Finite Group Databasebeta

← #20300

Subgroups — #20300

Order 256.

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

normal subgroups only

This group has 289 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#3374C211yesmaximal, characteristic, complemented
2#3372C211yesmaximal, characteristic, complemented
3#3368C211yesmaximal, characteristic
4#3333C211yesmaximal, characteristic
5#1711C211yesmaximal, characteristic, complemented
6#1704C211yesmaximal, characteristic, complemented
7#1702C211yesmaximal, characteristic, complemented
8#1699C211yesmaximal, characteristic, complemented
9#1698C211yesmaximal, characteristic, complemented
10#1644C211yesmaximal, characteristic, complemented
11#1642C211yesmaximal, characteristic
12#1636C211yesmaximal, characteristic, complemented
13#1626C211yesmaximal, characteristic, complemented
14#1591C211yesmaximal, characteristic, complemented
15#1590C211yesmaximal, characteristic, complemented
16#557C2211yescharacteristic
17#557C2211yescharacteristic
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20#491—21no—
21#491—21no—
22#490—21no—
23#487C2211yescharacteristic
24#487C2211yescharacteristic
25#487C2211yescharacteristic
26#486C2211yescharacteristic
27#486C2211yescharacteristic
28#486C2211yescharacteristic
29#480—21no—
30#479—21no—
31#479—21no—
32#478—21no—
33#477C2211yescharacteristic
34#477C2211yescharacteristic
35#475C2211yescharacteristic
36#474C2211yescharacteristic
37#468—21no—
38#462—21no—
39#462C2211yescharacteristic
40#461—21no—
41#461C2211yescharacteristic
42#460—21no—
43#460C2211yescharacteristic
44#460C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47#446C2211yescharacteristic
48#446C2211yescharacteristic
49#441—21no—
50#440—21no—
51#440—21no—
52C4×Q16C2211yescharacteristic
53C4×SD16C2211yescharacteristic
54#436C2211yescharacteristic
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64Q8⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×Q16D411yescharacteristic
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16D411yescharacteristic
74C2×M16—21no—
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80#127—21no—
81#127C2311yescharacteristic
82#125—21no—
83#125—21no—
84#125—21no—
85#125—41no—
86#125—41no—
87C22⋊Q8—21no—
88C22⋊Q8—41no—
89#121—21no—
90#121—41no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8C2311yescharacteristic
96C4×Q8C2311yescharacteristic
97C4×Q8C2311yescharacteristic
98C4×D4—21no—
99C4×D4—21no—
100C4×D4C2311yescharacteristic
101C2×C4⋊C4—21no—
102#107—21no—
103#107—21no—
104#107—21no—
105#107D411yescharacteristic
106#106—21no—
107#106D411yescharacteristic
108C4⋊C8—21no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111C4⋊C8C2311yescharacteristic
112C4⋊C8C2311yescharacteristic
113C4⋊C8C2311yescharacteristic
114C4⋊C8C2311yescharacteristic
115#103—21no—
116#103—21no—
117#103—21no—
118#103—21no—
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#103—41no—
125#103D411yescharacteristic
126#103D411yescharacteristic
127#103D411yescharacteristic
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—
140#97—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—81no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8C2×D411yescharacteristic
150C2×Q8C2×D411yescharacteristic
151C2×Q8C2×D411yescharacteristic
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4C2×D411yescharacteristic
158C22×C4—21no—
159C22×C4—41no—
160C22×C4—41no—
161Q16—41no—
162Q16—42no—
163SD16—41no—
164SD16—42no—
165SD16—42no—
166SD16—81no—
167M16—81no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8C2×D411yescharacteristic
174C2×C8C2×D411yescharacteristic
175C2×C8C2×D411yescharacteristic
176C2×C8C2×D411yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D411yescharacteristic
196C4⋊C4C2×D411yescharacteristic
197C4⋊C4C2×D411yescharacteristic
198C22⋊C4—21no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—41no—
205C42—41no—
206C42C2411yescharacteristic
207C23—21no—
208C23—41no—
209Q8—21no—
210Q8—21no—
211Q8—21no—
212Q8—21no—
213Q8—21no—
214Q8—21no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—81no—
220D4—21no—
221D4—21no—
222D4—41no—
223D4—41no—
224D4—41no—
225D4—41no—
226D4—81no—
227D4—82no—
228D4—161no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—82no—
242C2×C4—161no—
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4H(5,𝔽2)11yescharacteristic
246C8—21no—
247C8—21no—
248C8—21no—
249C8—21no—
250C8—41no—
251C8—41no—
252C8—41no—
253C8—41no—
254C8—41no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22D4211yescharacteristic
262C4—21no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—81no—
271C4—81no—
272C4—81no—
273C4#57511yescharacteristic
274C4#57511yescharacteristic
275C4#57711yescharacteristic
276C4#57811yescharacteristic
277C2—81no—
278C2—161no—
279C2#327511yescharacteristic
280C2#327711yescharacteristic
281C2#329011yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#550C22
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. #3374C2semidirect
2. #3372C2semidirect
5. #1711C2semidirect
6. #1704C2semidirect
7. #1702C2semidirect
8. #1699C2semidirect
9. #1698C2semidirect
10. #1644C2semidirect
12. #1636C2semidirect
13. #1626C2semidirect
14. #1591C2semidirect
15. #1590C2semidirect

Maximal subgroups

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

subgroupindex
#33742
#33722
#33682
#33332
#17112
#17042
#17022
#16992
#16982
#16442
#16422
#16362
#16262
#15912
#15902