Finite Group Databasebeta

← #27793

Subgroups — #27793

Order 256.

388 conjugacy classes of subgroups.

The subgroup lattice has 321 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#501C211yesmaximal, characteristic, complemented
2C2×#470C211yesmaximal, characteristic
3#2492C211yesmaximal, characteristic
4#2489C211yesmaximal, characteristic
5#2036C211yesmaximal, characteristic
6#2035C211yesmaximal, characteristic
7#2026C211yesmaximal, characteristic, complemented
8#2024C211yesmaximal, characteristic, complemented
9#1971C211yesmaximal, characteristic, complemented
10#1969C211yesmaximal, characteristic, complemented
11#1928C211yesmaximal, characteristic, complemented
12#1927C211yesmaximal, characteristic, complemented
13#1809C211yesmaximal, characteristic
14C4×#106C211yesmaximal, characteristic, complemented
15#1753C211yesmaximal, characteristic
16C22×Q16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×#125C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22#501—24no—
23#470—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×#97C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#400—21no—
39#400—21no—
40#397—21no—
41#393—21no—
42#391—21no—
43#391C2211yescharacteristic
44#391C2211yescharacteristic
45#391C2211yescharacteristic
46#391C2211yescharacteristic
47#388—21no—
48#388—21no—
49#382—21no—
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C23×C4C2311yescharacteristic
65C2×Q16—22no—
66C2×Q16—22no—
67C2×Q16—22no—
68C2×M16—22no—
69C2×M16—22no—
70C2×M16—22no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—22no—
76C4⋊Q8—41no—
77#125—22no—
78#125—41no—
79C22⋊Q8—22no—
80C22⋊Q8—22no—
81C22⋊Q8—41no—
82C22⋊Q8—41no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—22no—
106#106—22no—
107#106—22no—
108#106D414yes—
109#103—22no—
110#103—22no—
111#103—22no—
112#103—22no—
113#103—41no—
114#103—41no—
115C22⋊C8—22no—
116#97—22no—
117C4×C8—22no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95D411yescharacteristic
131#95D411yescharacteristic
132C24C2×D411yescharacteristic
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—81no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2411yescharacteristic
168Q16—44no—
169M16—44no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8C2×D412yes—
179C2×C8#4112yes—
180C2×C8#4112yes—
181C2×C8#4112yes—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4C2×D412yes—
198C4⋊C4C2×D412yes—
199C22⋊C4—22no—
200C22⋊C4—41no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—81no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—42no—
210C42—81no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23C22×D411yescharacteristic
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—84no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
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—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4#13712yes—
271C2×C4C22×D411yescharacteristic
272C2×C4C2×#4111yescharacteristic
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4H(5,𝔽2)11yescharacteristic
276C2×C42−1+411yescharacteristic
277C8—22no—
278C8—41no—
279C8—41no—
280C8#12414yes—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22#53211yescharacteristic
289C22D4211yescharacteristic
290C22#54711yescharacteristic
291C22C2×#13711yescharacteristic
292C22#57511yescharacteristic
293C22#57711yescharacteristic
294C22#57811yescharacteristic
295C4—22no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—42no—
300C4—41no—
301C4—42no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4C2×#12411yescharacteristic
309C4#54211yescharacteristic
310C4#54811yescharacteristic
311C4#55511yescharacteristic
312C2—81no—
313C2#249111yescharacteristic
314C2#317211yescharacteristic
315C2#317811yescharacteristic
316C2#327511yescharacteristic
317C2#327611yescharacteristic
318C2#334411yescharacteristic
319C2#334911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#501C2semidirect
7. #2026C2semidirect
8. #2024C2semidirect
9. #1971C2semidirect
10. #1969C2semidirect
11. #1928C2semidirect
12. #1927C2semidirect
14. C4×#106C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4702
#24922
#24892
#20362
#20352
#20262
#20242
#19712
#19692
#19282
#19272
#18092
C4×#1062
#17532