Finite Group Databasebeta

← #19221

Subgroups — #19221

Order 256.

303 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3248C211yesmaximal, characteristic, complemented
2#3247C211yesmaximal, characteristic
3#3167C211yesmaximal, characteristic, complemented
4#2966C211yesmaximal, characteristic, complemented
5#1704C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1702C211yesmaximal, characteristic, complemented
8#1701C211yesmaximal, characteristic, complemented
9#1672C211yesmaximal, characteristic, complemented
10#1671C211yesmaximal, characteristic, complemented
11#1666C211yesmaximal, characteristic, complemented
12#1665C211yesmaximal, characteristic, complemented
13C2×#334C211yesmaximal, characteristic, complemented
14#1552C211yesmaximal, characteristic, complemented
15#1551C211yesmaximal, characteristic, complemented
16#533C2211yescharacteristic
17#532C2211yescharacteristic
18#529C2211yescharacteristic
19#498C2211yescharacteristic
20#492C2211yescharacteristic
21#487C2211yescharacteristic
22#486C2211yescharacteristic
23#479—21no—
24#478—21no—
25#477C2211yescharacteristic
26#476C2211yescharacteristic
27#475C2211yescharacteristic
28#474C2211yescharacteristic
29#467—21no—
30#466—21no—
31#465—21no—
32#465—21no—
33#465—21no—
34#465—21no—
35#464—21no—
36#463—21no—
37#462C2211yescharacteristic
38#461C2211yescharacteristic
39#460C2211yescharacteristic
40#459C2211yescharacteristic
41#446C2211yescharacteristic
42C8×Q8C2211yescharacteristic
43C4×SD16—21no—
44C4×SD16—21no—
45#436C2211yescharacteristic
46#435—21no—
47C8×D4—21no—
48C8×D4C2211yescharacteristic
49#433—21no—
50#432C2211yescharacteristic
51C2×C4⋊C8C2211yescharacteristic
52C2×C4⋊C8C2211yescharacteristic
53#405—21no—
54C2×C4×C8C2211yescharacteristic
55#334C2211yescharacteristic
56#334C2211yescharacteristic
57#334C2211yescharacteristic
58#334C2211yescharacteristic
59#333C2211yescharacteristic
60#333C2211yescharacteristic
61#332C2211yescharacteristic
62#332C2211yescharacteristic
63#332C2211yescharacteristic
64#332C2211yescharacteristic
65#331C2211yescharacteristic
66#331C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×SD16—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×D8—21no—
74C2×M16—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8D411yescharacteristic
79C22×C8D411yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#127C2311yescharacteristic
84#126—21no—
85#125—21no—
86#125D411yescharacteristic
87#124—21no—
88#124D411yescharacteristic
89#123—21no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8D411yescharacteristic
93#121—21no—
94#121—21no—
95#121D411yescharacteristic
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8C2311yescharacteristic
99C4×D4—21no—
100C4×D4—21no—
101C4×D4—21no—
102C4×D4C2311yescharacteristic
103#117—21no—
104#117—21no—
105C2×C42C2311yescharacteristic
106#107—21no—
107#107—21no—
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8—21no—
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134C22⋊C8—21no—
135C22⋊C8—21no—
136C22⋊C8—21no—
137#97—21no—
138#97—21no—
139C4×C8—21no—
140C4×C8D411yescharacteristic
141C4×C8D411yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—82no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8C2×D411yescharacteristic
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—41no—
154C2×D4C2×D411yescharacteristic
155C22×C4—21no—
156C22×C4—41no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4#4111yescharacteristic
160Q16—81no—
161SD16—42no—
162SD16—42no—
163SD16—81no—
164SD16—81no—
165D8—81no—
166M16—42no—
167C2×C8—22no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—41no—
175C2×C8—42no—
176C2×C8—81no—
177C2×C8—81no—
178C2×C8C2×D411yescharacteristic
179C2×C8C2×D411yescharacteristic
180C2×C8C2×D411yescharacteristic
181C2×C8C2×D411yescharacteristic
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4C2×D411yescharacteristic
195C4⋊C4C2×D411yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C42—21no—
205C42—41no—
206C42C2×D411yescharacteristic
207C42C2×D411yescharacteristic
208C42#4111yescharacteristic
209C42C2411yescharacteristic
210C23—21no—
211C23—41no—
212C23#12111yescharacteristic
213Q8—21no—
214Q8—21no—
215Q8—41no—
216Q8—41no—
217Q8—41no—
218Q8—41no—
219Q8—81no—
220D4—21no—
221D4—21no—
222D4—42no—
223D4—41no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—81no—
228D4—82no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—82no—
243C2×C4#12111yescharacteristic
244C2×C4#12111yescharacteristic
245C2×C4#12111yescharacteristic
246C2×C4C22×D411yescharacteristic
247C2×C4C22×D411yescharacteristic
248C2×C4C2×#4111yescharacteristic
249C8—21no—
250C8—21no—
251C8—21no—
252C8—41no—
253C8—41no—
254C8—41no—
255C8—41no—
256C8—41no—
257C8—41no—
258C8#13512yes—
259C22—21no—
260C22—21no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22#47112yes—
268C22C2×#12111yescharacteristic
269C4—21no—
270C4—21no—
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4C2×#13511yescharacteristic
280C4#57511yescharacteristic
281C4#57711yescharacteristic
282C4#57711yescharacteristic
283C2—22no—
284C2—81no—
285C2—161no—
286C2#303511yescharacteristic
287C2#306011yescharacteristic
288C2C2×#47111yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3247C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #3248C2semidirect
3. #3167C2semidirect
4. #2966C2semidirect
5. #1704C2semidirect
6. #1703C2semidirect
7. #1702C2semidirect
8. #1701C2semidirect
9. #1672C2semidirect
10. #1671C2semidirect
11. #1666C2semidirect
12. #1665C2semidirect
13. C2×#334C2semidirect
14. #1552C2semidirect
15. #1551C2semidirect

Maximal subgroups

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

subgroupindex
#32482
#32472
#31672
#29662
#17042
#17032
#17022
#17012
#16722
#16712
#16662
#16652
C2×#3342
#15522
#15512