Finite Group Databasebeta

← #41011998

Subgroups — #41011998

Order 27008.

212 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 61 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#125×D211C211yesmaximal, characteristic
2#27213201C211yesmaximal, characteristic
3#27213098C211yesmaximal, characteristic
4#27212888C211yesmaximal, characteristic, complemented
5#27212887C211yesmaximal, characteristic, complemented
6#27212887C211yesmaximal, characteristic, complemented
7#330×C211C211yesmaximal, characteristic, complemented
8C2×#16809945—21no—
9#20269969C2211yescharacteristic
10#20269968C411yescharacteristic
11#20269968C411yescharacteristic
12#20269967—21no—
13#20269967—21no—
14C4⋊C4×D211—21no—
15C4⋊C4×D211—21no—
16C4⋊C4×D211C411yescharacteristic
17C4⋊C4×D211C411yescharacteristic
18C42×D211C2211yescharacteristic
19Dic211⋊C8—21no—
20(C2×D211)⋊C8—21no—
21#20269888C2211yescharacteristic
22#20269888C2211yescharacteristic
23C8×Dic211—21no—
24#125×C211C2211yescharacteristic
25#97×C211C2211yescharacteristic
26#97×C211C2211yescharacteristic
27C2×C4×D211—21no—
28C2×C4×D211—21no—
29C2×C4×D211C2×C411yescharacteristic
30C2×C4×D211D411yescharacteristic
31C2×C4×D211D411yescharacteristic
32#16809945—24no—
33C2×C211⋊C8—21no—
34C2×C211⋊C8—21no—
35C4⋊Dic211—21no—
36C4⋊Dic211—21no—
37C4⋊Dic211C2×C411yescharacteristic
38C4⋊Dic211C2×C411yescharacteristic
39Dic211⋊C4—21no—
40Dic211⋊C4—21no—
41Dic211⋊C4—21no—
42Dic211⋊C4—21no—
43Dic211⋊C4—21no—
44Dic211⋊C4—21no—
45C4×Dic211C2×C411yescharacteristic
46C4×Dic211D411yescharacteristic
47C4×Dic211D411yescharacteristic
48C2×C8×C211—21no—
49C2×C8×C211—21no—
50C4⋊C4×C211—21no—
51C4⋊C4×C211—21no—
52C4⋊C4×C211C2×C411yescharacteristic
53C4⋊C4×C211C2×C411yescharacteristic
54C42×C211C2311yescharacteristic
55C22×D211C22⋊C411yescharacteristic
56C2×Dic211—21no—
57C2×Dic211—21no—
58C2×Dic211C22⋊C411yescharacteristic
59C2×Dic211C22⋊C411yescharacteristic
60C2×Dic211C22⋊C411yescharacteristic
61C4×D211—22no—
62C4×D211—22no—
63C4×D211—41no—
64C4×D211—42no—
65C4×D211—42no—
66C211⋊C8—22no—
67C211⋊C8—41no—
68C2×C4×C211—21no—
69C2×C4×C211—21no—
70C2×C4×C211C22×C411yescharacteristic
71C2×C4×C211C2×D411yescharacteristic
72C2×C4×C211C2×D411yescharacteristic
73C8×C211—22no—
74C8×C211—41no—
75C22×C211C2×C22⋊C411yescharacteristic
76C2×D211—21no—
77C2×D211—21no—
78C2×D211#10412yes—
79C4×C211—21no—
80C4×C211—21no—
81C4×C211—21no—
82C4×C211—41no—
83C4×C211—41no—
84Dic211—21no—
85Dic211—21no—
86Dic211—41no—
87Dic211—41no—
88Dic211#10112yes—
89C2×C211C2×#10111yescharacteristic
90C2×C211C2×#10411yescharacteristic
91C2×C211#42111yescharacteristic
92D211—22no—
93C211#152811yescharacteristic, complemented, Sylow
94#1528—2111nomaximal, Sylow
95C2×#125—2111no—
96#432—2111no—
97C4×M16—2111no—
98#330—2111no—
99#330—2111no—
100#330—2111no—
101#330D21111yescharacteristic, complemented
102C2×M16—4221no—
103#125—4221no—
104#125—4221no—
105#125—2111no—
106#125—2111no—
107#125—2111no—
108#125C2×D21111yescharacteristic
109C2×C4⋊C4—4221no—
110C2×C4⋊C4—4221no—
111C2×C4⋊C4—2111no—
112C2×C4⋊C4—2111no—
113C2×C42—2111no—
114C4⋊C8—4221no—
115C22⋊C8—4221no—
116#97—2111no—
117#97—2111no—
118#97C2×D21111yescharacteristic
119#97C2×D21111yescharacteristic
120C4×C8—4221no—
121C22×C4—4221no—
122C22×C4—4221no—
123C22×C4—2111no—
124C22×C4—2111no—
125C22×C4—2111no—
126M16—4224no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—4221no—
130C2×C8—4221no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—4221no—
134C4⋊C4—4221no—
135C4⋊C4—4221no—
136C4⋊C4—4221no—
137C4⋊C4—4221no—
138C4⋊C4—4221no—
139C4⋊C4—4221no—
140C4⋊C4—4221no—
141C4⋊C4—2111no—
142C4⋊C4—2111no—
143C4⋊C4C4×D21111yescharacteristic
144C4⋊C4C4×D21111yescharacteristic
145C42—2111no—
146C42—2111no—
147C42—2111no—
148C42C22×D21111yescharacteristic
149C23—2111no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—8441no—
153C2×C4—8442no—
154C2×C4—8442no—
155C2×C4—4222no—
156C2×C4—4222no—
157C2×C4—4221no—
158C2×C4—4221no—
159C2×C4—2111no—
160C2×C4—2111no—
161C2×C4—2111no—
162C2×C4C2×C4×D21111yescharacteristic
163C2×C4D4×D21111yescharacteristic
164C2×C4D4×D21111yescharacteristic
165C8—22no—
166C8—41no—
167C8—8441no—
168C8—4222no—
169C22—4221no—
170C22—4221no—
171C22—2112no—
172C22C22⋊C4×D21111yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—8441no—
179C4—8441no—
180C4—4221no—
181C4—4221no—
182C4—2112no—
183C2—4222no—
184C2#101×D21111yescharacteristic
185C2#104×D21111yescharacteristic
186C2#2721316111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D211
commutatorC2×C4×C211C22×C4
FrattiniC42C22×D211
Fitting#330×C211C2
socleC22×C211C2×C22⋊C4
prime radicalC22×D211C22⋊C4
cosocleC42×C211C23
hypercenter#330D211
hypocenterC211#1528
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
4. #27212888C2semidirect
5. #27212887C2semidirect
6. #27212887C2semidirect
7. #330×C211C2semidirect
93. C211#1528semidirect
101. #330D211semidirect

Maximal subgroups

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

subgroupindex
#125×D2112
#272132012
#272130982
#272128882
#272128872
#330×C2112
#1528211