Finite Group Databasebeta

← #20044

Subgroups — #20044

Order 256.

289 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3385C211yesmaximal, characteristic, complemented
2#3369C211yesmaximal, characteristic, complemented
3#3335C211yesmaximal, characteristic
4#3302C211yesmaximal, characteristic
5#1718C211yesmaximal, characteristic, complemented
6#1700C211yesmaximal, characteristic, complemented
7#1692C211yesmaximal, characteristic, complemented
8#1683C211yesmaximal, characteristic, complemented
9#1675C211yesmaximal, characteristic, complemented
10#1645C211yesmaximal, characteristic, complemented
11#1644C211yesmaximal, characteristic, complemented
12#1638C211yesmaximal, characteristic, complemented
13#1635C211yesmaximal, characteristic
14#1585C211yesmaximal, characteristic, complemented
15#1585C211yesmaximal, characteristic, complemented
16Q82C2211yescharacteristic
17#556C2211yescharacteristic
18#554C2211yescharacteristic
19#547C2211yescharacteristic
20#501—21no—
21#500C2211yescharacteristic
22#498C2211yescharacteristic
23#489—21no—
24#487C2211yescharacteristic
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#487C2211yescharacteristic
28#485—21no—
29#483—21no—
30#481—21no—
31#479—21no—
32#477C2211yescharacteristic
33#476C2211yescharacteristic
34#476C2211yescharacteristic
35#476C2211yescharacteristic
36#475—21no—
37#475C2211yescharacteristic
38#474—21no—
39#474C2211yescharacteristic
40#467—21no—
41#465—21no—
42#464—21no—
43#462C2211yescharacteristic
44#462C2211yescharacteristic
45#452—21no—
46#446C2211yescharacteristic
47#446C2211yescharacteristic
48#441—21no—
49#440—21no—
50C4×SD16C2211yescharacteristic
51C4×SD16C2211yescharacteristic
52C8×D4C2211yescharacteristic
53C8×D4C2211yescharacteristic
54C2×#106—21no—
55#416—21no—
56#335C2211yescharacteristic
57#335C2211yescharacteristic
58#333C2211yescharacteristic
59#333C2211yescharacteristic
60#332C2211yescharacteristic
61#332C2211yescharacteristic
62Q8⋊C8C2211yescharacteristic
63Q8⋊C8C2211yescharacteristic
64D4⋊C8C2211yescharacteristic
65D4⋊C8C2211yescharacteristic
66#322C2211yescharacteristic
67C2×#41—21no—
68C2×Q16—21no—
69C2×Q16—21no—
70C2×SD16D411yescharacteristic
71C2×SD16D411yescharacteristic
72C22×C8—21no—
73C22×C8—21no—
74C4⋊Q8—21no—
75C4⋊Q8—21no—
76C4⋊Q8—41no—
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81#126—41no—
82#125—21no—
83#124—21no—
84#123—41no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8—41no—
89#121—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8C2311yescharacteristic
94C4×Q8C2311yescharacteristic
95C4×D4—21no—
96C4×D4C2311yescharacteristic
97C4×D4C2311yescharacteristic
98#117—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101#107—21no—
102#107—21no—
103#107—21no—
104#107—21no—
105#106—21no—
106#106—21no—
107#106—21no—
108#106—21no—
109#106—21no—
110#106—21no—
111#106D411yescharacteristic
112#106D411yescharacteristic
113C4⋊C8—21no—
114C4⋊C8—21no—
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
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—41no—
128#103—41no—
129#103D411yescharacteristic
130#103D411yescharacteristic
131#102—21no—
132#102—21no—
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×Q8C2×D411yescharacteristic
152C2×Q8C2×D411yescharacteristic
153C2×D4—41no—
154C2×D4C2×D411yescharacteristic
155C2×D4C2×D411yescharacteristic
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—41no—
160Q16—42no—
161Q16—81no—
162SD16—22no—
163SD16—41no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—42no—
170C2×C8—81no—
171C2×C8C2×D411yescharacteristic
172C2×C8C2×D411yescharacteristic
173C2×C8C2×D411yescharacteristic
174C2×C8C2×D411yescharacteristic
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4C2×D411yescharacteristic
192C4⋊C4C2×D411yescharacteristic
193C4⋊C4C2×D411yescharacteristic
194C4⋊C4C2×D411yescharacteristic
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—41no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—41no—
201C42—21no—
202C42—21no—
203C42—41no—
204C42—41no—
205C42C2411yescharacteristic
206C23—21no—
207C23—21no—
208Q8—21no—
209Q8—21no—
210Q8—21no—
211Q8—21no—
212Q8—41no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216Q8—81no—
217Q8—81no—
218Q8—161no—
219D4—21no—
220D4—21no—
221D4—21no—
222D4—82no—
223D4#13712yes—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—82no—
239C2×C4—161no—
240C2×C4C22×D411yescharacteristic
241C2×C4C22×D411yescharacteristic
242C2×C4H(5,𝔽2)11yescharacteristic
243C8—21no—
244C8—21no—
245C8—21no—
246C8—21no—
247C8—41no—
248C8—41no—
249C8—41no—
250C8—41no—
251C8—41no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22D4211yescharacteristic
259C4—21no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—81no—
267C4—81no—
268C4—81no—
269C4—81no—
270C4C2×#13711yescharacteristic
271C4#57511yescharacteristic
272C4#57711yescharacteristic
273C4#57811yescharacteristic
274C2—42no—
275C2—81no—
276C2#326611yescharacteristic
277C2#326911yescharacteristic
278C2#329011yescharacteristic
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. #3385C2semidirect
2. #3369C2semidirect
5. #1718C2semidirect
6. #1700C2semidirect
7. #1692C2semidirect
8. #1683C2semidirect
9. #1675C2semidirect
10. #1645C2semidirect
11. #1644C2semidirect
12. #1638C2semidirect
14. #1585C2semidirect
15. #1585C2semidirect

Maximal subgroups

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

subgroupindex
#33852
#33692
#33352
#33022
#17182
#17002
#16922
#16832
#16752
#16452
#16442
#16382
#16352
#15852