Finite Group Databasebeta

← #29760

Subgroups — #29760

Order 256.

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

normal subgroups only

This group has 455 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×#483C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic
3#2850C211yesmaximal, characteristic
4C2×#389C211yesmaximal, characteristic, complemented
5#2085C212yesmaximal, complemented
6#2068C212yesmaximal, complemented
7#2065C212yesmaximal, complemented
8#2042C212yesmaximal, complemented
9#1874C211yesmaximal, characteristic
10#1872C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic, complemented
12C2×#124C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C22×C4⋊C4—21no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C42C2211yescharacteristic
19#483—24no—
20#480—24no—
21C2×#107C2212yes—
22C2×#106C2212yes—
23C2×C4⋊C8C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2212yes—
27C2×#102C2212yes—
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30#391C2211yescharacteristic
31#390C2211yescharacteristic
32#389—21no—
33#389—21no—
34#389C2212yes—
35#389C2212yes—
36#388—21no—
37#388—21no—
38#386—21no—
39#386—21no—
40#382C2211yescharacteristic
41C2×#95—21no—
42#340C2212yes—
43#340C2212yes—
44#336C2212yes—
45C22×Q8C2311yescharacteristic
46C22×D4—21no—
47C22×D4C2311yescharacteristic
48C23×C4—21no—
49C23×C4—21no—
50C23×C4D411yescharacteristic
51C23×C4D411yescharacteristic
52C23×C4C2311yescharacteristic
53C22×C8C2312yes—
54C22×C8C2312yes—
55#124—42no—
56C22⋊Q8—22no—
57C22⋊Q8—22no—
58#121—22no—
59#121—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—22no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—42no—
67C2×C4⋊C4—44no—
68C2×C4⋊C4D414yes—
69C2×C4⋊C4C2312yes—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C42—42no—
80C2×C42D411yescharacteristic
81C2×C42D411yescharacteristic
82C2×C42C2311yescharacteristic
83C2×C42C2311yescharacteristic
84#107—24no—
85#106—24no—
86C4⋊C8—42no—
87#103—24no—
88#103—42no—
89#102—24no—
90#102—42no—
91C22⋊C8—22no—
92C22⋊C8—22no—
93#95—22no—
94#95—22no—
95#95—21no—
96#95—21no—
97#95—21no—
98#95—21no—
99C24—21no—
100C24C2×D411yescharacteristic
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—42no—
110C2×D4—42no—
111C2×D4—42no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D412yes—
136C22×C4C2×D411yescharacteristic
137C22×C4#4112yes—
138C22×C4C2411yescharacteristic
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C4⋊C4—24no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—42no—
154C4⋊C4—44no—
155C4⋊C4—44no—
156C4⋊C4—42no—
157C4⋊C4C2×D414yes—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C22⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C42—22no—
165C42—22no—
166C42—41no—
167C42—41no—
168C23—21no—
169C23—21no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23#12112yes—
180C23#13512yes—
181C23AGL(1,ℤ/8)12yes—
182C23C22×D411yescharacteristic
183Q8—42no—
184Q8—42no—
185D4—42no—
186D4—42no—
187D4—84no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—24no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—48no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—84no—
222C2×C4#12112yes—
223C2×C4#13712yes—
224C2×C4C22×D411yescharacteristic
225C2×C4C2×#4111yescharacteristic
226C2×C4H(5,𝔽2)12yes—
227C2×C42−1+412yes—
228C8—42no—
229C8—42no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22#46314yes—
244C22C2×#12111yescharacteristic
245C22#53411yescharacteristic
246C22#53611yescharacteristic
247C22C2×#13511yescharacteristic
248C22C2×AGL(1,ℤ/8)11yescharacteristic
249C22C2×#13711yescharacteristic
250C22#57511yescharacteristic
251C4—22no—
252C4—22no—
253C4—41no—
254C4—41no—
255C4—41no—
256C4—44no—
257C4—81no—
258C4—82no—
259C4—81no—
260C4—81no—
261C4#53712yes—
262C4#53912yes—
263C2—24no—
264C2—161no—
265C2#261811yescharacteristic
266C2C2×#46311yescharacteristic
267C2#303711yescharacteristic
268C2#319011yescharacteristic
269C2#319411yescharacteristic
270C2#323211yescharacteristic
271C2#323811yescharacteristic
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×#483C2semidirect
4. C2×#389C2semidirect
5. #2085C2semidirect
6. #2068C2semidirect
7. #2065C2semidirect
8. #2042C2semidirect
10. #1872C2semidirect
11. #1797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C2×#4802
#28502
C2×#3892
#20852
#20682
#20652
#20422
#18742
#18722
#17972