Finite Group Databasebeta

← #41013166

Subgroups — #41013166

Order 27008.

336 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#20269936C211yesmaximal, characteristic
2C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic
3C2×#20269887C211yesmaximal, characteristic, complemented
4C2×C8×Dic211C211yesmaximal, characteristic, complemented
5#27212900C211yesmaximal, characteristic, complemented
6#27212900C211yesmaximal, characteristic, complemented
7C2×C22⋊C8×C211C211yesmaximal, characteristic
8C22×C211⋊C8C2211yescharacteristic
9C22×C211⋊C8C2211yescharacteristic
10#20269936—22no—
11#20269936—22no—
12C2×C4⋊Dic211C412yes—
13C2×C22⋊Dic211C412yes—
14C2×C4×Dic211C2211yescharacteristic
15C2×C4×Dic211C2211yescharacteristic
16C22⋊(C211⋊C8)C414yescomplemented
17#20269887—22no—
18C8×Dic211—22no—
19C23×C4×C211C2211yescharacteristic
20C22×C8×C211C2211yescharacteristic
21C22×C8×C211C2211yescharacteristic
22C22⋊C8×C211C414yescomplemented
23C22×Dic211C2×C412yes—
24C22×Dic211C2×C412yes—
25C2×C211⋊C8—21no—
26C2×C211⋊C8—21no—
27C2×C211⋊C8—21no—
28C2×C211⋊C8—21no—
29C2×C211⋊C8C2×C412yes—
30C2×C211⋊C8C2×C412yes—
31C4⋊Dic211—24no—
32C22⋊Dic211—24no—
33C4×Dic211—21no—
34C4×Dic211—21no—
35C4×Dic211D412yes—
36C4×Dic211D412yes—
37C24×C211C2×C411yescharacteristic
38C22×C4×C211—21no—
39C22×C4×C211—21no—
40C22×C4×C211—21no—
41C22×C4×C211—21no—
42C22×C4×C211C2×C411yescharacteristic
43C22×C4×C211C2×C412yes—
44C22×C4×C211C2×C412yes—
45C22×C4×C211C2311yescharacteristic
46C2×C8×C211—21no—
47C2×C8×C211—21no—
48C2×C8×C211—21no—
49C2×C8×C211—21no—
50C2×C8×C211C2×C412yes—
51C2×C8×C211C2×C412yes—
52C2×Dic211—22no—
53C2×Dic211—22no—
54C2×Dic211—22no—
55C2×Dic211—22no—
56C2×Dic211—22no—
57C2×Dic211C22⋊C414yes—
58C211⋊C8—22no—
59C211⋊C8—22no—
60C23×C211—21no—
61C23×C211—21no—
62C23×C211—22no—
63C23×C211—22no—
64C23×C211C4212yes—
65C23×C211C22×C411yescharacteristic
66C2×C4×C211—21no—
67C2×C4×C211—21no—
68C2×C4×C211—22no—
69C2×C4×C211—22no—
70C2×C4×C211—22no—
71C2×C4×C211—22no—
72C2×C4×C211C4212yes—
73C2×C4×C211C22×C411yescharacteristic
74C2×C4×C211C22×C411yescharacteristic
75C2×C4×C211C2×D411yescharacteristic
76C2×C4×C211C2×D411yescharacteristic
77C2×C4×C211#4111yescharacteristic
78C2×C4×C211#4111yescharacteristic
79C8×C211—22no—
80C8×C211—22no—
81C22×C211—22no—
82C22×C211—22no—
83C22×C211—22no—
84C22×C211—42no—
85C22×C211—42no—
86C22×C211C2×C4211yescharacteristic
87C22×C211C2×C22⋊C411yescharacteristic
88C22×C211#11711yescharacteristic
89C22×C211#13112yes—
90C22×C211#13112yes—
91C4×C211—22no—
92C4×C211C4×D411yescharacteristic
93C4×C211C4×D411yescharacteristic
94C4×C211C4×D411yescharacteristic
95C4×C211C4×D411yescharacteristic
96Dic211—24no—
97Dic211—42no—
98C2×C211—42no—
99C2×C211C4×C22⋊C411yescharacteristic
100C2×C211#40511yescharacteristic
101C2×C211#40511yescharacteristic
102C2×C211#40812yes—
103C2×C211#43312yes—
