Finite Group Databasebeta

← #24067

Subgroups — #24067

Order 256.

588 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 141 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×#529C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic, complemented
3C2×#481C211yesmaximal, characteristic
4C2×#389C211yesmaximal, characteristic, complemented
5#2059C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#1874C211yesmaximal, characteristic
8#1872C211yesmaximal, characteristic, complemented
9#1865C212yesmaximal, complemented
10#1862C212yesmaximal, complemented
11#1817C211yesmaximal, characteristic, complemented
12#529—24no—
13#529—24no—
14C2×#126—21no—
15C2×#125C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×#123—21no—
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4C2211yescharacteristic
23C2×#117—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C42C2211yescharacteristic
28#484—24no—
29#481—24no—
30C2×#107C2212yes—
31C2×#106C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38#389—21no—
39#389—21no—
40#389C2212yes—
41#389C2212yes—
42C2×#95—21no—
43#340C2212yes—
44#340C2212yes—
45#336C2212yes—
46#336C2212yes—
47C22×Q8C2311yescharacteristic
48C22×D4—21no—
49C22×D4C2311yescharacteristic
50C23×C4—21no—
51C23×C4—21no—
52C23×C4—21no—
53C23×C4D411yescharacteristic
54C23×C4D411yescharacteristic
55C23×C4C2311yescharacteristic
56C22×C8C2312yes—
57C22×C8C2312yes—
58#126—28no—
59#125—22no—
60#125—22no—
61#124—22no—
62#124—22no—
63#123—28no—
64C22⋊Q8—22no—
65C22⋊Q8D414yes—
66#121—22no—
67#121D414yes—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×D4—28no—
71C4×D4—22no—
72C4×D4—22no—
73#117—28no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—24no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—44no—
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2312yes—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42D411yescharacteristic
99C2×C42D411yescharacteristic
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#107—24no—
103#106—24no—
104#103—24no—
105#103—24no—
106#102—24no—
107#102—24no—
108C22⋊C8—22no—
109C22⋊C8—22no—
110#95—22no—
111#95—22no—
112C24—21no—
113C24C2×D411yescharacteristic
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8C2×D412yes—
118C2×D4—24no—
119C2×D4—21no—
120C2×D4—22no—
121C2×D4—22no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4C2×D412yes—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—24no—
128C22×C4—24no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D412yes—
152C22×C4C2×D412yes—
153C22×C4C2×D411yescharacteristic
154C22×C4C2411yescharacteristic
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—22no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—44no—
172C4⋊C4—42no—
173C4⋊C4—44no—
174C4⋊C4C2×D412yes—
175C4⋊C4C2×D412yes—
176C4⋊C4#4114yes—
177C22⋊C4—24no—
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—24no—
181C22⋊C4—24no—
182C42—24no—
183C42—22no—
184C42—22no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—22no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23#12012yes—
201C23#13512yes—
202C23#13712yes—
203C23C22×D411yescharacteristic
204Q8—24no—
205Q8—42no—
206D4—24no—
207D4—48no—
208D4—42no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—44no—
229C2×C4—48no—
230C2×C4—48no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—44no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—84no—
244C2×C4#12012yes—
245C2×C4AGL(1,ℤ/8)12yes—
246C2×C4C22×D411yescharacteristic
247C2×C4C22×D411yescharacteristic
248C2×C4C2×#4112yes—
249C2×C42−1+412yes—
250C8—42no—
251C8—42no—
252C22—22no—
253C22—22no—
254C22—22no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22#45214yes—
266C22C2×#12011yescharacteristic
267C22#52511yescharacteristic
268C22#53611yescharacteristic
269C22C2×#13511yescharacteristic
270C22C2×AGL(1,ℤ/8)11yescharacteristic
271C22C2×#13711yescharacteristic
272C22#57511yescharacteristic
273C4—22no—
274C4—22no—
275C4—44no—
276C4—42no—
277C4—41no—
278C4—41no—
279C4—42no—
280C4—41no—
281C4—42no—
282C4—82no—
283C4#54712yes—
284C4#54712yes—
285C2—24no—
286C2—82no—
287C2#240611yescharacteristic
288C2C2×#45211yescharacteristic
289C2#300611yescharacteristic
290C2#310511yescharacteristic
291C2#311711yescharacteristic
292C2#323211yescharacteristic
293C2#323911yescharacteristic
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×#484C2semidirect
4. C2×#389C2semidirect
5. #2059C2semidirect
6. #2042C2semidirect
8. #1872C2semidirect
9. #1865C2semidirect
10. #1862C2semidirect
11. #1817C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4842
C2×#4812
C2×#3892
#20592
#20422
#18742
#18722
#18652
#18622
#18172