Finite Group Databasebeta

← #19547

Subgroups — #19547

Order 256.

276 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3238C211yesmaximal, characteristic
2#3233C211yesmaximal, characteristic
3#3230C211yesmaximal, characteristic, complemented
4#2993C211yesmaximal, characteristic
5#1704C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1696C211yesmaximal, characteristic, complemented
8#1695C211yesmaximal, characteristic, complemented
9#1684C211yesmaximal, characteristic, complemented
10#1683C211yesmaximal, characteristic, complemented
11#1676C211yesmaximal, characteristic, complemented
12#1675C211yesmaximal, characteristic, complemented
13#1572C211yesmaximal, characteristic, complemented
14#1559C211yesmaximal, characteristic, complemented
15#1548C211yesmaximal, characteristic, complemented
16#533C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18#529C2211yescharacteristic
19#501—21no—
20#500C2211yescharacteristic
21#500C2211yescharacteristic
22#499—21no—
23#489—21no—
24#488—21no—
25#487C2211yescharacteristic
26#487C2211yescharacteristic
27#485—21no—
28#484—21no—
29#484—21no—
30#483—21no—
31#482—21no—
32#480—21no—
33#477—21no—
34#477—21no—
35#477C2211yescharacteristic
36#477C2211yescharacteristic
37#477C2211yescharacteristic
38#476C2211yescharacteristic
39#476C2211yescharacteristic
40#476C2211yescharacteristic
41#475C2211yescharacteristic
42#474C2211yescharacteristic
43#446C2211yescharacteristic
44C8×Q8C2211yescharacteristic
45#436—21no—
46#436C2211yescharacteristic
47#435—21no—
48C8×D4C2211yescharacteristic
49#433—21no—
50#432C2211yescharacteristic
51#431C2211yescharacteristic
52#431C2211yescharacteristic
53#424—21no—
54#423C2211yescharacteristic
55#335C2211yescharacteristic
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#334C2211yescharacteristic
59#333C2211yescharacteristic
60#333C2211yescharacteristic
61#333C2211yescharacteristic
62#333C2211yescharacteristic
63#333C2211yescharacteristic
64#333C2211yescharacteristic
65#332C2211yescharacteristic
66#332C2211yescharacteristic
67C22×Q8—21no—
68C2×M16—21no—
69C2×M16—21no—
70C22×C8—21no—
71C4⋊Q8—21no—
72C4⋊Q8—21no—
73C4⋊Q8—21no—
74C4⋊Q8C2311yescharacteristic
75C4⋊Q8C2311yescharacteristic
76C4⋊Q8C2311yescharacteristic
77C4⋊Q8C2311yescharacteristic
78#126—21no—
79#125—21no—
80#125D411yescharacteristic
81#124D411yescharacteristic
82#123—21no—
83C22⋊Q8—21no—
84C22⋊Q8—21no—
85C22⋊Q8D411yescharacteristic
86#121D411yescharacteristic
87C4×Q8—21no—
88C4×Q8—21no—
89C4×Q8C2311yescharacteristic
90C4×D4—21no—
91C4×D4C2311yescharacteristic
92#117—21no—
93#117—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C42C2311yescharacteristic
97#107—21no—
98#107—21no—
99#107—21no—
100#107—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—
109C4⋊C8—21no—
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8D411yescharacteristic
113C4⋊C8D411yescharacteristic
114C4⋊C8C2311yescharacteristic
115C4⋊C8C2311yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
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#102—21no—
129#102—21no—
130#102—21no—
131#102—21no—
132C22⋊C8—21no—
133C22⋊C8—21no—
134C22⋊C8—21no—
135C22⋊C8—21no—
136C22⋊C8D411yescharacteristic
137C22⋊C8D411yescharacteristic
138#97—21no—
139#97—21no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8C2311yescharacteristic
143C4×C8C2311yescharacteristic
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—41no—
149C2×Q8—82no—
150C2×Q8C2×D411yescharacteristic
151C2×D4—21no—
152C2×D4C2×D411yescharacteristic
153C22×C4—21no—
154C22×C4—41no—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4#4111yescharacteristic
158M16—81no—
159M16—81no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—81no—
167C2×C8C2×D411yescharacteristic
168C2×C8C2×D411yescharacteristic
169C2×C8C2×D411yescharacteristic
170C2×C8C2×D411yescharacteristic
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
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—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4C2×D411yescharacteristic
188C4⋊C4C2×D411yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—41no—
195C42—21no—
196C42—41no—
197C42C2×D411yescharacteristic
198C42C2×D411yescharacteristic
199C42#4111yescharacteristic
200C42C2411yescharacteristic
201C23—21no—
202C23#12111yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—41no—
206Q8—41no—
207Q8—81no—
208Q8—81no—
209Q8—81no—
210Q8—82no—
211D4—21no—
212D4—21no—
213D4—81no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—82no—
229C2×C4#12111yescharacteristic
230C2×C4#12111yescharacteristic
231C2×C4#12111yescharacteristic
232C2×C4C22×D411yescharacteristic
233C2×C4C22×D411yescharacteristic
234C2×C4C2×#4111yescharacteristic
235C8—21no—
236C8—21no—
237C8—21no—
238C8—21no—
239C8—41no—
240C8—41no—
241C8—41no—
242C8—41no—
243C8—41no—
244C8—41no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22C2×#12111yescharacteristic
252C4—21no—
253C4—21no—
254C4—21no—
255C4—21no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—81no—
260C4—81no—
261C4—81no—
262C4—81no—
263C4#57711yescharacteristic
264C4#57811yescharacteristic
265C4#57811yescharacteristic
266C4#57811yescharacteristic
267C2—41no—
268C2—81no—
269C2#304611yescharacteristic
270C2#305911yescharacteristic
271C2#306811yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #3230C2semidirect
5. #1704C2semidirect
6. #1703C2semidirect
7. #1696C2semidirect
8. #1695C2semidirect
9. #1684C2semidirect
10. #1683C2semidirect
11. #1676C2semidirect
12. #1675C2semidirect
13. #1572C2semidirect
14. #1559C2semidirect
15. #1548C2semidirect

Maximal subgroups

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

subgroupindex
#32382
#32332
#32302
#29932
#17042
#17032
#16962
#16952
#16842
#16832
#16762
#16752
#15722
#15592
#15482