Finite Group Databasebeta

← #41012125

Subgroups — #41012125

Order 27008.

202 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 65 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#27213701C211yesmaximal, characteristic
2#27213202C211yesmaximal, characteristic
3#27213168C211yesmaximal, characteristic
4#27212895C211yesmaximal, characteristic, complemented
5#27212893C211yesmaximal, characteristic, complemented
6#27212891C211yesmaximal, characteristic, complemented
7#333×C211C211yesmaximal, characteristic, complemented
8C2×#16809945—21no—
9#20269977C2211yescharacteristic
10#20269968—21no—
11#20269957—21no—
12#20269953C412yes—
13Q8×Dic211C412yes—
14#20269942—21no—
15#20269935—21no—
16C42×D211C2211yescharacteristic
17C4⋊(C211⋊C8)C2211yescharacteristic
18C4⋊(C211⋊C8)C2211yescharacteristic
19Dic211⋊C8—21no—
20(C2×D211)⋊C8—21no—
21#20269887—21no—
22C4⋊Q8×C211C2211yescharacteristic
23C4⋊C8×C211C2211yescharacteristic
24C4⋊C8×C211C2211yescharacteristic
25C2×Dic422—21no—
26C2×C4×D211C2×C411yescharacteristic
27C2×C4×D211D411yescharacteristic
28C2×C4×D211D411yescharacteristic
29#16809945—42no—
30C2×C211⋊C8—21no—
31C2×C211⋊C8—21no—
32C4⋊Dic211—22no—
33C4⋊Dic211—21no—
34C4⋊Dic211C2×C412yes—
35Dic211⋊C4—22no—
36(C2×D211)⋊C4—22no—
37C4×Dic211—22no—
38C4×Dic211C2×C411yescharacteristic
39C4×Dic211D411yescharacteristic
40C4×Dic211D411yescharacteristic
41C2×Q8×C211C2×C412yes—
42C2×C8×C211—21no—
43C2×C8×C211—21no—
44C4⋊C4×C211—21no—
45C4⋊C4×C211—21no—
46C42×C211C2311yescharacteristic
47C22×D211C22⋊C411yescharacteristic
48Dic422—41no—
49Dic422—41no—
50C2×Dic211—22no—
51C2×Dic211C22⋊C411yescharacteristic
52C2×Dic211C22⋊C411yescharacteristic
53C2×Dic211C22⋊C411yescharacteristic
54C4×D211—21no—
55C4×D211—21no—
56C4×D211—21no—
57C4×D211—21no—
58C4×D211—41no—
59C211⋊C8—41no—
60C211⋊C8—41no—
61Q8×C211—42no—
62C2×C4×C211—22no—
63C2×C4×C211C22×C411yescharacteristic
64C2×C4×C211C2×D411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C8×C211—41no—
67C8×C211—41no—
68C22×C211C2×C22⋊C411yescharacteristic
69C2×D211—21no—
70C2×D211—21no—
71C2×D211—21no—
72C4×C211—21no—
73C4×C211—42no—
74C4×C211AGL(1,ℤ/8)11yescharacteristic
75C4×C211#13711yescharacteristic
76C4×C211#13711yescharacteristic
77C4×C211#13711yescharacteristic
78Dic211—21no—
79Dic211—21no—
80Dic211—21no—
81Dic211—42no—
82C2×C211#41311yescharacteristic
83C2×C211#41711yescharacteristic
84C2×C211#41911yescharacteristic
85D211—41no—
86C211#155911yescharacteristic, complemented, Sylow
87#1559—2111nomaximal, Sylow
88#533—2111no—
89#432—2111no—
90#423—2111no—
91#333—2111no—
92#333D21111yescharacteristic, complemented
93#332—2111no—
94#332—2111no—
95C2×M16—4221no—
96C4⋊Q8—2111no—
97C4⋊Q8C2×D21111yescharacteristic
98#125—4221no—
99C22⋊Q8—4221no—
100C22⋊Q8—2112no—
101C4×Q8—4221no—
102C4×Q8—2112no—
103#117—4221no—
104C2×C42—2111no—
105C4⋊C8—4221no—
106C4⋊C8—2111no—
107C4⋊C8—2111no—
108C4⋊C8C2×D21111yescharacteristic
109C4⋊C8C2×D21111yescharacteristic
110C22⋊C8—4221no—
111#97—4221no—
112C2×Q8—4221no—
113C2×Q8C4×D21112yes—
114C22×C4—2111no—
115C22×C4—2111no—
116C22×C4—2111no—
117M16—8442no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—4221no—
121C2×C8—4221no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—4222no—
125C4⋊C4—4222no—
126C4⋊C4—4221no—
127C4⋊C4—2112no—
128C22⋊C4—4222no—
129C42—4222no—
130C42—2111no—
131C42—2111no—
132C42—2111no—
133C42C22×D21111yescharacteristic
134C23—2111no—
135Q8—42no—
136Q8—8441no—
137Q8—8441no—
138C2×C4—22no—
139C2×C4—8441no—
140C2×C4—4222no—
141C2×C4—4221no—
142C2×C4—4221no—
143C2×C4—4221no—
144C2×C4—4221no—
145C2×C4—2111no—
146C2×C4—2111no—
147C2×C4—2111no—
148C2×C4C2×C4×D21111yescharacteristic
149C2×C4D4×D21111yescharacteristic
150C2×C4D4×D21111yescharacteristic
151C8—41no—
152C8—41no—
153C8—8441no—
154C8—8441no—
155C22—4221no—
156C22—4221no—
157C22—4221no—
158C22C22⋊C4×D21111yescharacteristic
159C4—21no—
160C4—42no—
161C4—8442no—
162C4—4221no—
163C4—4221no—
164C4—4221no—
165C4#2027000411yescharacteristic
166C4#2027000611yescharacteristic
167C4#2027000911yescharacteristic
168C4#2027000911yescharacteristic
169C2—8441no—
170C2#2721312711yescharacteristic
171C2#2721314611yescharacteristic
172C2#2721315311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#333×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC22×D211C22⋊C4
cosocleC42×C211C23
hypercenter#333D211
hypocenterC211#1559
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
4. #27212895C2semidirect
5. #27212893C2semidirect
6. #27212891C2semidirect
7. #333×C211C2semidirect
86. C211#1559semidirect
92. #333D211semidirect

Maximal subgroups

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

subgroupindex
#272137012
#272132022
#272131682
#272128952
#272128932
#272128912
#333×C2112
#1559211