Finite Group Databasebeta

← #20865

Subgroups — #20865

Order 256.

697 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3428C211yesmaximal, characteristic
2C2×#428C211yesmaximal, characteristic, complemented
3C2×#421C211yesmaximal, characteristic
4C2×#417C211yesmaximal, characteristic
5#2317C211yesmaximal, characteristic, complemented
6#1860C214yesmaximal, complemented
7#1859C214yesmaximal, complemented
8#1742C211yesmaximal, characteristic, complemented
9#1733C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C22×M16C2211yescharacteristic
12C22×M16C2211yescharacteristic
13#520—21no—
14#520—21no—
15#520C2212yes—
16#520C2212yes—
17#519—21no—
18#518—21no—
19#518—22no—
20C4×#41—22no—
21C4×#41—22no—
22C2×#117—22no—
23C2×#117C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C2×#117C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#428—22no—
28#428C2214yescomplemented
29C2×#107—21no—
30C2×#106—21no—
31#421C418yescomplemented
32C2×#104C2214yes—
33#417—22no—
34#417C2214yes—
35C2×#103—21no—
36C2×#102—21no—
37#389—22no—
38#385—22no—
39#384C2212yescomplemented
40#382C2212yescomplemented
41#343C2212yescomplemented
42#343C2212yescomplemented
43#339C2214yescomplemented
44C2×#41—21no—
45C2×#41—21no—
46C2×#41—22no—
47C2×#41—22no—
48C2×#41C2×C414yes—
49C2×#41D412yes—
50C2×#41Q814yes—
51C2×#41C2312yes—
52C22×Q8D411yescharacteristic
53C22×D4—21no—
54C22×D4D411yescharacteristic
55C23×C4—21no—
56C23×C4—21no—
57C23×C4C2311yescharacteristic
58C2×M16—21no—
59C2×M16—22no—
60C2×M16—21no—
61C2×M16—22no—
62C2×M16—22no—
63C2×M16C2×C414yescomplemented
64C2×M16C2312yes—
65C2×M16C2312yes—
66C22×C8—21no—
67C22×C8—21no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×D4—24no—
73C4×D4—24no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—24no—
79C4×D4—24no—
80#117—24no—
81#117—24no—
82#117—21no—
83#117—21no—
84#117—21no—
85#117—21no—
86#117—21no—
87#117—21no—
88#117—22no—
89#117—22no—
90#117—22no—
91#117C2×C414yescomplemented
92#117D412yes—
93#117C2312yes—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4D411yescharacteristic
105C2×C4⋊C4C2312yes—
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—24no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4D412yes—
112C2×C42—24no—
113C2×C42—21no—
114C2×C42D412yes—
115C2×C42C2312yes—
116#107—24no—
117#106—24no—
118#104C2×C4116yescomplemented
119#103—24no—
120#102—24no—
121#95—24no—
122C24—21no—
123C24C2×D411yescharacteristic
124#41—24no—
125#41—24no—
126#41—24no—
127#41—24no—
128#41—24no—
129#41—22no—
130#41—22no—
131#41C4⋊C418yescomplemented
132#41C22×C414yes—
133#41C2×Q814yes—
134C2×Q8—21no—
135C2×Q8—22no—
136C2×Q8C4⋊C414yescomplemented
137C2×Q8C2×D412yes—
138C2×D4—24no—
139C2×D4—22no—
140C2×D4—24no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4C4⋊C414yescomplemented
145C2×D4C2×D412yes—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—82no—
160C22×C4C22×C412yes—
161C22×C4C2×D412yes—
162C22×C4C2×D412yes—
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×Q812yes—
169C22×C4#4112yes—
170C22×C4#4112yes—
171C22×C4C2411yescharacteristic
172M16—24no—
173M16—22no—
174M16—22no—
175M16—22no—
176M16—42no—
177M16C22×C414yes—
178C2×C8—24no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—42no—
182C4⋊C4—24no—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—84no—
196C4⋊C4C2×D412yes—
197C4⋊C4C2×D412yes—
198C22⋊C4—24no—
199C22⋊C4—24no—
200C22⋊C4—24no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C42—24no—
206C42—24no—
207C42—22no—
208C42—22no—
209C42—22no—
210C42—22no—
211C42—22no—
212C42C22×C414yes—
213C23—22no—
214C23—24no—
215C23—41no—
216C23—41no—
217C23—81no—
218C23—81no—
219C23C4×D412yes—
220C23#12011yescharacteristic
221C23#12011yescharacteristic
222C23C22⋊Q812yes—
223C23C22×D411yescharacteristic
224Q8—22no—
225Q8—24no—
226Q8C2×C4⋊C414yes—
227D4—24no—
228D4—24no—
229D4—28no—
230D4—22no—
231D4—24no—
232D4C2×C4⋊C414yes—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—28no—
238C2×C4—24no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4C2×C4⋊C412yes—
256C2×C4C2×C4⋊C412yes—
257C2×C4C4×D412yes—
258C2×C4C4×D412yes—
259C2×C4C4×D412yes—
260C2×C4#12011yescharacteristic
261C2×C4#12011yescharacteristic
262C2×C4C22⋊Q812yes—
263C2×C4C22⋊Q812yes—
264C2×C4C22⋊Q812yes—
265C2×C4#12312yes—
266C2×C4#12312yes—
267C2×C4C23×C411yescharacteristic
268C2×C4C22×D411yescharacteristic
269C2×C4C22×D411yescharacteristic
270C2×C4C22×Q811yescharacteristic
271C2×C4C2×#4111yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C8—24no—
274C8—42no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—84no—
287C22#38512yes—
288C22#38512yes—
289C22C2×C4×D411yescharacteristic
290C22C2×#12011yescharacteristic
291C22C2×C22⋊Q811yescharacteristic
292C4—24no—
293C4—44no—
294C4—42no—
295C4—42no—
296C4—42no—
297C4—42no—
298C4—82no—
299C4#38512yes—
300C4#38512yes—
301C4C22×C4⋊C411yescharacteristic
302C4C2×C4×D411yescharacteristic
303C4C2×C22⋊Q811yescharacteristic
304C4C2×#12311yescharacteristic
305C2—21no—
306C2—22no—
307C2—21no—
308C2—22no—
309C2—44no—
310C2C2×#38511yescharacteristic
311C2#302111yescharacteristic
312C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#428C2semidirect
5. #2317C2semidirect
6. #1860C2semidirect
7. #1859C2semidirect
8. #1742C2semidirect
9. #1733C2semidirect
28. #428C22semidirect
31. #421C4semidirect
39. #384C22semidirect
40. #382C22semidirect
41. #343C22semidirect
42. #343C22semidirect
43. #339C22semidirect
63. C2×M16C2×C4semidirect
91. #117C2×C4semidirect
118. #104C2×C4semidirect
131. #41C4⋊C4semidirect
136. C2×Q8C4⋊C4semidirect
144. C2×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#34282
C2×#4282
C2×#4212
C2×#4172
#23172
#18602
#18592
#17422
#17332