Finite Group Databasebeta

← #19766

Subgroups — #19766

Order 256.

325 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3384C211yesmaximal, characteristic, complemented
2#3335C211yesmaximal, characteristic, complemented
3#3298C211yesmaximal, characteristic, complemented
4#3283C211yesmaximal, characteristic
5#1716C211yesmaximal, characteristic, complemented
6#1690C211yesmaximal, characteristic, complemented
7#1679C211yesmaximal, characteristic, complemented
8#1671C211yesmaximal, characteristic, complemented
9#1671C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1651C211yesmaximal, characteristic, complemented
12#1643C211yesmaximal, characteristic, complemented
13#1635C211yesmaximal, characteristic, complemented
14#1582C211yesmaximal, characteristic, complemented
15#1580C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#554C2211yescharacteristic
18D4×Q8C2211yescharacteristic
19#547C2211yescharacteristic
20#547C2211yescharacteristic
21#501—21no—
22#498C2211yescharacteristic
23#498C2211yescharacteristic
24#498C2211yescharacteristic
25#494C2211yescharacteristic
26#487C2211yescharacteristic
27#487C2211yescharacteristic
28#483—21no—
29#483—21no—
30#481—21no—
31#478—21no—
32#476—21no—
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#475C2211yescharacteristic
36#475C2211yescharacteristic
37#474C2211yescharacteristic
38#467—21no—
39#465—21no—
40#465—21no—
41#461C2211yescharacteristic
42#461C2211yescharacteristic
43#461C2211yescharacteristic
44#452—21no—
45#452—21no—
46#448—21no—
47#446C2211yescharacteristic
48#442—21no—
49C4×SD16C2211yescharacteristic
50C4×D8C2211yescharacteristic
51C8×D4C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54C2×#106—21no—
55C2×#103—21no—
56#334C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63D4⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69C22×Q8—21no—
70#135—44no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16D411yescharacteristic
77C2×D8D411yescharacteristic
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C4⋊Q8—21no—
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86#125—21no—
87#123—41no—
88#123—41no—
89C22⋊Q8—21no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95#121—21no—
96#121—21no—
97C4×Q8C2311yescharacteristic
98C4×D4—21no—
99C4×D4—21no—
100C4×D4C2311yescharacteristic
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106#107—21no—
107#107D411yescharacteristic
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115#106—21no—
116#106—41no—
117#106D411yescharacteristic
118C4⋊C8—21no—
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
122C4⋊C8C2311yescharacteristic
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103—41no—
131#103D411yescharacteristic
132#102—21no—
133#102—21no—
134#102—21no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C22⋊C8—21no—
141#97—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C4×C8C2311yescharacteristic
145#41—42no—
146#41—42no—
147#41—81no—
148#41—81no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—41no—
154C2×Q8—42no—
155C2×Q8—81no—
156C2×Q8C2×D411yescharacteristic
157C2×D4—41no—
158C2×D4—41no—
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169Q16—42no—
170Q16—81no—
171SD16—41no—
172SD16—42no—
173SD16—42no—
174SD16—81no—
175D8SD1614yescomplemented
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—81no—
184C2×C8C2×D411yescharacteristic
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4C2×D411yescharacteristic
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C42—21no—
214C42C2411yescharacteristic
215C23—21no—
216C23—21no—
217C23—21no—
218Q8—21no—
219Q8—21no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—81no—
227Q8—82no—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—21no—
232D4—82no—
233D4—82no—
234D4C2×SD1612yes—
235D4C2×SD1612yes—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—161no—
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C2×C4H(5,𝔽2)11yescharacteristic
258C8—21no—
259C8—21no—
260C8—21no—
261C8—41no—
262C8—41no—
263C8—41no—
264C8—41no—
265C8—41no—
266C8C2×SD1612yes—
267C22—22no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22D4211yescharacteristic
277C4—21no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4C22×SD1611yescharacteristic
288C4#57511yescharacteristic
289C4#57711yescharacteristic
290C4#57711yescharacteristic
291C2—42no—
292C2—42no—
293C2—81no—
294C2#326711yescharacteristic
295C2D4×SD1611yescharacteristic
296C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3283C2
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. #3384C2semidirect
2. #3335C2semidirect
3. #3298C2semidirect
5. #1716C2semidirect
6. #1690C2semidirect
7. #1679C2semidirect
8. #1671C2semidirect
9. #1671C2semidirect
10. #1654C2semidirect
11. #1651C2semidirect
12. #1643C2semidirect
13. #1635C2semidirect
14. #1582C2semidirect
15. #1580C2semidirect
175. D8SD16semidirect

Maximal subgroups

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

subgroupindex
#33842
#33352
#32982
#32832
#17162
#16902
#16792
#16712
#16542
#16512
#16432
#16352
#15822
#15802