Finite Group Databasebeta

← #1532×C211 #41009152

Subgroups — #1532×C211 #41009152

Order 27008.

218 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 88 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#529×C211C211yesmaximal, characteristic
2#432×C211C211yesmaximal, characteristic, complemented
3C2×#97×C211C211yesmaximal, characteristic, complemented
4#330×C211C212yesmaximal, complemented
5#329×C211C212yesmaximal, complemented
6C22×C8×C211—21no—
7#126×C211—21no—
8#125×C211C2211yescharacteristic
9#124×C211C2211yescharacteristic
10#123×C211—21no—
11C22⋊Q8×C211C411yescharacteristic
12#121×C211C411yescharacteristic
13C4×Q8×C211C411yescharacteristic
14C4×D4×C211—21no—
15C4×D4×C211C411yescharacteristic
16#117×C211—21no—
17C2×C42×C211C2211yescharacteristic
18C4⋊C8×C211—21no—
19C22⋊C8×C211—21no—
20#97×C211—21no—
21#97×C211C2212yes—
22#97×C211C2212yes—
23C2×Q8×C211C2×C411yescharacteristic
24C2×D4×C211—21no—
25C2×D4×C211C2×C411yescharacteristic
26C22×C4×C211—21no—
27C22×C4×C211C2×C411yescharacteristic
28C22×C4×C211D411yescharacteristic
29C22×C4×C211D411yescharacteristic
30C2×C8×C211—24no—
31C2×C8×C211—22no—
32C2×C8×C211—22no—
33C4⋊C4×C211—21no—
34C4⋊C4×C211—21no—
35C4⋊C4×C211—21no—
36C4⋊C4×C211—21no—
37C4⋊C4×C211C2×C411yescharacteristic
38C4⋊C4×C211C2×C411yescharacteristic
39C22⋊C4×C211—21no—
40C22⋊C4×C211—21no—
41C22⋊C4×C211—21no—
42C22⋊C4×C211—21no—
43C22⋊C4×C211—21no—
44C42×C211—21no—
45C42×C211C2×C411yescharacteristic
46C42×C211D411yescharacteristic
47C42×C211D411yescharacteristic
48C42×C211C2311yescharacteristic
49C23×C211—21no—
50C23×C211C22⋊C411yescharacteristic
51Q8×C211—41no—
52D4×C211—41no—
53D4×C211—42no—
54C2×C4×C211—22no—
55C2×C4×C211—21no—
56C2×C4×C211—21no—
57C2×C4×C211—21no—
58C2×C4×C211—41no—
59C2×C4×C211—41no—
60C2×C4×C211—42no—
61C2×C4×C211C22⋊C411yescharacteristic
62C2×C4×C211C22⋊C411yescharacteristic
63C2×C4×C211C22⋊C411yescharacteristic
64C2×C4×C211C22×C411yescharacteristic
65C2×C4×C211C2×D411yescharacteristic
66C2×C4×C211C2×D411yescharacteristic
67C8×C211—24no—
68C8×C211—42no—
69C22×C211—21no—
70C22×C211—21no—
71C22×C211—41no—
72C22×C211—41no—
73C22×C211—41no—
74C22×C211#10412yes—
75C22×C211C2×C22⋊C411yescharacteristic
76C4×C211—21no—
77C4×C211—21no—
78C4×C211—21no—
79C4×C211—21no—
80C4×C211—21no—
81C4×C211—21no—
82C4×C211—41no—
83C4×C211—41no—
84C4×C211—41no—
85C2×C211—22no—
86C2×C211—81no—
87C2×C211#41311yescharacteristic
88C2×C211C2×#10411yescharacteristic
89C2×C211#42111yescharacteristic
90C211#153211yescharacteristic, direct product factor, Sylow
91#1532C21111yesmaximal, characteristic, direct product factor, Sylow
92#529C2×C21111yescharacteristic
93#432C2×C21111yescharacteristic, complemented
94C2×#97C2×C21111yescharacteristic, complemented
95#330C2×C21112yescomplemented
96#329C2×C21112yescomplemented
97C22×C8—21no—
98#126—21no—
99#125C22×C21111yescharacteristic
100#124C22×C21111yescharacteristic
101#123—21no—
102C22⋊Q8C4×C21111yescharacteristic
103#121C4×C21111yescharacteristic
104C4×Q8C4×C21111yescharacteristic
105C4×D4—21no—
106C4×D4C4×C21111yescharacteristic
107#117—21no—
108C2×C42C22×C21111yescharacteristic
109C4⋊C8—21no—
110C22⋊C8—21no—
111#97—21no—
112#97C22×C21112yes—
113#97C22×C21112yes—
114C2×Q8C2×C4×C21111yescharacteristic
115C2×D4—21no—
116C2×D4C2×C4×C21111yescharacteristic
117C22×C4—21no—
118C22×C4C2×C4×C21111yescharacteristic
119C22×C4D4×C21111yescharacteristic
120C22×C4D4×C21111yescharacteristic
121C2×C8—24no—
122C2×C8—22no—
123C2×C8—22no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4C2×C4×C21111yescharacteristic
129C4⋊C4C2×C4×C21111yescharacteristic
130C22⋊C4—21no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C42—21no—
136C42C2×C4×C21111yescharacteristic
137C42D4×C21111yescharacteristic
138C42D4×C21111yescharacteristic
139C42C23×C21111yescharacteristic
140C23—21no—
141C23C22⋊C4×C21111yescharacteristic
142Q8—41no—
143D4—41no—
144D4—42no—
145C2×C4—22no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4C22⋊C4×C21111yescharacteristic
153C2×C4C22⋊C4×C21111yescharacteristic
154C2×C4C22⋊C4×C21111yescharacteristic
155C2×C4C22×C4×C21111yescharacteristic
156C2×C4C2×D4×C21111yescharacteristic
157C2×C4C2×D4×C21111yescharacteristic
158C8—24no—
159C8—42no—
160C22—21no—
161C22—21no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#104×C21112yes—
166C22C2×C22⋊C4×C21111yescharacteristic
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—41no—
174C4—41no—
175C4—41no—
176C2—22no—
177C2—81no—
178C2#413×C21111yescharacteristic
179C2C2×#104×C21111yescharacteristic
180C2#421×C21111yescharacteristic
subgroupG/H
center, socleC22×C211C2×C22⋊C4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC42C23×C211
prime radical#121×C211C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C2×C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #432×C211C2semidirect
3. C2×#97×C211C2semidirect
4. #330×C211C2semidirect
5. #329×C211C2semidirect
90. C211#1532direct (N × H)
91. #1532C211direct (N × H)
93. #432C2×C211semidirect
94. C2×#97C2×C211semidirect
95. #330C2×C211semidirect
96. #329C2×C211semidirect

Maximal subgroups

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

subgroupindex
#529×C2112
#432×C2112
C2×#97×C2112
#330×C2112
#329×C2112
#1532211