Finite Group Databasebeta

← #20283

Subgroups — #20283

Order 256.

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

normal subgroups only

This group has 301 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#3399C211yesmaximal, characteristic
2#3372C211yesmaximal, characteristic, complemented
3#3365C211yesmaximal, characteristic, complemented
4#3333C211yesmaximal, characteristic, complemented
5#1717C211yesmaximal, characteristic, complemented
6#1704C211yesmaximal, characteristic, complemented
7#1701C211yesmaximal, characteristic, complemented
8#1688C211yesmaximal, characteristic, complemented
9#1678C211yesmaximal, characteristic, complemented
10#1642C211yesmaximal, characteristic, complemented
11#1630C211yesmaximal, characteristic, complemented
12#1626C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14#1593C211yesmaximal, characteristic, complemented
15#1583C211yesmaximal, characteristic, complemented
16#559C2211yescharacteristic
17#557C2211yescharacteristic
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20#496—21no—
21#493C2211yescharacteristic
22#492C2211yescharacteristic
23#491—21no—
24#488—21no—
25#487C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#480—21no—
30#479—21no—
31#478—21no—
32#477C2211yescharacteristic
33#477C2211yescharacteristic
34#475C2211yescharacteristic
35#468—21no—
36#464—21no—
37#463—21no—
38#461—21no—
39#461C2211yescharacteristic
40#460C2211yescharacteristic
41#460C2211yescharacteristic
42#460C2211yescharacteristic
43#459—21no—
44#459C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47#446C2211yescharacteristic
48C8×Q8C2211yescharacteristic
49#442—21no—
50#440—21no—
51C4×Q16C2211yescharacteristic
52C4×SD16—21no—
53C4×SD16C2211yescharacteristic
54#436C2211yescharacteristic
55#417—21no—
56#334C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#331C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16D411yescharacteristic
69C2×SD16—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—41no—
74C2×SD16D411yescharacteristic
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×M16—21no—
79C4⋊Q8—21no—
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82#127—21no—
83#127C2311yescharacteristic
84#127C2311yescharacteristic
85#125—21no—
86#125—21no—
87#125—41no—
88#124—21no—
89#124—21no—
90#123—41no—
91C22⋊Q8—21no—
92C22⋊Q8—41no—
93#121—21no—
94#121—41no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8C2311yescharacteristic
98C4×Q8C2311yescharacteristic
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4C2311yescharacteristic
105C2×C4⋊C4—21no—
106#107—21no—
107#107—21no—
108#107D411yescharacteristic
109#106—21no—
110#106D411yescharacteristic
111C4⋊C8—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—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—41no—
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—
163C22×C4—41no—
164Q16—41no—
165SD16—41no—
166SD16—42no—
167SD16—42no—
168SD16—81no—
169SD16—81no—
170SD16—82no—
171D8—42no—
172D8—81no—
173D8—81no—
174M16—81no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—41no—
210C42C2411yescharacteristic
211C23—21no—
212C23—41no—
213C23—41no—
214Q8—21no—
215Q8—21no—
216Q8—21no—
217Q8—21no—
218Q8—21no—
219Q8—21no—
220Q8—41no—
221Q8—41no—
222Q8—81no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—81no—
234D4—81no—
235D4—82no—
236D4—161no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—82no—
249C2×C4—161no—
250C2×C4—161no—
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C2×C4H(5,𝔽2)11yescharacteristic
254C8—21no—
255C8—21no—
256C8—21no—
257C8—21no—
258C8—41no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22D4211yescharacteristic
273C4—21no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—81no—
282C4—81no—
283C4#57511yescharacteristic
284C4#57511yescharacteristic
285C4#57711yescharacteristic
286C4#57711yescharacteristic
287C2—81no—
288C2—161no—
289C2—161no—
290C2#327111yescharacteristic
291C2#327511yescharacteristic
292C2#328811yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3399C2
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
2. #3372C2semidirect
3. #3365C2semidirect
4. #3333C2semidirect
5. #1717C2semidirect
6. #1704C2semidirect
7. #1701C2semidirect
8. #1688C2semidirect
9. #1678C2semidirect
10. #1642C2semidirect
11. #1630C2semidirect
12. #1626C2semidirect
13. #1622C2semidirect
14. #1593C2semidirect
15. #1583C2semidirect

Maximal subgroups

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

subgroupindex
#33992
#33722
#33652
#33332
#17172
#17042
#17012
#16882
#16782
#16422
#16302
#16262
#16222
#15932
#15832