Finite Group Databasebeta

← #18684

Subgroups — #18684

Order 256.

503 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 115 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#3544C211yesmaximal, characteristic, complemented
2#3217C212yesmaximal, complemented
3C2×#492C211yesmaximal, characteristic, complemented
4#1624C212yesmaximal, complemented
5#1623C212yesmaximal, complemented
6#1622C214yesmaximal, complemented
7#1558C211yesmaximal, characteristic, complemented
8#1494C212yesmaximal, complemented
9C22×SD16—22no—
10#563—22no—
11Q82C2211yescharacteristic, complemented
12#555C2211yescharacteristic, complemented
13#554C2212yescomplemented
14#552—22no—
15#547—22no—
16#544—22no—
17#542—22no—
18#536—22no—
19C2×C4⋊Q8C2211yescharacteristic
20C2×#127C2211yescharacteristic
21#529C2212yes—
22#496—22no—
23#492—22no—
24#492C2212yes—
25#492C2212yes—
26#490—22no—
27#460C2214yes—
28#459C2214yes—
29#450—22no—
30#447—22no—
31#432C2212yes—
32C2×C4×C8C2211yescharacteristic
33#333C2212yescomplemented
34#333C2212yescomplemented
35Q8⋊C8C2214yescomplemented
36D4⋊C8C2214yescomplemented
37C2×#41—22no—
38C22×Q8D412yes—
39C22×D4—21no—
40C22×D4—22no—
41C2×SD16—24no—
42C2×SD16—24no—
43C2×SD16—24no—
44C2×SD16—48no—
45C2×D8—24no—
46C22×C8—21no—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C4⋊Q8D412yes—
52C4⋊Q8C2311yescharacteristic
53C4⋊Q8C2311yescharacteristic
54#127—22no—
55#127—21no—
56#127C2312yes—
57#126—24no—
58#126—22no—
59#125—24no—
60#125—22no—
61#125D412yes—
62#124—22no—
63#124—22no—
64#124D412yes—
65#123—24no—
66#123—24no—
67#123—22no—
68C22⋊Q8—24no—
69C22⋊Q8—24no—
70C22⋊Q8—24no—
71C22⋊Q8—22no—
72C22⋊Q8D412yes—
73#121—22no—
74#121D412yes—
75C4×Q8—22no—
76C4×Q8C2312yes—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4C2312yes—
80#117—22no—
81#117—22no—
82#117—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C42C2311yescharacteristic
86C4⋊C8C2314yes—
87#102—24no—
88#102—24no—
89C22⋊C8—22no—
90#97—22no—
91C4×C8—21no—
92C4×C8C2312yes—
93C24—41no—
94#41—84no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—44no—
98C2×Q8C2×D412yes—
99C2×Q8C2×D412yes—
100C2×Q8C2×D412yes—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—24no—
104C2×D4—22no—
105C2×D4—44no—
106C2×D4—42no—
107C2×D4—41no—
108C2×D4—84no—
109C2×D4C2×D412yes—
110C22×C4—22no—
111C22×C4—24no—
112C22×C4—22no—
113C22×C4C2×D412yes—
114C22×C4C2×D411yescharacteristic
115SD16—48no—
116SD16—48no—
117SD16—84no—
118D8—84no—
119C2×C8—24no—
120C2×C8—22no—
121C2×C8—24no—
122C4⋊C4—24no—
123C4⋊C4—24no—
124C4⋊C4—24no—
125C4⋊C4—24no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4C2×D412yes—
133C4⋊C4C2×D412yes—
134C22⋊C4—24no—
135C22⋊C4—24no—
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C42—22no—
142C42—22no—
143C42—22no—
144C42C2×D411yescharacteristic
145C42C2×D412yes—
146C42C2411yescharacteristic
147C23—22no—
148C23—42no—
149C23—84no—
150C23—82no—
151C23#12011yescharacteristic
152Q8—24no—
153Q8—42no—
154Q8—42no—
155Q8—82no—
156Q8—82no—
157D4—24no—
158D4—44no—
159D4—44no—
160D4—44no—
161D4—44no—
162D4—82no—
163D4—84no—
164D4—82no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—44no—
173C2×C4—44no—
174C2×C4—84no—
175C2×C4#12012yes—
176C2×C4#12011yescharacteristic
177C2×C4C22×D411yescharacteristic
178C2×C4C22×D412yes—
179C8—24no—
180C8—44no—
181C22—22no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—84no—
186C22—82no—
187C22—164no—
188C22#45312yes—
189C22C2×#12011yescharacteristic
190C4—22no—
191C4—21no—
192C4—21no—
193C4—42no—
194C4—42no—
195C4—42no—
196C4—42no—
197C4—42no—
198C4#57714yes—
199C2—22no—
200C2—82no—
201C2—162no—
202C2#301012yes—
203C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3544C2semidirect
2. #3217C2semidirect
3. C2×#492C2semidirect
4. #1624C2semidirect
5. #1623C2semidirect
6. #1622C2semidirect
7. #1558C2semidirect
8. #1494C2semidirect
11. Q82C22semidirect
12. #555C22semidirect
13. #554C22semidirect
33. #333C22semidirect
34. #333C22semidirect
35. Q8⋊C8C22semidirect
36. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35442
#32172
C2×#4922
#16242
#16232
#16222
#15582
#14942