Finite Group Databasebeta

← #21589

Subgroups — #21589

Order 256.

422 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#485C211yesmaximal, characteristic
2#2323C211yesmaximal, characteristic
3#2274C211yesmaximal, characteristic
4#1928C211yesmaximal, characteristic, complemented
5#1928C211yesmaximal, characteristic, complemented
6#1920C211yesmaximal, characteristic, complemented
7#1918C211yesmaximal, characteristic, complemented
8#1872C211yesmaximal, characteristic
9#1852C211yesmaximal, characteristic
10#1825C211yesmaximal, characteristic
11#1776C211yesmaximal, characteristic, complemented
12C4×#107C211yesmaximal, characteristic, complemented
13#1763C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×#117C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21#485C418yescomplemented
22C2×#107C2211yescharacteristic
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×#97C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#391C2211yescharacteristic
35#389—21no—
36#389C2211yescharacteristic
37#388—21no—
38#388—21no—
39#387—21no—
40#380C2211yescharacteristic
41C4×C4⋊C4—21no—
42C4×C4⋊C4—21no—
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C22⋊C4—21no—
49C4×C22⋊C4—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C22×Q8C2311yescharacteristic
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22⋊Q8—22no—
68C22⋊Q8C2×C414yes—
69C4×Q8—22no—
70C4×Q8—22no—
71#117—22no—
72#117—22no—
73#117—22no—
74#117C2×C414yes—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4D411yescharacteristic
94C2×C22⋊C4D411yescharacteristic
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—22no—
106#107C2×C414yescomplemented
107#106C2×C414yescomplemented
108#103—22no—
109#103—22no—
110#103C2×C414yescomplemented
111#103C2×C414yescomplemented
112C22⋊C8—22no—
113C22⋊C8C2×C414yes—
114#97—22no—
115C4×C8—22no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125C24C2×D411yescharacteristic
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8C22×C412yes—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—24no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4C22×C412yes—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4#4112yes—
154C22×C4C2411yescharacteristic
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8C22×C412yes—
166C2×C8C22×C412yes—
167C4⋊C4—24no—
168C4⋊C4—24no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—42no—
181C4⋊C4—81no—
182C4⋊C4C22×C412yes—
183C4⋊C4C22×C412yes—
184C4⋊C4C22×C412yes—
185C4⋊C4#4112yes—
186C4⋊C4#4112yes—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—81no—
192C42—24no—
193C42—21no—
194C42—21no—
195C42—42no—
196C42—81no—
197C42—81no—
198C42—81no—
199C42—81no—
200C42—81no—
201C42#4112yes—
202C42#4112yes—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23C4×D412yes—
210C23C22×D411yescharacteristic
211Q8—22no—
212Q8—22no—
213Q8—41no—
214Q8—41no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
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—41no—
254C2×C4—41no—
255C2×C4—84no—
256C2×C4C4×D412yes—
257C2×C4#12312yes—
258C2×C4#12312yes—
259C2×C4C23×C411yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C8—22no—
266C8—22no—
267C8—41no—
268C8—41no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22C2×C4×D411yescharacteristic
277C22C2×#12311yescharacteristic
278C22#52511yescharacteristic
279C22#57611yescharacteristic
280C22#57711yescharacteristic
281C22#57811yescharacteristic
282C22#57811yescharacteristic
283C4—22no—
284C4—22no—
285C4—22no—
286C4—41no—
287C4—41no—
288C4—42no—
289C4—42no—
290C4—42no—
291C4—42no—
292C4—42no—
293C4—42no—
294C4—81no—
295C4—81no—
296C4C4×#4111yescharacteristic
297C4C4×#4111yescharacteristic
298C4#52911yescharacteristic
299C4#52911yescharacteristic
300C2—42no—
301C2C4×#12311yescharacteristic
302C2#294711yescharacteristic
303C2#295111yescharacteristic
304C2#309711yescharacteristic
305C2#309811yescharacteristic
306C2#311811yescharacteristic
307C2#312711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #1928C2semidirect
5. #1928C2semidirect
6. #1920C2semidirect
7. #1918C2semidirect
11. #1776C2semidirect
12. C4×#107C2semidirect
13. #1763C2semidirect
14. C4×#103C2semidirect
21. #485C4semidirect
106. #107C2×C4semidirect
107. #106C2×C4semidirect
110. #103C2×C4semidirect
111. #103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
#23232
#22742
#19282
#19202
#19182
#18722
#18522
#18252
#17762
C4×#1072
#17632
C4×#1032
#17502