Finite Group Databasebeta

← #18270

Subgroups — #18270

Order 256.

500 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 115 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#3536C211yesmaximal, characteristic
2#3092C211yesmaximal, characteristic, complemented
3C2×#480C211yesmaximal, characteristic, complemented
4#2502C211yesmaximal, characteristic, complemented
5#1610C214yesmaximal, complemented
6#1609C214yesmaximal, complemented
7#1507C211yesmaximal, characteristic, complemented
8#1501C211yesmaximal, characteristic, complemented
9#1472C211yesmaximal, characteristic, complemented
10#562—22no—
11#554—21no—
12#548—21no—
13#547—21no—
14#546—21no—
15#546—22no—
16#543—21no—
17#542—21no—
18#539C2212yes—
19#538—21no—
20#537—21no—
21#537C2212yes—
22#525C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26#484—21no—
27#483C2212yescomplemented
28#481—21no—
29#480—22no—
30#480C2214yes—
31#480C2212yescomplemented
32#428—21no—
33C2×#107—21no—
34#417—21no—
35C2×#102—21no—
36#407C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38#398C2214yescomplemented
39#391—21no—
40#385—22no—
41#384—21no—
42C4×C22⋊C4C2211yescharacteristic
43#328C2212yescomplemented
44#327C2214yes—
45#327C2212yescomplemented
46#324C2214yescomplemented
47C2×#41—21no—
48C2×#41—22no—
49C2×#41D411yescharacteristic
50C22×Q8D411yescharacteristic
51C22×D4—21no—
52C22×D4D411yescharacteristic
53C23×C4—21no—
54C23×C4C2311yescharacteristic
55C2×M16—21no—
56C22×C8—21no—
57C4⋊Q8—21no—
58#126—22no—
59#125—22no—
60#124—22no—
61#124—21no—
62#124—22no—
63#123—22no—
64#123—22no—
65#123—22no—
66#123—21no—
67#123—22no—
68C22⋊Q8—21no—
69C22⋊Q8—22no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73C22⋊Q8—21no—
74C22⋊Q8—21no—
75C22⋊Q8—21no—
76C22⋊Q8—21no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8D411yescharacteristic
80C22⋊Q8C2311yescharacteristic
81#121—21no—
82#121—22no—
83#121—22no—
84#121—22no—
85#121—22no—
86#121—21no—
87#121—21no—
88#121—22no—
89#121D412yes—
90#121D411yescharacteristic
91#121C2312yes—
92#121C2311yescharacteristic
93#120—22no—
94#120—21no—
95C4×Q8—21no—
96C4×D4—22no—
97C4×D4—21no—
98C4×D4—21no—
99C4×D4—21no—
100C4×D4—22no—
101#117D411yescharacteristic
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4D411yescharacteristic
111C2×C4⋊C4C2312yes—
112C2×C4⋊C4C2312yes—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4D412yes—
117C2×C42—21no—
118#107—24no—
119#107—22no—
120#106—22no—
121#103—22no—
122#102—24no—
123#102—22no—
124C22⋊C8—21no—
125C22⋊C8—21no—
126C22⋊C8C2312yes—
127C22⋊C8C2312yes—
128#95—24no—
129#95C2312yes—
130C24—21no—
131C24C2×D411yescharacteristic
132#41—42no—
133#41—84no—
134#41—82no—
135C2×Q8—22no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—42no—
139C2×Q8C2×D411yescharacteristic
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—21no—
146C2×D4—22no—
147C2×D4—21no—
148C2×D4—42no—
149C2×D4—82no—
150C2×D4C2×D411yescharacteristic
151C2×D4C2×D412yes—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—42no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—82no—
163C22×C4C2×D412yes—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D412yes—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171M16—42no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—42no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—84no—
191C4⋊C4C2×D411yescharacteristic
192C4⋊C4C2×D412yes—
193C4⋊C4C2×D411yescharacteristic
194C22⋊C4—24no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—22no—
200C22⋊C4—21no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—21no—
207C22⋊C4—22no—
208C22⋊C4—21no—
209C42—22no—
210C42—21no—
211C42—42no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—21no—
216C23—41no—
217C23—42no—
218C23—81no—
219C23—81no—
220C23#12011yescharacteristic
221C23#12011yescharacteristic
222C23C22×D411yescharacteristic
223Q8—41no—
224Q8—42no—
225Q8—81no—
226Q8—82no—
227D4—42no—
228D4—41no—
229D4—44no—
230D4—42no—
231D4—42no—
232D4—42no—
233D4—82no—
234D4—81no—
235D4—82no—
236D4—82no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—81no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4#12011yescharacteristic
264C2×C4#12011yescharacteristic
265C2×C4C22×D411yescharacteristic
266C2×C4C22×D411yescharacteristic
267C8—42no—
268C8—42no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—44no—
277C22—41no—
278C22—42no—
279C22—41no—
280C22—42no—
281C22—41no—
282C22—42no—
283C22—162no—
284C22#45612yes—
285C22C2×#12011yescharacteristic
286C22#57612yes—
287C4—21no—
288C4—21no—
289C4—21no—
290C4—44no—
291C4—42no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—42no—
297C4—82no—
298C4#45812yes—
299C2—22no—
300C2—22no—
301C2—41no—
302C2—81no—
303C2—82no—
304C2#300811yescharacteristic
305C2C2×#45611yescharacteristic
306C2C2×#45811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3536C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3092C2semidirect
3. C2×#480C2semidirect
4. #2502C2semidirect
5. #1610C2semidirect
6. #1609C2semidirect
7. #1507C2semidirect
8. #1501C2semidirect
9. #1472C2semidirect
27. #483C22semidirect
31. #480C22semidirect
38. #398C22semidirect
43. #328C22semidirect
45. #327C22semidirect
46. #324C22semidirect

Maximal subgroups

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

subgroupindex
#35362
#30922
C2×#4802
#25022
#16102
#16092
#15072
#15012
#14722