Finite Group Databasebeta

← #29749

Subgroups — #29749

Order 256.

452 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 113 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×#467C211yesmaximal, characteristic, complemented
2#3007C212yesmaximal
3C2×#408C211yesmaximal, characteristic
4#2078C212yesmaximal, complemented
5#2052C214yesmaximal, complemented
6#2020C212yesmaximal, complemented
7#1901C212yesmaximal, complemented
8#1810C211yesmaximal, characteristic, complemented
9C22×#41C2211yescharacteristic
10C2×#137C2214yes—
11C22×Q16C2211yescharacteristic, complemented
12C22×M16C2211yescharacteristic
13C23×C8C2211yescharacteristic
14C2×C22⋊Q8C2212yes—
15#467—22no—
16#467—22no—
17#452—24no—
18#451—22no—
19#450—22no—
20C2×#108C2212yescomplemented
21C2×#107C2211yescharacteristic, complemented
22#417C2214yes—
23C2×#103C2212yescomplemented
24C2×#101C2214yescomplemented
25#408—22no—
26#408C2212yes—
27#408C2212yes—
28#407C2212yes—
29C2×C22⋊C8—22no—
30#343C2214yescomplemented
31#341C2212yescomplemented
32C2×#41—21no—
33C2×#41—21no—
34C2×#41—42no—
35C2×#41—42no—
36C2×#41D412yes—
37C2×#41C2312yes—
38C22×Q8D411yescharacteristic
39C22×Q8C2312yes—
40C22×D4—21no—
41C22×D4D411yescharacteristic
42C23×C4—21no—
43C23×C4C2311yescharacteristic
44#137—44no—
45#137—44no—
46C2×Q16—24no—
47C2×Q16—22no—
48C2×Q16—42no—
49C2×SD16—24no—
50C2×M16—21no—
51C2×M16—42no—
52C2×M16C2312yes—
53C2×M16C2314yes—
54C22×C8—21no—
55C22×C8—42no—
56C22×C8D412yes—
57C22×C8D412yes—
58C22×C8C2311yescharacteristic
59C22×C8C2311yescharacteristic
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C2×C4⋊C4—22no—
65C2×C4⋊C4C2312yes—
66C2×C22⋊C4—22no—
67#108—42no—
68#107—22no—
69#103—24no—
70#103—24no—
71#102—24no—
72#101—44no—
73C22⋊C8—22no—
74C22⋊C8—22no—
75C22⋊C8—24no—
76C22⋊C8—24no—
77C24—21no—
78C24#4111yescharacteristic
79#41—24no—
80#41—24no—
81#41—44no—
82#41—44no—
83#41—42no—
84#41—42no—
85C2×Q8—21no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8—24no—
89C2×Q8—24no—
90C2×Q8—42no—
91C2×Q8C2×D412yes—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4—21no—
95C2×D4—42no—
96C2×D4—44no—
97C2×D4C2×D412yes—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4#4111yescharacteristic
112C22×C4C2411yescharacteristic
113Q16—44no—
114Q16—48no—
115SD16—48no—
116M16—24no—
117M16—24no—
118M16—24no—
119M16—42no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—24no—
124C2×C8—42no—
125C2×C8—42no—
126C2×C8—42no—
127C2×C8C2×D412yes—
128C2×C8C2×D412yes—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C23—22no—
136C23—22no—
137C23—22no—
138C23—41no—
139C23—42no—
140C23—42no—
141C23—42no—
142C23#12111yescharacteristic
143C23#12111yescharacteristic
144C23C2×#4111yescharacteristic
145Q8—24no—
146Q8—42no—
147Q8—44no—
148Q8—42no—
149Q8—42no—
150D4—24no—
151D4—42no—
152D4—44no—
153D4—44no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—44no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—44no—
169C2×C4—82no—
170C2×C4#12111yescharacteristic
171C2×C4#12111yescharacteristic
172C2×C4C22×D411yescharacteristic
173C2×C4C22×D411yescharacteristic
174C2×C4H(5,𝔽2)12yes—
175C2×C4H(5,𝔽2)12yes—
176C8—22no—
177C8—22no—
178C8—44no—
179C8—42no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—44no—
188C22—42no—
189C22—42no—
190C22—42no—
191C22—84no—
192C22#47312yes—
193C22C2×#12111yescharacteristic
194C22#53812yes—
195C4—21no—
196C4—21no—
197C4—42no—
198C4—82no—
199C4—82no—
200C4#53411yescharacteristic
201C4#53412yes—
202C4#53511yescharacteristic
203C2—22no—
204C2—22no—
205C2—41no—
206C2—82no—
207C2#261711yescharacteristic
208C2C2×#47311yescharacteristic
209C2#306711yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#467C2semidirect
4. #2078C2semidirect
5. #2052C2semidirect
6. #2020C2semidirect
7. #1901C2semidirect
8. #1810C2semidirect
11. C22×Q16C22semidirect
20. C2×#108C22semidirect
21. C2×#107C22semidirect
23. C2×#103C22semidirect
24. C2×#101C22semidirect
30. #343C22semidirect
31. #341C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
#30072
C2×#4082
#20782
#20522
#20202
#19012
#18102