Finite Group Databasebeta

← #29336

Subgroups — #29336

Order 256.

434 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 123 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#2850C211yesmaximal, characteristic
2C2×#389C211yesmaximal, characteristic, complemented
3#2101C212yesmaximal, complemented
4#2042C212yesmaximal, complemented
5#1936C212yesmaximal, complemented
6#1935C212yesmaximal, complemented
7#1874C211yesmaximal, characteristic
8#1874C211yesmaximal, characteristic
9#1872C211yesmaximal, characteristic, complemented
10#1872C211yesmaximal, characteristic, complemented
11#1749C211yesmaximal, characteristic, complemented
12C2×#124C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#121C2211yescharacteristic
15C22×C4⋊C4—21no—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2211yescharacteristic
18C22×C42C2211yescharacteristic
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×C22⋊C8C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25C2×#97C2212yes—
26#391C2211yescharacteristic
27#390C2211yescharacteristic
28#389—21no—
29#389—21no—
30#389C2212yes—
31#389C2212yes—
32#388—21no—
33#388—21no—
34#386—21no—
35#386—21no—
36#382C2211yescharacteristic
37C2×#95—21no—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C23×C4D411yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2312yes—
53#124—42no—
54C22⋊Q8—22no—
55C22⋊Q8—22no—
56#121—22no—
57#121—22no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—24no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4—44no—
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C42—42no—
77C2×C42D411yescharacteristic
78C2×C42D411yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81#103—24no—
82#103—42no—
83#102—24no—
84#102—42no—
85C22⋊C8—22no—
86C22⋊C8—22no—
87#97—42no—
88#95—22no—
89#95—22no—
90#95—21no—
91#95—21no—
92#95—21no—
93#95—21no—
94C24—21no—
95C24C2×D411yescharacteristic
96C2×Q8—22no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×D4—22no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—22no—
104C2×D4—42no—
105C2×D4—42no—
106C2×D4—42no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—22no—
117C22×C4—41no—
118C22×C4—41no—
119C22×C4—41no—
120C22×C4—41no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4—42no—
125C22×C4—42no—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4#4112yes—
132C22×C4#4112yes—
133C22×C4C2411yescharacteristic
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—42no—
149C4⋊C4—44no—
150C4⋊C4—44no—
151C4⋊C4—42no—
152C4⋊C4#4114yes—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C42—22no—
160C42—22no—
161C42—41no—
162C42—41no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23#12412yes—
175C23AGL(1,ℤ/8)12yes—
176C23#13712yes—
177C23C22×D411yescharacteristic
178Q8—42no—
179Q8—42no—
180D4—42no—
181D4—42no—
182D4—84no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—24no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—48no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—84no—
218C2×C4#12412yes—
219C2×C4C2×#4112yes—
220C2×C4C2×#4111yescharacteristic
221C2×C4C2×#4111yescharacteristic
222C2×C4H(5,𝔽2)12yes—
223C8—42no—
224C8—42no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22#48914yes—
239C22#52511yescharacteristic
240C22C2×#12411yescharacteristic
241C22#53411yescharacteristic
242C22C2×AGL(1,ℤ/8)11yescharacteristic
243C22C2×#13711yescharacteristic
244C22#57511yescharacteristic
245C22#57511yescharacteristic
246C4—22no—
247C4—44no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—41no—
252C4—81no—
253C4—81no—
254C4—81no—
255C4—82no—
256C4#55312yes—
257C4#55312yes—
258C2—24no—
259C2—161no—
260C2#259111yescharacteristic
261C2#311411yescharacteristic
262C2#311711yescharacteristic
263C2C2×#48911yescharacteristic
264C2#314211yescharacteristic
265C2#319411yescharacteristic
266C2#319611yescharacteristic
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
2. C2×#389C2semidirect
3. #2101C2semidirect
4. #2042C2semidirect
5. #1936C2semidirect
6. #1935C2semidirect
9. #1872C2semidirect
10. #1872C2semidirect
11. #1749C2semidirect

Maximal subgroups

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

subgroupindex
#28502
C2×#3892
#21012
#20422
#19362
#19352
#18742
#18722
#17492