Finite Group Databasebeta

← #20141

Subgroups — #20141

Order 256.

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

normal subgroups only

This group has 305 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#3400C211yesmaximal, characteristic, complemented
2#3394C211yesmaximal, characteristic, complemented
3#3368C211yesmaximal, characteristic, complemented
4#3304C211yesmaximal, characteristic, complemented
5#1715C211yesmaximal, characteristic, complemented
6#1702C211yesmaximal, characteristic, complemented
7#1686C211yesmaximal, characteristic, complemented
8#1682C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1653C211yesmaximal, characteristic, complemented
11#1652C211yesmaximal, characteristic, complemented
12#1621C211yesmaximal, characteristic
13#1620C211yesmaximal, characteristic, complemented
14#1587C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16#559C2211yescharacteristic
17#557C2211yescharacteristic
18#556C2211yescharacteristic
19#550C2211yescharacteristic
20#498C2211yescharacteristic
21#497—21no—
22#495—21no—
23#493C2211yescharacteristic
24#492C2211yescharacteristic
25#492C2211yescharacteristic
26#490—21no—
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#482—21no—
30#478—21no—
31#475C2211yescharacteristic
32#474C2211yescharacteristic
33#465—21no—
34#464—21no—
35#464—21no—
36#463—21no—
37#462C2211yescharacteristic
38#462C2211yescharacteristic
39#461—21no—
40#460—21no—
41#460C2211yescharacteristic
42#460C2211yescharacteristic
43#460C2211yescharacteristic
44#459C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47#446C2211yescharacteristic
48C8×Q8C2211yescharacteristic
49#440—21no—
50#440—21no—
51C4×Q16C2211yescharacteristic
52C4×SD16—21no—
53C4×SD16C2211yescharacteristic
54C8×D4C2211yescharacteristic
55#416—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×Q16—21no—
70C2×Q16D411yescharacteristic
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—21no—
78C2×SD16D411yescharacteristic
79C2×D8—21no—
80C2×D8—41no—
81C22×C8—21no—
82C4⋊Q8—21no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#127—21no—
86#127C2311yescharacteristic
87#127C2311yescharacteristic
88#126—41no—
89#125—21no—
90#125—41no—
91#124—21no—
92#124—21no—
93#124—21no—
94#123—41no—
95C22⋊Q8—21no—
96#121—21no—
97#121—41no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8C2311yescharacteristic
102C4×Q8C2311yescharacteristic
103C4×Q8C2311yescharacteristic
104C4×D4—21no—
105C4×D4—21no—
106C4×D4—21no—
107C4×D4C2311yescharacteristic
108#117—21no—
109#107—21no—
110#107D411yescharacteristic
111#106—21no—
112#106—41no—
113#106D411yescharacteristic
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121#103—21no—
122#103—21no—
123#103—21no—
124#103D411yescharacteristic
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#102D411yescharacteristic
137C22⋊C8—21no—
138#97—21no—
139#97—21no—
140C4×C8—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8C2×D411yescharacteristic
149C2×Q8C2×D411yescharacteristic
150C2×Q8C2×D411yescharacteristic
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4C2×D411yescharacteristic
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—41no—
163Q16—41no—
164Q16—81no—
165Q16—81no—
166SD16—41no—
167SD16—42no—
168SD16—42no—
169SD16—42no—
170SD16—81no—
171SD16—81no—
172SD16—81no—
173SD16—81no—
174D8—82no—
175D8—81no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—81no—
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—41no—
212C42—41no—
213C42C2411yescharacteristic
214C23—21no—
215C23—41no—
216C23—41no—
217Q8—21no—
218Q8—21no—
219Q8—21no—
220Q8—21no—
221Q8—21no—
222Q8—21no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—41no—
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—161no—
241C2×C4—21no—
242C2×C4—21no—
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—82no—
253C2×C4—161no—
254C2×C4C22×D411yescharacteristic
255C2×C4C22×D411yescharacteristic
256C2×C4H(5,𝔽2)11yescharacteristic
257C8—22no—
258C8—21no—
259C8—21no—
260C8—21no—
261C8—21no—
262C8—41no—
263C8—41no—
264C8—41no—
265C8—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22D4211yescharacteristic
276C4—21no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—81no—
285C4—81no—
286C4#57511yescharacteristic
287C4#57511yescharacteristic
288C4#57611yescharacteristic
289C4#57711yescharacteristic
290C2—81no—
291C2—161no—
292C2—161no—
293C2#326811yescharacteristic
294C2#327111yescharacteristic
295C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#1621C2
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. #3400C2semidirect
2. #3394C2semidirect
3. #3368C2semidirect
4. #3304C2semidirect
5. #1715C2semidirect
6. #1702C2semidirect
7. #1686C2semidirect
8. #1682C2semidirect
9. #1665C2semidirect
10. #1653C2semidirect
11. #1652C2semidirect
13. #1620C2semidirect
14. #1587C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#34002
#33942
#33682
#33042
#17152
#17022
#16862
#16822
#16652
#16532
#16522
#16212
#16202
#15872
#15852