Finite Group Databasebeta

← #26535

Subgroups — #26535

Order 256.

361 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2570C211yesmaximal, characteristic, complemented
2#2367C211yesmaximal, characteristic
3#2095C211yesmaximal, characteristic, complemented
4#2092C211yesmaximal, characteristic
5#2067C211yesmaximal, characteristic
6#2066C211yesmaximal, characteristic, complemented
7#1983C211yesmaximal, characteristic
8#1958C211yesmaximal, characteristic
9#1949C211yesmaximal, characteristic, complemented
10#1939C211yesmaximal, characteristic, complemented
11#1863C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic
13#1805C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic
15C4×#107C211yesmaximal, characteristic, complemented
16C2×#124C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#400—21no—
36#398—21no—
37#391C2211yescharacteristic
38#390C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#388—21no—
42#388—21no—
43#387—21no—
44#387—21no—
45#385—21no—
46#384C2211yescharacteristic
47#382C2211yescharacteristic
48#382C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66#124—22no—
67#124Q814yes—
68C4×Q8—22no—
69C4×Q8—41no—
70C4×D4—22no—
71C4×D4—41no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4D411yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96#107—22no—
97#107—22no—
98#107—41no—
99#106—41no—
100C4⋊C8—22no—
101C4⋊C8—41no—
102C4⋊C8—41no—
103#103—22no—
104#103—41no—
105#103—41no—
106#103—41no—
107#102—22no—
108#102—41no—
109#102—41no—
110#102—41no—
111C4×C8—22no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95D411yescharacteristic
123#95D411yescharacteristic
124C24—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8C2×Q812yes—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4C2×Q812yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—42no—
146C22×C4—81no—
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2×D411yescharacteristic
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2411yescharacteristic
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8#4112yes—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4#4112yes—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—81no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—42no—
195C42—81no—
196C42—81no—
197C42C2×Q812yes—
198C42#4112yes—
199C23—22no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23C22×D411yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—41no—
210Q8—41no—
211D4—22no—
212D4—22no—
213D4—41no—
214D4—41no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
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—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—84no—
252C2×C4C22⋊Q812yes—
253C2×C4C22⋊Q812yes—
254C2×C4#13512yes—
255C2×C4C22×Q811yescharacteristic
256C2×C4C2×#4111yescharacteristic
257C2×C4C2×#4111yescharacteristic
258C2×C4C2×#4111yescharacteristic
259C2×C4H(5,𝔽2)11yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C8—22no—
262C8—41no—
263C8—41no—
264C8—41no—
265C22—42no—
266C22—42no—
267C22—42no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22C2×C22⋊Q811yescharacteristic
273C22#54611yescharacteristic
274C22#55011yescharacteristic
275C22C2×#13511yescharacteristic
276C22#57511yescharacteristic
277C22#57611yescharacteristic
278C22#57811yescharacteristic
279C4—22no—
280C4—22no—
281C4—22no—
282C4—42no—
283C4—41no—
284C4—42no—
285C4—41no—
286C4—41no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4—81no—
291C4—81no—
292C4#52911yescharacteristic
293C4#54011yescharacteristic
294C4#55411yescharacteristic
295C4#55411yescharacteristic
296C2—82no—
297C2#245411yescharacteristic
298C2#307611yescharacteristic
299C2#308311yescharacteristic
300C2#329711yescharacteristic
301C2#332711yescharacteristic
302C2#339311yescharacteristic
303C2#340311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2570C2semidirect
3. #2095C2semidirect
6. #2066C2semidirect
9. #1949C2semidirect
10. #1939C2semidirect
11. #1863C2semidirect
13. #1805C2semidirect
15. C4×#107C2semidirect

Maximal subgroups

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

subgroupindex
#25702
#23672
#20952
#20922
#20672
#20662
#19832
#19582
#19492
#19392
#18632
#18622
#18052
#18022
C4×#1072