Finite Group Databasebeta

← #32427

Subgroups — #32427

Order 256.

340 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2565C211yesmaximal, characteristic, complemented
2#2512C211yesmaximal, characteristic
3#2093C211yesmaximal, characteristic
4#2073C211yesmaximal, characteristic
5#2072C211yesmaximal, characteristic, complemented
6#2059C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic
8#2025C211yesmaximal, characteristic
9#1968C211yesmaximal, characteristic
10#1949C211yesmaximal, characteristic, complemented
11#1928C211yesmaximal, characteristic, complemented
12#1916C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic
14#1863C211yesmaximal, characteristic, complemented
15#1834C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×#126—21no—
18C2×#125C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C2×#107C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×C4×C8C2211yescharacteristic
31#398—21no—
32#398—21no—
33#397—21no—
34#396—21no—
35#391C2211yescharacteristic
36#391C2211yescharacteristic
37#390C2211yescharacteristic
38#390C2211yescharacteristic
39#390C2211yescharacteristic
40#389C2211yescharacteristic
41#388—21no—
42#387—21no—
43#387—21no—
44#384C2211yescharacteristic
45#384C2211yescharacteristic
46#382C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×D4C2311yescharacteristic
61C2×SD16—22no—
62C2×SD16—22no—
63C2×SD16—41no—
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68#126—44no—
69#125—41no—
70#125—41no—
71C4×Q8—41no—
72C4×Q8—41no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D411yescharacteristic
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C2×C42C2311yescharacteristic
100#107—41no—
101#107—41no—
102#106—41no—
103C4⋊C8—22no—
104#103—22no—
105#103—22no—
106#103—41no—
107#102—41no—
108#102—41no—
109#102—41no—
110C4×C8—22no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95D411yescharacteristic
120#95D411yescharacteristic
121C24—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—21no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8#4112yes—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2411yescharacteristic
148SD16—44no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8#4112yes—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
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—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—81no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—81no—
188C22⋊C4—81no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—42no—
196C42—42no—
197C23—41no—
198C23—41no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23C22×D411yescharacteristic
205Q8—22no—
206Q8—22no—
207Q8—41no—
208Q8—41no—
209D4—41no—
210D4—41no—
211D4—41no—
212D4—41no—
213C2×C4—21no—
214C2×C4—21no—
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—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—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#13512yes—
251C2×C4AGL(1,ℤ/8)12yes—
252C2×C4C2×#4111yescharacteristic
253C2×C4C2×#4111yescharacteristic
254C2×C4H(5,𝔽2)11yescharacteristic
255C2×C4H(5,𝔽2)11yescharacteristic
256C2×C42−1+411yescharacteristic
257C2×C42−1+411yescharacteristic
258C8—22no—
259C8—41no—
260C8—41no—
261C8—41no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22#53511yescharacteristic
270C22#54711yescharacteristic
271C22#54811yescharacteristic
272C22C2×#13511yescharacteristic
273C22C2×AGL(1,ℤ/8)11yescharacteristic
274C22#57711yescharacteristic
275C22#57811yescharacteristic
276C4—22no—
277C4—22no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—42no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4#53911yescharacteristic
290C4#54211yescharacteristic
291C4#55111yescharacteristic
292C4#56311yescharacteristic
293C2—161no—
294C2#272311yescharacteristic
295C2#320711yescharacteristic
296C2#322511yescharacteristic
297C2#333711yescharacteristic
298C2#334311yescharacteristic
299C2#336411yescharacteristic
300C2#336811yescharacteristic
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. #2565C2semidirect
5. #2072C2semidirect
6. #2059C2semidirect
10. #1949C2semidirect
11. #1928C2semidirect
12. #1916C2semidirect
14. #1863C2semidirect
15. #1834C2semidirect

Maximal subgroups

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

subgroupindex
#25652
#25122
#20932
#20732
#20722
#20592
#20552
#20252
#19682
#19492
#19282
#19162
#18802
#18632
#18342