Finite Group Databasebeta

← #18393

Subgroups — #18393

Order 256.

455 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 113 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#3491C211yesmaximal, characteristic
2#3232C212yesmaximal, complemented
3#2611C211yesmaximal, characteristic, complemented
4#1615C212yesmaximal, complemented
5#1614C211yesmaximal, characteristic, complemented
6#1613C211yesmaximal, characteristic
7#1609C212yesmaximal, complemented
8#1608C211yesmaximal, characteristic, complemented
9#1607C211yesmaximal, characteristic
10#1500C212yesmaximal, complemented
11#1458C211yesmaximal, characteristic, complemented
12#552—22no—
13#552—22no—
14#551—22no—
15#551—22no—
16#543C2211yescharacteristic
17#539—21no—
18#539C2212yes—
19#534C2211yescharacteristic
20#529—21no—
21#529C2212yes—
22C2×#125C2211yescharacteristic, complemented
23C2×#123—21no—
24#518—21no—
25C2×#117C2211yescharacteristic
26#499—22no—
27#488—22no—
28#484C2212yescomplemented
29#483C2212yescomplemented
30#481C2212yes—
31#480C2212yes—
32#479—22no—
33#478—22no—
34#431C2212yescomplemented
35#400C2211yescharacteristic, complemented
36#399C2211yescharacteristic
37#398C2211yescharacteristic, complemented
38#397C2211yescharacteristic
39#389—21no—
40#388—21no—
41#387—21no—
42#385—21no—
43#376C2211yescharacteristic, complemented
44#328C2212yescomplemented
45#328C2212yescomplemented
46#327C2212yes—
47#327C2212yes—
48#324C2212yescomplemented
49#323C2211yescharacteristic
50#323C2211yescharacteristic
51C22×D4—21no—
52C23×C4D411yescharacteristic
53#126—22no—
54#126—22no—
55#126—22no—
56#126—22no—
57#125—21no—
58#125—22no—
59#125—22no—
60#125—22no—
61#125—22no—
62#125D412yes—
63#124—21no—
64#124D412yes—
65#123—22no—
66#123—22no—
67#123—22no—
68#123—22no—
69#123—22no—
70#123—22no—
71#123—22no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—21no—
76C22⋊Q8—22no—
77C22⋊Q8C2312yes—
78#121—21no—
79#121C2312yes—
80#120—22no—
81C4×Q8—21no—
82C4×Q8D412yes—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—21no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4D412yes—
89#117—22no—
90#117—22no—
91#117—22no—
92#117—22no—
93#117—22no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4D411yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C22⋊C4—22no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4D411yescharacteristic
111C2×C22⋊C4C2311yescharacteristic
112C2×C42—21no—
113C2×C42D411yescharacteristic
114C2×C42C2311yescharacteristic
115#107—22no—
116#107—22no—
117#106—22no—
118#106—22no—
119C4⋊C8—22no—
120#103—22no—
121#103—22no—
122#102—22no—
123#102—22no—
124C22⋊C8C2312yes—
125C22⋊C8C2312yes—
126C4×C8—22no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95C2311yescharacteristic
133#95C2311yescharacteristic
134C24—21no—
135C24C2×D411yescharacteristic
136C2×Q8—21no—
137C2×Q8C2×D412yes—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—21no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4C2×D412yes—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—41no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2411yescharacteristic
163C2×C8—22no—
164C2×C8—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D412yes—
186C22⋊C4—22no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C42—22no—
201C42—21no—
202C42—21no—
203C42—22no—
204C42—42no—
205C42C2×D412yes—
206C42C2×D412yes—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—81no—
214C23—81no—
215C23#13512yes—
216C23C22×D411yescharacteristic
217Q8—42no—
218Q8—81no—
219D4—42no—
220D4—44no—
221D4—42no—
222D4—42no—
223D4—81no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4#12011yescharacteristic
258C2×C4#12011yescharacteristic
259C2×C4#12012yes—
260C2×C4C22×D412yes—
261C8—42no—
262C8—42no—
263C22—21no—
264C22—21no—
265C22—22no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—81no—
273C22—162no—
274C22C2×#12011yescharacteristic
275C22C2×#13511yescharacteristic
276C22#57511yescharacteristic
277C4—21no—
278C4—21no—
279C4—22no—
280C4—22no—
281C4—42no—
282C4—41no—
283C4—42no—
284C4—41no—
285C4—42no—
286C4—41no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4—81no—
291C2—21no—
292C2—21no—
293C2—42no—
294C2—82no—
295C2#300211yescharacteristic
296C2#301811yescharacteristic
297C2#302811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#534C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3232C2semidirect
3. #2611C2semidirect
4. #1615C2semidirect
5. #1614C2semidirect
7. #1609C2semidirect
8. #1608C2semidirect
10. #1500C2semidirect
11. #1458C2semidirect
22. C2×#125C22semidirect
28. #484C22semidirect
29. #483C22semidirect
34. #431C22semidirect
35. #400C22semidirect
37. #398C22semidirect
43. #376C22semidirect
44. #328C22semidirect
45. #328C22semidirect
48. #324C22semidirect

Maximal subgroups

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

subgroupindex
#34912
#32322
#26112
#16152
#16142
#16132
#16092
#16082
#16072
#15002
#14582