Finite Group Databasebeta

← #19982

Subgroups — #19982

Order 256.

327 conjugacy classes of subgroups.

The subgroup lattice has 299 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3392C211yesmaximal, characteristic, complemented
2#3381C211yesmaximal, characteristic, complemented
3#3335C211yesmaximal, characteristic, complemented
4#3284C211yesmaximal, characteristic
5#1705C211yesmaximal, characteristic, complemented
6#1672C211yesmaximal, characteristic, complemented
7#1671C211yesmaximal, characteristic, complemented
8#1666C211yesmaximal, characteristic, complemented
9#1665C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic, complemented
11#1653C211yesmaximal, characteristic, complemented
12#1631C211yesmaximal, characteristic, complemented
13#1622C211yesmaximal, characteristic, complemented
14C8×SD16C211yesmaximal, characteristic, complemented
15C8×SD16C211yesmaximal, characteristic, complemented
16C2×#135—21no—
17#557C2211yescharacteristic
18#554C2211yescharacteristic
19#550C2211yescharacteristic
20#547C2211yescharacteristic
21#499—21no—
22#498C2211yescharacteristic
23#498C2211yescharacteristic
24#498C2211yescharacteristic
25#492—21no—
26#492C2211yescharacteristic
27#492C2211yescharacteristic
28#492C2211yescharacteristic
29#483—21no—
30#481—21no—
31#479—21no—
32#478—21no—
33#477C2211yescharacteristic
34#476C2211yescharacteristic
35#475C2211yescharacteristic
36#474C2211yescharacteristic
37#465—21no—
38#465—21no—
39#462C2211yescharacteristic
40#461—21no—
41#461C2211yescharacteristic
42#460—21no—
43#460C2211yescharacteristic
44#459C2211yescharacteristic
45#452—21no—
46#449—21no—
47C8×Q8C2211yescharacteristic
48C8×Q8C2211yescharacteristic
49C4×SD16—21no—
50C4×SD16—21no—
51C4×SD16C2211yescharacteristic
52C4×SD16C2211yescharacteristic
53C8×D4C2211yescharacteristic
54C8×D4C2211yescharacteristic
55C2×#106—21no—
56#334C2211yescharacteristic
57#334C2211yescharacteristic
58#333C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#331C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66C82C2211yescharacteristic
67C2×#41—21no—
68C2×#41—21no—
69#135—44no—
70C2×Q16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16—21no—
76C2×SD16—21no—
77C2×SD16—41no—
78C2×SD16D411yescharacteristic
79C2×SD16D411yescharacteristic
80C2×D8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C4⋊Q8—21no—
84C4⋊Q8C2311yescharacteristic
85C4⋊Q8C2311yescharacteristic
86C4⋊Q8C2311yescharacteristic
87#127—21no—
88#127C2311yescharacteristic
89#125—21no—
90#125—21no—
91#125—41no—
92#123—41no—
93C22⋊Q8—21no—
94C22⋊Q8—21no—
95C22⋊Q8—41no—
96#121—21no—
97#121—21no—
98#121—41no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8C2311yescharacteristic
102C4×Q8C2311yescharacteristic
103C4×D4—21no—
104C4×D4—21no—
105C4×D4C2311yescharacteristic
106C4×D4C2311yescharacteristic
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109#107—21no—
110#107—21no—
111#106—21no—
112#106—21no—
113#106—21no—
114#106—21no—
115#106—21no—
116#106—21no—
117#106—41no—
118#106D411yescharacteristic
119#106D411yescharacteristic
120C4⋊C8—21no—
121C4⋊C8—21no—
122C4⋊C8C2311yescharacteristic
123C4⋊C8C2311yescharacteristic
124C4⋊C8C2311yescharacteristic
125C4⋊C8C2311yescharacteristic
126#103—21no—
127#103—21no—
128#103—21no—
129#103—21no—
130#103D411yescharacteristic
131#103D411yescharacteristic
132#102—21no—
133#102—21no—
134#102—21no—
135#102—21no—
136#102D411yescharacteristic
137#102D411yescharacteristic
138C22⋊C8—21no—
139C22⋊C8—21no—
140C4×C8—21no—
141C4×C8—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×Q8C2×D411yescharacteristic
153C2×Q8C2×D411yescharacteristic
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4C2×D411yescharacteristic
161C2×D4C2×D411yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167Q16—42no—
168SD16—22no—
169SD16—42no—
170SD16—42no—
171SD16—81no—
172SD16—81no—
173SD16—82no—
174SD16—81no—
175SD16—81no—
176SD16SD1614yescomplemented
177D8—42no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8C2×D411yescharacteristic
186C2×C8C2×D411yescharacteristic
187C2×C8C2×D411yescharacteristic
188C2×C8C2×D411yescharacteristic
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⋊C4—41no—
203C4⋊C4C2×D411yescharacteristic
204C4⋊C4C2×D411yescharacteristic
205C4⋊C4C2×D411yescharacteristic
206C4⋊C4C2×D411yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—21no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C42—21no—
213C42—21no—
214C42—41no—
215C42C2411yescharacteristic
216C23—21no—
217C23—21no—
218C23—41no—
219Q8—21no—
220Q8—21no—
221Q8—21no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—81no—
227Q8C2×SD1612yes—
228D4—21no—
229D4—21no—
230D4—21no—
231D4—41no—
232D4—41no—
233D4—41no—
234D4—41no—
235D4—82no—
236D4—81no—
237D4—82no—
238D4—161no—
239D4C2×SD1612yes—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—161no—
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D411yescharacteristic
258C2×C4H(5,𝔽2)11yescharacteristic
259C8—22no—
260C8—21no—
261C8—21no—
262C8—41no—
263C8—41no—
264C8—41no—
265C8—41no—
266C8C2×SD1612yes—
267C8#13512yescomplemented
268C22—22no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22—81no—
276C22—81no—
277C22D4211yescharacteristic
278C4—22no—
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4C22×SD1611yescharacteristic
289C4C2×#13511yescharacteristic
290C4#57711yescharacteristic
291C4#57711yescharacteristic
292C2—42no—
293C2—81no—
294C2—161no—
295C2D4×SD1611yescharacteristic
296C2#328411yescharacteristic
297C2#329111yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3284C2
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. #3392C2semidirect
2. #3381C2semidirect
3. #3335C2semidirect
5. #1705C2semidirect
6. #1672C2semidirect
7. #1671C2semidirect
8. #1666C2semidirect
9. #1665C2semidirect
10. #1654C2semidirect
11. #1653C2semidirect
12. #1631C2semidirect
13. #1622C2semidirect
14. C8×SD16C2semidirect
15. C8×SD16C2semidirect
176. SD16SD16semidirect
267. C8#135semidirect

Maximal subgroups

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

subgroupindex
#33922
#33812
#33352
#32842
#17052
#16722
#16712
#16662
#16652
#16542
#16532
#16312
#16222
C8×SD162