Finite Group Databasebeta

← #20043

Subgroups — #20043

Order 256.

300 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3398C211yesmaximal, characteristic, complemented
2#3386C211yesmaximal, characteristic, complemented
3#3302C211yesmaximal, characteristic
4#3298C211yesmaximal, characteristic
5#1723C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1700C211yesmaximal, characteristic, complemented
8#1673C211yesmaximal, characteristic, complemented
9#1669C211yesmaximal, characteristic, complemented
10#1654C211yesmaximal, characteristic
11#1653C211yesmaximal, characteristic, complemented
12#1650C211yesmaximal, characteristic, complemented
13#1649C211yesmaximal, characteristic, complemented
14#1585C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16#557C2211yescharacteristic
17#556C2211yescharacteristic
18D4×Q8C2211yescharacteristic
19#547C2211yescharacteristic
20#501—21no—
21#500C2211yescharacteristic
22#500C2211yescharacteristic
23#498C2211yescharacteristic
24#497—21no—
25#494C2211yescharacteristic
26#487C2211yescharacteristic
27#487C2211yescharacteristic
28#485—21no—
29#483—21no—
30#478—21no—
31#477—21no—
32#475C2211yescharacteristic
33#475C2211yescharacteristic
34#474C2211yescharacteristic
35#474C2211yescharacteristic
36#467—21no—
37#465—21no—
38#464—21no—
39#462—21no—
40#462C2211yescharacteristic
41#462C2211yescharacteristic
42#461C2211yescharacteristic
43#461C2211yescharacteristic
44#452—21no—
45#448—21no—
46#446C2211yescharacteristic
47#446C2211yescharacteristic
48#441—21no—
49#440—21no—
50C4×SD16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54#416—21no—
55C2×#103—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#332C2211yescharacteristic
59#332C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C22×Q8—21no—
69C2×Q16—21no—
70C2×Q16—21no—
71C2×Q16—21no—
72C2×Q16—21no—
73C2×SD16—21no—
74C2×SD16—21no—
75C2×SD16D411yescharacteristic
76C2×SD16D411yescharacteristic
77C22×C8—21no—
78C22×C8—21no—
79C4⋊Q8—21no—
80C4⋊Q8—21no—
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83C4⋊Q8C2311yescharacteristic
84C4⋊Q8C2311yescharacteristic
85#126—41no—
86#125—21no—
87#125—41no—
88#124—21no—
89#123—41no—
90C22⋊Q8—21no—
91C22⋊Q8—21no—
92C22⋊Q8—21no—
93C22⋊Q8—41no—
94#121—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8C2311yescharacteristic
99C4×Q8C2311yescharacteristic
100C4×D4—21no—
101C4×D4C2311yescharacteristic
102C4×D4C2311yescharacteristic
103#117—21no—
104C2×C4⋊C4—21no—
105#107—21no—
106#107—21no—
107#107—21no—
108#107—21no—
109#107—41no—
110#106—21no—
111#106—21no—
112#106—41no—
113#106D411yescharacteristic
114#106D411yescharacteristic
115C4⋊C8—21no—
116C4⋊C8—21no—
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
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—21no—
130#103—21no—
131#103D411yescharacteristic
132#103D411yescharacteristic
133#102—21no—
134#102—21no—
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139#97—21no—
140#97—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×Q8—41no—
151C2×Q8—42no—
152C2×Q8—81no—
153C2×Q8C2×D411yescharacteristic
154C2×Q8C2×D411yescharacteristic
155C2×D4—41no—
156C2×D4C2×D411yescharacteristic
157C2×D4C2×D411yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—41no—
161C22×C4—41no—
162Q16—42no—
163Q16—81no—
164Q16—81no—
165Q16—81no—
166SD16—22no—
167SD16—41no—
168SD16—81no—
169SD16—81no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—81no—
176C2×C8—81no—
177C2×C8C2×D411yescharacteristic
178C2×C8C2×D411yescharacteristic
179C2×C8C2×D411yescharacteristic
180C2×C8C2×D411yescharacteristic
181C4⋊C4—21no—
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—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4C2×D411yescharacteristic
198C4⋊C4C2×D411yescharacteristic
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—41no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C42—21no—
208C42—21no—
209C42—41no—
210C42—41no—
211C42C2411yescharacteristic
212C23—21no—
213C23—21no—
214Q8—21no—
215Q8—21no—
216Q8—21no—
217Q8—21no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224Q8—41no—
225Q8—41no—
226Q8—82no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—82no—
231D4#13712yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4C22×D411yescharacteristic
249C2×C4C22×D411yescharacteristic
250C2×C4H(5,𝔽2)11yescharacteristic
251C8—22no—
252C8—21no—
253C8—21no—
254C8—21no—
255C8—21no—
256C8—41no—
257C8—41no—
258C8—41no—
259C8—41no—
260C22—22no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—41no—
266C22D4211yescharacteristic
267C4—21no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—41no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4C2×#13711yescharacteristic
279C4#57511yescharacteristic
280C4#57611yescharacteristic
281C4#57711yescharacteristic
282C2—42no—
283C2—81no—
284C2#326611yescharacteristic
285C2#326811yescharacteristic
286C2#329311yescharacteristic
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. #3398C2semidirect
2. #3386C2semidirect
5. #1723C2semidirect
6. #1700C2semidirect
7. #1700C2semidirect
8. #1673C2semidirect
9. #1669C2semidirect
11. #1653C2semidirect
12. #1650C2semidirect
13. #1649C2semidirect
14. #1585C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33982
#33862
#33022
#32982
#17232
#17002
#16732
#16692
#16542
#16532
#16502
#16492
#15852