Finite Group Databasebeta

← #30839

Subgroups — #30839

Order 256.

321 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#3163C211yesmaximal, characteristic, complemented
2#3136C211yesmaximal, characteristic
3#3096C211yesmaximal, characteristic
4#3083C211yesmaximal, characteristic, complemented
5#2092C211yesmaximal, characteristic, complemented
6#2079C211yesmaximal, characteristic, complemented
7#2063C211yesmaximal, characteristic, complemented
8#2061C211yesmaximal, characteristic, complemented
9#2037C211yesmaximal, characteristic
10#2027C211yesmaximal, characteristic, complemented
11#1952C211yesmaximal, characteristic, complemented
12#1931C211yesmaximal, characteristic, complemented
13#1929C211yesmaximal, characteristic, complemented
14#1922C211yesmaximal, characteristic, complemented
15#1860C211yesmaximal, characteristic, complemented
16#544C2211yescharacteristic
17#544C2211yescharacteristic
18#535C2211yescharacteristic
19#520C2211yescharacteristic
20#501—21no—
21#501—21no—
22#499—21no—
23#498—21no—
24#490—21no—
25#489—21no—
26#488—21no—
27#486—21no—
28#482—21no—
29#482—21no—
30#481—21no—
31#480—21no—
32#479—21no—
33#478—21no—
34#477—21no—
35#474—21no—
36C2×#108C2211yescharacteristic
37#428C2211yescharacteristic
38#428C2211yescharacteristic
39#428C2211yescharacteristic
40#428C2211yescharacteristic
41#427C2211yescharacteristic
42C2×#107C2211yescharacteristic
43C2×#106C2211yescharacteristic
44#424C2211yescharacteristic
45C2×#104C2211yescharacteristic
46#418C2211yescharacteristic
47#418C2211yescharacteristic
48#417C2211yescharacteristic
49#417C2211yescharacteristic
50#416C2211yescharacteristic
51C2×#102C2211yescharacteristic
52C2×#102C2211yescharacteristic
53C2×#100C2211yescharacteristic
54#408C2211yescharacteristic
55#405C2211yescharacteristic
56#405C2211yescharacteristic
57#390C2211yescharacteristic
58#382C2211yescharacteristic
59#343C2211yescharacteristic
60#343C2211yescharacteristic
61#343C2211yescharacteristic
62#341C2211yescharacteristic
63#339C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#336C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C2×M16C2311yescharacteristic
70C2×M16C2311yescharacteristic
71C2×M16C2311yescharacteristic
72C2×M16C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77C4⋊Q8—21no—
78C4⋊Q8—21no—
79#127—21no—
80#125—21no—
81#125—21no—
82#125—21no—
83#125—21no—
84#124—21no—
85C22⋊Q8—41no—
86C22⋊Q8—41no—
87#121—21no—
88#121—21no—
89#120—41no—
90C4×Q8—21no—
91C4×D4—21no—
92C4×D4—41no—
93#117—21no—
94#117C2311yescharacteristic
95#117C2311yescharacteristic
96#117C2311yescharacteristic
97C2×C4⋊C4—21no—
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C22⋊C4—41no—
102C2×C42C2311yescharacteristic
103#108—22no—
104#107—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107—21no—
110#107D411yescharacteristic
111#107D411yescharacteristic
112#106—21no—
113#106—21no—
114#106—21no—
115#106—21no—
116#106—21no—
117#106—21no—
118#106D411yescharacteristic
119#106D411yescharacteristic
120C4⋊C8—21no—
121C4⋊C8—21no—
122#104—22no—
123#103—21no—
124#103—21no—
125#103D411yescharacteristic
126#103D411yescharacteristic
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102—21no—
135#102—21no—
136#102—21no—
137#102D411yescharacteristic
138#102D411yescharacteristic
139#100—22no—
140C22⋊C8—21no—
141C22⋊C8—21no—
142#97—21no—
143#97—21no—
144C4×C8—21no—
145C4×C8—21no—
146#95—41no—
147C24—21no—
148#41—21no—
149#41—21no—
150#41—21no—
151#41—21no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8C2×D411yescharacteristic
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4C2×D411yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—41no—
171C22×C4C2411yescharacteristic
172M16—21no—
173M16—21no—
174M16—21no—
175M16—21no—
176M16—21no—
177M16—21no—
178M16—21no—
179M16#4112yes—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
191C2×C8C2×D411yescharacteristic
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C4⋊C4C2×D411yescharacteristic
208C4⋊C4C2×D411yescharacteristic
209C4⋊C4C2×D411yescharacteristic
210C4⋊C4C2×D411yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—82no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—41no—
224C23—21no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—81no—
229C23—81no—
230C23H(5,𝔽2)11yescharacteristic
231Q8—21no—
232Q8—21no—
233Q8—161no—
234Q8—161no—
235D4—21no—
236D4—21no—
237D4—41no—
238D4—41no—
239D4—41no—
240D4—41no—
241D4—41no—
242D4—41no—
243D4—161no—
244D4—161no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
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—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—161no—
264C2×C4C22×D411yescharacteristic
265C2×C4C22×D411yescharacteristic
266C2×C4C2×#4111yescharacteristic
267C2×C42−1+411yescharacteristic
268C2×C42−1+411yescharacteristic
269C2×C42−1+411yescharacteristic
270C8—21no—
271C8—21no—
272C8—21no—
273C8—21no—
274C8—22no—
275C8—41no—
276C8—41no—
277C8—41no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—82no—
284C22—81no—
285C22—81no—
286C22—81no—
287C22—161no—
288C22#54211yescharacteristic
289C22D4211yescharacteristic
290C22#56311yescharacteristic
291C4—41no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—81no—
299C4—81no—
300C4—81no—
301C4—81no—
302C4#53611yescharacteristic
303C4#53611yescharacteristic
304C4#54711yescharacteristic
305C4#54811yescharacteristic
306C2—21no—
307C2—21no—
308C2—81no—
309C2—161no—
310C2#267411yescharacteristic
311C2#328811yescharacteristic
312C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC22×C4C24
prime radical#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3163C2semidirect
4. #3083C2semidirect
5. #2092C2semidirect
6. #2079C2semidirect
7. #2063C2semidirect
8. #2061C2semidirect
10. #2027C2semidirect
11. #1952C2semidirect
12. #1931C2semidirect
13. #1929C2semidirect
14. #1922C2semidirect
15. #1860C2semidirect

Maximal subgroups

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

subgroupindex
#31632
#31362
#30962
#30832
#20922
#20792
#20632
#20612
#20372
#20272
#19522
#19312
#19292
#19222
#18602