Finite Group Databasebeta

← #1595×C211 #41009215

Subgroups — #1595×C211 #41009215

Order 27008.

368 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 68 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#534×C211C211yesmaximal, characteristic, complemented
2#447×C211C212yesmaximal, complemented
3#392×C211C211yesmaximal, characteristic, complemented
4#327×C211C212yesmaximal, complemented
5#323×C211C211yesmaximal, characteristic, complemented
6C22×D4×C211—21no—
7C22×D4×C211—21no—
8C22×D4×C211C2211yescharacteristic, complemented
9C2×D8×C211—22no—
10#123×C211—22no—
11#121×C211—21no—
12#121×C211C2212yes—
13#120×C211—22no—
14C2×C22⋊C4×C211—21no—
15C2×C22⋊C4×C211C2211yescharacteristic
16#102×C211—22no—
17C22⋊C8×C211C2212yescomplemented
18#95×C211C2211yescharacteristic, complemented
19C24×C211—21no—
20C24×C211—21no—
21C24×C211D411yescharacteristic
22C2×D4×C211—22no—
23C2×D4×C211—22no—
24C2×D4×C211—21no—
25C2×D4×C211—22no—
26C2×D4×C211—42no—
27C2×D4×C211—42no—
28C2×D4×C211—42no—
29C2×D4×C211—42no—
30C2×D4×C211—42no—
31C2×D4×C211—42no—
32C2×D4×C211D412yescomplemented
33C22×C4×C211—21no—
34C22×C4×C211—21no—
35C22×C4×C211D411yescharacteristic
36C22×C4×C211C2311yescharacteristic
37D8×C211—44no—
38C2×C8×C211—22no—
39C4⋊C4×C211—21no—
40C4⋊C4×C211D412yescomplemented
41C22⋊C4×C211—22no—
42C22⋊C4×C211—22no—
43C22⋊C4×C211—22no—
44C22⋊C4×C211—42no—
45C23×C211—21no—
46C23×C211—21no—
47C23×C211—22no—
48C23×C211—22no—
49C23×C211—41no—
50C23×C211—41no—
51C23×C211—41no—
52C23×C211—41no—
53C23×C211—41no—
54C23×C211—42no—
55C23×C211—41no—
56C23×C211—41no—
57C23×C211—41no—
58C23×C211—81no—
59C23×C211—81no—
60C23×C211D812yescomplemented
61C23×C211C2×D411yescharacteristic
62D4×C211—24no—
63D4×C211—42no—
64D4×C211—42no—
65D4×C211—44no—
66D4×C211—48no—
67D4×C211—42no—
68D4×C211—42no—
69D4×C211—81no—
70C2×C4×C211—22no—
71C2×C4×C211—21no—
72C2×C4×C211—22no—
73C2×C4×C211—41no—
74C2×C4×C211—41no—
75C2×C4×C211—42no—
76C2×C4×C211—41no—
77C2×C4×C211—41no—
78C2×C4×C211C2×D412yes—
79C8×C211—42no—
80C22×C211—21no—
81C22×C211—21no—
82C22×C211—22no—
83C22×C211—42no—
84C22×C211—42no—
85C22×C211—42no—
86C22×C211—41no—
87C22×C211—41no—
88C22×C211—42no—
89C22×C211—42no—
90C22×C211—42no—
91C22×C211—42no—
92C22×C211—81no—
93C22×C211—81no—
94C22×C211—81no—
95C22×C211—84no—
96C22×C211—81no—
97C22×C211—81no—
98C22×C211#12011yescharacteristic
99C22×C211C2×D811yescharacteristic
100C22×C211AGL(1,ℤ/8)11yescharacteristic
101C4×C211—22no—
102C4×C211—42no—
103C4×C211—42no—
104C4×C211—41no—
105C2×C211—21no—
106C2×C211—21no—
107C2×C211—42no—
108C2×C211—82no—
109C2×C211—82no—
110C2×C211#44711yescharacteristic
111C2×C211#45311yescharacteristic
112C2×C211B(4,𝔽2)11yescharacteristic
113C211#159511yescharacteristic, direct product factor, Sylow
114#1595C21111yesmaximal, characteristic, direct product factor, Sylow
115#534C2×C21111yescharacteristic, complemented
116#447C2×C21112yescomplemented
117#392C2×C21111yescharacteristic, complemented
118#327C2×C21112yescomplemented
119#323C2×C21111yescharacteristic, complemented
120C22×D4—21no—
121C22×D4—21no—
122C22×D4C22×C21111yescharacteristic, complemented
123C2×D8—22no—
124#123—22no—
125#121—21no—
126#121C22×C21112yes—
127#120—22no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4C22×C21111yescharacteristic
130#102—22no—
131C22⋊C8C22×C21112yescomplemented
132#95C22×C21111yescharacteristic, complemented
133C24—21no—
134C24—21no—
135C24D4×C21111yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—22no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4D4×C21112yescomplemented
147C22×C4—21no—
148C22×C4—21no—
149C22×C4D4×C21111yescharacteristic
150C22×C4C23×C21111yescharacteristic
151D8—44no—
152C2×C8—22no—
153C4⋊C4—21no—
154C4⋊C4D4×C21112yescomplemented
155C22⋊C4—22no—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—42no—
159C23—21no—
160C23—21no—
161C23—22no—
162C23—22no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—42no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—81no—
173C23—81no—
174C23D8×C21112yescomplemented
175C23C2×D4×C21111yescharacteristic
176D4—24no—
177D4—42no—
178D4—42no—
179D4—44no—
180D4—48no—
181D4—42no—
182D4—42no—
183D4—81no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4C2×D4×C21112yes—
193C8—42no—
194C22—21no—
195C22—21no—
196C22—22no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—41no—
201C22—41no—
202C22—42no—
203C22—42no—
204C22—42no—
205C22—42no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22—84no—
210C22—81no—
211C22—81no—
212C22#120×C21111yescharacteristic
213C22C2×D8×C21111yescharacteristic
214C22AGL(1,ℤ/8)×C21111yescharacteristic
215C4—22no—
216C4—42no—
217C4—42no—
218C4—41no—
219C2—21no—
220C2—21no—
221C2—42no—
222C2—82no—
223C2—82no—
224C2#447×C21111yescharacteristic
225C2#453×C21111yescharacteristic
226C2B(4,𝔽2)×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC22×C4C23×C211
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. #534×C211C2semidirect
2. #447×C211C2semidirect
3. #392×C211C2semidirect
4. #327×C211C2semidirect
5. #323×C211C2semidirect
8. C22×D4×C211C22semidirect
17. C22⋊C8×C211C22semidirect
18. #95×C211C22semidirect
32. C2×D4×C211D4semidirect
40. C4⋊C4×C211D4semidirect
60. C23×C211D8semidirect
113. C211#1595direct (N × H)
114. #1595C211direct (N × H)
115. #534C2×C211semidirect
116. #447C2×C211semidirect
117. #392C2×C211semidirect
118. #327C2×C211semidirect
119. #323C2×C211semidirect
122. C22×D4C22×C211semidirect
131. C22⋊C8C22×C211semidirect
132. #95C22×C211semidirect
146. C2×D4D4×C211semidirect
154. C4⋊C4D4×C211semidirect
174. C23D8×C211semidirect

Maximal subgroups

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

subgroupindex
#534×C2112
#447×C2112
#392×C2112
#327×C2112
#323×C2112
#1595211