—
119C42—4224no—
120C42—4222no—
121C42C4×D21111yescharacteristic
122C42C4×D21111yescharacteristic
123C42C22⋊D21111yescharacteristic
124C42C22⋊D21111yescharacteristic
125C23C2×D84411yescharacteristic
126Q8—8442no—
127Q8—8442no—
128C2×C4—22no—
129C2×C4—41no—
130C2×C4—8442no—
131C2×C4—8442no—
132C2×C4—4224no—
133C2×C4—4222no—
134C2×C4—4222no—
135C2×C4(C2×D211)⋊C411yescharacteristic
136C2×C4(C2×D211)⋊C411yescharacteristic
137C2×C4(C2×D211)⋊C411yescharacteristic
138C2×C4(C2×D211)⋊C411yescharacteristic
139C2×C4C2×C4×D21111yescharacteristic
140C2×C4C2×C22⋊D21111yescharacteristic
141C2×C4C2×Dic42212yes—
142C2×C4#1680996212yes—
143C8—24no—
144C22—21no—
145C22—21no—
146C22C2×(C2×D211)⋊C411yescharacteristic
147C22#2026996511yescharacteristic
148C22#2026997711yescharacteristic
149C4—21no—
150C4—21no—
151C4—21no—
152C4—8444no—
153C4—8442no—
154C4—8442no—
155C4#2026990312yes—
156C4C4×Dic42212yes—
157C4#2026995612yes—
158C2—21no—
159C2—21no—
160C2#2721305211yescharacteristic
161C2C2×#2026990311yescharacteristic
162C2#2721316211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
FittingC4×M16×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC23×C211C2×D4
cosocleC22×C4×C211C23
hypercenterC4×M16D211
hypocenterC211#1986
LCSG ▷ C2×C4×C211 ▷ C2×C211 ▷ C211
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
5. #20269977C4semidirect
6. #20269977C4semidirect
7. #20269969C4semidirect
37. M16×C211Q8semidirect
81. C211#1986semidirect
109. M16Dic422semidirect

Maximal subgroups

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

subgroupindex
#272137032
C4×C4⋊Dic2112
#272129072
C4×M16×C2112
#1986211