Finite Group Databasebeta

← #29694

Subgroups — #29694

Order 256.

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

normal subgroups only

This group has 652 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×#532C211yesmaximal, characteristic
2C2×#478C212yesmaximal, complemented
3#2851C211yesmaximal, characteristic, complemented
4#2058C211yesmaximal, characteristic, complemented
5#2057C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic, complemented
8#2034C212yesmaximal, complemented
9#2033C212yesmaximal, complemented
10#1804C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13#532—28no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#127C2211yescharacteristic
16C2×#125C2211yescharacteristic
17C2×#124—21no—
18C2×#121—21no—
19C2×#121C2212yes—
20C2×C4×D4—21no—
21C2×C4×D4C2212yes—
22C22×C42C2211yescharacteristic
23#478—28no—
24C2×#107C2212yes—
25C2×#106C2212yes—
26C2×C4⋊C8C2212yes—
27C2×#102C2212yes—
28C2×#102C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×C22⋊C8C2212yes—
32#389C2211yescharacteristic
33#389C2211yescharacteristic
34#387—21no—
35#387—21no—
36#385—21no—
37#385—21no—
38#382C2211yescharacteristic
39#380C2211yescharacteristic
40#380C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43#336C2211yescharacteristic
44#336C2211yescharacteristic
45C2×#41—28no—
46C2×#41—44no—
47C2×#41—44no—
48C22×Q8—21no—
49C22×D4—22no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4C2312yes—
53C23×C4—22no—
54C23×C4D411yescharacteristic
55C23×C4D411yescharacteristic
56C23×C4C2311yescharacteristic
57C22×C8C2312yes—
58C22×C8C2312yes—
59C4⋊Q8—24no—
60#127—24no—
61#125—24no—
62#124—28no—
63#121—28no—
64#121—24no—
65#121—24no—
66C4×D4—28no—
67C4×D4—24no—
68C4×D4D418yes—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2312yes—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C42—24no—
84C2×C42D411yescharacteristic
85C2×C42D411yescharacteristic
86C2×C42C2311yescharacteristic
87C2×C42C2311yescharacteristic
88#107—24no—
89#106—24no—
90C4⋊C8—24no—
91#102—24no—
92#102—24no—
93#102—42no—
94#102—42no—
95C22⋊C8—42no—
96#95—21no—
97#95—21no—
98#95—21no—
99#95—21no—
100#95—21no—
101#95—21no—
102C24—22no—
103C24C2×D411yescharacteristic
104#41—432no—
105C2×Q8—24no—
106C2×Q8—42no—
107C2×Q8—42no—
108C2×D4—24no—
109C2×D4—28no—
110C2×D4—24no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—22no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4C2×D414yes—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—28no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4—44no—
140C22×C4—44no—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2411yescharacteristic
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4C2×D414yes—
166C22⋊C4—28no—
167C22⋊C4—28no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C42—22no—
171C42—22no—
172C42—22no—
173C42C2×D414yes—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—24no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23—42no—
185C23—42no—
186C23—42no—
187C23AGL(1,ℤ/8)12yes—
188C23C22×D411yescharacteristic
189Q8—48no—
190D4—24no—
191D4—24no—
192D4—48no—
193D4—48no—
194D4—416no—
195D4—42no—
196D4—42no—
197C2×C4—28no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—44no—
211C2×C4—416no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4#12012yes—
229C2×C4#12012yes—
230C2×C4#13512yes—
231C2×C4AGL(1,ℤ/8)12yes—
232C2×C4C22×D412yes—
233C2×C4H(5,𝔽2)11yescharacteristic
234C2×C42−1+411yescharacteristic
235C2×C42−1+412yes—
236C8—42no—
237C8—42no—
238C22—22no—
239C22—22no—
240C22—22no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—44no—
246C22—44no—
247C22—44no—
248C22—82no—
249C22—82no—
250C22—82no—
251C22—82no—
252C22C2×#12011yescharacteristic
253C22#53611yescharacteristic
254C22#53711yescharacteristic
255C22C2×#13511yescharacteristic
256C22C2×AGL(1,ℤ/8)11yescharacteristic
257C22C2×AGL(1,ℤ/8)11yescharacteristic
258C22#57511yescharacteristic
259C4—22no—
260C4—22no—
261C4—22no—
262C4—22no—
263C4—41no—
264C4—44no—
265C4—81no—
266C4—81no—
267C4—81no—
268C4—81no—
269C4#53612yes—
270C4#53712yes—
271C2—42no—
272C2—84no—
273C2#261611yescharacteristic
274C2#300211yescharacteristic
275C2#300311yescharacteristic
276C2#322811yescharacteristic
277C2#323411yescharacteristic
278C2#324711yescharacteristic
279C2#325111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#532C2
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×#478C2semidirect
3. #2851C2semidirect
4. #2058C2semidirect
5. #2057C2semidirect
6. #2041C2semidirect
7. #2041C2semidirect
8. #2034C2semidirect
9. #2033C2semidirect
10. #1804C2semidirect
11. #1797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
C2×#4782
#28512
#20582
#20572
#20412
#20342
#20332
#18042
#17972