Finite Group Databasebeta

← #20235

Subgroups — #20235

Order 256.

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

normal subgroups only

This group has 287 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#3372C211yesmaximal, characteristic, complemented
2#3364C211yesmaximal, characteristic, complemented
3#3353C211yesmaximal, characteristic
4#3331C211yesmaximal, characteristic
5#1718C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1699C211yesmaximal, characteristic, complemented
8#1694C211yesmaximal, characteristic, complemented
9#1684C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic
11#1653C211yesmaximal, characteristic, complemented
12#1644C211yesmaximal, characteristic, complemented
13#1638C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1583C211yesmaximal, characteristic, complemented
16#557C2211yescharacteristic
17#556C2211yescharacteristic
18#554C2211yescharacteristic
19#547C2211yescharacteristic
20#500C2211yescharacteristic
21#498C2211yescharacteristic
22#491—21no—
23#488—21no—
24#487C2211yescharacteristic
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#487C2211yescharacteristic
28#484—21no—
29#483—21no—
30#482—21no—
31#479—21no—
32#479—21no—
33#477C2211yescharacteristic
34#476C2211yescharacteristic
35#475C2211yescharacteristic
36#474C2211yescharacteristic
37#474C2211yescharacteristic
38#470—21no—
39#468—21no—
40#464—21no—
41#462C2211yescharacteristic
42#462C2211yescharacteristic
43#461—21no—
44#461C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47C8×Q8C2211yescharacteristic
48#440—21no—
49C4×Q16—21no—
50C4×SD16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52#436C2211yescharacteristic
53#436C2211yescharacteristic
54#428—21no—
55#417—21no—
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×Q16—21no—
70C2×SD16—21no—
71C2×SD16D411yescharacteristic
72C2×SD16D411yescharacteristic
73C2×M16—21no—
74C2×M16—21no—
75C4⋊Q8—21no—
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80#126—41no—
81#125—21no—
82#125—21no—
83#125—41no—
84#124—21no—
85#123—41no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—41no—
90#121—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8C2311yescharacteristic
95C4×Q8C2311yescharacteristic
96C4×D4—21no—
97C4×D4C2311yescharacteristic
98C4×D4C2311yescharacteristic
99#117—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102#107—21no—
103#107—21no—
104#107—21no—
105#107—21no—
106#107—21no—
107#106—21no—
108#106—21no—
109#106—41no—
110#106D411yescharacteristic
111#106D411yescharacteristic
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#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—41no—
130#103D411yescharacteristic
131#103D411yescharacteristic
132#102—21no—
133#102—21no—
134#102—21no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—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×Q8—21no—
149C2×Q8—21no—
150C2×Q8C2×D411yescharacteristic
151C2×Q8C2×D411yescharacteristic
152C2×D4—41no—
153C2×D4C2×D411yescharacteristic
154C2×D4C2×D411yescharacteristic
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—41no—
158C22×C4—41no—
159Q16—42no—
160Q16—81no—
161SD16—41no—
162SD16—41no—
163SD16—42no—
164M16—42no—
165M16—81no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8C2×D411yescharacteristic
172C2×C8C2×D411yescharacteristic
173C2×C8C2×D411yescharacteristic
174C2×C8C2×D411yescharacteristic
175C4⋊C4—21no—
176C4⋊C4—21no—
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—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4C2×D411yescharacteristic
193C4⋊C4C2×D411yescharacteristic
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D411yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C42—21no—
203C42—21no—
204C42—41no—
205C42—41no—
206C42C2411yescharacteristic
207C23—21no—
208C23—21no—
209Q8—21no—
210Q8—21no—
211Q8—21no—
212Q8—21no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—81no—
220D4—21no—
221D4—21no—
222D4—21no—
223D4—21no—
224D4—82no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4—161no—
241C2×C4C22×D411yescharacteristic
242C2×C4C22×D411yescharacteristic
243C2×C4H(5,𝔽2)11yescharacteristic
244C8—21no—
245C8—21no—
246C8—21no—
247C8—21no—
248C8—41no—
249C8—41no—
250C8—41no—
251C8—41no—
252C8—41no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22D4211yescharacteristic
260C4—21no—
261C4—41no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—41no—
267C4—81no—
268C4—81no—
269C4—81no—
270C4—81no—
271C4#57511yescharacteristic
272C4#57511yescharacteristic
273C4#57711yescharacteristic
274C4#57811yescharacteristic
275C2—81no—
276C2—81no—
277C2#326911yescharacteristic
278C2#327511yescharacteristic
279C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#547C22
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. #3372C2semidirect
2. #3364C2semidirect
5. #1718C2semidirect
6. #1703C2semidirect
7. #1699C2semidirect
8. #1694C2semidirect
9. #1684C2semidirect
11. #1653C2semidirect
12. #1644C2semidirect
13. #1638C2semidirect
14. #1592C2semidirect
15. #1583C2semidirect

Maximal subgroups

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

subgroupindex
#33722
#33642
#33532
#33312
#17182
#17032
#16992
#16942
#16842
#16542
#16532
#16442
#16382
#15922
#15832