Finite Group Databasebeta

← #20274

Subgroups — #20274

Order 256.

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

normal subgroups only

This group has 313 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#3366C211yesmaximal, characteristic, complemented
3#3333C211yesmaximal, characteristic, complemented
4#3304C211yesmaximal, characteristic, complemented
5#1717C211yesmaximal, characteristic, complemented
6#1701C211yesmaximal, characteristic, complemented
7#1698C211yesmaximal, characteristic, complemented
8#1686C211yesmaximal, characteristic, complemented
9#1677C211yesmaximal, characteristic, complemented
10#1643C211yesmaximal, characteristic, complemented
11#1638C211yesmaximal, characteristic, complemented
12#1621C211yesmaximal, characteristic, complemented
13#1621C211yesmaximal, characteristic, complemented
14#1593C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16#556C2211yescharacteristic
17#554C2211yescharacteristic
18#550C2211yescharacteristic
19#550C2211yescharacteristic
20#496—21no—
21#493C2211yescharacteristic
22#492C2211yescharacteristic
23#490—21no—
24#487C2211yescharacteristic
25#486C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#482—21no—
29#480—21no—
30#478—21no—
31#477C2211yescharacteristic
32#476C2211yescharacteristic
33#475C2211yescharacteristic
34#468—21no—
35#465—21no—
36#463—21no—
37#461C2211yescharacteristic
38#460—21no—
39#460C2211yescharacteristic
40#460C2211yescharacteristic
41#459—21no—
42#459C2211yescharacteristic
43#459C2211yescharacteristic
44#452—21no—
45#452—21no—
46#446C2211yescharacteristic
47#446C2211yescharacteristic
48#442—21no—
49#440—21no—
50C4×Q16C2211yescharacteristic
51C4×D8C2211yescharacteristic
52#436C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#417—21no—
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×Q16D411yescharacteristic
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—41no—
78C2×D8D411yescharacteristic
79C2×M16—21no—
80C22×C8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127—21no—
84#127—21no—
85#127C2311yescharacteristic
86#127C2311yescharacteristic
87#126—41no—
88#125—21no—
89#124—21no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—41no—
93#121—21no—
94#121—21no—
95#121—41no—
96#121—41no—
97C4×Q8—21no—
98C4×Q8C2311yescharacteristic
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4C2311yescharacteristic
104C4×D4C2311yescharacteristic
105#117—21no—
106C2×C4⋊C4—21no—
107#107—21no—
108#107D411yescharacteristic
109#107D411yescharacteristic
110#106—21no—
111#106—21no—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118#103—21no—
119#103—21no—
120#103—21no—
121#103—41no—
122#103D411yescharacteristic
123#103D411yescharacteristic
124#102—21no—
125#102—21no—
126#102—21no—
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#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139#97—21no—
140#97—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—42no—
146#41—81no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C2×D411yescharacteristic
151C2×Q8C2×D411yescharacteristic
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4C2×D411yescharacteristic
163C2×D4C2×D411yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169Q16—41no—
170SD16—42no—
171SD16—81no—
172SD16—81no—
173SD16—81no—
174SD16—81no—
175D8—41no—
176D8—42no—
177D8—81no—
178D8—82no—
179M16—81no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—81no—
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C42—21no—
211C42—21no—
212C42—41no—
213C42C2411yescharacteristic
214C23—21no—
215C23—21no—
216C23—41no—
217C23—41no—
218Q8—21no—
219Q8—21no—
220Q8—21no—
221Q8—21no—
222Q8—41no—
223Q8—41no—
224Q8—81no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—41no—
237D4—81no—
238D4—81no—
239D4—82no—
240D4—82no—
241D4—161no—
242D4—161no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—161no—
258C2×C4C22×D411yescharacteristic
259C2×C4C22×D411yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C8—21no—
262C8—21no—
263C8—21no—
264C8—21no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C8—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22D4211yescharacteristic
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—81no—
291C4—81no—
292C4#57511yescharacteristic
293C4#57511yescharacteristic
294C4#57611yescharacteristic
295C4#57711yescharacteristic
296C2—81no—
297C2—81no—
298C2—161no—
299C2—161no—
300C2#327111yescharacteristic
301C2#327311yescharacteristic
302C2#328811yescharacteristic
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. #3366C2semidirect
3. #3333C2semidirect
4. #3304C2semidirect
5. #1717C2semidirect
6. #1701C2semidirect
7. #1698C2semidirect
8. #1686C2semidirect
9. #1677C2semidirect
10. #1643C2semidirect
11. #1638C2semidirect
12. #1621C2semidirect
13. #1621C2semidirect
14. #1593C2semidirect
15. #1581C2semidirect

Maximal subgroups

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

subgroupindex
#33972
#33662
#33332
#33042
#17172
#17012
#16982
#16862
#16772
#16432
#16382
#16212
#15932
#15812