Finite Group Databasebeta

← #19588

Subgroups — #19588

Order 256.

284 conjugacy classes of subgroups.

The subgroup lattice has 277 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3248C211yesmaximal, characteristic
2#3238C211yesmaximal, characteristic
3#3229C211yesmaximal, characteristic, complemented
4#2994C211yesmaximal, characteristic
5#1704C211yesmaximal, characteristic, complemented
6#1703C211yesmaximal, characteristic, complemented
7#1692C211yesmaximal, characteristic, complemented
8#1691C211yesmaximal, characteristic, complemented
9#1684C211yesmaximal, characteristic, complemented
10#1683C211yesmaximal, characteristic, complemented
11#1672C211yesmaximal, characteristic, complemented
12#1671C211yesmaximal, characteristic, complemented
13#1573C211yesmaximal, characteristic, complemented
14#1552C211yesmaximal, characteristic, complemented
15C2×#333C211yesmaximal, characteristic, complemented
16#533C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18#529C2211yescharacteristic
19#501—21no—
20#498C2211yescharacteristic
21#498C2211yescharacteristic
22#489—21no—
23#487C2211yescharacteristic
24#487C2211yescharacteristic
25#483—21no—
26#483—21no—
27#483—21no—
28#480—21no—
29#479—21no—
30#477C2211yescharacteristic
31#477C2211yescharacteristic
32#476C2211yescharacteristic
33#476C2211yescharacteristic
34#475—21no—
35#475—21no—
36#475C2211yescharacteristic
37#474C2211yescharacteristic
38#467—21no—
39#465—21no—
40#464—21no—
41#462C2211yescharacteristic
42#461C2211yescharacteristic
43#446C2211yescharacteristic
44C8×Q8C2211yescharacteristic
45#436C2211yescharacteristic
46#435—21no—
47C8×D4—21no—
48C8×D4C2211yescharacteristic
49#433—21no—
50#432C2211yescharacteristic
51#431C2211yescharacteristic
52#424—21no—
53C2×C4⋊C8C2211yescharacteristic
54C2×C4⋊C8C2211yescharacteristic
55#335C2211yescharacteristic
56#335C2211yescharacteristic
57#334C2211yescharacteristic
58#334C2211yescharacteristic
59#333C2211yescharacteristic
60#333C2211yescharacteristic
61#333C2211yescharacteristic
62#333C2211yescharacteristic
63#332C2211yescharacteristic
64#332C2211yescharacteristic
65#332C2211yescharacteristic
66#332C2211yescharacteristic
67C22×Q8—21no—
68C2×Q16—21no—
69C2×SD16—21no—
70C2×M16—21no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—21no—
74C4⋊Q8—21no—
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77C4⋊Q8C2311yescharacteristic
78C4⋊Q8C2311yescharacteristic
79C4⋊Q8C2311yescharacteristic
80C4⋊Q8C2311yescharacteristic
81#126—21no—
82#125—21no—
83#125D411yescharacteristic
84#124D411yescharacteristic
85#123—21no—
86C22⋊Q8—21no—
87C22⋊Q8—21no—
88C22⋊Q8D411yescharacteristic
89#121D411yescharacteristic
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8C2311yescharacteristic
93C4×D4—21no—
94C4×D4C2311yescharacteristic
95#117—21no—
96#117—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C42C2311yescharacteristic
100#107—21no—
101#107—21no—
102#107—21no—
103#106—21no—
104#106—21no—
105#106—21no—
106#106—21no—
107#106—21no—
108#106—21no—
109#106—21no—
110C4⋊C8—21no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8—21no—
114C4⋊C8D411yescharacteristic
115C4⋊C8D411yescharacteristic
116C4⋊C8C2311yescharacteristic
117C4⋊C8C2311yescharacteristic
118C4⋊C8C2311yescharacteristic
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121C4⋊C8C2311yescharacteristic
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#102—21no—
132#102—21no—
133#102—21no—
134C22⋊C8—21no—
135C22⋊C8—21no—
136C22⋊C8—21no—
137C22⋊C8D411yescharacteristic
138C22⋊C8D411yescharacteristic
139#97—21no—
140#97—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
158Q16—81no—
159SD16—81no—
160M16—81no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—81no—
168C2×C8—81no—
169C2×C8—81no—
170C2×C8C2×D411yescharacteristic
171C2×C8C2×D411yescharacteristic
172C2×C8C2×D411yescharacteristic
173C2×C8C2×D411yescharacteristic
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—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4C2×D411yescharacteristic
191C4⋊C4C2×D411yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—21no—
196C22⋊C4—21no—
197C22⋊C4—41no—
198C42—21no—
199C42—41no—
200C42C2×D411yescharacteristic
201C42C2×D411yescharacteristic
202C42#4111yescharacteristic
203C42C2411yescharacteristic
204C23—21no—
205C23#12111yescharacteristic
206Q8—21no—
207Q8—21no—
208Q8—41no—
209Q8—41no—
210Q8—41no—
211Q8—41no—
212Q8—81no—
213Q8—81no—
214Q8—82no—
215D4—21no—
216D4—21no—
217D4—42no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—82no—
233C2×C4#12111yescharacteristic
234C2×C4#12111yescharacteristic
235C2×C4#12111yescharacteristic
236C2×C4C22×D411yescharacteristic
237C2×C4C22×D411yescharacteristic
238C2×C4C2×#4111yescharacteristic
239C8—21no—
240C8—21no—
241C8—21no—
242C8—21no—
243C8—41no—
244C8—41no—
245C8—41no—
246C8—41no—
247C8—41no—
248C8—41no—
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22#47112yes—
255C22C2×#12111yescharacteristic
256C4—21no—
257C4—21no—
258C4—21no—
259C4—21no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—81no—
264C4—81no—
265C4—81no—
266C4—81no—
267C4#57711yescharacteristic
268C4#57711yescharacteristic
269C4#57811yescharacteristic
270C4#57811yescharacteristic
271C2—22no—
272C2—81no—
273C2#304611yescharacteristic
274C2#306311yescharacteristic
275C2C2×#47111yescharacteristic
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. #3229C2semidirect
5. #1704C2semidirect
6. #1703C2semidirect
7. #1692C2semidirect
8. #1691C2semidirect
9. #1684C2semidirect
10. #1683C2semidirect
11. #1672C2semidirect
12. #1671C2semidirect
13. #1573C2semidirect
14. #1552C2semidirect
15. C2×#333C2semidirect

Maximal subgroups

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

subgroupindex
#32482
#32382
#32292
#29942
#17042
#17032
#16922
#16912
#16842
#16832
#16722
#16712
#15732
#15522
C2×#3332