Finite Group Databasebeta

← #18771

Subgroups — #18771

Order 256.

617 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C2×#550C211yesmaximal, characteristic
2#3208C211yesmaximal, characteristic, complemented
3#3104C211yesmaximal, characteristic, complemented
4#3077C211yesmaximal, characteristic, complemented
5#1641C214yesmaximal, complemented
6#1631C214yesmaximal, complemented
7#1545C211yesmaximal, characteristic, complemented
8#1487C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C22×#41—21no—
11#550—22no—
12#550—22no—
13#550—22no—
14#550C2214yes—
15#533C2211yescharacteristic
16C2×#127—21no—
17C2×#127C2211yescharacteristic
18C2×#121—22no—
19C2×#121—21no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23#492C2212yescomplemented
24#486C2212yescomplemented
25#484—21no—
26#478—22no—
27#477C2212yescomplemented
28#476C2212yescomplemented
29#474C2212yescomplemented
30#462C2212yescomplemented
31#460—22no—
32#451—21no—
33#450—22no—
34C4×Q16—22no—
35#431C2211yescharacteristic
36#431C2211yescharacteristic
37#428—21no—
38#423C2211yescharacteristic
39#418—21no—
40#331C2214yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C2×#41—24no—
44C2×#41—24no—
45C2×#41—48no—
46C22×Q8D411yescharacteristic
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4D411yescharacteristic
52C22×D4D412yes—
53C23×C4—22no—
54C23×C4—21no—
55C2×Q16—22no—
56C2×SD16—24no—
57C2×M16—21no—
58C4⋊Q8C2311yescharacteristic
59C4⋊Q8C2311yescharacteristic
60#127—24no—
61#127—24no—
62#127—21no—
63#127—21no—
64#127D412yes—
65#127C2312yes—
66#125—21no—
67C22⋊Q8—21no—
68C22⋊Q8—21no—
69#121—24no—
70#121—28no—
71#121—28no—
72#121—24no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×Q8D412yes—
78C4×Q8C2312yes—
79C4×D4—24no—
80C4×D4—24no—
81C4×D4—21no—
82C4×D4—21no—
83C4×D4D412yes—
84C4×D4C2312yes—
85#117—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D411yescharacteristic
89C2×C22⋊C4—22no—
90C2×C22⋊C4—21no—
91C2×C42—21no—
92C2×C42C2311yescharacteristic
93#107—22no—
94#107—22no—
95#106—22no—
96C4⋊C8—21no—
97C4⋊C8—21no—
98C4⋊C8C2312yes—
99C4⋊C8C2312yes—
100#103—22no—
101#103—22no—
102#102—22no—
103#102—22no—
104#102—24no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C4×C8—21no—
108C4×C8C2312yes—
109C24—21no—
110C24—22no—
111#41—416no—
112#41—48no—
113#41—48no—
114C2×Q8—24no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8C2×D412yes—
119C2×D4—28no—
120C2×D4—28no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—48no—
130C2×D4—44no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—82no—
134C2×D4—84no—
135C2×D4C2×D412yes—
136C2×D4C2×D414yes—
137C22×C4—21no—
138C22×C4—28no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—48no—
144C22×C4—44no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148Q16—44no—
149SD16—84no—
150M16—42no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4C2×D412yes—
165C4⋊C4C2×D412yes—
166C22⋊C4—28no—
167C22⋊C4—22no—
168C22⋊C4—22no—
169C22⋊C4—41no—
170C42—22no—
171C42—22no—
172C42—41no—
173C42C2×D411yescharacteristic
174C42C2×D411yescharacteristic
175C42C2×D411yescharacteristic
176C42C2411yescharacteristic
177C23—22no—
178C23—24no—
179C23—44no—
180C23—42no—
181C23—44no—
182C23—42no—
183C23—42no—
184C23—44no—
185C23#12011yescharacteristic
186Q8—24no—
187Q8—22no—
188Q8—42no—
189Q8—42no—
190Q8AGL(1,ℤ/8)14yes—
191D4—22no—
192D4—22no—
193D4—44no—
194D4—416no—
195D4—44no—
196D4—42no—
197D4—44no—
198D4—42no—
199D4—42no—
200D4—44no—
201D4—82no—
202D4—88no—
203C2×C4—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—416no—
210C2×C4—44no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—44no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4#12011yescharacteristic
217C2×C4#12011yescharacteristic
218C2×C4#12011yescharacteristic
219C2×C4#13512yes—
220C2×C4#13712yes—
221C2×C4C22×D411yescharacteristic
222C2×C4C22×D411yescharacteristic
223C2×C4C22×D411yescharacteristic
224C8—42no—
225C8—42no—
226C8—42no—
227C22—21no—
228C22—21no—
229C22—42no—
230C22—44no—
231C22—44no—
232C22—42no—
233C22—44no—
234C22—42no—
235C22—84no—
236C22—88no—
237C22#45212yes—
238C22C2×#12011yescharacteristic
239C4—24no—
240C4—21no—
241C4—21no—
242C4—21no—
243C4—42no—
244C4—42no—
245C4—81no—
246C4—81no—
247C4#45212yes—
248C4C2×#13511yescharacteristic
249C4C2×AGL(1,ℤ/8)12yes—
250C4C2×#13711yescharacteristic
251C2—22no—
252C2—82no—
253C2—84no—
254C2C2×#45211yescharacteristic
255C2#300311yescharacteristic
256C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#550C2
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. #3208C2semidirect
3. #3104C2semidirect
4. #3077C2semidirect
5. #1641C2semidirect
6. #1631C2semidirect
7. #1545C2semidirect
8. #1487C2semidirect
9. #1478C2semidirect
23. #492C22semidirect
24. #486C22semidirect
27. #477C22semidirect
28. #476C22semidirect
29. #474C22semidirect
30. #462C22semidirect
40. #331C22semidirect
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
C2×#5502
#32082
#31042
#30772
#16412
#16312
#15452
#14872
#14782