Finite Group Databasebeta

← #2132×C211 #41009752

Subgroups — #2132×C211 #41009752

Order 27008.

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

normal subgroups only

This group has 250 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#532×C211C211yesmaximal, characteristic
2#413×C211C212yesmaximal, complemented
3#356×C211C212yesmaximal, complemented
4#355×C211C212yesmaximal, complemented
5C2×#41×C211C411yescharacteristic
6C2×#41×C211C2211yescharacteristic
7C2×M16×C211—22no—
8C4⋊Q8×C211C2211yescharacteristic
9#127×C211C411yescharacteristic
10#124×C211C411yescharacteristic
11#124×C211C2211yescharacteristic
12#121×C211—22no—
13C4×D4×C211—22no—
14C2×C42×C211C411yescharacteristic
15#101×C211C2214yes—
16#100×C211—22no—
17#41×C211—41no—
18#41×C211—41no—
19#41×C211—41no—
20#41×C211—41no—
21C2×Q8×C211C2×C411yescharacteristic
22C2×Q8×C211C2311yescharacteristic
23C2×D4×C211—22no—
24C2×D4×C211—21no—
25C2×D4×C211C2×C411yescharacteristic
26C2×D4×C211C2×C411yescharacteristic
27C2×D4×C211D412yes—
28C22×C4×C211—21no—
29C22×C4×C211—21no—
30C22×C4×C211C2×C411yescharacteristic
31C22×C4×C211D412yes—
32M16×C211—24no—
33M16×C211—42no—
34C2×C8×C211—42no—
35C4⋊C4×C211—22no—
36C22⋊C4×C211—22no—
37C22⋊C4×C211—22no—
38C42×C211—21no—
39C42×C211C2×C411yescharacteristic
40C42×C211C2×C411yescharacteristic
41C23×C211—21no—
42C23×C211C22⋊C412yes—
43C23×C211C22⋊C411yescharacteristic
44Q8×C211—41no—
45Q8×C211—41no—
46D4×C211—41no—
47D4×C211—42no—
48D4×C211—41no—
49D4×C211—44no—
50D4×C211—42no—
51C2×C4×C211—21no—
52C2×C4×C211—22no—
53C2×C4×C211—21no—
54C2×C4×C211—22no—
55C2×C4×C211—21no—
56C2×C4×C211—21no—
57C2×C4×C211—21no—
58C2×C4×C211—42no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211C22⋊C411yescharacteristic
62C2×C4×C211C22×C411yescharacteristic
63C2×C4×C211C2×D412yes—
64C8×C211—44no—
65C22×C211—21no—
66C22×C211—22no—
67C22×C211—41no—
68C22×C211—42no—
69C22×C211—42no—
70C22×C211—41no—
71C22×C211C2×C22⋊C411yescharacteristic
72C4×C211—22no—
73C4×C211—22no—
74C4×C211—21no—
75C4×C211—41no—
76C4×C211—41no—
77C4×C211#9911yescharacteristic
78C4×C211#9911yescharacteristic
79C2×C211—21no—
80C2×C211—41no—
81C2×C211—42no—
82C2×C211—81no—
83C2×C211C2×#9911yescharacteristic
84C211#213211yescharacteristic, direct product factor, Sylow
85#2132C21111yesmaximal, characteristic, direct product factor, Sylow
86#532C2×C21111yescharacteristic
87#413C2×C21112yescomplemented
88#356C2×C21112yescomplemented
89#355C2×C21112yescomplemented
90C2×#41C4×C21111yescharacteristic
91C2×#41C22×C21111yescharacteristic
92C2×M16—22no—
93C4⋊Q8C22×C21111yescharacteristic
94#127C4×C21111yescharacteristic
95#124C4×C21111yescharacteristic
96#124C22×C21111yescharacteristic
97#121—22no—
98C4×D4—22no—
99C2×C42C4×C21111yescharacteristic
100#101C22×C21114yes—
101#100—22no—
102#41—41no—
103#41—41no—
104#41—41no—
105#41—41no—
106C2×Q8C2×C4×C21111yescharacteristic
107C2×Q8C23×C21111yescharacteristic
108C2×D4—22no—
109C2×D4—21no—
110C2×D4C2×C4×C21111yescharacteristic
111C2×D4C2×C4×C21111yescharacteristic
112C2×D4D4×C21112yes—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4C2×C4×C21111yescharacteristic
116C22×C4D4×C21112yes—
117M16—24no—
118M16—42no—
119C2×C8—42no—
120C4⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C42—21no—
124C42C2×C4×C21111yescharacteristic
125C42C2×C4×C21111yescharacteristic
126C23—21no—
127C23C22⋊C4×C21112yes—
128C23C22⋊C4×C21111yescharacteristic
129Q8—41no—
130Q8—41no—
131D4—41no—
132D4—42no—
133D4—41no—
134D4—44no—
135D4—42no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—42no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4C22⋊C4×C21111yescharacteristic
147C2×C4C22×C4×C21111yescharacteristic
148C2×C4C2×D4×C21112yes—
149C8—44no—
150C22—21no—
151C22—22no—
152C22—41no—
153C22—42no—
154C22—42no—
155C22—41no—
156C22C2×C22⋊C4×C21111yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—41no—
162C4#99×C21111yescharacteristic
163C4#99×C21111yescharacteristic
164C2—21no—
165C2—41no—
166C2—42no—
167C2—81no—
168C2C2×#99×C21111yescharacteristic
subgroupG/H
centerC4×C211#99
commutatorC2×C4C22×C4×C211
Frattini, cosocleC2×Q8C23×C211
socleC2×C211C2×#99
prime radical#532×C211C2
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C211 ◁ C2×C4×C211 ◁ C2×#41×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #413×C211C2semidirect
3. #356×C211C2semidirect
4. #355×C211C2semidirect
84. C211#2132direct (N × H)
85. #2132C211direct (N × H)
87. #413C2×C211semidirect
88. #356C2×C211semidirect
89. #355C2×C211semidirect

Maximal subgroups

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

subgroupindex
#532×C2112
#413×C2112
#356×C2112
#355×C2112
#2132211