Finite Group Databasebeta

← #25716

Subgroups — #25716

Order 256.

529 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 529 conjugacy classes of subgroups — too many to draw. Only its 133 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×#552C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic, complemented
3C2×#477C211yesmaximal, characteristic, complemented
4#2547C211yesmaximal, characteristic
5#2036C212yesmaximal, complemented
6#2035C212yesmaximal, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic
9#1854C212yesmaximal, complemented
10#1852C212yesmaximal, complemented
11#1763C211yesmaximal, characteristic, complemented
12#552—24no—
13#552—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#126—21no—
16C2×#125—21no—
17C2×#125C2211yescharacteristic
18C2×#124—21no—
19C2×#123—21no—
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117—21no—
25C2×#117C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#501—24no—
28#477—24no—
29C2×#107C2212yes—
30C2×#107C2212yes—
31C2×#106C2212yes—
32C2×C4⋊C8C2211yescharacteristic
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×C22⋊C8C2212yes—
36C2×C22⋊C8C2212yes—
37C2×#97C2211yescharacteristic
38#397—21no—
39#397—21no—
40#391—21no—
41#380C2212yes—
42#380C2212yes—
43C4×C4⋊C4C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#340C2212yes—
47C22×Q8—21no—
48C22×Q8C2311yescharacteristic
49C22×D4D411yescharacteristic
50C23×C4—21no—
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C22×C8C2312yes—
54C22×C8C2312yes—
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57#126—28no—
58#125—28no—
59#125—22no—
60#125D414yes—
61#124—44no—
62#123—28no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C4×Q8—22no—
68C4×Q8D414yes—
69C4×D4—22no—
70C4×D4—22no—
71#117—28no—
72#117—22no—
73#117—22no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4D411yescharacteristic
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100#107—24no—
101#107—24no—
102#106—24no—
103C4⋊C8—22no—
104#103—24no—
105#103—24no—
106C22⋊C8—24no—
107C22⋊C8—24no—
108#97—22no—
109#95—21no—
110#95—22no—
111#95—22no—
112C24C2×D411yescharacteristic
113C24C2×D411yescharacteristic
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8C2×D412yes—
121C2×D4—22no—
122C2×D4—22no—
123C2×D4—41no—
124C22×C4—24no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2411yescharacteristic
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8#4114yes—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4C2×D412yes—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D412yes—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—44no—
176C22⋊C4—44no—
177C42—24no—
178C42—24no—
179C42—42no—
180C42C2×D412yes—
181C42C2×D412yes—
182C23—21no—
183C23—22no—
184C23—21no—
185C23—22no—
186C23—21no—
187C23—22no—
188C23—21no—
189C23—21no—
190C23—22no—
191C23AGL(1,ℤ/8)12yes—
192C23C22×D411yescharacteristic
193Q8—22no—
194Q8—22no—
195Q8—42no—
196Q8—84no—
197D4—44no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—48no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4#12012yes—
229C2×C4#12012yes—
230C2×C4C22×D411yescharacteristic
231C2×C4C22×D411yescharacteristic
232C2×C4C2×#4112yes—
233C2×C42−1+412yes—
234C8—24no—
235C8—42no—
236C22—24no—
237C22—22no—
238C22—42no—
239C22—41no—
240C22—41no—
241C22—42no—
242C22—42no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22C2×#12011yescharacteristic
247C22#54712yes—
248C22C2×AGL(1,ℤ/8)11yescharacteristic
249C22#57511yescharacteristic
250C22#57812yes—
251C4—22no—
252C4—22no—
253C4—41no—
254C4—41no—
255C4—42no—
256C4—42no—
257C4—42no—
258C4—42no—
259C4—82no—
260C4—82no—
261C4#53211yescharacteristic
262C4#53611yescharacteristic
263C4#54812yes—
264C2—42no—
265C2—81no—
266C2#243211yescharacteristic
267C2#300611yescharacteristic
268C2#301111yescharacteristic
269C2#334312yes—
270C2#334912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#501C2semidirect
3. C2×#477C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #1989C2semidirect
9. #1854C2semidirect
10. #1852C2semidirect
11. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5522
C2×#5012
C2×#4772
#25472
#20362
#20352
#19892
#19552
#18542
#18522
#17632