Finite Group Databasebeta

← #1501×C211 #41009121

Subgroups — #1501×C211 #41009121

Order 27008.

304 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 96 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#525×C211C211yesmaximal, characteristic
2C2×C22⋊C8×C211C211yesmaximal, characteristic, complemented
3C4×C22⋊C4×C211C211yesmaximal, characteristic, complemented
4#328×C211C212yesmaximal, complemented
5#327×C211C212yesmaximal, complemented
6C2×#41×C211C411yescharacteristic, complemented
7C23×C4×C211C2211yescharacteristic
8C22×C8×C211—21no—
9#123×C211—21no—
10C22⋊Q8×C211C411yescharacteristic, complemented
11C22⋊Q8×C211C2211yescharacteristic
12#121×C211C411yescharacteristic, complemented
13#121×C211C2211yescharacteristic
14#120×C211—21no—
15C4×D4×C211—21no—
16C4×D4×C211—21no—
17#117×C211C411yescharacteristic, complemented
18C2×C22⋊C4×C211—21no—
19C2×C42×C211—21no—
20C22⋊C8×C211—21no—
21C22⋊C8×C211C2212yes—
22#95×C211C2212yes—
23C24×C211D411yescharacteristic
24#41×C211—42no—
25C2×Q8×C211C2×C411yescharacteristic
26C2×D4×C211—21no—
27C2×D4×C211—21no—
28C2×D4×C211—21no—
29C2×D4×C211C2×C411yescharacteristic
30C22×C4×C211—22no—
31C22×C4×C211—21no—
32C22×C4×C211—42no—
33C22×C4×C211C2×C411yescharacteristic
34C22×C4×C211C2×C411yescharacteristic
35C22×C4×C211D411yescharacteristic
36C22×C4×C211D411yescharacteristic
37C22×C4×C211D411yescharacteristic
38C22×C4×C211C2311yescharacteristic
39C2×C8×C211—22no—
40C2×C8×C211—42no—
41C4⋊C4×C211—21no—
42C4⋊C4×C211—21no—
43C4⋊C4×C211C2×C411yescharacteristic
44C4⋊C4×C211C2×C411yescharacteristic
45C22⋊C4×C211—24no—
46C22⋊C4×C211—21no—
47C22⋊C4×C211—21no—
48C22⋊C4×C211—21no—
49C22⋊C4×C211—21no—
50C22⋊C4×C211—21no—
51C42×C211—21no—
52C42×C211—42no—
53C23×C211—22no—
54C23×C211—22no—
55C23×C211—21no—
56C23×C211—41no—
57C23×C211C22⋊C411yescharacteristic
58C23×C211C22⋊C411yescharacteristic
59C23×C211C2×D411yescharacteristic
60Q8×C211—41no—
61D4×C211—41no—
62D4×C211—42no—
63D4×C211—42no—
64D4×C211—42no—
65C2×C4×C211—22no—
66C2×C4×C211—22no—
67C2×C4×C211—22no—
68C2×C4×C211—24no—
69C2×C4×C211—21no—
70C2×C4×C211—21no—
71C2×C4×C211—21no—
72C2×C4×C211—41no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211—42no—
76C2×C4×C211—42no—
77C2×C4×C211—42no—
78C2×C4×C211C22⋊C411yescharacteristic
79C2×C4×C211C22⋊C411yescharacteristic
80C2×C4×C211C22×C411yescharacteristic
81C2×C4×C211C2×D411yescharacteristic
82C8×C211—42no—
83C22×C211—21no—
84C22×C211—21no—
85C22×C211—21no—
86C22×C211—21no—
87C22×C211—21no—
88C22×C211—21no—
89C22×C211—21no—
90C22×C211—44no—
91C22×C211—41no—
92C22×C211—41no—
93C22×C211—41no—
94C22×C211#10412yes—
95C22×C211C2×C22⋊C411yescharacteristic
96C22×C211#13512yes—
97C4×C211—21no—
98C4×C211—21no—
99C4×C211—21no—
100C4×C211—44no—
101C4×C211—41no—
102C4×C211—41no—
103C4×C211—41no—
104C4×C211#9912yes—
105C2×C211—22no—
106C2×C211—22no—
107C2×C211—41no—
108C2×C211—81no—
109C2×C211C2×#9911yescharacteristic
110C2×C211#41611yescharacteristic
111C2×C211C2×#10411yescharacteristic
112C211#150111yescharacteristic, direct product factor, Sylow
113#1501C21111yesmaximal, characteristic, direct product factor, Sylow
