Finite Group Databasebeta

← #20178

Subgroups — #20178

Order 256.

308 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3384C211yesmaximal, characteristic, complemented
2#3331C211yesmaximal, characteristic, complemented
3#3330C211yesmaximal, characteristic
4#3298C211yesmaximal, characteristic, complemented
5#1718C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1699C211yesmaximal, characteristic, complemented
8#1691C211yesmaximal, characteristic, complemented
9#1683C211yesmaximal, characteristic, complemented
10#1649C211yesmaximal, characteristic, complemented
11#1647C211yesmaximal, characteristic, complemented
12#1643C211yesmaximal, characteristic, complemented
13#1635C211yesmaximal, characteristic, complemented
14#1592C211yesmaximal, characteristic, complemented
15#1581C211yesmaximal, characteristic, complemented
16#554C2211yescharacteristic
17D4×Q8C2211yescharacteristic
18#547C2211yescharacteristic
19#547C2211yescharacteristic
20#501—21no—
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#487C2211yescharacteristic
24#487C2211yescharacteristic
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#484—21no—
28#483—21no—
29#483—21no—
30#478—21no—
31#476—21no—
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#475C2211yescharacteristic
35#474C2211yescharacteristic
36#474C2211yescharacteristic
37#470—21no—
38#468—21no—
39#465—21no—
40#462C2211yescharacteristic
41#461C2211yescharacteristic
42#461C2211yescharacteristic
43#452—21no—
44#449—21no—
45#448—21no—
46#446C2211yescharacteristic
47#442—21no—
48C4×SD16C2211yescharacteristic
49C4×D8C2211yescharacteristic
50#436C2211yescharacteristic
51#436C2211yescharacteristic
52C8×D4C2211yescharacteristic
53#417—21no—
54#417—21no—
55C2×#103—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
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—
70C2×Q16—21no—
71C2×SD16—21no—
72C2×SD16—21no—
73C2×SD16D411yescharacteristic
74C2×D8D411yescharacteristic
75C2×M16—21no—
76C2×M16—21no—
77C22×C8—21no—
78C4⋊Q8—21no—
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81C4⋊Q8C2311yescharacteristic
82C4⋊Q8C2311yescharacteristic
83#125—21no—
84#123—41no—
85#123—41no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—21no—
89C22⋊Q8—21no—
90C22⋊Q8—41no—
91C22⋊Q8—41no—
92#121—21no—
93#121—21no—
94C4×Q8C2311yescharacteristic
95C4×D4—21no—
96C4×D4—21no—
97C4×D4C2311yescharacteristic
98C4×D4C2311yescharacteristic
99C4×D4C2311yescharacteristic
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103#107—21no—
104#107—21no—
105#107—41no—
106#107D411yescharacteristic
107#106—21no—
108#106—21no—
109#106—21no—
110#106—21no—
111#106—21no—
112#106D411yescharacteristic
113C4⋊C8—21no—
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118#103—21no—
119#103—21no—
120#103—21no—
121#103—21no—
122#103—21no—
123#103—21no—
124#103—21no—
125#103—21no—
126#103—21no—
127#103—21no—
128#103—41no—
129#103D411yescharacteristic
130#102—21no—
131#102—21no—
132#102—21no—
133#102—21no—
134#102D411yescharacteristic
135#102D411yescharacteristic
136#102D411yescharacteristic
137C22⋊C8—21no—
138C22⋊C8—21no—
139C22⋊C8—21no—
140#97—21no—
141C4×C8C2311yescharacteristic
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144#41—42no—
145#41—42no—
146#41—81no—
147#41—81no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—41no—
153C2×Q8—42no—
154C2×Q8—81no—
155C2×Q8C2×D411yescharacteristic
156C2×D4—41no—
157C2×D4—41no—
158C2×D4C2×D411yescharacteristic
159C2×D4C2×D411yescharacteristic
160C2×D4C2×D411yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168Q16—81no—
169SD16—41no—
170SD16—81no—
171SD16—81no—
172D8—22no—
173M16—81no—
174M16—81no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—81no—
181C2×C8C2×D411yescharacteristic
182C2×C8C2×D411yescharacteristic
183C2×C8C2×D411yescharacteristic
184C2×C8C2×D411yescharacteristic
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4C2×D411yescharacteristic
200C4⋊C4C2×D411yescharacteristic
201C4⋊C4C2×D411yescharacteristic
202C4⋊C4C2×D411yescharacteristic
203C22⋊C4—21no—
204C22⋊C4—21no—
205C22⋊C4—21no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C42—21no—
211C42C2411yescharacteristic
212C23—21no—
213C23—21no—
214C23—21no—
215Q8—21no—
216Q8—21no—
217Q8—41no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—81no—
224Q8—82no—
225D4—21no—
226D4—21no—
227D4—21no—
228D4—21no—
229D4—21no—
230D4—82no—
231D4—82no—
232D4#13712yes—
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—41no—
247C2×C4—41no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4—161no—
252C2×C4C22×D411yescharacteristic
253C2×C4C22×D411yescharacteristic
254C2×C4H(5,𝔽2)11yescharacteristic
255C8—21no—
256C8—21no—
257C8—21no—
258C8—21no—
259C8—41no—
260C8—41no—
261C8—41no—
262C8—41no—
263C8—41no—
264C22—22no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22D4211yescharacteristic
274C4—21no—
275C4—41no—
276C4—41no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4—81no—
281C4—81no—
282C4—81no—
283C4—81no—
284C4C2×#13711yescharacteristic
285C4#57511yescharacteristic
286C4#57611yescharacteristic
287C4#57711yescharacteristic
288C2—42no—
289C2—81no—
290C2—81no—
291C2#326611yescharacteristic
292C2#327311yescharacteristic
293C2#328911yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radical#3330C2
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. #3331C2semidirect
4. #3298C2semidirect
5. #1718C2semidirect
6. #1700C2semidirect
7. #1699C2semidirect
8. #1691C2semidirect
9. #1683C2semidirect
10. #1649C2semidirect
11. #1647C2semidirect
12. #1643C2semidirect
13. #1635C2semidirect
14. #1592C2semidirect
15. #1581C2semidirect

Maximal subgroups

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

subgroupindex
#33842
#33312
#33302
#32982
#17182
#17002
#16992
#16912
#16832
#16492
#16472
#16432
#16352
#15922
#15812