Finite Group Databasebeta

← #27807

Subgroups — #27807

Order 256.

355 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3163C211yesmaximal, characteristic, complemented
2#3137C211yesmaximal, characteristic, complemented
3#3136C211yesmaximal, characteristic
4#3057C211yesmaximal, characteristic
5#2038C211yesmaximal, characteristic, complemented
6#2037C211yesmaximal, characteristic
7#2028C211yesmaximal, characteristic, complemented
8#2027C211yesmaximal, characteristic, complemented
9#1977C211yesmaximal, characteristic, complemented
10#1968C211yesmaximal, characteristic, complemented
11#1931C211yesmaximal, characteristic, complemented
12#1930C211yesmaximal, characteristic, complemented
13#1815C211yesmaximal, characteristic, complemented
14#1772C211yesmaximal, characteristic, complemented
15#1764C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×#131C2211yescharacteristic
18#544C2211yescharacteristic
19#544C2211yescharacteristic
20#536C2211yescharacteristic
21#535C2211yescharacteristic
22#501—21no—
23#501—21no—
24#499—21no—
25#498—21no—
26#491—21no—
27#490—21no—
28#489—21no—
29#489—21no—
30#488—21no—
31#488—21no—
32#487—21no—
33#486—21no—
34#468—21no—
35#468—21no—
36#465—21no—
37#465—21no—
38C2×#108C2211yescharacteristic
39#428C2211yescharacteristic
40#428C2211yescharacteristic
41#427C2211yescharacteristic
42C2×#106C2211yescharacteristic
43C2×#104C2211yescharacteristic
44#419C2211yescharacteristic
45#419C2211yescharacteristic
46#418C2211yescharacteristic
47#418C2211yescharacteristic
48#417C2211yescharacteristic
49#417C2211yescharacteristic
50#416C2211yescharacteristic
51C2×#103C2211yescharacteristic
52C2×#102C2211yescharacteristic
53C2×#101C2211yescharacteristic
54C2×#100C2211yescharacteristic
55#405C2211yescharacteristic
56#405C2211yescharacteristic
57#405C2211yescharacteristic
58C2×C4×C8C2211yescharacteristic
59#391C2211yescharacteristic
60#390C2211yescharacteristic
61#343C2211yescharacteristic
62#343C2211yescharacteristic
63#341C2211yescharacteristic
64#339C2211yescharacteristic
65#339C2211yescharacteristic
66#339C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C22×D4C2311yescharacteristic
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—22no—
73C2×SD16—22no—
74#131—22no—
75#131—22no—
76C2×M16—21no—
77C2×M16C2311yescharacteristic
78C2×M16C2311yescharacteristic
79C2×M16C2311yescharacteristic
80C2×M16C2311yescharacteristic
81C22×C8—21no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C4⋊Q8—21no—
85C4⋊Q8—21no—
86C4⋊Q8—21no—
87C4⋊Q8—21no—
88C4⋊Q8—21no—
89#127—21no—
90#125—21no—
91#125—21no—
92#125—21no—
93#125—21no—
94#124—21no—
95#124—21no—
96#123—41no—
97C22⋊Q8—21no—
98C22⋊Q8—21no—
99C22⋊Q8—41no—
100C22⋊Q8—41no—
101#121—21no—
102#121—21no—
103#120—41no—
104#117C2311yescharacteristic
105#117C2311yescharacteristic
106#117C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C22⋊C4—41no—
110C2×C42C2311yescharacteristic
111#108—22no—
112#107—21no—
113#107—21no—
114#107D411yescharacteristic
115#107D411yescharacteristic
116#106—21no—
117#106—21no—
118#106—21no—
119#106—21no—
120#106D411yescharacteristic
121#106D411yescharacteristic
122#104—22no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103—21no—
131#103D411yescharacteristic
132#103D411yescharacteristic
133#102—21no—
134#102—21no—
135#102—21no—
136#102—21no—
137#102—21no—
138#102—21no—
139#102—21no—
140#102—21no—
141#102D411yescharacteristic
142#102D411yescharacteristic
143#101—22no—
144#100—22no—
145#97—21no—
146#97—21no—
147#97—21no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8—22no—
152#95—41no—
153C24—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—41no—
165C2×Q8—41no—
166C2×Q8C2×D411yescharacteristic
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—41no—
175C2×D4—41no—
176C2×D4C2×D411yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4C2411yescharacteristic
183SD16—42no—
184SD16—42no—
185M16—21no—
186M16—21no—
187M16—21no—
188M16—21no—
189M16—21no—
190M16—21no—
191M16—21no—
192M16—42no—
193M16#4112yes—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—42no—
201C2×C8C2×D411yescharacteristic
202C2×C8C2×D411yescharacteristic
203C2×C8C2×D411yescharacteristic
204C2×C8C2×D411yescharacteristic
205C2×C8#4112yes—
206C2×C8#4112yes—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4C2×D411yescharacteristic
220C4⋊C4C2×D411yescharacteristic
221C4⋊C4C2×D411yescharacteristic
222C4⋊C4C2×D411yescharacteristic
223C4⋊C4C2×D411yescharacteristic
224C4⋊C4C2×D411yescharacteristic
225C22⋊C4—21no—
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—41no—
232C22⋊C4—82no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C23—21no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—81no—
243C23—81no—
244C23H(5,𝔽2)11yescharacteristic
245Q8—21no—
246Q8—21no—
247Q8—41no—
248Q8—41no—
249Q8—41no—
250Q8—41no—
251Q8—161no—
252Q8—161no—
253D4—21no—
254D4—21no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—41no—
260D4—41no—
261D4—161no—
262D4—161no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4C22×D411yescharacteristic
283C2×C4C22×D411yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C4C2×#4111yescharacteristic
287C2×C42−1+411yescharacteristic
288C8—21no—
289C8—21no—
290C8—22no—
291C8—41no—
292C8—41no—
293C8—41no—
294C8#12412yes—
295C8#12412yes—
296C22—21no—
297C22—21no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—82no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—161no—
306C22#54211yescharacteristic
307C22D4211yescharacteristic
308C22#55511yescharacteristic
309C4—42no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4C2×#12411yescharacteristic
321C4#53211yescharacteristic
322C4#54711yescharacteristic
323C4#54811yescharacteristic
324C2—21no—
325C2—21no—
326C2—81no—
327C2—161no—
328C2#249111yescharacteristic
329C2#329011yescharacteristic
330C2#329111yescharacteristic
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
2. #3137C2semidirect
5. #2038C2semidirect
7. #2028C2semidirect
8. #2027C2semidirect
9. #1977C2semidirect
10. #1968C2semidirect
11. #1931C2semidirect
12. #1930C2semidirect
13. #1815C2semidirect
14. #1772C2semidirect
15. #1764C2semidirect

Maximal subgroups

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

subgroupindex
#31632
#31372
#31362
#30572
#20382
#20372
#20282
#20272
#19772
#19682
#19312
#19302
#18152
#17722
#17642