Finite Group Databasebeta

← #1495×C211 #41009115

Subgroups — #1495×C211 #41009115

Order 27008.

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

normal subgroups only

This group has 216 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
1#529×C211C211yesmaximal, characteristic
2#431×C211C211yesmaximal, characteristic, complemented
3#423×C211C211yesmaximal, characteristic, complemented
4Q8⋊C8×C211C212yesmaximal, complemented
5D4⋊C8×C211C212yesmaximal, complemented
6C2×M16×C211—21no—
7#126×C211—21no—
8#125×C211C411yescharacteristic
9#124×C211C411yescharacteristic
10#123×C211—21no—
11C22⋊Q8×C211C411yescharacteristic
12#121×C211C411yescharacteristic
13C4×Q8×C211C2211yescharacteristic
14C4×D4×C211—21no—
15C4×D4×C211C2211yescharacteristic
16#117×C211—21no—
17C2×C42×C211C2211yescharacteristic
18C4⋊C8×C211—21no—
19C4⋊C8×C211—21no—
20C4⋊C8×C211C2212yes—
21C22⋊C8×C211—21no—
22C4×C8×C211C2212yes—
23C2×Q8×C211C2×C411yescharacteristic
24C2×D4×C211—21no—
25C2×D4×C211C2×C411yescharacteristic
26C22×C4×C211—21no—
27C22×C4×C211C2×C411yescharacteristic
28C22×C4×C211D411yescharacteristic
29C22×C4×C211D411yescharacteristic
30M16×C211—42no—
31C2×C8×C211—22no—
32C2×C8×C211—22no—
33C4⋊C4×C211—21no—
34C4⋊C4×C211—21no—
35C4⋊C4×C211—21no—
36C4⋊C4×C211—21no—
37C4⋊C4×C211C2×C411yescharacteristic
38C4⋊C4×C211C2×C411yescharacteristic
39C22⋊C4×C211—21no—
40C22⋊C4×C211—21no—
41C22⋊C4×C211—21no—
42C22⋊C4×C211—21no—
43C22⋊C4×C211—21no—
44C42×C211—21no—
45C42×C211C2×C411yescharacteristic
46C42×C211D411yescharacteristic
47C42×C211D411yescharacteristic
48C42×C211C2311yescharacteristic
49C23×C211—21no—
50C23×C211C22⋊C411yescharacteristic
51Q8×C211—22no—
52D4×C211—22no—
53D4×C211—42no—
54C2×C4×C211—22no—
55C2×C4×C211—21no—
56C2×C4×C211—21no—
57C2×C4×C211—21no—
58C2×C4×C211—21no—
59C2×C4×C211—21no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211C22⋊C411yescharacteristic
64C2×C4×C211C22⋊C411yescharacteristic
65C2×C4×C211C22⋊C411yescharacteristic
66C2×C4×C211C22×C411yescharacteristic
67C2×C4×C211C2×D411yescharacteristic
68C2×C4×C211C2×D411yescharacteristic
69C8×C211—42no—
70C8×C211—42no—
71C22×C211—21no—
72C22×C211—21no—
73C22×C211—21no—
74C22×C211—41no—
75C22×C211—41no—
76C22×C211—41no—
77C22×C211C2×C22⋊C411yescharacteristic
78C4×C211—21no—
79C4×C211—21no—
80C4×C211—21no—
81C4×C211—21no—
82C4×C211—41no—
83C4×C211—41no—
84C4×C211—41no—
85C4×C211#13112yes—
86C4×C211AGL(1,ℤ/8)11yescharacteristic
87C4×C211#13711yescharacteristic
88C2×C211—41no—
89C2×C211—81no—
90C2×C211#40811yescharacteristic
91C2×C211#41711yescharacteristic
92C2×C211#42111yescharacteristic
93C211#149511yescharacteristic, direct product factor, Sylow
94#1495C21111yesmaximal, characteristic, direct product factor, Sylow
95#529C2×C21111yescharacteristic
96#431C2×C21111yescharacteristic, complemented
97#423C2×C21111yescharacteristic, complemented
98Q8⋊C8C2×C21112yescomplemented
99D4⋊C8C2×C21112yescomplemented
100C2×M16—21no—
101#126—21no—
102#125C4×C21111yescharacteristic
103#124C4×C21111yescharacteristic
104#123—21no—
105C22⋊Q8C4×C21111yescharacteristic
106#121C4×C21111yescharacteristic
107C4×Q8C22×C21111yescharacteristic
108C4×D4—21no—
109C4×D4C22×C21111yescharacteristic
110#117—21no—
111C2×C42C22×C21111yescharacteristic
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8C22×C21112yes—
115C22⋊C8—21no—
116C4×C8C22×C21112yes—
117C2×Q8C2×C4×C21111yescharacteristic
118C2×D4—21no—
119C2×D4C2×C4×C21111yescharacteristic
120C22×C4—21no—
121C22×C4C2×C4×C21111yescharacteristic
122C22×C4D4×C21111yescharacteristic
123C22×C4D4×C21111yescharacteristic
124M16—42no—
125C2×C8—22no—
126C2×C8—22no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4C2×C4×C21111yescharacteristic
132C4⋊C4C2×C4×C21111yescharacteristic
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C22⋊C4—21no—
138C42—21no—
139C42C2×C4×C21111yescharacteristic
140C42D4×C21111yescharacteristic
141C42D4×C21111yescharacteristic
142C42C23×C21111yescharacteristic
143C23—21no—
144C23C22⋊C4×C21111yescharacteristic
145Q8—22no—
146D4—22no—
147D4—42no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4C22⋊C4×C21111yescharacteristic
158C2×C4C22⋊C4×C21111yescharacteristic
159C2×C4C22⋊C4×C21111yescharacteristic
160C2×C4C22×C4×C21111yescharacteristic
161C2×C4C2×D4×C21111yescharacteristic
162C2×C4C2×D4×C21111yescharacteristic
163C8—42no—
164C8—42no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22C2×C22⋊C4×C21111yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—41no—
179C4#131×C21112yes—
180C4AGL(1,ℤ/8)×C21111yescharacteristic
181C4#137×C21111yescharacteristic
182C2—41no—
183C2—81no—
184C2#408×C21111yescharacteristic
185C2#417×C21111yescharacteristic
186C2#421×C21111yescharacteristic
subgroupG/H
centerC2×C4×C211C2×D4
commutatorC2×C4C22×C4×C211
Frattini, cosocleC42C23×C211
socleC22×C211C2×C22⋊C4
prime radical#121×C211C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C211 ◁ C2×C42×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #431×C211C2semidirect
3. #423×C211C2semidirect
4. Q8⋊C8×C211C2semidirect
5. D4⋊C8×C211C2semidirect
93. C211#1495direct (N × H)
94. #1495C211direct (N × H)
96. #431C2×C211semidirect
97. #423C2×C211semidirect
98. Q8⋊C8C2×C211semidirect
99. D4⋊C8C2×C211semidirect

Maximal subgroups

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

subgroupindex
#529×C2112
#431×C2112
#423×C2112
Q8⋊C8×C2112
D4⋊C8×C2112
#1495211