Finite Group Databasebeta

← #29113

Subgroups — #29113

Order 256.

455 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 119 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#2857C211yesmaximal, characteristic, complemented
2#2297C211yesmaximal, characteristic
3#2072C211yesmaximal, characteristic, complemented
4#2071C211yesmaximal, characteristic, complemented
5#2024C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic, complemented
7#1936C212yesmaximal, complemented
8#1935C212yesmaximal, complemented
9#1825C212yesmaximal, complemented
10#1788C212yesmaximal
11#1749C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8—21no—
15C2×#117C2211yescharacteristic
16C22×C42C2211yescharacteristic
17C2×#103C2212yes—
18C2×#103C2212yes—
19C2×#102C2212yes—
20C2×#102C2212yes—
21C2×C22⋊C8C2212yes—
22C2×#97C2212yes—
23#391—21no—
24#391C2211yescharacteristic
25#391C2211yescharacteristic
26#390C2211yescharacteristic
27#390C2211yescharacteristic
28#389—21no—
29#389C2212yes—
30#386—21no—
31#386—21no—
32#384C2211yescharacteristic
33#380—21no—
34#380C2212yes—
35C4×C22⋊C4—21no—
36C4×C22⋊C4—21no—
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C2×#41—22no—
44C2×#41—22no—
45C2×#41—24no—
46C2×#41—42no—
47C22×Q8—21no—
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4D411yescharacteristic
53C23×C4D411yescharacteristic
54C23×C4C2311yescharacteristic
55C22×C8C2312yes—
56C22×C8C2312yes—
57C4⋊Q8—42no—
58C22⋊Q8—44no—
59#117—24no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C42—21no—
75C2×C42—42no—
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80#103—24no—
81#103—42no—
82#102—24no—
83#102—42no—
84C22⋊C8—24no—
85#97—42no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—21no—
90#95—22no—
91#95—22no—
92C24—21no—
93C24C2×D411yescharacteristic
94#41—48no—
95#41—48no—
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—42no—
100C2×Q8—42no—
101C2×Q8—42no—
102C2×Q8#4112yes—
103C2×D4—24no—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—42no—
108C2×D4—42no—
109C2×D4#4112yes—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—22no—
113C22×C4—24no—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—41no—
122C22×C4—41no—
123C22×C4—41no—
124C22×C4—41no—
125C22×C4—42no—
126C22×C4—42no—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4#4114yes—
133C22×C4C2411yescharacteristic
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—22no—
139C2×C8—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—42no—
147C4⋊C4—44no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—44no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C42—22no—
157C42—22no—
158C42—41no—
159C42—41no—
160C42—42no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23#12412yes—
175C23C22×D411yescharacteristic
176Q8—24no—
177Q8—42no—
178Q8—84no—
179D4—24no—
180D4—48no—
181D4—42no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—48no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—42no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4#12412yes—
215C2×C4AGL(1,ℤ/8)12yes—
216C2×C4#13712yes—
217C2×C4C2×#4111yescharacteristic
218C2×C4C2×#4111yescharacteristic
219C2×C4C2×#4112yes—
220C2×C4H(5,𝔽2)12yes—
221C8—42no—
222C8—42no—
223C22—22no—
224C22—22no—
225C22—22no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22#52511yescharacteristic
238C22C2×#12411yescharacteristic
239C22#53411yescharacteristic
240C22C2×AGL(1,ℤ/8)11yescharacteristic
241C22C2×#13711yescharacteristic
242C22#57511yescharacteristic
243C22#57511yescharacteristic
244C4—22no—
245C4—22no—
246C4—22no—
247C4—42no—
248C4—41no—
249C4—41no—
250C4—82no—
251C4—81no—
252C4—81no—
253C4—82no—
254C4#55112yes—
255C4#55112yes—
256C2—42no—
257C2—82no—
258C2#256111yescharacteristic
259C2#311511yescharacteristic
260C2#311611yescharacteristic
261C2#313511yescharacteristic
262C2#314111yescharacteristic
263C2#319411yescharacteristic
264C2#319611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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. #2857C2semidirect
3. #2072C2semidirect
4. #2071C2semidirect
5. #2024C2semidirect
6. #2023C2semidirect
7. #1936C2semidirect
8. #1935C2semidirect
9. #1825C2semidirect
11. #1749C2semidirect

Maximal subgroups

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

subgroupindex
#28572
#22972
#20722
#20712
#20242
#20232
#19362
#19352
#18252
#17882
#17492