Finite Group Databasebeta

← #29040

Subgroups — #29040

Order 256.

387 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2494C211yesmaximal, characteristic, complemented
2#2427C211yesmaximal, characteristic
3#2101C211yesmaximal, characteristic, complemented
4#2042C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic
7#1958C211yesmaximal, characteristic, complemented
8#1942C211yesmaximal, characteristic, complemented
9#1933C211yesmaximal, characteristic, complemented
10#1928C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic
12#1892C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15C8×C22⋊C4C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#103C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C4×C8C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#401—21no—
35#397—21no—
36#397—21no—
37#394—21no—
38#391C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#389C2211yescharacteristic
42#388—21no—
43#387—21no—
44#386—21no—
45#382C2211yescharacteristic
46#380—21no—
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C22⋊C4C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4C2311yescharacteristic
64C22×C8—22no—
65C22×C8—41no—
66C22×C8—41no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71#124—22no—
72#124—41no—
73C22⋊Q8—22no—
74C22⋊Q8—41no—
75#121—22no—
76#121—41no—
77C4×D4—22no—
78C4×D4—41no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4D411yescharacteristic
97C2×C22⋊C4D411yescharacteristic
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—22no—
102#107—41no—
103#103—22no—
104#103—22no—
105#103—41no—
106#103—41no—
107#102—22no—
108#102—22no—
109#102—22no—
110#102—41no—
111C22⋊C8—22no—
112C4×C8—41no—
113C4×C8—41no—
114#95—21no—
115#95—21no—
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#95D411yescharacteristic
125#95D411yescharacteristic
126C24—21no—
127C24C2×D411yescharacteristic
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—81no—
142C2×D4#4112yes—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
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—42no—
157C22×C4—42no—
158C22×C4—81no—
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2411yescharacteristic
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—44no—
176C2×C8#4112yes—
177C2×C8#4112yes—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4#4112yes—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—81no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—81no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—22no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23#13512yes—
222C23C22×D411yescharacteristic
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227D4—22no—
228D4—22no—
229D4—41no—
230D4—41no—
231D4—84no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
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—41no—
271C2×C4—84no—
272C2×C4#13512yes—
273C2×C4C2×#4111yescharacteristic
274C2×C4C2×#4111yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4H(5,𝔽2)11yescharacteristic
278C2×C4H(5,𝔽2)11yescharacteristic
279C8—22no—
280C8—22no—
281C8—41no—
282C8—41no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22#53211yescharacteristic
298C22#54611yescharacteristic
299C22#54611yescharacteristic
300C22C2×#13511yescharacteristic
301C22C2×#13511yescharacteristic
302C22#57611yescharacteristic
303C22#57811yescharacteristic
304C4—22no—
305C4—41no—
306C4—42no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4#52911yescharacteristic
317C4#53811yescharacteristic
318C4#55311yescharacteristic
319C4#55311yescharacteristic
320C2—42no—
321C2—82no—
322C2#255911yescharacteristic
323C2#316811yescharacteristic
324C2#317511yescharacteristic
325C2#329711yescharacteristic
326C2#329711yescharacteristic
327C2#330011yescharacteristic
328C2#330211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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. #2494C2semidirect
3. #2101C2semidirect
4. #2042C2semidirect
5. #2036C2semidirect
7. #1958C2semidirect
8. #1942C2semidirect
9. #1933C2semidirect
10. #1928C2semidirect
12. #1892C2semidirect
13. #1862C2semidirect
14. C4×#102C2semidirect
15. C8×C22⋊C4C2semidirect

Maximal subgroups

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

subgroupindex
#24942
#24272
#21012
#20422
#20362
#20332
#19582
#19422
#19332
#19282
#18932
#18922
#18622
C4×#1022
C8×C22⋊C42