Finite Group Databasebeta

← #19578

Subgroups — #19578

Order 256.

309 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3251C211yesmaximal, characteristic, complemented
2#3212C211yesmaximal, characteristic, complemented
3#3207C211yesmaximal, characteristic, complemented
4#2994C211yesmaximal, characteristic, complemented
5#1701C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1688C211yesmaximal, characteristic, complemented
8#1685C211yesmaximal, characteristic, complemented
9#1682C211yesmaximal, characteristic, complemented
10#1679C211yesmaximal, characteristic, complemented
11#1667C211yesmaximal, characteristic, complemented
12#1666C211yesmaximal, characteristic, complemented
13#1574C211yesmaximal, characteristic, complemented
14#1547C211yesmaximal, characteristic, complemented
15#1540C211yesmaximal, characteristic, complemented
16#532C2211yescharacteristic
17#532C2211yescharacteristic
18#529C2211yescharacteristic
19#495—21no—
20#493C2211yescharacteristic
21#492C2211yescharacteristic
22#488—21no—
23#487C2211yescharacteristic
24#486C2211yescharacteristic
25#478—21no—
26#475C2211yescharacteristic
27#474C2211yescharacteristic
28#469—21no—
29#468—21no—
30#463—21no—
31#463—21no—
32#463—21no—
33#462C2211yescharacteristic
34#461C2211yescharacteristic
35#460C2211yescharacteristic
36#460C2211yescharacteristic
37#459C2211yescharacteristic
38#459C2211yescharacteristic
39#452—21no—
40#449—21no—
41#449—21no—
42#449—21no—
43#446C2211yescharacteristic
44C8×Q8C2211yescharacteristic
45#436C2211yescharacteristic
46#435—21no—
47C8×D4—21no—
48C8×D4C2211yescharacteristic
49#433—21no—
50#431C2211yescharacteristic
51#431C2211yescharacteristic
52#424—21no—
53#423C2211yescharacteristic
54C2×C4⋊C8C2211yescharacteristic
55#335C2211yescharacteristic
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#334C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62#332C2211yescharacteristic
63#331C2211yescharacteristic
64#331C2211yescharacteristic
65#331C2211yescharacteristic
66#331C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C2×Q16—21no—
70C2×SD16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×D8—21no—
76C2×D8—21no—
77C2×D8—21no—
78C2×D8—21no—
79C2×M16—21no—
80C2×M16—21no—
81C22×C8—21no—
82C22×C8—21no—
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#127C2311yescharacteristic
86#127C2311yescharacteristic
87#126—21no—
88#125D411yescharacteristic
89#124—21no—
90#124—21no—
91#124D411yescharacteristic
92#123—21no—
93C22⋊Q8D411yescharacteristic
94#121—21no—
95#121—21no—
96#121—21no—
97#121—21no—
98#121D411yescharacteristic
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4—21no—
102C4×D4—21no—
103C4×D4—21no—
104C4×D4—21no—
105C4×D4C2311yescharacteristic
106#117—21no—
107C2×C42C2311yescharacteristic
108#107—21no—
109#106—21no—
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8D411yescharacteristic
115C4⋊C8D411yescharacteristic
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#102—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—
137C22⋊C8D411yescharacteristic
138C22⋊C8D411yescharacteristic
139#97—21no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—81no—
145#41—81no—
146#41—81no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8C2×D411yescharacteristic
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4C2×D411yescharacteristic
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4#4111yescharacteristic
167Q16—81no—
168SD16—81no—
169SD16—81no—
170SD16—81no—
171SD16—81no—
172SD16—81no—
173D8—81no—
174D8—81no—
175D8—81no—
176D8—81no—
177M16—81no—
178M16—81no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—81no—
186C2×C8—81no—
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
189C2×C8C2×D411yescharacteristic
190C2×C8C2×D411yescharacteristic
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C42—21no—
211C42C2×D411yescharacteristic
212C42C2×D411yescharacteristic
213C42#4111yescharacteristic
214C42C2411yescharacteristic
215C23—21no—
216C23—41no—
217C23—41no—
218C23#12111yescharacteristic
219Q8—21no—
220Q8—21no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225D4—21no—
226D4—21no—
227D4—42no—
228D4—41no—
229D4—41no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—41no—
236D4—81no—
237D4—81no—
238D4—82no—
239D4—82no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4#12111yescharacteristic
255C2×C4#12111yescharacteristic
256C2×C4#12111yescharacteristic
257C2×C4C22×D411yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C8—21no—
261C8—21no—
262C8—21no—
263C8—21no—
264C8—41no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—41no—
269C8—41no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22C2×#12111yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—21no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—81no—
291C4—81no—
292C4#57611yescharacteristic
293C4#57611yescharacteristic
294C4#57711yescharacteristic
295C4#57711yescharacteristic
296C2—41no—
297C2—81no—
298C2—161no—
299C2—161no—
300C2#304511yescharacteristic
301C2#306211yescharacteristic
302C2#306711yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
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. #3251C2semidirect
2. #3212C2semidirect
3. #3207C2semidirect
4. #2994C2semidirect
5. #1701C2semidirect
6. #1700C2semidirect
7. #1688C2semidirect
8. #1685C2semidirect
9. #1682C2semidirect
10. #1679C2semidirect
11. #1667C2semidirect
12. #1666C2semidirect
13. #1574C2semidirect
14. #1547C2semidirect
15. #1540C2semidirect

Maximal subgroups

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

subgroupindex
#32512
#32122
#32072
#29942
#17012
#17002
#16882
#16852
#16822
#16792
#16672
#16662
#15742
#15472
#15402