Finite Group Databasebeta

← #1510×C211 #41009130

Subgroups — #1510×C211 #41009130

Order 27008.

326 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 92 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×#121×C211C211yesmaximal, characteristic
2#407×C211C211yesmaximal, characteristic, complemented
3#381×C211C211yesmaximal, characteristic, complemented
4#327×C211C214yesmaximal, complemented
5C22×D4×C211—21no—
6C22×D4×C211C411yescharacteristic, complemented
7C23×C4×C211C2211yescharacteristic
8C2×M16×C211—21no—
9#121×C211—21no—
10#121×C211—21no—
11#121×C211C412yescomplemented
12#121×C211C2212yes—
13C2×C4⋊C4×C211C411yescharacteristic, complemented
14C2×C22⋊C4×C211—21no—
15C2×C22⋊C4×C211—21no—
16C22⋊C8×C211—21no—
17C22⋊C8×C211C2212yescomplemented
18#95×C211—21no—
19#95×C211C2212yescomplemented
20C24×C211—21no—
21C24×C211D411yescharacteristic
22C2×D4×C211—22no—
23C2×D4×C211—22no—
24C2×D4×C211—21no—
25C2×D4×C211—42no—
26C2×D4×C211—44no—
27C2×D4×C211C2×C412yes—
28C22×C4×C211—21no—
29C22×C4×C211—22no—
30C22×C4×C211—42no—
31C22×C4×C211C2×C411yescharacteristic
32C22×C4×C211C2×C411yescharacteristic
33C22×C4×C211D411yescharacteristic
34C22×C4×C211D411yescharacteristic
35C22×C4×C211D411yescharacteristic
36C22×C4×C211C2311yescharacteristic
37M16×C211—42no—
38C2×C8×C211—22no—
39C4⋊C4×C211—21no—
40C4⋊C4×C211C2×C412yes—
41C22⋊C4×C211—22no—
42C22⋊C4×C211—22no—
43C22⋊C4×C211—42no—
44C23×C211—22no—
45C23×C211—22no—
46C23×C211—22no—
47C23×C211—41no—
48C23×C211—42no—
49C23×C211—42no—
50C23×C211—42no—
51C23×C211C22⋊C411yescharacteristic
52C23×C211C22⋊C411yescharacteristic
53C23×C211C2×D411yescharacteristic
54D4×C211—42no—
55D4×C211—42no—
56D4×C211—44no—
57D4×C211—44no—
58C2×C4×C211—22no—
59C2×C4×C211—22no—
60C2×C4×C211—22no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211—42no—
66C2×C4×C211—42no—
67C2×C4×C211—42no—
68C2×C4×C211—42no—
69C2×C4×C211C22⋊C411yescharacteristic
70C2×C4×C211C22⋊C411yescharacteristic
71C2×C4×C211C22×C411yescharacteristic
72C2×C4×C211C2×D411yescharacteristic
73C8×C211—42no—
74C22×C211—21no—
75C22×C211—21no—
76C22×C211—21no—
77C22×C211—21no—
78C22×C211—21no—
79C22×C211—21no—
80C22×C211—21no—
81C22×C211—44no—
82C22×C211—42no—
83C22×C211—42no—
84C22×C211—42no—
85C22×C211—84no—
86C22×C211#9912yes—
87C22×C211C2×C22⋊C411yescharacteristic
88C22×C211AGL(1,ℤ/8)12yes—
89C4×C211—21no—
90C4×C211—21no—
91C4×C211—21no—
92C4×C211—21no—
93C4×C211—42no—
94C4×C211—82no—
95C2×C211—22no—
96C2×C211—22no—
97C2×C211—41no—
98C2×C211—82no—
99C2×C211C2×#9911yescharacteristic
100C2×C211#41811yescharacteristic
101C2×C211#42111yescharacteristic
102C211#151011yescharacteristic, direct product factor, Sylow
103#1510C21111yesmaximal, characteristic, direct product factor, Sylow
104C2×#121C2×C21111yescharacteristic
105#407C2×C21111yescharacteristic, complemented
106#381C2×C21111yescharacteristic, complemented
107#327C2×C21114yescomplemented
108C22×D4—21no—
109C22×D4C4×C21111yescharacteristic, complemented
110C23×C4C22×C21111yescharacteristic
111C2×M16—21no—
112#121—21no—
113#121—21no—
114#121C4×C21112yescomplemented
115#121C22×C21112yes—
116C2×C4⋊C4C4×C21111yescharacteristic, complemented
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C22⋊C8—21no—
120C22⋊C8C22×C21112yescomplemented
121#95—21no—
122#95C22×C21112yescomplemented
123C24—21no—
124C24D4×C21111yescharacteristic
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—21no—
128C2×D4—42no—
129C2×D4—44no—
130C2×D4C2×C4×C21112yes—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4C2×C4×C21111yescharacteristic
135C22×C4C2×C4×C21111yescharacteristic
136C22×C4D4×C21111yescharacteristic
137C22×C4D4×C21111yescharacteristic
138C22×C4D4×C21111yescharacteristic
139C22×C4C23×C21111yescharacteristic
140M16—42no—
141C2×C8—22no—
142C4⋊C4—21no—
143C4⋊C4C2×C4×C21112yes—
144C22⋊C4—22no—
145C22⋊C4—22no—
146C22⋊C4—42no—
147C23—22no—
148C23—22no—
149C23—22no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23C22⋊C4×C21111yescharacteristic
155C23C22⋊C4×C21111yescharacteristic
156C23C2×D4×C21111yescharacteristic
157D4—42no—
158D4—42no—
159D4—44no—
160D4—44no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4C22⋊C4×C21111yescharacteristic
173C2×C4C22⋊C4×C21111yescharacteristic
174C2×C4C22×C4×C21111yescharacteristic
175C2×C4C2×D4×C21111yescharacteristic
176C8—42no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—44no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—84no—
189C22#99×C21112yes—
190C22C2×C22⋊C4×C21111yescharacteristic
191C22AGL(1,ℤ/8)×C21112yes—
192C4—21no—
193C4—21no—
194C4—21no—
195C4—21no—
196C4—42no—
197C4—82no—
198C2—22no—
199C2—22no—
200C2—41no—
201C2—82no—
202C2C2×#99×C21111yescharacteristic
203C2#418×C21111yescharacteristic
204C2#421×C21111yescharacteristic
subgroupG/H
center, socleC22×C211C2×C22⋊C4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC2×#121×C211C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #407×C211C2semidirect
3. #381×C211C2semidirect
4. #327×C211C2semidirect
6. C22×D4×C211C4semidirect
11. #121×C211C4semidirect
13. C2×C4⋊C4×C211C4semidirect
17. C22⋊C8×C211C22semidirect
19. #95×C211C22semidirect
102. C211#1510direct (N × H)
103. #1510C211direct (N × H)
105. #407C2×C211semidirect
106. #381C2×C211semidirect
107. #327C2×C211semidirect
109. C22×D4C4×C211semidirect
114. #121C4×C211semidirect
116. C2×C4⋊C4C4×C211semidirect
120. C22⋊C8C22×C211semidirect
122. #95C22×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#121×C2112
#407×C2112
#381×C2112
#327×C2112
#1510211