Finite Group Databasebeta

← C2×#2186 #33514

Subgroups — C2×#2186 #33514

Order 256.

599 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 135 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×#135C211yesmaximal, characteristic, complemented
2C22×SD32C211yesmaximal, characteristic, complemented
3C22×D16C211yesmaximal, characteristic, complemented
4C2×#466C211yesmaximal, characteristic, complemented
5#2186C218yesmaximal, direct product factor
6C2×#358C211yesmaximal, characteristic, complemented
7C2×#357C211yesmaximal, characteristic, complemented
8C2×C22⋊C16C211yesmaximal, characteristic, complemented
9C22×#41—21no—
10C2×#135—24no—
11C2×#135—24no—
12C2×#135—22no—
13C2×#135C2214yescomplemented
14C22×Q16C2211yescharacteristic
15C22×SD16—21no—
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×#121—21no—
20C2×SD32—24no—
21C2×SD32C2214yescomplemented
22C2×D16—24no—
23C2×D16C2214yescomplemented
24C22×C16C2211yescharacteristic
25C22×C16C2211yescharacteristic
26#466—22no—
27#466C2214yescomplemented
28C2×#107C2211yescharacteristic
29C2×#102—21no—
30#358C2214yescomplemented
31#357C2214yescomplemented
32C22⋊C16C2214yescomplemented
33C2×#41—28no—
34C2×#41—24no—
35C2×#41—44no—
36C2×#41—42no—
37C22×Q8—21no—
38C22×D4—21no—
39C22×D4—21no—
40C22×D4—41no—
41C22×D4—41no—
42C23×C4—41no—
43C23×C4D411yescharacteristic
44#135—216no—
45#135—28no—
46#135—28no—
47C2×Q16—21no—
48C2×Q16—22no—
49C2×Q16D414yescomplemented
50C2×Q16C2312yes—
51C2×SD16—28no—
52C2×SD16—24no—
53C2×D8—21no—
54C2×D8—22no—
55C2×D8—22no—
56C2×D8—22no—
57C2×D8—21no—
58C2×D8D414yescomplemented
59C2×D8C2312yes—
60C2×D8C2312yes—
61C22×C8—22no—
62C22×C8—22no—
63C22×C8D411yescharacteristic
64C22×C8D412yes—
65C22×C8C2312yes—
66C22×C8C2311yescharacteristic
67#121—24no—
68#121—42no—
69C2×C4⋊C4—21no—
70C2×C22⋊C4—41no—
71SD32—28no—
72D16—28no—
73C2×C16—22no—
74C2×C16—22no—
75C2×C16C2312yes—
76C2×C16C2312yes—
77#107—21no—
78#107C2312yes—
79#102—24no—
80C24—41no—
81C24—41no—
82C24D811yescharacteristic
83#41—216no—
84#41—48no—
85#41—416no—
86C2×Q8—24no—
87C2×Q8—22no—
88C2×Q8—41no—
89C2×Q8—42no—
90C2×D4—24no—
91C2×D4—22no—
92C2×D4—22no—
93C2×D4—48no—
94C2×D4—44no—
95C2×D4—44no—
96C2×D4—42no—
97C2×D4—41no—
98C2×D4—41no—
99C2×D4—42no—
100C2×D4—42no—
101C2×D4—84no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—48no—
105C22×C4—41no—
106C22×C4—44no—
107C22×C4—41no—
108C22×C4D812yes—
109C22×C4D811yescharacteristic
110C22×C4C2×D412yes—
111C22×C4C2×D411yescharacteristic
112Q16—24no—
113Q16—22no—
114Q16C2×D414yescomplemented
115SD16—216no—
116D8—24no—
117D8—22no—
118D8—22no—
119D8—22no—
120D8—42no—
121D8C2×D414yescomplemented
122C2×C8—22no—
123C2×C8—24no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8C2×D412yes—
127C2×C8C2×D411yescharacteristic
128C2×C8C2×D412yes—
129C2×C8C2×D412yes—
130C2×C8C2411yescharacteristic
131C4⋊C4—22no—
132C4⋊C4—41no—
133C22⋊C4—44no—
134C16—22no—
135C16—22no—
136C23—22no—
137C23—22no—
138C23—22no—
139C23—42no—
140C23—44no—
141C23—42no—
142C23—82no—
143C23—82no—
144C23—82no—
145C23—82no—
146C23—82no—
147C23C2×D812yes—
148C23C2×D811yescharacteristic
149Q8—24no—
150Q8—42no—
151Q8—44no—
152D4—24no—
153D4—416no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—44no—
158D4—88no—
159D4—82no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—416no—
165C2×C4—42no—
166C2×C4—44no—
167C2×C4—42no—
168C2×C4—82no—
169C2×C4#12012yes—
170C2×C4C2×D812yes—
171C2×C4C2×D811yescharacteristic
172C2×C4AGL(1,ℤ/8)12yes—
173C2×C4C22×D411yescharacteristic
174C8—22no—
175C8#12012yes—
176C8C22×D411yescharacteristic
177C8C22×D411yescharacteristic
178C22—22no—
179C22—22no—
180C22—22no—
181C22—44no—
182C22—44no—
183C22—82no—
184C22—82no—
185C22—88no—
186C22—82no—
187C22—82no—
188C22—82no—
189C22—164no—
190C22#44712yes—
191C22#50812yes—
192C22#50912yes—
193C22C22×D811yescharacteristic
194C4—22no—
195C4—44no—
196C4—82no—
197C4#44712yes—
198C4C2×#12011yescharacteristic
199C4C2×AGL(1,ℤ/8)11yescharacteristic
200C2—42no—
201C2—84no—
202C2—162no—
203C2#218614yesdirect product factor
204C2C2×#44711yescharacteristic
205C2C2×#50811yescharacteristic
206C2C2×#50911yescharacteristic
subgroupG/H
center, socleC23C2×D8
commutator, Frattini, cosocleC2×C8C24
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#135C2semidirect
2. C22×SD32C2semidirect
3. C22×D16C2semidirect
4. C2×#466C2semidirect
5. #2186C2direct (N × H)
6. C2×#358C2semidirect
7. C2×#357C2semidirect
8. C2×C22⋊C16C2semidirect
13. C2×#135C22semidirect
21. C2×SD32C22semidirect
23. C2×D16C22semidirect
27. #466C22semidirect
30. #358C22semidirect
31. #357C22semidirect
32. C22⋊C16C22semidirect
49. C2×Q16D4semidirect
58. C2×D8D4semidirect
114. Q16C2×D4semidirect
121. D8C2×D4semidirect
203. C2#2186direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#1352
C22×SD322
C22×D162
C2×#4662
#21862
C2×#3582
C2×#3572
C2×C22⋊C162