Finite Group Databasebeta

← #21643

Subgroups — #21643

Order 256.

409 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#488C211yesmaximal, characteristic
2#2330C211yesmaximal, characteristic, complemented
3#2326C211yesmaximal, characteristic
4#1992C211yesmaximal, characteristic, complemented
5#1959C211yesmaximal, characteristic
6#1936C211yesmaximal, characteristic
7#1935C211yesmaximal, characteristic, complemented
8#1934C211yesmaximal, characteristic
9#1933C211yesmaximal, characteristic, complemented
10#1834C211yesmaximal, characteristic, complemented
11#1763C211yesmaximal, characteristic, complemented
12#1762C211yesmaximal, characteristic
13C4×#103C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic
15#1726C211yesmaximal, characteristic, complemented
16C2×#125C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×C4×Q8C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20#488C418yescomplemented
21C2×#103C2211yescharacteristic
22C2×#103C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#97C2211yescharacteristic
30C2×#97C2211yescharacteristic
31C2×#97C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#391C2211yescharacteristic
36#390C2211yescharacteristic
37#389—21no—
38#389—21no—
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#388—21no—
42#388—21no—
43#387—21no—
44#387—21no—
45#385—21no—
46#385—21no—
47#384C2211yescharacteristic
48#384C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51#376C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60C22×Q8C2311yescharacteristic
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67#125—22no—
68#125C2×C414yescomplemented
69#124—22no—
70#124C2×C414yes—
71C4×Q8—22no—
72C4×Q8—22no—
73C4×D4—22no—
74C4×D4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
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⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42—21no—
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#103—22no—
102#103—22no—
103#103C2×C414yescomplemented
104#103C2×C414yescomplemented
105#102—22no—
106#102—22no—
107#102C2×C414yes—
108#102C2×C414yes—
109#97—22no—
110#97—22no—
111#97—22no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8C2×C414yescomplemented
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95D411yescharacteristic
122#95D411yescharacteristic
123#95D411yescharacteristic
124#95D411yescharacteristic
125C24—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8C22×C412yes—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4C22×C412yes—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—42no—
146C22×C4—81no—
147C22×C4—81no—
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
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×C8C22×C412yes—
164C2×C8C22×C412yes—
165C2×C8#4112yes—
166C2×C8#4112yes—
167C4⋊C4—24no—
168C4⋊C4—24no—
169C4⋊C4—21no—
170C4⋊C4—21no—
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—42no—
180C4⋊C4—81no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C4⋊C4C22×C412yes—
187C4⋊C4C22×C412yes—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—42no—
191C22⋊C4—81no—
192C22⋊C4—81no—
193C42—21no—
194C42—21no—
195C42—42no—
196C42—81no—
197C42—81no—
198C42C22×C412yes—
199C42#4112yes—
200C42#4112yes—
201C42#4112yes—
202C23—22no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23C22×D411yescharacteristic
210Q8—22no—
211Q8—22no—
212Q8—41no—
213Q8—41no—
214D4—22no—
215D4—22no—
216D4—41no—
217D4—41no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
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—84no—
254C2×C4C4×D412yes—
255C2×C4C4×D412yes—
256C2×C4#12412yes—
257C2×C4#12412yes—
258C2×C4C23×C411yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4C2×#4111yescharacteristic
261C2×C4C2×#4111yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C8—22no—
265C8—22no—
266C8—22no—
267C8—22no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22C2×C4×D411yescharacteristic
276C22C2×#12411yescharacteristic
277C22#53211yescharacteristic
278C22#57611yescharacteristic
279C22#57711yescharacteristic
280C22#57711yescharacteristic
281C22#57811yescharacteristic
282C4—22no—
283C4—22no—
284C4—22no—
285C4—22no—
286C4—41no—
287C4—41no—
288C4—42no—
289C4—42no—
290C4—42no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4—81no—
295C4C4×#4111yescharacteristic
296C4C4×#4111yescharacteristic
297C4#52911yescharacteristic
298C4#52911yescharacteristic
299C2—82no—
300C2C4×#12411yescharacteristic
301C2#295111yescharacteristic
302C2#295211yescharacteristic
303C2#313611yescharacteristic
304C2#313711yescharacteristic
305C2#317811yescharacteristic
306C2#317911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2330C2semidirect
4. #1992C2semidirect
7. #1935C2semidirect
9. #1933C2semidirect
10. #1834C2semidirect
11. #1763C2semidirect
13. C4×#103C2semidirect
15. #1726C2semidirect
20. #488C4semidirect
68. #125C2×C4semidirect
103. #103C2×C4semidirect
104. #103C2×C4semidirect
114. C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4882
#23302
#23262
#19922
#19592
#19362
#19352
#19342
#19332
#18342
#17632
#17622
C4×#1032
C4×#1022
#17262