114#525C2×C21111yescharacteristic
115C2×C22⋊C8C2×C21111yescharacteristic, complemented
116C4×C22⋊C4C2×C21111yescharacteristic, complemented
117#328C2×C21112yescomplemented
118#327C2×C21112yescomplemented
119C2×#41C4×C21111yescharacteristic, complemented
120C23×C4C22×C21111yescharacteristic
121C22×C8—21no—
122#123—21no—
123C22⋊Q8C4×C21111yescharacteristic, complemented
124C22⋊Q8C22×C21111yescharacteristic
125#121C4×C21111yescharacteristic, complemented
126#121C22×C21111yescharacteristic
127#120—21no—
128C4×D4—21no—
129C4×D4—21no—
130#117C4×C21111yescharacteristic, complemented
131C2×C22⋊C4—21no—
132C2×C42—21no—
133C22⋊C8—21no—
134C22⋊C8C22×C21112yes—
135#95C22×C21112yes—
136C24D4×C21111yescharacteristic
137#41—42no—
138C2×Q8C2×C4×C21111yescharacteristic
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4C2×C4×C21111yescharacteristic
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—42no—
146C22×C4C2×C4×C21111yescharacteristic
147C22×C4C2×C4×C21111yescharacteristic
148C22×C4D4×C21111yescharacteristic
149C22×C4D4×C21111yescharacteristic
150C22×C4D4×C21111yescharacteristic
151C22×C4C23×C21111yescharacteristic
152C2×C8—22no—
153C2×C8—42no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4C2×C4×C21111yescharacteristic
157C4⋊C4C2×C4×C21111yescharacteristic
158C22⋊C4—24no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C42—21no—
165C42—42no—
166C23—22no—
167C23—22no—
168C23—21no—
169C23—41no—
170C23C22⋊C4×C21111yescharacteristic
171C23C22⋊C4×C21111yescharacteristic
172C23C2×D4×C21111yescharacteristic
173Q8—41no—
174D4—41no—
175D4—42no—
176D4—42no—
177D4—42no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4C22⋊C4×C21111yescharacteristic
192C2×C4C22⋊C4×C21111yescharacteristic
193C2×C4C22×C4×C21111yescharacteristic
194C2×C4C2×D4×C21111yescharacteristic
195C8—42no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22#104×C21112yes—
208C22C2×C22⋊C4×C21111yescharacteristic
209C22#135×C21112yes—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—44no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4#99×C21112yes—
218C2—22no—
219C2—22no—
220C2—41no—
221C2—81no—
222C2C2×#99×C21111yescharacteristic
223C2#416×C21111yescharacteristic
224C2C2×#104×C21111yescharacteristic
subgroupG/H
centerC2×C4×C211C22⋊C4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
socleC22×C211C2×C22⋊C4
prime radical#525×C211C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C8×C211C2semidirect
3. C4×C22⋊C4×C211C2semidirect
4. #328×C211C2semidirect
5. #327×C211C2semidirect
6. C2×#41×C211C4semidirect
10. C22⋊Q8×C211C4semidirect
12. #121×C211C4semidirect
17. #117×C211C4semidirect
112. C211#1501direct (N × H)
113. #1501C211direct (N × H)
115. C2×C22⋊C8C2×C211semidirect
116. C4×C22⋊C4C2×C211semidirect
117. #328C2×C211semidirect
118. #327C2×C211semidirect
119. C2×#41C4×C211semidirect
123. C22⋊Q8C4×C211semidirect
125. #121C4×C211semidirect
130. #117C4×C211semidirect

Maximal subgroups

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

subgroupindex
#525×C2112
C2×C22⋊C8×C2112
C4×C22⋊C4×C2112
#328×C2112
#327×C2112
#1501211