104C211#175011yescharacteristic, complemented, Sylow
105#1750—2111nomaximal, Sylow
106C2×#117—2111no—
107C2×C22⋊C8—2111no—
108C2×C22⋊C8D21111yescharacteristic
109C2×#97—2111no—
110C2×C4×C8—2111no—
111#336—2111no—
112#336—2111no—
113C23×C4C2×D21111yescharacteristic
114C22×C8—2111no—
115C22×C8—2111no—
116C22×C8C2×D21111yescharacteristic
117C22×C8C2×D21111yescharacteristic
118#117—4222no—
119#117—4222no—
120C2×C4⋊C4—2112no—
121C2×C22⋊C4—2112no—
122C2×C42—2111no—
123C2×C42—2111no—
124C22⋊C8—2114no—
125C22⋊C8Dic21114yescomplemented
126#97—4222no—
127C4×C8—4222no—
128C24C4×D21111yescharacteristic
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—2112no—
134C22×C4—2112no—
135C22×C4C4×D21111yescharacteristic
136C22×C4C4×D21112yes—
137C22×C4C2×Dic21112yes—
138C22×C4C22×D21111yescharacteristic
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—4221no—
144C2×C8—4221no—
145C2×C8—4221no—
146C2×C8—4221no—
147C2×C8—2112no—
148C2×C8—2112no—
149C2×C8C2×Dic21112yes—
150C2×C8C2×Dic21112yes—
151C4⋊C4—4224no—
152C22⋊C4—4224no—
153C42—4221no—
154C42—4221no—
155C42—2112no—
156C42—2112no—
157C23—21no—
158C23—21no—
159C23—22no—
160C23—22no—
161C23C4×Dic21112yes—
162C23C2×C4×D21111yescharacteristic
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—4222no—
170C2×C4—4222no—
171C2×C4—4222no—
172C2×C4—4222no—
173C2×C4—4222no—
174C2×C4—2114no—
175C2×C4C4×Dic21112yes—
176C2×C4C2×C4×D21111yescharacteristic
177C2×C4C22×Dic21111yescharacteristic
178C2×C4D4×D21111yescharacteristic
179C2×C4D4×D21111yescharacteristic
180C2×C4#1680996111yescharacteristic
181C2×C4#1680996111yescharacteristic
182C8—22no—
183C8—22no—
184C8—4222no—
185C8—4222no—
186C22—22no—
187C22—22no—
188C22—22no—
189C22—42no—
190C22—42no—
191C22C2×C4×Dic21111yescharacteristic
192C22C22⋊C4×D21111yescharacteristic
193C22#2026993311yescharacteristic
194C22#2026998312yes—
195C22#2026998512yes—
196C4—22no—
197C4—8442no—
198C4—4224no—
199C4(C22⋊D211)⋊C411yescharacteristic
200C4(C22⋊D211)⋊C411yescharacteristic
201C4D4×Dic21111yescharacteristic
202C4D4×Dic21111yescharacteristic
203C2—42no—
204C2C22⋊C4×Dic21111yescharacteristic
205C2#2721310111yescharacteristic
206C2#2721310411yescharacteristic
207C2#2721311112yes—
208C2#2721320412yes—
subgroupG/H
center, FrattiniC22×C4C22×D211
commutatorC22×C211C2×C42
FittingC2×C22⋊C8×C211C2
socleC23×C211C22×C4
prime radicalC24×C211C2×C4
cosocleC22×C4×C211C23
hypercenterC2×C22⋊C8D211
hypocenterC211#1750
LCSG ▷ C22×C211 ▷ C211
UCSC1 ◁ C22×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×#20269887C2semidirect
4. C2×C8×Dic211C2semidirect
5. #27212900C2semidirect
6. #27212900C2semidirect
16. C22⋊(C211⋊C8)C4semidirect
22. C22⋊C8×C211C4semidirect
104. C211#1750semidirect
125. C22⋊C8Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×#202699362
C2×C22⋊(C211⋊C8)2
C2×#202698872
C2×C8×Dic2112
#272129002
C2×C22⋊C8×C2112
#1750211