Finite Group Databasebeta

← #2133×C211 #41009753

Subgroups — #2133×C211 #41009753

Order 27008.

230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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#536×C211C211yesmaximal, characteristic
2C2×#101×C211C212yesmaximal, complemented
3#356×C211C212yesmaximal, complemented
4#355×C211C212yesmaximal, complemented
5C2×#41×C211C411yescharacteristic
6C22×Q8×C211C2211yescharacteristic
7C2×M16×C211—22no—
8C4⋊Q8×C211C411yescharacteristic
9C4⋊Q8×C211C2211yescharacteristic
10#124×C211C411yescharacteristic
11#124×C211C2211yescharacteristic
12#123×C211—22no—
13C22⋊Q8×C211—22no—
14#117×C211C411yescharacteristic
15#101×C211—22no—
16#101×C211C2214yes—
17#41×C211—42no—
18C2×Q8×C211—42no—
19C2×Q8×C211C2×C411yescharacteristic
20C2×Q8×C211C2×C411yescharacteristic
21C2×Q8×C211D412yes—
22C2×Q8×C211C2311yescharacteristic
23C2×D4×C211—21no—
24C2×D4×C211C2×C411yescharacteristic
25C22×C4×C211—21no—
26C22×C4×C211C2×C411yescharacteristic
27C22×C4×C211D412yes—
28M16×C211—24no—
29M16×C211—42no—
30C2×C8×C211—42no—
31C4⋊C4×C211—21no—
32C4⋊C4×C211—22no—
33C4⋊C4×C211—22no—
34C22⋊C4×C211—21no—
35C22⋊C4×C211—22no—
36C22⋊C4×C211—22no—
37C42×C211C2×C411yescharacteristic
38C42×C211C2×C411yescharacteristic
39C23×C211—21no—
40C23×C211C22⋊C411yescharacteristic
41Q8×C211—41no—
42Q8×C211—42no—
43Q8×C211—41no—
44Q8×C211—41no—
45D4×C211—41no—
46D4×C211—42no—
47C2×C4×C211—22no—
48C2×C4×C211—22no—
49C2×C4×C211—22no—
50C2×C4×C211—21no—
51C2×C4×C211—21no—
52C2×C4×C211—21no—
53C2×C4×C211—42no—
54C2×C4×C211C22⋊C412yes—
55C2×C4×C211C22⋊C411yescharacteristic
56C2×C4×C211C22×C411yescharacteristic
57C2×C4×C211C2×D412yes—
58C8×C211—44no—
59C22×C211—41no—
60C22×C211—42no—
61C22×C211—41no—
62C22×C211#9912yes—
63C22×C211C2×C22⋊C411yescharacteristic
64C4×C211—22no—
65C4×C211—21no—
66C4×C211—22no—
67C4×C211—21no—
68C4×C211—41no—
69C4×C211—41no—
70C4×C211—41no—
71C2×C211—21no—
72C2×C211—22no—
73C2×C211—81no—
74C2×C211C2×#9911yescharacteristic
75C211#213311yescharacteristic, direct product factor, Sylow
76#2133C21111yesmaximal, characteristic, direct product factor, Sylow
77#536C2×C21111yescharacteristic
78C2×#101C2×C21112yescomplemented
79#356C2×C21112yescomplemented
80#355C2×C21112yescomplemented
81C2×#41C4×C21111yescharacteristic
82C22×Q8C22×C21111yescharacteristic
83C2×M16—22no—
84C4⋊Q8C4×C21111yescharacteristic
85C4⋊Q8C22×C21111yescharacteristic
86#124C4×C21111yescharacteristic
87#124C22×C21111yescharacteristic
88#123—22no—
89C22⋊Q8—22no—
90#117C4×C21111yescharacteristic
91#101—22no—
92#101C22×C21114yes—
93#41—42no—
94C2×Q8—42no—
95C2×Q8C2×C4×C21111yescharacteristic
96C2×Q8C2×C4×C21111yescharacteristic
97C2×Q8D4×C21112yes—
98C2×Q8C23×C21111yescharacteristic
99C2×D4—21no—
100C2×D4C2×C4×C21111yescharacteristic
101C22×C4—21no—
102C22×C4C2×C4×C21111yescharacteristic
103C22×C4D4×C21112yes—
104M16—24no—
105M16—42no—
106C2×C8—42no—
107C4⋊C4—21no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C22⋊C4—21no—
111C22⋊C4—22no—
112C22⋊C4—22no—
113C42C2×C4×C21111yescharacteristic
114C42C2×C4×C21111yescharacteristic
115C23—21no—
116C23C22⋊C4×C21111yescharacteristic
117Q8—41no—
118Q8—42no—
119Q8—41no—
120Q8—41no—
121D4—41no—
122D4—42no—
123C2×C4—22no—
124C2×C4—22no—
125C2×C4—22no—
126C2×C4—21no—
127C2×C4—21no—
128C2×C4—21no—
129C2×C4—42no—
130C2×C4C22⋊C4×C21112yes—
131C2×C4C22⋊C4×C21111yescharacteristic
132C2×C4C22×C4×C21111yescharacteristic
133C2×C4C2×D4×C21112yes—
134C8—44no—
135C22—41no—
136C22—42no—
137C22—41no—
138C22#99×C21112yes—
139C22C2×C22⋊C4×C21111yescharacteristic
140C4—22no—
141C4—21no—
142C4—22no—
143C4—21no—
144C4—41no—
145C4—41no—
146C4—41no—
147C2—21no—
148C2—22no—
149C2—81no—
150C2C2×#99×C21111yescharacteristic
subgroupG/H
center, socleC2×C211C2×#99
commutatorC2×C4C22×C4×C211
Frattini, cosocleC2×Q8C23×C211
prime radicalC2×#41×C211C4
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C211 ◁ C23×C211 ◁ C22×Q8×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#101×C211C2semidirect
3. #356×C211C2semidirect
4. #355×C211C2semidirect
75. C211#2133direct (N × H)
76. #2133C211direct (N × H)
78. C2×#101C2×C211semidirect
79. #356C2×C211semidirect
80. #355C2×C211semidirect

Maximal subgroups

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

subgroupindex
#536×C2112
C2×#101×C2112
#356×C2112
#355×C2112
#2133211