Finite Group Databasebeta

← #1809×C211 #41009429

Subgroups — #1809×C211 #41009429

Order 27008.

316 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C22×M16×C211C211yesmaximal, characteristic
2C2×#107×C211C211yesmaximal, characteristic, complemented
3C2×#106×C211C211yesmaximal, characteristic, complemented
4#380×C211C212yesmaximal
5#340×C211C212yesmaximal, complemented
6C23×C4×C211C2211yescharacteristic
7C2×M16×C211—22no—
8C2×M16×C211—22no—
9C2×M16×C211C414yescomplemented
10C22×C8×C211C2211yescharacteristic
11C22×C8×C211C2211yescharacteristic
12C2×C4⋊C4×C211C2212yes—
13C2×C4⋊C4×C211C2212yes—
14C2×C22⋊C4×C211—22no—
15#107×C211—22no—
16#106×C211—22no—
17#95×C211—22no—
18C24×C211D411yescharacteristic
19C22×C4×C211—21no—
20C22×C4×C211—21no—
21C22×C4×C211—21no—
22C22×C4×C211—21no—
23C22×C4×C211—22no—
24C22×C4×C211—22no—
25C22×C4×C211C2×C412yes—
26C22×C4×C211D411yescharacteristic
27C22×C4×C211Q812yes—
28C22×C4×C211C2311yescharacteristic
29M16×C211—28no—
30C2×C8×C211—21no—
31C2×C8×C211—21no—
32C2×C8×C211C2×C412yes—
33C2×C8×C211C2×C412yes—
34C2×C8×C211D412yes—
35C2×C8×C211D412yes—
36C4⋊C4×C211—22no—
37C4⋊C4×C211—22no—
38C4⋊C4×C211—22no—
39C4⋊C4×C211—22no—
40C22⋊C4×C211—44no—
41C23×C211—21no—
42C23×C211—21no—
43C23×C211—22no—
44C23×C211—22no—
45C23×C211C4⋊C412yes—
46C23×C211C2×D411yescharacteristic
47C2×C4×C211—21no—
48C2×C4×C211—21no—
49C2×C4×C211—22no—
50C2×C4×C211—22no—
51C2×C4×C211—22no—
52C2×C4×C211—22no—
53C2×C4×C211—42no—
54C2×C4×C211—42no—
55C2×C4×C211—42no—
56C2×C4×C211—42no—
57C2×C4×C211—42no—
58C2×C4×C211—42no—
59C2×C4×C211C4⋊C412yes—
60C2×C4×C211C22×C411yescharacteristic
61C2×C4×C211C2×D411yescharacteristic
62C2×C4×C211C2×D411yescharacteristic
63C2×C4×C211C2×Q811yescharacteristic
64C2×C4×C211#4111yescharacteristic
65C2×C4×C211#4111yescharacteristic
66C8×C211—22no—
67C8×C211C22⋊C414yescomplemented
68C22×C211—22no—
69C22×C211—24no—
70C22×C211—42no—
71C22×C211—41no—
72C22×C211—41no—
73C22×C211C2×C4⋊C411yescharacteristic
74C22×C211#12111yescharacteristic
75C22×C211#12111yescharacteristic
76C22×C211AGL(1,ℤ/8)12yes—
77C22×C211#13712yes—
78C4×C211—22no—
79C4×C211—82no—
80C4×C211—82no—
81C4×C211C2×C22⋊C411yescharacteristic
82C4×C211#11711yescharacteristic
83C4×C211C22⋊Q811yescharacteristic
84C4×C211C22⋊Q811yescharacteristic
85C2×C211—42no—
86C2×C211#38011yescharacteristic
87C2×C211#42812yes—
88C2×C211#46812yes—
89C2×C211#46911yescharacteristic
90C2×C211#47011yescharacteristic
91C211#180911yescharacteristic, direct product factor, Sylow
92#1809C21111yesmaximal, characteristic, direct product factor, Sylow
93C22×M16C2×C21111yescharacteristic
94C2×#107C2×C21111yescharacteristic, complemented
95C2×#106C2×C21111yescharacteristic, complemented
96#380C2×C21112yes—
97#340C2×C21112yescomplemented
98C23×C4C22×C21111yescharacteristic
99C2×M16—22no—
100C2×M16—22no—
101C2×M16C4×C21114yescomplemented
102C22×C8C22×C21111yescharacteristic
103C22×C8C22×C21111yescharacteristic
104C2×C4⋊C4C22×C21112yes—
105C2×C4⋊C4C22×C21112yes—
106C2×C22⋊C4—22no—
107#107—22no—
108#106—22no—
109#95—22no—
110C24D4×C21111yescharacteristic
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4C2×C4×C21112yes—
118C22×C4D4×C21111yescharacteristic
119C22×C4Q8×C21112yes—
120C22×C4C23×C21111yescharacteristic
121M16—28no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8C2×C4×C21112yes—
125C2×C8C2×C4×C21112yes—
126C2×C8D4×C21112yes—
127C2×C8D4×C21112yes—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C22⋊C4—44no—
133C23—21no—
134C23—21no—
135C23—22no—
136C23—22no—
137C23C4⋊C4×C21112yes—
138C23C2×D4×C21111yescharacteristic
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—22no—
142C2×C4—22no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—42no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—42no—
151C2×C4C4⋊C4×C21112yes—
152C2×C4C22×C4×C21111yescharacteristic
153C2×C4C2×D4×C21111yescharacteristic
154C2×C4C2×D4×C21111yescharacteristic
155C2×C4C2×Q8×C21111yescharacteristic
156C2×C4#41×C21111yescharacteristic
157C2×C4#41×C21111yescharacteristic
158C8—22no—
159C8C22⋊C4×C21114yescomplemented
160C22—22no—
161C22—24no—
162C22—42no—
163C22—41no—
164C22—41no—
165C22C2×C4⋊C4×C21111yescharacteristic
166C22#121×C21111yescharacteristic
167C22#121×C21111yescharacteristic
168C22AGL(1,ℤ/8)×C21112yes—
169C22#137×C21112yes—
170C4—22no—
171C4—82no—
172C4—82no—
173C4C2×C22⋊C4×C21111yescharacteristic
174C4#117×C21111yescharacteristic
175C4C22⋊Q8×C21111yescharacteristic
176C4C22⋊Q8×C21111yescharacteristic
177C2—42no—
178C2#380×C21111yescharacteristic
179C2#428×C21112yes—
180C2#468×C21112yes—
181C2#469×C21111yescharacteristic
182C2#470×C21111yescharacteristic
subgroupG/H
center, socleC23×C211C2×D4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC22×C4C23×C211
prime radicalC24×C211D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C211 ◁ C23×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#107×C211C2semidirect
3. C2×#106×C211C2semidirect
5. #340×C211C2semidirect
9. C2×M16×C211C4semidirect
67. C8×C211C22⋊C4semidirect
91. C211#1809direct (N × H)
92. #1809C211direct (N × H)
94. C2×#107C2×C211semidirect
95. C2×#106C2×C211semidirect
97. #340C2×C211semidirect
101. C2×M16C4×C211semidirect
159. C8C22⋊C4×C211semidirect

Maximal subgroups

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

subgroupindex
C22×M16×C2112
C2×#107×C2112
C2×#106×C2112
#380×C2112
#340×C2112
#1809211