Finite Group Databasebeta

← #17771

Subgroups — #17771

Order 256.

495 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 179 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#3508C211yesmaximal, characteristic
2#2990C212yesmaximal, complemented
3#1557C212yesmaximal, complemented
4#1555C211yesmaximal, characteristic, complemented
5#1553C212yesmaximal, complemented
6#1552C212yesmaximal, complemented
7#1547C212yesmaximal, complemented
8#1540C212yesmaximal, complemented
9C2×#331C211yesmaximal, characteristic, complemented
10#559—22no—
11#550—22no—
12#546—22no—
13#546—22no—
14#540C414yes—
15#535C412yes—
16#533C2211yescharacteristic
17#532C2212yes—
18#532C2212yes—
19C2×#127C2211yescharacteristic
20#518C412yes—
21C4×#41C2211yescharacteristic
22#446—22no—
23#435—22no—
24#435—22no—
25C8×D4—22no—
26#432C2212yes—
27#432C2212yescomplemented
28#431C2212yescomplemented
29#423C2212yescomplemented
30C2×C4⋊C8C2212yes—
31C2×C4⋊C8C2212yescomplemented
32#332C2212yescomplemented
33#332C2212yescomplemented
34#332C2212yescomplemented
35#332C2212yescomplemented
36#331C2211yescharacteristic
37#331C2212yes—
38#331C2211yescharacteristic
39#331C2212yes—
40#331C2211yescharacteristic
41#331C2211yescharacteristic
42C2×#41—22no—
43C2×#41—22no—
44C2×#41C2×C411yescharacteristic
45C22×D4—22no—
46C22×D4C2×C412yes—
47C2×M16—22no—
48C22×C8—22no—
49C22×C8—22no—
50C4⋊Q8C2312yes—
51#127—22no—
52#127—22no—
53#127C2×C412yes—
54#127C2311yescharacteristic
55#127C2311yescharacteristic
56#125C2×C412yes—
57#124—22no—
58#124—22no—
59#124—22no—
60#123—24no—
61#123—24no—
62C22⋊Q8—22no—
63#121—22no—
64#121—22no—
65#121—24no—
66#121—24no—
67#121C2×C414yes—
68#121C2×C414yes—
69#120—24no—
70C4×Q8—22no—
71C4×Q8D412yes—
72C4×D4—24no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4C2×C414yes—
76C4×D4C2×C414yes—
77C4×D4D412yes—
78C4×D4D412yes—
79C4×D4D412yes—
80#117C2×C412yes—
81#117C2×C411yescharacteristic
82#117C2×C412yes—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C42C2311yescharacteristic
86C2×C42C2312yes—
87C4⋊C8—22no—
88C4⋊C8—22no—
89C4⋊C8—22no—
90C4⋊C8C2312yes—
91C4⋊C8C2312yes—
92C4⋊C8C2312yes—
93C4⋊C8C2312yes—
94C22⋊C8—22no—
95C22⋊C8—22no—
96C22⋊C8—22no—
97#97—22no—
98#97—22no—
99C4×C8—22no—
100C24—22no—
101#41—42no—
102#41—82no—
103#41—82no—
104#41—82no—
105#41—82no—
106C2×Q8—22no—
107C2×Q8C22⋊C411yescharacteristic
108C2×D4—24no—
109C2×D4—24no—
110C2×D4—22no—
111C2×D4—24no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—44no—
115C2×D4—82no—
116C2×D4—82no—
117C2×D4C22⋊C412yes—
118C2×D4C22⋊C411yescharacteristic
119C2×D4C22×C412yes—
120C2×D4C22×C412yes—
121C22×C4—24no—
122C22×C4—24no—
123C22×C4C22×C411yescharacteristic
124C22×C4C22×C412yes—
125C22×C4C2×D412yes—
126C22×C4C2×D412yes—
127C22×C4C2×D412yes—
128M16—82no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—82no—
134C2×C8—82no—
135C4⋊C4—24no—
136C4⋊C4—22no—
137C4⋊C4C22⋊C412yes—
138C4⋊C4C22⋊C412yes—
139C4⋊C4C22⋊C412yes—
140C4⋊C4C22×C414yes—
141C22⋊C4—22no—
142C22⋊C4—24no—
143C22⋊C4—24no—
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4C22⋊C411yescharacteristic
147C22⋊C4C22⋊C411yescharacteristic
148C22⋊C4C22⋊C412yes—
149C22⋊C4C22⋊C412yes—
150C42—22no—
151C42C22×C412yes—
152C42C22×C411yescharacteristic
153C42C2×D412yes—
154C42C2×D412yes—
155C42C2×D412yes—
156C42C2411yescharacteristic
157C23—22no—
158C23—22no—
159C23—44no—
160C23—84no—
161C23C2×C22⋊C412yes—
162C23C2×C22⋊C411yescharacteristic
163Q8—41no—
164Q8—42no—
165Q8—42no—
166D4—24no—
167D4—41no—
168D4—42no—
169D4—42no—
170D4—44no—
171D4—42no—
172D4—42no—
173D4—42no—
174D4—42no—
175D4—84no—
176D4—84no—
177D4—84no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—42no—
187C2×C4—84no—
188C2×C4—84no—
189C2×C4C2×C22⋊C411yescharacteristic
190C2×C4C2×C22⋊C412yes—
191C2×C4C2×C22⋊C412yes—
192C2×C4C2×C22⋊C412yes—
193C2×C4C2×C22⋊C412yes—
194C2×C4C23×C411yescharacteristic
195C2×C4C22×D412yes—
196C8—42no—
197C8—42no—
198C8—42no—
199C8—42no—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—22no—
204C22—42no—
205C22—42no—
206C22—44no—
207C22—44no—
208C22—164no—
209C22#41312yes—
210C22C22×C22⋊C411yescharacteristic
211C4—21no—
212C4—21no—
213C4—22no—
214C4—22no—
215C4—22no—
216C4—22no—
217C4—44no—
218C4#57612yes—
219C4#57712yes—
220C2—22no—
221C2—42no—
222C2—82no—
223C2—82no—
224C2C2×#41311yescharacteristic
225C2#289712yes—
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC2×C4C23×C4
Frattini, cosocleC42C24
prime radical#3508C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2990C2semidirect
3. #1557C2semidirect
4. #1555C2semidirect
5. #1553C2semidirect
6. #1552C2semidirect
7. #1547C2semidirect
8. #1540C2semidirect
9. C2×#331C2semidirect
27. #432C22semidirect
28. #431C22semidirect
29. #423C22semidirect
31. C2×C4⋊C8C22semidirect
32. #332C22semidirect
33. #332C22semidirect
34. #332C22semidirect
35. #332C22semidirect

Maximal subgroups

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

subgroupindex
#35082
#29902
#15572
#15552
#15532
#15522
#15472
#15402
C2×#3312