Finite Group Databasebeta

← #18756

Subgroups — #18756

Order 256.

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

normal subgroups only

This group has 612 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
1C2×#547C211yesmaximal, characteristic, complemented
2#3226C211yesmaximal, characteristic, complemented
3#3100C211yesmaximal, characteristic, complemented
4#3036C211yesmaximal, characteristic, complemented
5#1647C214yesmaximal, complemented
6#1625C214yesmaximal, complemented
7#1546C211yesmaximal, characteristic, complemented
8#1486C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic, complemented
10C22×#41—21no—
11C2×AGL(1,ℤ/8)—21no—
12#547—22no—
13#547—22no—
14#547—22no—
15#547C2214yes—
16#532C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#123—22no—
19C2×C22⋊Q8—21no—
20C2×#121—21no—
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C22×C4⋊C4—21no—
24#500C2212yescomplemented
25#486C2212yescomplemented
26#480—21no—
27#480—22no—
28#474—22no—
29#474C2212yescomplemented
30#463—22no—
31#461C2212yescomplemented
32#459C2212yescomplemented
33#459C2212yescomplemented
34#447—21no—
35C4×D8—22no—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#423C2211yescharacteristic
39#417—21no—
40#332C2214yescomplemented
41D4⋊C8C2214yescomplemented
42D4⋊C8C2214yescomplemented
43C2×#41—24no—
44C2×#41—24no—
45C2×#41—21no—
46C2×#41—44no—
47C2×#41—44no—
48C22×Q8—21no—
49C22×D4—22no—
50C22×D4D411yescharacteristic
51C22×D4D411yescharacteristic
52C23×C4—22no—
53C23×C4—21no—
54C23×C4—21no—
55AGL(1,ℤ/8)—44no—
56C2×SD16—22no—
57C2×D8—22no—
58C2×D8—22no—
59C2×M16—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8D412yes—
63C4⋊Q8C2311yescharacteristic
64C4⋊Q8C2312yes—
65#127C2311yescharacteristic
66#124—21no—
67#123—28no—
68#123—28no—
69C22⋊Q8—24no—
70C22⋊Q8—24no—
71#121—24no—
72#121—24no—
73#121—21no—
74#121—21no—
75C4×D4—21no—
76C4×D4—21no—
77C4×D4—21no—
78C4×D4—21no—
79C4×D4—21no—
80C4×D4—21no—
81C4×D4D412yes—
82C4×D4D412yes—
83C4×D4C2312yes—
84C4×D4C2312yes—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D412yes—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42C2311yescharacteristic
96#107—22no—
97#107—24no—
98C4⋊C8—21no—
99C4⋊C8—21no—
100C4⋊C8C2312yes—
101C4⋊C8C2312yes—
102#103—22no—
103#102—22no—
104#102—22no—
105#102—22no—
106#102—24no—
107C22⋊C8—21no—
108C22⋊C8—21no—
109C4×C8—21no—
110C4×C8C2312yes—
111C24—21no—
112C24—21no—
113#41—48no—
114#41—416no—
115#41—48no—
116#41—42no—
117#41—42no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—21no—
121C2×Q8—42no—
122C2×Q8—42no—
123C2×D4—28no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—22no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—42no—
131C2×D4—48no—
132C2×D4—41no—
133C2×D4D814yescomplemented
134C2×D4C2×D412yes—
135C2×D4C2×D412yes—
136C22×C4—28no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—44no—
145C22×C4—48no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—41no—
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152SD16—44no—
153D8—44no—
154D8—44no—
155M16—42no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C4⋊C4—28no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4C2×D412yes—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D414yes—
170C22⋊C4—28no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—41no—
176C22⋊C4—41no—
177C42C2×D411yescharacteristic
178C42C2×D411yescharacteristic
179C42C2×D411yescharacteristic
180C42C2411yescharacteristic
181C23—22no—
182C23—24no—
183C23—22no—
184C23—42no—
185C23—42no—
186C23—42no—
187C23—42no—
188C23—42no—
189C23—41no—
190C23#12011yescharacteristic
191Q8—42no—
192Q8—42no—
193Q8—44no—
194Q8—41no—
195Q8—41no—
196D4—22no—
197D4—22no—
198D4—22no—
199D4—24no—
200D4—42no—
201D4—42no—
202D4—416no—
203D4—41no—
204D4—41no—
205D4—42no—
206D4—82no—
207D4C2×D814yes—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—41no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—416no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—44no—
221C2×C4—44no—
222C2×C4—41no—
223C2×C4—82no—
224C2×C4#12011yescharacteristic
225C2×C4#12011yescharacteristic
226C2×C4#12011yescharacteristic
227C2×C4C2×D812yes—
228C2×C4AGL(1,ℤ/8)12yes—
229C2×C4C22×D411yescharacteristic
230C2×C4C22×D411yescharacteristic
231C2×C4C22×D411yescharacteristic
232C8—42no—
233C8—42no—
234C8—42no—
235C22—21no—
236C22—21no—
237C22—24no—
238C22—48no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—84no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22#44712yes—
249C22C2×#12011yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—44no—
255C4—42no—
256C4—81no—
257C4#44712yes—
258C4C22×D811yescharacteristic
259C4C2×AGL(1,ℤ/8)11yescharacteristic
260C4#57512yes—
261C2—22no—
262C2—44no—
263C2—82no—
264C2—161no—
265C2C2×#44711yescharacteristic
266C2#300511yescharacteristic
267C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
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. C2×#547C2semidirect
2. #3226C2semidirect
3. #3100C2semidirect
4. #3036C2semidirect
5. #1647C2semidirect
6. #1625C2semidirect
7. #1546C2semidirect
8. #1486C2semidirect
9. #1478C2semidirect
24. #500C22semidirect
25. #486C22semidirect
29. #474C22semidirect
31. #461C22semidirect
32. #459C22semidirect
33. #459C22semidirect
40. #332C22semidirect
41. D4⋊C8C22semidirect
42. D4⋊C8C22semidirect
133. C2×D4D8semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
#32262
#31002
#30362
#16472
#16252
#15462
#14862
#14782