Finite Group Databasebeta

← #2633×C211 #41010253

Subgroups — #2633×C211 #41010253

Order 27008.

512 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 192 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×#125×C211C211yesmaximal, characteristic, complemented
2C2×C4×D4×C211C211yesmaximal, characteristic
3#399×C211C212yesmaximal, complemented
4#398×C211C212yesmaximal, complemented
5#397×C211C212yesmaximal, complemented
6#394×C211C212yesmaximal
7#385×C211C214yesmaximal, complemented
8#383×C211C211yesmaximal, characteristic, complemented
9C22×D4×C211C2211yescharacteristic
10C23×C4×C211C2212yes—
11#125×C211—24no—
12C4×D4×C211—24no—
13C2×C4⋊C4×C211C2212yes—
14C2×C4⋊C4×C211C2212yes—
15C2×C4⋊C4×C211C2211yescharacteristic
16C2×C4⋊C4×C211C2211yescharacteristic
17C2×C4⋊C4×C211C2211yescharacteristic
18C2×C4⋊C4×C211C2214yes—
19C2×C22⋊C4×C211C2214yes—
20C2×C22⋊C4×C211C2212yes—
21C2×C22⋊C4×C211C2214yes—
22C2×C42×C211C2211yescharacteristic
23#95×C211C2212yes—
24#95×C211C2212yes—
25#95×C211C2212yes—
26#95×C211C2214yes—
27C24×C211C2312yes—
28C2×D4×C211—24no—
29C2×D4×C211—22no—
30C22×C4×C211—24no—
31C22×C4×C211—44no—
32C22×C4×C211C2314yes—
33C22×C4×C211C2312yes—
34C22×C4×C211C2312yes—
35C22×C4×C211C2312yes—
36C22×C4×C211C2311yescharacteristic
37C22×C4×C211C2311yescharacteristic
38C22×C4×C211C2311yescharacteristic
39C4⋊C4×C211—22no—
40C4⋊C4×C211—22no—
41C4⋊C4×C211—22no—
42C4⋊C4×C211—24no—
43C4⋊C4×C211—24no—
44C4⋊C4×C211—44no—
45C22⋊C4×C211—28no—
46C22⋊C4×C211—24no—
47C22⋊C4×C211—44no—
48C42×C211D414yes—
49C23×C211—22no—
50C23×C211—22no—
51C23×C211—24no—
52C23×C211—24no—
53C23×C211#4114yes—
54C23×C211C2411yescharacteristic
55D4×C211—28no—
56C2×C4×C211—22no—
57C2×C4×C211—22no—
58C2×C4×C211—22no—
59C2×C4×C211—22no—
60C2×C4×C211—22no—
61C2×C4×C211—22no—
62C2×C4×C211—24no—
63C2×C4×C211—24no—
64C2×C4×C211—22no—
65C2×C4×C211—24no—
66C2×C4×C211—22no—
67C2×C4×C211—24no—
68C2×C4×C211—24no—
69C2×C4×C211—48no—
70C2×C4×C211C2×D412yes—
71C2×C4×C211C2×D412yes—
72C2×C4×C211C2×D412yes—
73C22×C211—24no—
74C22×C211—28no—
75C22×C211—42no—
76C22×C211—42no—
77C22×C211—44no—
78C22×C211C22×D411yescharacteristic
79C22×C211C2×#4112yes—
80C22×C211H(5,𝔽2)12yes—
81C22×C2112−1+412yes—
82C4×C211—22no—
83C4×C211—22no—
84C4×C211—22no—
85C4×C211—42no—
86C4×C211—42no—
87C4×C211—42no—
88C4×C211—44no—
89C2×C211—44no—
90C2×C211#52511yescharacteristic
91C2×C211#53511yescharacteristic
92C2×C211#53611yescharacteristic
93C2×C211#53914yes—
94C211#263311yescharacteristic, direct product factor, Sylow
95#2633C21111yesmaximal, characteristic, direct product factor, Sylow
96C2×#125C2×C21111yescharacteristic, complemented
97C2×C4×D4C2×C21111yescharacteristic
98#399C2×C21112yescomplemented
99#398C2×C21112yescomplemented
100#397C2×C21112yescomplemented
101#394C2×C21112yes—
102#385C2×C21114yescomplemented
103#383C2×C21111yescharacteristic, complemented
104C22×D4C22×C21111yescharacteristic
105C23×C4C22×C21112yes—
106#125—24no—
107C4×D4—24no—
108C2×C4⋊C4C22×C21111yescharacteristic
109C2×C4⋊C4C22×C21112yes—
110C2×C4⋊C4C22×C21112yes—
111C2×C4⋊C4C22×C21111yescharacteristic
112C2×C4⋊C4C22×C21111yescharacteristic
113C2×C4⋊C4C22×C21114yes—
114C2×C22⋊C4C22×C21112yes—
115C2×C22⋊C4C22×C21114yes—
116C2×C22⋊C4C22×C21114yes—
117C2×C42C22×C21111yescharacteristic
118#95C22×C21112yes—
119#95C22×C21112yes—
120#95C22×C21112yes—
121#95C22×C21114yes—
122C24C23×C21112yes—
123C2×D4—24no—
124C2×D4—22no—
125C22×C4—24no—
126C22×C4—44no—
127C22×C4C23×C21114yes—
128C22×C4C23×C21112yes—
129C22×C4C23×C21112yes—
130C22×C4C23×C21112yes—
131C22×C4C23×C21111yescharacteristic
132C22×C4C23×C21111yescharacteristic
133C22×C4C23×C21111yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—44no—
140C22⋊C4—28no—
141C22⋊C4—24no—
142C22⋊C4—44no—
143C42D4×C21114yes—
144C23—22no—
145C23—22no—
146C23—24no—
147C23—24no—
148C23#41×C21114yes—
149C23C24×C21111yescharacteristic
150D4—28no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—48no—
165C2×C4C2×D4×C21112yes—
166C2×C4C2×D4×C21112yes—
167C2×C4C2×D4×C21112yes—
168C22—24no—
169C22—28no—
170C22—42no—
171C22—42no—
172C22—44no—
173C22C22×D4×C21111yescharacteristic
174C22C2×#41×C21112yes—
175C22H(5,𝔽2)×C21112yes—
176C222−1+4×C21112yes—
177C4—22no—
178C4—22no—
179C4—22no—
180C4—42no—
181C4—42no—
182C4—42no—
183C4—44no—
184C2—44no—
185C2#525×C21111yescharacteristic
186C2#535×C21111yescharacteristic
187C2#536×C21111yescharacteristic
188C2#539×C21114yes—
subgroupG/H
center, socleC23×C211C24
commutator, Frattini, cosocleC23C24×C211
prime radicalC22×D4×C211C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C211 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#125×C211C2semidirect
3. #399×C211C2semidirect
4. #398×C211C2semidirect
5. #397×C211C2semidirect
7. #385×C211C2semidirect
8. #383×C211C2semidirect
94. C211#2633direct (N × H)
95. #2633C211direct (N × H)
96. C2×#125C2×C211semidirect
98. #399C2×C211semidirect
99. #398C2×C211semidirect
100. #397C2×C211semidirect
102. #385C2×C211semidirect
103. #383C2×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#125×C2112
C2×C4×D4×C2112
#399×C2112
#398×C2112
#397×C2112
#394×C2112
#385×C2112
#383×C2112
#2633211