Finite Group Databasebeta

← #41011966

Subgroups — #41011966

Order 27008.

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

normal subgroups only

This group has 340 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
1C2×C22⋊(C211⋊C8)C211yesmaximal, characteristic, complemented
2#27213006C211yesmaximal, characteristic, complemented
3#27212880C214yesmaximal, complemented
4C2×#121×C211C211yesmaximal, characteristic
5C22×C211⋊C8—21no—
6C2×C4⋊Dic211—21no—
7C2×C22⋊Dic211—21no—
8C22⋊(C211⋊C8)—21no—
9C22⋊(C211⋊C8)C2212yescomplemented
10C211⋊#95C2212yescomplemented
11C22×D4×C211—21no—
12C22×D4×C211C411yescharacteristic, complemented
13C23×C4×C211C2211yescharacteristic
14#121×C211—21no—
15#121×C211—21no—
16#121×C211C412yescomplemented
17#121×C211C2212yes—
18C2×C4⋊C4×C211C411yescharacteristic, complemented
19C2×C22⋊C4×C211—21no—
20C22×Dic211—22no—
21C2×C211⋊C8—22no—
22C2×C211⋊C8—42no—
23C4⋊Dic211—42no—
24C22⋊Dic211—42no—
25C24×C211—21no—
26C24×C211D411yescharacteristic
27C2×D4×C211—22no—
28C2×D4×C211—22no—
29C2×D4×C211—21no—
30C2×D4×C211—42no—
31C2×D4×C211—42no—
32C2×D4×C211—42no—
33C2×D4×C211C2×C412yes—
34C22×C4×C211—21no—
35C22×C4×C211—42no—
36C22×C4×C211C2×C411yescharacteristic
37C22×C4×C211C2×C411yescharacteristic
38C22×C4×C211D411yescharacteristic
39C22×C4×C211D411yescharacteristic
40C22×C4×C211D411yescharacteristic
41C22×C4×C211C2311yescharacteristic
42C4⋊C4×C211—21no—
43C4⋊C4×C211C2×C412yes—
44C22⋊C4×C211—22no—
45C22⋊C4×C211—22no—
46C2×Dic211—42no—
47C2×Dic211—42no—
48C2×Dic211—42no—
49C211⋊C8—42no—
50C23×C211—22no—
51C23×C211—41no—
52C23×C211—41no—
53C23×C211—42no—
54C23×C211—42no—
55C23×C211—42no—
56C23×C211C22⋊C411yescharacteristic
57C23×C211C22⋊C411yescharacteristic
58C23×C211D812yes—
59C23×C211SD1612yes—
60C23×C211C2×D411yescharacteristic
61D4×C211—42no—
62D4×C211—44no—
63D4×C211—42no—
64D4×C211—42no—
65D4×C211—42no—
66C2×C4×C211—22no—
67C2×C4×C211—22no—
68C2×C4×C211—22no—
69C2×C4×C211—22no—
70C2×C4×C211—41no—
71C2×C4×C211—41no—
72C2×C4×C211—41no—
73C2×C4×C211—42no—
74C2×C4×C211C22⋊C411yescharacteristic
75C2×C4×C211C22⋊C411yescharacteristic
76C2×C4×C211C22×C411yescharacteristic
77C2×C4×C211C2×D411yescharacteristic
78C22×C211—21no—
79C22×C211—21no—
80C22×C211—21no—
81C22×C211—21no—
82C22×C211—21no—
83C22×C211—21no—
84C22×C211—42no—
85C22×C211—42no—
86C22×C211—42no—
87C22×C211—42no—
88C22×C211—42no—
89C22×C211—84no—
90C22×C211#10212yes—
91C22×C211#10212yes—
92C22×C211C2×C22⋊C411yescharacteristic
93C22×C211C2×D811yescharacteristic
94C22×C211C2×SD1611yescharacteristic
95C4×C211—21no—
96C4×C211—21no—
97C4×C211—21no—
98C4×C211—42no—
99C4×C211#9912yes—
100Dic211—82no—
101C2×C211—21no—
102C2×C211—22no—
103C2×C211—22no—
104C2×C211—21no—
105C2×C211—82no—
106C2×C211C2×#9911yescharacteristic
107C2×C211C2×#10211yescharacteristic
108C2×C211#42111yescharacteristic
