Finite Group Databasebeta

← #1472×C211 #41009092

Subgroups — #1472×C211 #41009092

Order 27008.

270 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 96 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×C22⋊Q8×C211C211yesmaximal, characteristic
2#407×C211C211yesmaximal, characteristic
3C2×C22⋊C8×C211C211yesmaximal, characteristic
4#324×C211C214yesmaximal, complemented
5C22×Q8×C211C411yescharacteristic
6C23×C4×C211C2211yescharacteristic
7C2×M16×C211—21no—
8C22×C8×C211—21no—
9C22⋊Q8×C211—21no—
10C22⋊Q8×C211—21no—
11C22⋊Q8×C211—21no—
12C22⋊Q8×C211—21no—
13C2×C4⋊C4×C211C411yescharacteristic
14C2×C4⋊C4×C211C2212yes—
15C2×C22⋊C4×C211C412yes—
16C22⋊C8×C211—21no—
17C22⋊C8×C211—21no—
18C22⋊C8×C211C2212yes—
19C22⋊C8×C211C2212yes—
20C24×C211C2×C411yescharacteristic
21C2×Q8×C211—21no—
22C2×Q8×C211—21no—
23C2×Q8×C211—42no—
24C22×C4×C211—42no—
25C22×C4×C211C2×C412yes—
26C22×C4×C211C2×C411yescharacteristic
27C22×C4×C211C2×C412yes—
28C22×C4×C211D411yescharacteristic
29C22×C4×C211D411yescharacteristic
30C22×C4×C211D411yescharacteristic
31C22×C4×C211D411yescharacteristic
32C22×C4×C211C2311yescharacteristic
33M16×C211—42no—
34C2×C8×C211—22no—
35C2×C8×C211—22no—
36C2×C8×C211—42no—
37C4⋊C4×C211—21no—
38C4⋊C4×C211—21no—
39C4⋊C4×C211—22no—
40C4⋊C4×C211—22no—
41C22⋊C4×C211—22no—
42C22⋊C4×C211—22no—
43C23×C211—22no—
44C23×C211—22no—
45C23×C211—41no—
46C23×C211C22⋊C411yescharacteristic
47C23×C211C22⋊C411yescharacteristic
48C23×C211C22×C411yescharacteristic
49Q8×C211—42no—
50Q8×C211—42no—
51C2×C4×C211—22no—
52C2×C4×C211—22no—
53C2×C4×C211—22no—
54C2×C4×C211—22no—
55C2×C4×C211—22no—
56C2×C4×C211—24no—
57C2×C4×C211—41no—
58C2×C4×C211—41no—
59C2×C4×C211—41no—
60C2×C4×C211—42no—
61C2×C4×C211C22⋊C411yescharacteristic
62C2×C4×C211C22⋊C411yescharacteristic
63C2×C4×C211C2×D411yescharacteristic
64C2×C4×C211C2×D411yescharacteristic
65C8×C211—42no—
66C8×C211—42no—
67C22×C211—21no—
68C22×C211—21no—
69C22×C211—21no—
70C22×C211—21no—
71C22×C211—21no—
72C22×C211—21no—
73C22×C211—21no—
74C22×C211—44no—
75C22×C211#10112yes—
76C22×C211C2×C22⋊C411yescharacteristic
77C22×C211#13112yes—
78C4×C211—21no—
79C4×C211—21no—
80C4×C211—21no—
81C4×C211—42no—
82C4×C211—42no—
83C4×C211#9912yes—
84C2×C211—22no—
85C2×C211—22no—
86C2×C211—41no—
87C2×C211#40811yescharacteristic
88C2×C211C2×#9911yescharacteristic
89C2×C211C2×#10111yescharacteristic
90C211#147211yescharacteristic, direct product factor, Sylow
91#1472C21111yesmaximal, characteristic, direct product factor, Sylow
92C2×C22⋊Q8C2×C21111yescharacteristic
93#407C2×C21111yescharacteristic
94C2×C22⋊C8C2×C21111yescharacteristic
95#324C2×C21114yescomplemented
96C22×Q8C4×C21111yescharacteristic
97C23×C4C22×C21111yescharacteristic
98C2×M16—21no—
99C22×C8—21no—
100C22⋊Q8—21no—
101C22⋊Q8—21no—
102C22⋊Q8—21no—
103C22⋊Q8—21no—
104C2×C4⋊C4C4×C21111yescharacteristic
105C2×C4⋊C4C22×C21112yes—
106C2×C22⋊C4C4×C21112yes—
107C22⋊C8—21no—
108C22⋊C8—21no—
109C22⋊C8C22×C21112yes—
110C22⋊C8C22×C21112yes—
111C24C2×C4×C21111yescharacteristic
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—42no—
115C22×C4—42no—
116C22×C4C2×C4×C21112yes—
117C22×C4C2×C4×C21111yescharacteristic
118C22×C4C2×C4×C21112yes—
119C22×C4D4×C21111yescharacteristic
120C22×C4D4×C21111yescharacteristic
121C22×C4D4×C21111yescharacteristic
122C22×C4D4×C21111yescharacteristic
123C22×C4C23×C21111yescharacteristic
124M16—42no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—42no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—22no—
134C23—22no—
135C23—22no—
136C23—41no—
137C23C22⋊C4×C21111yescharacteristic
138C23C22⋊C4×C21111yescharacteristic
139C23C22×C4×C21111yescharacteristic
140Q8—42no—
141Q8—42no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—24no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4C22⋊C4×C21111yescharacteristic
153C2×C4C22⋊C4×C21111yescharacteristic
154C2×C4C2×D4×C21111yescharacteristic
155C2×C4C2×D4×C21111yescharacteristic
156C8—42no—
157C8—42no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—44no—
166C22#101×C21112yes—
167C22C2×C22⋊C4×C21111yescharacteristic
168C22#131×C21112yes—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—42no—
173C4—42no—
174C4#99×C21112yes—
175C2—22no—
176C2—22no—
177C2—41no—
178C2#408×C21111yescharacteristic
179C2C2×#99×C21111yescharacteristic
180C2C2×#101×C21111yescharacteristic
subgroupG/H
center, socleC22×C211C2×C22⋊C4
commutatorC23C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC24×C211C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
4. #324×C211C2semidirect
90. C211#1472direct (N × H)
91. #1472C211direct (N × H)
95. #324C2×C211semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×C2112
#407×C2112
C2×C22⋊C8×C2112
#324×C2112
#1472211