Finite Group Databasebeta

← #18683

Subgroups — #18683

Order 256.

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

normal subgroups only

This group has 529 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#3543C211yesmaximal, characteristic
2#3238C212yesmaximal, complemented
3C2×#492C211yesmaximal, characteristic, complemented
4#1631C214yesmaximal, complemented
5#1630C212yesmaximal, complemented
6#1629C212yesmaximal, complemented
7#1555C211yesmaximal, characteristic, complemented
8#1494C212yesmaximal, complemented
9C22×SD16—22no—
10#560—22no—
11#559C2211yescharacteristic
12#550C2212yes—
13#546—22no—
14D42—22no—
15D42C2211yescharacteristic
16#540—22no—
17#535—22no—
18#534—22no—
19C2×C4⋊Q8C2211yescharacteristic, complemented
20C2×#127C2211yescharacteristic
21#529C2212yes—
22#501—22no—
23#492—22no—
24#492C2212yes—
25#492C2212yescomplemented
26#489—22no—
27#483—22no—
28#480—22no—
29#477C2214yescomplemented
30#476C2214yescomplemented
31#432C2212yescomplemented
32C2×C4×C8C2211yescharacteristic, complemented
33#331C2212yes—
34#331C2212yescomplemented
35Q8⋊C8C2214yescomplemented
36D4⋊C8C2214yescomplemented
37C2×#41—22no—
38C22×Q8—21no—
39C22×D4—22no—
40C22×D4—22no—
41C22×D4—22no—
42C22×D4D412yes—
43C2×SD16—24no—
44C2×SD16—24no—
45C2×SD16—48no—
46C22×C8—21no—
47C4⋊Q8—22no—
48C4⋊Q8—21no—
49C4⋊Q8C2312yes—
50#127—22no—
51#127—22no—
52#127D412yes—
53#127C2311yescharacteristic
54#127C2311yescharacteristic
55#126—22no—
56#125D412yes—
57#124—22no—
58#124D412yes—
59#123—22no—
60#123—24no—
61#123—22no—
62#123—22no—
63C22⋊Q8D412yes—
64#121—24no—
65#121—24no—
66#121—24no—
67#121—24no—
68#121—22no—
69#121—22no—
70#121D412yes—
71#120—22no—
72#120—24no—
73#120—24no—
74#120—22no—
75C4×Q8C2312yes—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4C2312yes—
80#117—22no—
81C2×C4⋊C4—22no—
82C2×C22⋊C4—22no—
83C2×C42C2311yescharacteristic
84#107—24no—
85#106—24no—
86C4⋊C8C2314yes—
87#103—24no—
88#102—24no—
89C22⋊C8—22no—
90#97—22no—
91C4×C8—21no—
92C4×C8C2312yes—
93C24—22no—
94C24—22no—
95#41—84no—
96C2×Q8—22no—
97C2×Q8—44no—
98C2×Q8—41no—
99C2×Q8C2×D412yes—
100C2×D4—22no—
101C2×D4—24no—
102C2×D4—24no—
103C2×D4—24no—
104C2×D4—22no—
105C2×D4—24no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—44no—
109C2×D4—84no—
110C2×D4—84no—
111C2×D4—84no—
112C2×D4C2×D412yes—
113C2×D4C2×D412yes—
114C2×D4C2×D412yes—
115C22×C4—24no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—41no—
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121SD16—48no—
122SD16—48no—
123C2×C8—24no—
124C2×C8—22no—
125C2×C8—24no—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—24no—
131C4⋊C4—42no—
132C4⋊C4C2×D412yes—
133C4⋊C4C2×D412yes—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—24no—
137C22⋊C4—24no—
138C22⋊C4—22no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C42—22no—
144C42C2×D411yescharacteristic
145C42C2×D412yes—
146C42C2411yescharacteristic
147C23—22no—
148C23—22no—
149C23—22no—
150C23—44no—
151C23—84no—
152C23—84no—
153C23—82no—
154C23#12011yescharacteristic
155Q8—24no—
156Q8—44no—
157Q8—82no—
158D4—24no—
159D4—42no—
160D4—42no—
161D4—44no—
162D4—44no—
163D4—44no—
164D4—82no—
165D4—84no—
166D4—84no—
167D4—84no—
168D4—84no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—44no—
175C2×C4—42no—
176C2×C4—82no—
177C2×C4—84no—
178C2×C4—82no—
179C2×C4#12012yes—
180C2×C4#12011yescharacteristic
181C2×C4C22×D411yescharacteristic
182C2×C4C22×D412yes—
183C8—24no—
184C8—44no—
185C22—22no—
186C22—42no—
187C22—44no—
188C22—42no—
189C22—44no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—164no—
194C22—164no—
195C22#45312yes—
196C22C2×#12011yescharacteristic
197C4—22no—
198C4—21no—
199C4—21no—
200C4—42no—
201C4—42no—
202C4—42no—
203C4—82no—
204C4#57714yes—
205C2—22no—
206C2—82no—
207C2—82no—
208C2—82no—
209C2#300912yes—
210C2C2×#45311yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3543C2
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
2. #3238C2semidirect
3. C2×#492C2semidirect
4. #1631C2semidirect
5. #1630C2semidirect
6. #1629C2semidirect
7. #1555C2semidirect
8. #1494C2semidirect
19. C2×C4⋊Q8C22semidirect
25. #492C22semidirect
29. #477C22semidirect
30. #476C22semidirect
31. #432C22semidirect
32. C2×C4×C8C22semidirect
34. #331C22semidirect
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
#35432
#32382
C2×#4922
#16312
#16302
#16292
#15552
#14942