Finite Group Databasebeta

← #18914

Subgroups — #18914

Order 256.

477 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 477 conjugacy classes of subgroups — too many to draw. Only its 129 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#3502C211yesmaximal, characteristic, complemented
2#3208C211yesmaximal, characteristic
3#3208C211yesmaximal, characteristic
4C2×#460C211yesmaximal, characteristic
5#1626C214yesmaximal, complemented
6#1623C214yesmaximal, complemented
7#1542C211yesmaximal, characteristic, complemented
8#1481C211yesmaximal, characteristic, complemented
9#1479C211yesmaximal, characteristic, complemented
10C22×SD16—21no—
11Q82—21no—
12Q82C2212yescomplemented
13#557—21no—
14#557—22no—
15#557—22no—
16#557C2212yescomplemented
17#544—22no—
18#544—21no—
19#533—22no—
20#533C2211yescharacteristic
21#533C2211yescharacteristic
22C2×#127C2211yescharacteristic
23#519—21no—
24C2×C4×Q8C2211yescharacteristic
25#492—21no—
26#492C2212yes—
27#486—21no—
28#486C2212yes—
29#460—22no—
30#460—22no—
31#460C2214yes—
32#460C2214yes—
33#450—22no—
34#450—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#102—21no—
39#333C2212yescomplemented
40#332C2212yescomplemented
41Q8⋊C8C2214yescomplemented
42Q8⋊C8C2214yescomplemented
43C22×Q8D411yescharacteristic
44C22×D4—21no—
45C22×D4—21no—
46C22×D4—21no—
47C2×SD16—24no—
48C2×SD16—24no—
49C2×SD16—24no—
50C2×SD16—44no—
51C22×C8—21no—
52C4⋊Q8—24no—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4⋊Q8—24no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8D411yescharacteristic
59C4⋊Q8D411yescharacteristic
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2311yescharacteristic
62#127—21no—
63#127—21no—
64#127—21no—
65#127C2312yes—
66#125—24no—
67#125—24no—
68#125—22no—
69#125—24no—
70#125—22no—
71#125—21no—
72#125—21no—
73C22⋊Q8—24no—
74C22⋊Q8—21no—
75C22⋊Q8—21no—
76C22⋊Q8D412yes—
77C22⋊Q8D412yes—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8—24no—
81C4×Q8—22no—
82C4×Q8—22no—
83C4×Q8—22no—
84C4×Q8—22no—
85C4×Q8—21no—
86C4×Q8—21no—
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8D412yes—
90C4×Q8D412yes—
91C4×Q8C2312yes—
92C4×Q8C2312yes—
93#117—24no—
94#117—21no—
95#117—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4D411yescharacteristic
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C4⋊C8—21no—
101C4⋊C8—21no—
102C4⋊C8C2312yes—
103C4⋊C8C2312yes—
104#102—24no—
105#102—24no—
106#102—24no—
107C22⋊C8—21no—
108C22⋊C8—21no—
109C4×C8—21no—
110C4×C8C2312yes—
111C24—41no—
112C2×Q8—24no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—21no—
117C2×Q8SD1614yescomplemented
118C2×Q8C2×D412yes—
119C2×Q8C2×D412yes—
120C2×Q8C2×D412yes—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—42no—
128C2×D4—42no—
129C2×D4—82no—
130C2×D4—82no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136SD16—48no—
137SD16—84no—
138SD16—84no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—42no—
143C4⋊C4—24no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—22no—
161C4⋊C4C2×D412yes—
162C4⋊C4C2×D412yes—
163C4⋊C4C2×D412yes—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C42—24no—
167C42—24no—
168C42—22no—
169C42—22no—
170C42—22no—
171C42—22no—
172C42C2×D411yescharacteristic
173C42C2×D411yescharacteristic
174C42C2×D411yescharacteristic
175C42C2411yescharacteristic
176C23—42no—
177C23—82no—
178C23—82no—
179C23—82no—
180C23#12011yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—24no—
184Q8—42no—
185Q8—42no—
186Q8—81no—
187Q8—84no—
188Q8C2×SD1614yes—
189D4—44no—
190D4—42no—
191D4—42no—
192D4—44no—
193D4—82no—
194D4—82no—
195D4—82no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—41no—
205C2×C4—81no—
206C2×C4#12011yescharacteristic
207C2×C4#12011yescharacteristic
208C2×C4#12011yescharacteristic
209C2×C4C2×SD1612yes—
210C2×C4#13712yes—
211C2×C4C22×D411yescharacteristic
212C2×C4C22×D411yescharacteristic
213C2×C4C22×D411yescharacteristic
214C8—42no—
215C8—42no—
216C8—42no—
217C22—21no—
218C22—21no—
219C22—82no—
220C22—82no—
221C22—82no—
222C22—164no—
223C22#44812yes—
224C22C2×#12011yescharacteristic
225C4—24no—
226C4—21no—
227C4—21no—
228C4—21no—
229C4—42no—
230C4—42no—
231C4—42no—
232C4—42no—
233C4—42no—
234C4#44812yes—
235C4C22×SD1611yescharacteristic
236C4C2×#13711yescharacteristic
237C4#57712yes—
238C2—22no—
239C2—162no—
240C2C2×#44811yescharacteristic
241C2#301011yescharacteristic
242C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#127C22
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. #3502C2semidirect
5. #1626C2semidirect
6. #1623C2semidirect
7. #1542C2semidirect
8. #1481C2semidirect
9. #1479C2semidirect
12. Q82C22semidirect
16. #557C22semidirect
39. #333C22semidirect
40. #332C22semidirect
41. Q8⋊C8C22semidirect
42. Q8⋊C8C22semidirect
117. C2×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#35022
#32082
C2×#4602
#16262
#16232
#15422
#14812
#14792