Finite Group Databasebeta

← #1527×C211 #41009147

Subgroups — #1527×C211 #41009147

Order 27008.

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

normal subgroups only

This group has 264 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×#124×C211C211yesmaximal, characteristic
2#432×C211C211yesmaximal, characteristic, complemented
3C4×M16×C211C211yesmaximal, characteristic, complemented
4#329×C211C214yesmaximal, complemented
5C22×Q8×C211C411yescharacteristic
6C22×D4×C211C411yescharacteristic
7C2×M16×C211—21no—
8#124×C211—21no—
9#124×C211—21no—
10#124×C211C412yes—
11#124×C211C2212yes—
12C2×C22⋊C4×C211—21no—
13C2×C22⋊C4×C211—21no—
14C2×C42×C211C2211yescharacteristic
15C4⋊C8×C211—21no—
16C22⋊C8×C211—21no—
17#97×C211C2212yescomplemented
18#97×C211C2212yescomplemented
19C4×C8×C211—21no—
20C24×C211—21no—
21C2×Q8×C211—21no—
22C2×Q8×C211—42no—
23C2×Q8×C211C2×C412yes—
24C2×D4×C211—21no—
25C2×D4×C211—42no—
26C2×D4×C211C2×C412yes—
27C22×C4×C211—21no—
28C22×C4×C211C2×C411yescharacteristic
29C22×C4×C211D411yescharacteristic
30C22×C4×C211D411yescharacteristic
31M16×C211—24no—
32C2×C8×C211—22no—
33C2×C8×C211—22no—
34C22⋊C4×C211—22no—
35C22⋊C4×C211—22no—
36C22⋊C4×C211—22no—
37C22⋊C4×C211—22no—
38C42×C211C2×C411yescharacteristic
39C42×C211D411yescharacteristic
40C42×C211D411yescharacteristic
41C42×C211C2311yescharacteristic
42C23×C211—22no—
43C23×C211—42no—
44C23×C211—42no—
45C23×C211—42no—
46C23×C211C22⋊C411yescharacteristic
47Q8×C211—42no—
48Q8×C211—42no—
49D4×C211—42no—
50D4×C211—42no—
51C2×C4×C211—22no—
52C2×C4×C211—22no—
53C2×C4×C211—22no—
54C2×C4×C211—41no—
55C2×C4×C211—42no—
56C2×C4×C211C22⋊C411yescharacteristic
57C2×C4×C211C22⋊C411yescharacteristic
58C2×C4×C211C22⋊C411yescharacteristic
59C2×C4×C211C22×C411yescharacteristic
60C2×C4×C211C2×D411yescharacteristic
61C2×C4×C211C2×D411yescharacteristic
62C8×C211—24no—
63C8×C211—42no—
64C22×C211—21no—
65C22×C211—21no—
66C22×C211—42no—
67C22×C211—42no—
68C22×C211—42no—
69C22×C211—84no—
70C22×C211#10412yes—
71C22×C211C2×C22⋊C411yescharacteristic
72C4×C211—21no—
73C4×C211—21no—
74C4×C211—21no—
75C4×C211—21no—
76C4×C211—21no—
77C4×C211—42no—
78C4×C211#10012yes—
79C2×C211—22no—
80C2×C211—82no—
81C2×C211C2×#10011yescharacteristic
82C2×C211C2×#10411yescharacteristic
83C2×C211#42111yescharacteristic
84C211#152711yescharacteristic, direct product factor, Sylow
85#1527C21111yesmaximal, characteristic, direct product factor, Sylow
86C2×#124C2×C21111yescharacteristic
87#432C2×C21111yescharacteristic, complemented
88C4×M16C2×C21111yescharacteristic, complemented
89#329C2×C21114yescomplemented
90C22×Q8C4×C21111yescharacteristic
91C22×D4C4×C21111yescharacteristic
92C2×M16—21no—
93#124—21no—
94#124—21no—
95#124C4×C21112yes—
96#124C22×C21112yes—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C42C22×C21111yescharacteristic
100C4⋊C8—21no—
101C22⋊C8—21no—
102#97C22×C21112yescomplemented
103#97C22×C21112yescomplemented
104C4×C8—21no—
105C24—21no—
106C2×Q8—21no—
107C2×Q8—42no—
108C2×Q8C2×C4×C21112yes—
109C2×D4—21no—
110C2×D4—42no—
111C2×D4C2×C4×C21112yes—
112C22×C4—21no—
113C22×C4C2×C4×C21111yescharacteristic
114C22×C4D4×C21111yescharacteristic
115C22×C4D4×C21111yescharacteristic
116M16—24no—
117C2×C8—22no—
118C2×C8—22no—
119C22⋊C4—22no—
120C22⋊C4—22no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C42C2×C4×C21111yescharacteristic
124C42D4×C21111yescharacteristic
125C42D4×C21111yescharacteristic
126C42C23×C21111yescharacteristic
127C23—22no—
128C23—42no—
129C23—42no—
130C23—42no—
131C23C22⋊C4×C21111yescharacteristic
132Q8—42no—
133Q8—42no—
134D4—42no—
135D4—42no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4C22⋊C4×C21111yescharacteristic
142C2×C4C22⋊C4×C21111yescharacteristic
143C2×C4C22⋊C4×C21111yescharacteristic
144C2×C4C22×C4×C21111yescharacteristic
145C2×C4C2×D4×C21111yescharacteristic
146C2×C4C2×D4×C21111yescharacteristic
147C8—24no—
148C8—42no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22—84no—
155C22#104×C21112yes—
156C22C2×C22⋊C4×C21111yescharacteristic
157C4—21no—
158C4—21no—
159C4—21no—
160C4—21no—
161C4—21no—
162C4—42no—
163C4#100×C21112yes—
164C2—22no—
165C2—82no—
166C2C2×#100×C21111yescharacteristic
167C2C2×#104×C21111yescharacteristic
168C2#421×C21111yescharacteristic
subgroupG/H
center, socleC22×C211C2×C22⋊C4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC42C23×C211
prime radicalC22×D4×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. C4×M16×C211C2semidirect
4. #329×C211C2semidirect
17. #97×C211C22semidirect
18. #97×C211C22semidirect
84. C211#1527direct (N × H)
85. #1527C211direct (N × H)
87. #432C2×C211semidirect
88. C4×M16C2×C211semidirect
89. #329C2×C211semidirect
102. #97C22×C211semidirect
103. #97C22×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#124×C2112
#432×C2112
C4×M16×C2112
#329×C2112
#1527211