Finite Group Databasebeta

← #29644

Subgroups — #29644

Order 256.

532 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 131 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×#533C211yesmaximal, characteristic, complemented
2C2×#482C212yesmaximal
3#2854C211yesmaximal, characteristic
4#2059C212yesmaximal, complemented
5#2058C211yesmaximal, characteristic, complemented
6#2057C211yesmaximal, characteristic, complemented
7#2025C212yesmaximal, complemented
8#2023C211yesmaximal, characteristic, complemented
9#2023C211yesmaximal, characteristic, complemented
10#1825C212yesmaximal, complemented
11#1817C211yesmaximal, characteristic, complemented
12#533—28no—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2212yes—
16C2×#123—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—21no—
19C2×#121C2211yescharacteristic
20C2×C4×Q8—21no—
21C2×C4×Q8—21no—
22C2×#117C2212yes—
23C22×C42C2211yescharacteristic
24#482—28no—
25C2×#107C2212yes—
26C2×#106C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×C22⋊C8C2212yes—
30#390C2211yescharacteristic
31#390C2211yescharacteristic
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#388—21no—
35#388—21no—
36#386—21no—
37#380C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2211yescharacteristic
43#336C2211yescharacteristic
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8—21no—
47C22×D4—21no—
48C22×D4C2311yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C22×C8C2312yes—
53C22×C8C2312yes—
54C4⋊Q8—24no—
55C4⋊Q8—24no—
56#125—24no—
57#125—24no—
58#123—44no—
59C22⋊Q8—28no—
60C22⋊Q8—28no—
61#121—24no—
62C4×Q8—28no—
63C4×Q8—28no—
64#117—24no—
65#117D418yes—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2312yes—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C42—22no—
85C2×C42—24no—
86C2×C42D411yescharacteristic
87C2×C42D411yescharacteristic
88C2×C42C2311yescharacteristic
89C2×C42C2311yescharacteristic
90#107—24no—
91#106—24no—
92#102—24no—
93#102—24no—
94C22⋊C8—24no—
95#95—21no—
96#95—21no—
97#95—21no—
98C24—21no—
99C24C2×D411yescharacteristic
100C2×Q8—24no—
101C2×Q8—24no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×Q8—42no—
105C2×Q8—42no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—42no—
111C2×D4—42no—
112C2×D4—42no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—41no—
126C22×C4—41no—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D414yes—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2411yescharacteristic
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C4⋊C4—28no—
141C4⋊C4—28no—
142C4⋊C4—24no—
143C4⋊C4—24no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4C2×D414yes—
153C4⋊C4C2×D414yes—
154C22⋊C4—28no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C42—22no—
162C42—22no—
163C42—28no—
164C42—22no—
165C42—22no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23#12012yes—
180C23C22×D411yescharacteristic
181Q8—48no—
182Q8—48no—
183D4—42no—
184D4—42no—
185D4—84no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—24no—
197C2×C4—24no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4#12012yes—
218C2×C4#13512yes—
219C2×C4AGL(1,ℤ/8)12yes—
220C2×C4C22×D412yes—
221C2×C42−1+412yes—
222C2×C42−1+411yescharacteristic
223C2×C42−1+411yescharacteristic
224C8—42no—
225C8—42no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22C2×#12011yescharacteristic
241C22#53611yescharacteristic
242C22#53611yescharacteristic
243C22C2×#13511yescharacteristic
244C22C2×AGL(1,ℤ/8)11yescharacteristic
245C22#57511yescharacteristic
246C22#57511yescharacteristic
247C4—22no—
248C4—22no—
249C4—22no—
250C4—41no—
251C4—44no—
252C4—41no—
253C4—44no—
254C4—81no—
255C4—81no—
256C4—81no—
257C4#53612yes—
258C4#53612yes—
259C2—42no—
260C2—161no—
261C2#261411yescharacteristic
262C2#300211yescharacteristic
263C2#300611yescharacteristic
264C2#322811yescharacteristic
265C2#323311yescharacteristic
266C2#323511yescharacteristic
267C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#533C2semidirect
4. #2059C2semidirect
5. #2058C2semidirect
6. #2057C2semidirect
7. #2025C2semidirect
8. #2023C2semidirect
9. #2023C2semidirect
10. #1825C2semidirect
11. #1817C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4822
#28542
#20592
#20582
#20572
#20252
#20232
#18252
#18172