109C211#151911yescharacteristic, complemented, Sylow
110#1519—2111nomaximal, Sylow
111C2×#121D21111yescharacteristic
112C2×C22⋊C8—2111no—
113#380—2111no—
114#327—2114no—
115C22×D4—21no—
116C22×D4Dic21111yescharacteristic, complemented
117C23×C4C2×D21111yescharacteristic
118C22×C8—4221no—
119#121—21no—
120#121—21no—
121#121Dic21112yescomplemented
122#121C2×D21112yes—
123C2×C4⋊C4—4221no—
124C2×C4⋊C4Dic21111yescharacteristic, complemented
125C2×C22⋊C4—21no—
126C2×C22⋊C4—4221no—
127C22⋊C8—4221no—
128C22⋊C8—2112no—
129#95—2112no—
130C24—21no—
131C24C22⋊D21111yescharacteristic
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—21no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4C2×Dic21112yes—
139C22×C4—21no—
140C22×C4—42no—
141C22×C4—4222no—
142C22×C4C2×Dic21111yescharacteristic
143C22×C4C2×Dic21111yescharacteristic
144C22×C4C22⋊D21111yescharacteristic
145C22×C4C22⋊D21111yescharacteristic
146C22×C4C22⋊D21111yescharacteristic
147C22×C4C22×D21111yescharacteristic
148C2×C8—8442no—
149C2×C8—4222no—
150C4⋊C4—21no—
151C4⋊C4—8442no—
152C4⋊C4C2×Dic21112yes—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—8442no—
156C23—22no—
157C23—41no—
158C23—41no—
159C23—42no—
160C23—42no—
161C23—42no—
162C23C22⋊Dic21111yescharacteristic
163C23C22⋊Dic21111yescharacteristic
164C23#1680994712yes—
165C23#1680995012yes—
166C23C2×C22⋊D21111yescharacteristic
167D4—42no—
168D4—44no—
169D4—42no—
170D4—42no—
171D4—42no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—8442no—
181C2×C4—8442no—
182C2×C4—8442no—
183C2×C4C22⋊Dic21111yescharacteristic
184C2×C4C22⋊Dic21111yescharacteristic
185C2×C4C22×Dic21111yescharacteristic
186C2×C4C2×C22⋊D21111yescharacteristic
187C8—8442no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—84no—
200C22#2026989812yes—
201C22#2026989812yes—
202C22C2×C22⋊Dic21111yescharacteristic
203C22C2×#1680994711yescharacteristic
204C22C2×#1680995011yescharacteristic
205C4—21no—
206C4—21no—
207C4—21no—
208C4—42no—
209C4—16882no—
210C4#2026989212yes—
211C2—21no—
212C2—22no—
213C2—22no—
214C2—21no—
215C2—82no—
216C2C2×#2026989211yescharacteristic
217C2C2×#2026989811yescharacteristic
218C2#2721316311yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting, prime radicalC2×#121×C211C2
socleC22×C211C2×C22⋊C4
cosocleC22×C4×C211C23
hypercenterC2×#121D211
hypocenterC211#1519
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊(C211⋊C8)C2semidirect
2. #27213006C2semidirect
3. #27212880C2semidirect
9. C22⋊(C211⋊C8)C22semidirect
10. C211⋊#95C22semidirect
12. C22×D4×C211C4semidirect
16. #121×C211C4semidirect
18. C2×C4⋊C4×C211C4semidirect
109. C211#1519semidirect
116. C22×D4Dic211semidirect
121. #121Dic211semidirect
124. C2×C4⋊C4Dic211semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊(C211⋊C8)2
#272130062
#272128802
C2×#121×C2112
#1519211