Finite Group Databasebeta

← #18898

Subgroups — #18898

Order 256.

489 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 121 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#3507C211yesmaximal, characteristic, complemented
2#3208C211yesmaximal, characteristic, complemented
3#3205C211yesmaximal, characteristic
4C2×#460C211yesmaximal, characteristic, complemented
5#1628C214yesmaximal, complemented
6#1622C214yesmaximal, complemented
7#1542C211yesmaximal, characteristic, complemented
8#1480C211yesmaximal, characteristic, complemented
9#1479C211yesmaximal, characteristic, complemented
10C22×SD16—21no—
11#559—21no—
12#557—21no—
13#556—22no—
14#556C2212yescomplemented
15#554C2212yescomplemented
16#552—22no—
17#539—22no—
18#536—21no—
19#533C2211yescharacteristic
20#532C2211yescharacteristic
21C2×#127C2211yescharacteristic
22#529—22no—
23#520—21no—
24C2×C4×Q8C2211yescharacteristic
25#493—21no—
26#492C2212yes—
27#486—21no—
28#486C2212yes—
29#460—22no—
30#460—22no—
31#460C2214yes—
32#459C2214yes—
33#450—22no—
34#447—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#102—21no—
39#333C2212yescomplemented
40#332C2212yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C2×#41—21no—
44C22×Q8D411yescharacteristic
45C22×D4—21no—
46C22×D4—21no—
47C22×D4—21no—
48C2×SD16—24no—
49C2×SD16—24no—
50C2×SD16—44no—
51C2×D8—24no—
52C22×C8—21no—
53C4⋊Q8—22no—
54C4⋊Q8D411yescharacteristic
55C4⋊Q8C2311yescharacteristic
56C4⋊Q8C2311yescharacteristic
57#127—21no—
58#127—21no—
59#127—21no—
60#127D411yescharacteristic
61#127C2312yes—
62#126—24no—
63#126—24no—
64#125—24no—
65#125—22no—
66#125—21no—
67#124—22no—
68#124—22no—
69#124—21no—
70#123—24no—
71#123—24no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C22⋊Q8—24no—
75C22⋊Q8—21no—
76C22⋊Q8D412yes—
77#121—21no—
78#121D412yes—
79C4×Q8—22no—
80C4×Q8—22no—
81C4×Q8—21no—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8D412yes—
85C4×Q8D412yes—
86C4×Q8C2312yes—
87C4×D4—24no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4—21no—
91C4×D4C2312yes—
92#117—24no—
93#117—21no—
94#117—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—22no—
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—
112#41—42no—
113#41—42no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8C2×D412yes—
120C2×Q8C2×D412yes—
121C2×D4—22no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—82no—
133C2×D4—82no—
134C2×D4C2×D412yes—
135C22×C4—21no—
136C22×C4—24no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142SD16—48no—
143SD16—84no—
144D8—84no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—42no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4C2×D412yes—
162C4⋊C4C2×D412yes—
163C4⋊C4C2×D412yes—
164C22⋊C4—24no—
165C22⋊C4—24no—
166C22⋊C4—24no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C42—24no—
171C42—22no—
172C42—22no—
173C42—22no—
174C42C2×D411yescharacteristic
175C42C2×D411yescharacteristic
176C42C2×D411yescharacteristic
177C42C2411yescharacteristic
178C23—22no—
179C23—42no—
180C23—82no—
181C23—82no—
182C23—82no—
183C23#12011yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—42no—
187Q8—42no—
188Q8—42no—
189Q8—42no—
190D4—24no—
191D4—44no—
192D4—44no—
193D4—42no—
194D4—42no—
195D4—44no—
196D4—42no—
197D4—81no—
198D4—82no—
199D4—82no—
200D4—82no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—81no—
212C2×C4—84no—
213C2×C4#12011yescharacteristic
214C2×C4#12011yescharacteristic
215C2×C4#12011yescharacteristic
216C2×C4#13512yes—
217C2×C4AGL(1,ℤ/8)12yes—
218C2×C4C22×D411yescharacteristic
219C2×C4C22×D411yescharacteristic
220C2×C4C22×D411yescharacteristic
221C8—42no—
222C8—42no—
223C8—42no—
224C22—21no—
225C22—21no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—82no—
230C22—82no—
231C22—82no—
232C22—164no—
233C22#44912yes—
234C22C2×#12011yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—42no—
240C4—42no—
241C4—42no—
242C4—42no—
243C4#44912yes—
244C4C2×#13511yescharacteristic
245C4C2×AGL(1,ℤ/8)11yescharacteristic
246C4#57712yes—
247C2—22no—
248C2—82no—
249C2—162no—
250C2C2×#44911yescharacteristic
251C2#301011yescharacteristic
252C2#302111yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3205C2
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. #3507C2semidirect
2. #3208C2semidirect
4. C2×#460C2semidirect
5. #1628C2semidirect
6. #1622C2semidirect
7. #1542C2semidirect
8. #1480C2semidirect
9. #1479C2semidirect
14. #556C22semidirect
15. #554C22semidirect
39. #333C22semidirect
40. #332C22semidirect
41. Q8⋊C8C22semidirect
42. D4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35072
#32082
#32052
C2×#4602
#16282
#16222
#15422
#14802
#14792