Finite Group Databasebeta

← #20255

Subgroups — #20255

Order 256.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3397C211yesmaximal, characteristic, complemented
2#3332C211yesmaximal, characteristic, complemented
3#3330C211yesmaximal, characteristic, complemented
4#3302C211yesmaximal, characteristic, complemented
5#1719C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1699C211yesmaximal, characteristic, complemented
8#1685C211yesmaximal, characteristic, complemented
9#1681C211yesmaximal, characteristic, complemented
10#1650C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1641C211yesmaximal, characteristic, complemented
13#1625C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#550C2211yescharacteristic
18#547C2211yescharacteristic
19#547C2211yescharacteristic
20#500C2211yescharacteristic
21#496—21no—
22#492C2211yescharacteristic
23#487C2211yescharacteristic
24#487C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#485—21no—
28#484—21no—
29#481—21no—
30#475C2211yescharacteristic
31#474C2211yescharacteristic
32#474C2211yescharacteristic
33#469—21no—
34#468—21no—
35#467—21no—
36#463—21no—
37#462C2211yescharacteristic
38#461C2211yescharacteristic
39#461C2211yescharacteristic
40#460—21no—
41#459C2211yescharacteristic
42#459C2211yescharacteristic
43#452—21no—
44#449—21no—
45#449—21no—
46#446C2211yescharacteristic
47#440—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#417—21no—
54#417—21no—
55#416—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×#41—21no—
70C2×Q16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16D411yescharacteristic
76C2×D8—21no—
77C2×D8—21no—
78C2×D8D411yescharacteristic
79C2×M16—21no—
80C2×M16—21no—
81C22×C8—21no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#127—21no—
86#127C2311yescharacteristic
87#126—41no—
88#124—21no—
89#123—41no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—21no—
94#121—21no—
95#121—21no—
96#121—21no—
97#121—41no—
98C4×Q8—21no—
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103C4×D4C2311yescharacteristic
104#117—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107#107—21no—
108#107—21no—
109#107—41no—
110#107D411yescharacteristic
111#106—21no—
112#106D411yescharacteristic
113C4⋊C8—21no—
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118#103—21no—
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103D411yescharacteristic
127#102—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—41no—
134#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140#97—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—42no—
146#41—42no—
147#41—81no—
148#41—81no—
149#41—81no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8C2×D411yescharacteristic
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4C2×D411yescharacteristic
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170Q16—81no—
171SD16—41no—
172SD16—42no—
173SD16—81no—
174SD16—81no—
175SD16—81no—
176D8—41no—
177D8—81no—
178D8—81no—
179M16—81no—
180M16—81no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—81no—
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
191C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C42—21no—
214C42—41no—
215C42C2411yescharacteristic
216C23—21no—
217C23—21no—
218C23—21no—
219C23—41no—
220Q8—21no—
221Q8—21no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
227Q8—41no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—21no—
233D4—21no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—41no—
238D4—81no—
239D4—82no—
240D4—82no—
241D4—82no—
242D4—161no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
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—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4C22×D411yescharacteristic
261C2×C4C22×D411yescharacteristic
262C2×C4H(5,𝔽2)11yescharacteristic
263C8—21no—
264C8—21no—
265C8—21no—
266C8—21no—
267C8—41no—
268C8—41no—
269C8—41no—
270C8—41no—
271C8—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22D4211yescharacteristic
285C4—21no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—81no—
292C4—81no—
293C4—81no—
294C4#57511yescharacteristic
295C4#57511yescharacteristic
296C4#57611yescharacteristic
297C4#57611yescharacteristic
298C2—81no—
299C2—81no—
300C2—81no—
301C2—161no—
302C2#326811yescharacteristic
303C2#327311yescharacteristic
304C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3397C2semidirect
2. #3332C2semidirect
3. #3330C2semidirect
4. #3302C2semidirect
5. #1719C2semidirect
6. #1700C2semidirect
7. #1699C2semidirect
8. #1685C2semidirect
9. #1681C2semidirect
10. #1650C2semidirect
11. #1647C2semidirect
12. #1641C2semidirect
13. #1625C2semidirect
14. #1592C2semidirect
15. #1581C2semidirect

Maximal subgroups

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

subgroupindex
#33972
#33322
#33302
#33022
#17192
#17002
#16992
#16852
#16812
#16502
#16472
#16412
#16252
#15922
#15812