Finite Group Databasebeta

← #41011954

Subgroups — #41011954

Order 27008.

284 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 81 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#27213009C211yesmaximal, characteristic, complemented
3#27212883C212yesmaximal, complemented
4#27212880C212yesmaximal, complemented
5#525×C211C211yesmaximal, characteristic
6C22×C211⋊C8—21no—
7C2×C22⋊Dic211—21no—
8C22⋊(C211⋊C8)—21no—
9C22⋊(C211⋊C8)C2212yes—
10C211⋊#95—21no—
11C211⋊#95C2212yes—
12C2×#41×C211C411yescharacteristic, complemented
13C23×C4×C211C2211yescharacteristic
14#123×C211—21no—
15C22⋊Q8×C211C411yescharacteristic, complemented
16C22⋊Q8×C211C2211yescharacteristic
17#121×C211C411yescharacteristic, complemented
18#121×C211C2211yescharacteristic
19#120×C211—21no—
20C4×D4×C211—21no—
21C4×D4×C211—21no—
22#117×C211C411yescharacteristic, complemented
23C22×Dic211—22no—
24C2×C211⋊C8—22no—
25C2×C211⋊C8—42no—
26C22⋊Dic211—42no—
27C24×C211D411yescharacteristic
28#41×C211—42no—
29C2×Q8×C211C2×C411yescharacteristic
30C2×D4×C211—21no—
31C2×D4×C211—21no—
32C2×D4×C211—21no—
33C2×D4×C211C2×C411yescharacteristic
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×C211—21no—
44C4⋊C4×C211C2×C411yescharacteristic
45C4⋊C4×C211C2×C411yescharacteristic
46C22⋊C4×C211—21no—
47C22⋊C4×C211—21no—
48C22⋊C4×C211—21no—
49C22⋊C4×C211—21no—
50C22⋊C4×C211—21no—
51C42×C211—21no—
52C2×Dic211—42no—
53C2×Dic211—42no—
54C2×Dic211—42no—
55C211⋊C8—42no—
56C23×C211—22no—
57C23×C211—22no—
58C23×C211—21no—
59C23×C211—41no—
60C23×C211C22⋊C411yescharacteristic
61C23×C211C22⋊C411yescharacteristic
62C23×C211C2×D411yescharacteristic
63Q8×C211—41no—
64D4×C211—41no—
65D4×C211—42no—
66D4×C211—42no—
67D4×C211—42no—
68C2×C4×C211—22no—
69C2×C4×C211—22no—
70C2×C4×C211—21no—
71C2×C4×C211—21no—
72C2×C4×C211—21no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211—41no—
76C2×C4×C211—41no—
77C2×C4×C211—42no—
78C2×C4×C211C22⋊C411yescharacteristic
79C2×C4×C211C22⋊C411yescharacteristic
80C2×C4×C211C22×C411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C22×C211—21no—
83C22×C211—21no—
84C22×C211—21no—
85C22×C211—21no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211—21no—
89C22×C211—44no—
90C22×C211—41no—
91C22×C211—41no—
92C22×C211—41no—
93C22×C211#9912yes—
94C22×C211C2×C22⋊C411yescharacteristic
95C22×C211#13512yes—
96C4×C211—21no—
97C4×C211—21no—
98C4×C211—21no—
99C4×C211—21no—
100C4×C211—41no—
101C4×C211—41no—
102C4×C211—41no—
103Dic211—82no—
104C2×C211—22no—
105C2×C211—22no—
106C2×C211—41no—
107C2×C211—81no—
108C2×C211C2×#9911yescharacteristic
109C2×C211#41611yescharacteristic
110C2×C211#42111yescharacteristic
111C211#151611yescharacteristic, complemented, Sylow
112#1516—2111nomaximal, Sylow
113#525D21111yescharacteristic
114C2×C22⋊C8—2111no—
115#381—2111no—
116#328—2112no—
117#327—2112no—
118C2×#41Dic21111yescharacteristic, complemented
119C23×C4C2×D21111yescharacteristic
120C22×C8—4221no—
121#123—21no—
122C22⋊Q8Dic21111yescharacteristic, complemented
123C22⋊Q8C2×D21111yescharacteristic
124#121Dic21111yescharacteristic, complemented
125#121C2×D21111yescharacteristic
126#120—21no—
127C4×D4—21no—
128C4×D4—21no—
129#117Dic21111yescharacteristic, complemented
130C2×C22⋊C4—4221no—
131C22⋊C8—4221no—
132C22⋊C8—2112no—
133#95—4221no—
134#95—2112no—
135C24C22⋊D21111yescharacteristic
136#41—42no—
137C2×Q8C2×Dic21111yescharacteristic
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4C2×Dic21111yescharacteristic
142C22×C4—21no—
143C22×C4—42no—
144C22×C4—4222no—
145C22×C4C2×Dic21111yescharacteristic
146C22×C4C2×Dic21111yescharacteristic
147C22×C4C22⋊D21111yescharacteristic
148C22×C4C22⋊D21111yescharacteristic
149C22×C4C22⋊D21111yescharacteristic
150C22×C4C22×D21111yescharacteristic
151C2×C8—8442no—
152C2×C8—4222no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4C2×Dic21111yescharacteristic
156C4⋊C4C2×Dic21111yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—8442no—
163C42—21no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—41no—
168C23C22⋊Dic21111yescharacteristic
169C23C22⋊Dic21111yescharacteristic
170C23C2×C22⋊D21111yescharacteristic
171Q8—41no—
172D4—41no—
173D4—42no—
174D4—42no—
175D4—42no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—8442no—
187C2×C4—8442no—
188C2×C4—8442no—
189C2×C4C22⋊Dic21111yescharacteristic
190C2×C4C22⋊Dic21111yescharacteristic
191C2×C4C22×Dic21111yescharacteristic
192C2×C4C2×C22⋊D21111yescharacteristic
193C8—8442no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#2026989212yes—
206C22C2×C22⋊Dic21111yescharacteristic
207C22#2026999712yes—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—16882no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—81no—
220C2C2×#2026989211yescharacteristic
221C2#2721314211yescharacteristic
222C2#2721316311yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic211
commutatorC2×C4×C211C22×C4
FrattiniC22×C4C22×D211
Fitting, prime radical#525×C211C2
socleC22×C211C2×C22⋊C4
cosocleC22×C4×C211C23
hypercenter#525D211
hypocenterC211#1516
LCSG ▷ C2×C4×C211 ▷ C22×C211 ▷ C211
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊(C211⋊C8)C2semidirect
2. #27213009C2semidirect
3. #27212883C2semidirect
4. #27212880C2semidirect
12. C2×#41×C211C4semidirect
15. C22⋊Q8×C211C4semidirect
17. #121×C211C4semidirect
22. #117×C211C4semidirect
111. C211#1516semidirect
118. C2×#41Dic211semidirect
122. C22⋊Q8Dic211semidirect
124. #121Dic211semidirect
129. #117Dic211semidirect

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
#272130092
#272128832
#272128802
#525×C2112
#1516211