Finite Group Databasebeta

← #1607×C211 #41009227

Subgroups — #1607×C211 #41009227

Order 27008.

280 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 68 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#534×C211C211yesmaximal, characteristic
2#480×C211C212yesmaximal, complemented
3#397×C211C211yesmaximal, characteristic, complemented
4#327×C211C212yesmaximal, complemented
5#323×C211C211yesmaximal, characteristic, complemented
6C22×D4×C211—21no—
7#123×C211—22no—
8#121×C211—21no—
9#121×C211C2212yes—
10#120×C211—22no—
11C2×C4⋊C4×C211C2211yescharacteristic, complemented
12C2×C22⋊C4×C211—21no—
13C2×C22⋊C4×C211C2211yescharacteristic
14#107×C211—22no—
15#102×C211—22no—
16C22⋊C8×C211C2212yescomplemented
17#95×C211—21no—
18#95×C211C2211yescharacteristic, complemented
19C24×C211—21no—
20C24×C211D411yescharacteristic
21C2×D4×C211—22no—
22C2×D4×C211—22no—
23C2×D4×C211—21no—
24C2×D4×C211—42no—
25C2×D4×C211—42no—
26C2×D4×C211D412yes—
27C22×C4×C211—21no—
28C22×C4×C211—21no—
29C22×C4×C211—21no—
30C22×C4×C211D411yescharacteristic
31C22×C4×C211C2311yescharacteristic
32C2×C8×C211—22no—
33C4⋊C4×C211—21no—
34C4⋊C4×C211—22no—
35C4⋊C4×C211D412yes—
36C22⋊C4×C211—22no—
37C22⋊C4×C211—22no—
38C22⋊C4×C211—22no—
39C22⋊C4×C211—42no—
40C23×C211—21no—
41C23×C211—21no—
42C23×C211—22no—
43C23×C211—41no—
44C23×C211—41no—
45C23×C211—42no—
46C23×C211—81no—
47C23×C211—81no—
48C23×C211D812yes—
49C23×C211C2×D411yescharacteristic
50D4×C211—42no—
51D4×C211—44no—
52D4×C211—42no—
53D4×C211—42no—
54D4×C211—81no—
55C2×C4×C211—21no—
56C2×C4×C211—22no—
57C2×C4×C211—41no—
58C2×C4×C211—42no—
59C2×C4×C211—41no—
60C2×C4×C211—41no—
61C2×C4×C211—41no—
62C2×C4×C211—41no—
63C2×C4×C211—41no—
64C2×C4×C211—41no—
65C2×C4×C211—41no—
66C2×C4×C211C2×D412yes—
67C8×C211—42no—
68C22×C211—21no—
69C22×C211—21no—
70C22×C211—22no—
71C22×C211—42no—
72C22×C211—41no—
73C22×C211—41no—
74C22×C211—42no—
75C22×C211—42no—
76C22×C211—42no—
77C22×C211—81no—
78C22×C211—84no—
79C22×C211#12011yescharacteristic
80C22×C211C2×D811yescharacteristic
81C22×C211AGL(1,ℤ/8)11yescharacteristic
82C4×C211—22no—
83C4×C211—42no—
84C4×C211—41no—
85C4×C211—81no—
86C4×C211—81no—
87C2×C211—21no—
88C2×C211—21no—
89C2×C211—42no—
90C2×C211—82no—
91C2×C211#44711yescharacteristic
92C2×C211#45511yescharacteristic
93C2×C211#45811yescharacteristic
94C211#160711yescharacteristic, direct product factor, Sylow
95#1607C21111yesmaximal, characteristic, direct product factor, Sylow
96#534C2×C21111yescharacteristic
97#480C2×C21112yescomplemented
98#397C2×C21111yescharacteristic, complemented
99#327C2×C21112yescomplemented
100#323C2×C21111yescharacteristic, complemented
101C22×D4—21no—
102#123—22no—
103#121—21no—
104#121C22×C21112yes—
105#120—22no—
106C2×C4⋊C4C22×C21111yescharacteristic, complemented
107C2×C22⋊C4—21no—
108C2×C22⋊C4C22×C21111yescharacteristic
109#107—22no—
110#102—22no—
111C22⋊C8C22×C21112yescomplemented
112#95—21no—
113#95C22×C21111yescharacteristic, complemented
114C24—21no—
115C24D4×C21111yescharacteristic
116C2×D4—22no—
117C2×D4—22no—
118C2×D4—21no—
119C2×D4—42no—
120C2×D4—42no—
121C2×D4D4×C21112yes—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4D4×C21111yescharacteristic
126C22×C4C23×C21111yescharacteristic
127C2×C8—22no—
128C4⋊C4—21no—
129C4⋊C4—22no—
130C4⋊C4D4×C21112yes—
131C22⋊C4—22no—
132C22⋊C4—22no—
133C22⋊C4—22no—
134C22⋊C4—42no—
135C23—21no—
136C23—21no—
137C23—22no—
138C23—41no—
139C23—41no—
140C23—42no—
141C23—81no—
142C23—81no—
143C23D8×C21112yes—
144C23C2×D4×C21111yescharacteristic
145D4—42no—
146D4—44no—
147D4—42no—
148D4—42no—
149D4—81no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4C2×D4×C21112yes—
162C8—42no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22—81no—
173C22—84no—
174C22#120×C21111yescharacteristic
175C22C2×D8×C21111yescharacteristic
176C22AGL(1,ℤ/8)×C21111yescharacteristic
177C4—22no—
178C4—42no—
179C4—41no—
180C4—81no—
181C4—81no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—82no—
186C2#447×C21111yescharacteristic
187C2#455×C21111yescharacteristic
188C2#458×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutator, Frattini, cosocleC22×C4C23×C211
prime radical#534×C211C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C211 ◁ C22×C4×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #480×C211C2semidirect
3. #397×C211C2semidirect
4. #327×C211C2semidirect
5. #323×C211C2semidirect
11. C2×C4⋊C4×C211C22semidirect
16. C22⋊C8×C211C22semidirect
18. #95×C211C22semidirect
94. C211#1607direct (N × H)
95. #1607C211direct (N × H)
97. #480C2×C211semidirect
98. #397C2×C211semidirect
99. #327C2×C211semidirect
100. #323C2×C211semidirect
106. C2×C4⋊C4C22×C211semidirect
111. C22⋊C8C22×C211semidirect
113. #95C22×C211semidirect

Maximal subgroups

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

subgroupindex
#534×C2112
#480×C2112
#397×C2112
#327×C2112
#323×C2112
#1607211