Finite Group Databasebeta

← #1826×C211 #41009446

Subgroups — #1826×C211 #41009446

Order 27008.

306 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 108 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×#117×C211C211yesmaximal, characteristic
2#407×C211C211yesmaximal, characteristic
3C4×C22⋊C4×C211C211yesmaximal, characteristic
4#339×C211C214yesmaximal, complemented
5C23×C4×C211C2211yescharacteristic
6C2×M16×C211C2212yes—
7#117×C211—21no—
8#117×C211—21no—
9#117×C211C412yescomplemented
10#117×C211C2212yes—
11C2×C4⋊C4×C211C411yescharacteristic, complemented
12C2×C4⋊C4×C211C411yescharacteristic, complemented
13C2×C22⋊C4×C211—21no—
14C2×C22⋊C4×C211—21no—
15C2×C42×C211—21no—
16C2×C42×C211C2212yes—
17C22⋊C8×C211—21no—
18C22⋊C8×C211—21no—
19#95×C211—21no—
20C24×C211D411yescharacteristic
21C22×C4×C211—24no—
22C22×C4×C211—21no—
23C22×C4×C211—21no—
24C22×C4×C211—22no—
25C22×C4×C211C2×C411yescharacteristic
26C22×C4×C211C2×C411yescharacteristic
27C22×C4×C211D411yescharacteristic
28C22×C4×C211D411yescharacteristic
29C22×C4×C211D411yescharacteristic
30C22×C4×C211C2311yescharacteristic
31M16×C211—24no—
32C2×C8×C211—22no—
33C4⋊C4×C211—21no—
34C4⋊C4×C211—21no—
35C4⋊C4×C211C2×C412yes—
36C4⋊C4×C211C2×C412yes—
37C22⋊C4×C211—24no—
38C22⋊C4×C211—22no—
39C22⋊C4×C211—22no—
40C42×C211—22no—
41C42×C211—22no—
42C42×C211—24no—
43C23×C211—22no—
44C23×C211—22no—
45C23×C211—41no—
46C23×C211C22⋊C411yescharacteristic
47C23×C211C22⋊C411yescharacteristic
48C23×C211C2×D411yescharacteristic
49C2×C4×C211—22no—
50C2×C4×C211—22no—
51C2×C4×C211—22no—
52C2×C4×C211—22no—
53C2×C4×C211—22no—
54C2×C4×C211—24no—
55C2×C4×C211—22no—
56C2×C4×C211—22no—
57C2×C4×C211—42no—
58C2×C4×C211—42no—
59C2×C4×C211—42no—
60C2×C4×C211—42no—
61C2×C4×C211C22⋊C411yescharacteristic
62C2×C4×C211C22⋊C411yescharacteristic
63C2×C4×C211C22×C411yescharacteristic
64C2×C4×C211C2×D411yescharacteristic
65C2×C4×C211#4114yes—
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#10412yes—
76C22×C211C2×C22⋊C411yescharacteristic
77C22×C211#12312yes—
78C4×C211—21no—
79C4×C211—21no—
80C4×C211—44no—
81C4×C211—42no—
82C4×C211—42no—
83C4×C211#11712yes—
84C4×C211#12312yes—
85C2×C211—22no—
86C2×C211—22no—
87C2×C211—41no—
88C2×C211#38111yescharacteristic
89C2×C211C2×#10411yescharacteristic
90C2×C211#42111yescharacteristic
91C211#182611yescharacteristic, direct product factor, Sylow
92#1826C21111yesmaximal, characteristic, direct product factor, Sylow
93C2×#117C2×C21111yescharacteristic
94#407C2×C21111yescharacteristic
95C4×C22⋊C4C2×C21111yescharacteristic
96#339C2×C21114yescomplemented
97C23×C4C22×C21111yescharacteristic
98C2×M16C22×C21112yes—
99#117—21no—
100#117—21no—
101#117C4×C21112yescomplemented
102#117C22×C21112yes—
103C2×C4⋊C4C4×C21111yescharacteristic, complemented
104C2×C4⋊C4C4×C21111yescharacteristic, complemented
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C42—21no—
108C2×C42C22×C21112yes—
109C22⋊C8—21no—
110C22⋊C8—21no—
111#95—21no—
112C24D4×C21111yescharacteristic
113C22×C4—24no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4C2×C4×C21111yescharacteristic
118C22×C4C2×C4×C21111yescharacteristic
119C22×C4D4×C21111yescharacteristic
120C22×C4D4×C21111yescharacteristic
121C22×C4D4×C21111yescharacteristic
122C22×C4C23×C21111yescharacteristic
123M16—24no—
124C2×C8—22no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4C2×C4×C21112yes—
128C4⋊C4C2×C4×C21112yes—
129C22⋊C4—24no—
130C22⋊C4—22no—
131C22⋊C4—22no—
132C42—22no—
133C42—22no—
134C42—24no—
135C23—22no—
136C23—22no—
137C23—41no—
138C23C22⋊C4×C21111yescharacteristic
139C23C22⋊C4×C21111yescharacteristic
140C23C2×D4×C21111yescharacteristic
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—24no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4C22⋊C4×C21111yescharacteristic
154C2×C4C22⋊C4×C21111yescharacteristic
155C2×C4C22×C4×C21111yescharacteristic
156C2×C4C2×D4×C21111yescharacteristic
157C2×C4#41×C21114yes—
158C8—42no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—44no—
167C22#104×C21112yes—
168C22C2×C22⋊C4×C21111yescharacteristic
169C22#123×C21112yes—
170C4—21no—
171C4—21no—
172C4—44no—
173C4—42no—
174C4—42no—
175C4#117×C21112yes—
176C4#123×C21112yes—
177C2—22no—
178C2—22no—
179C2—41no—
180C2#381×C21111yescharacteristic
181C2C2×#104×C21111yescharacteristic
182C2#421×C21111yescharacteristic
subgroupG/H
centerC2×C4×C211C2×D4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
socleC22×C211C2×C22⋊C4
prime radicalC24×C211D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
4. #339×C211C2semidirect
9. #117×C211C4semidirect
11. C2×C4⋊C4×C211C4semidirect
12. C2×C4⋊C4×C211C4semidirect
91. C211#1826direct (N × H)
92. #1826C211direct (N × H)
96. #339C2×C211semidirect
101. #117C4×C211semidirect
103. C2×C4⋊C4C4×C211semidirect
104. C2×C4⋊C4C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#117×C2112
#407×C2112
C4×C22⋊C4×C2112
#339×C2112
#1826211