Finite Group Databasebeta

← #27147

Subgroups — #27147

Order 256.

340 conjugacy classes of subgroups.

The subgroup lattice has 291 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2512C211yesmaximal, characteristic
2#2369C211yesmaximal, characteristic, complemented
3#2100C211yesmaximal, characteristic, complemented
4#2065C211yesmaximal, characteristic
5#2058C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic
7#1947C211yesmaximal, characteristic, complemented
8#1936C211yesmaximal, characteristic
9#1926C211yesmaximal, characteristic
10#1925C211yesmaximal, characteristic
11#1925C211yesmaximal, characteristic
12#1919C211yesmaximal, characteristic, complemented
13#1918C211yesmaximal, characteristic, complemented
14#1917C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×#126—21no—
18C2×#107C2211yescharacteristic
19C2×#107C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×#106C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#97C2211yescharacteristic
28#397—21no—
29#396—21no—
30#391—21no—
31#391—21no—
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#390C2211yescharacteristic
35#389—21no—
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#388—21no—
40#384C2211yescharacteristic
41C4×C4⋊C4—21no—
42C4×C4⋊C4C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45C4×C4⋊C4C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×D4C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C4⋊Q8—22no—
66C4⋊Q8—22no—
67#126—44no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4D411yescharacteristic
75C2×C4⋊C4D411yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2311yescharacteristic
83C2×C22⋊C4—21no—
84C2×C22⋊C4—21no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96#107—22no—
97#107—22no—
98#106—22no—
99#106—22no—
100C4⋊C8—22no—
101#102—22no—
102#102—22no—
103#102—22no—
104#102—22no—
105#97—22no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95D411yescharacteristic
114#95D411yescharacteristic
115C24—21no—
116C2×Q8—24no—
117C2×Q8—81no—
118C2×Q8—81no—
119C2×D4—21no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2411yescharacteristic
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8#4112yes—
151C4⋊C4—24no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—81no—
168C4⋊C4—81no—
169C4⋊C4—81no—
170C4⋊C4#4112yes—
171C4⋊C4#4112yes—
172C4⋊C4#4112yes—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—81no—
178C22⋊C4—81no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—42no—
186C42—42no—
187C42—81no—
188C42—81no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23C22×D411yescharacteristic
197Q8—84no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—41no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
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—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
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×C4AGL(1,ℤ/8)12yes—
240C2×C4#13712yes—
241C2×C4C2×#4111yescharacteristic
242C2×C4C2×#4111yescharacteristic
243C2×C4C2×#4111yescharacteristic
244C2×C4C2×#4111yescharacteristic
245C2×C42−1+411yescharacteristic
246C2×C42−1+411yescharacteristic
247C8—22no—
248C8—41no—
249C8—41no—
250C8—41no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22—81no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22#52511yescharacteristic
259C22#54711yescharacteristic
260C22#54811yescharacteristic
261C22C2×AGL(1,ℤ/8)11yescharacteristic
262C22C2×#13711yescharacteristic
263C22#57611yescharacteristic
264C22#57711yescharacteristic
265C4—22no—
266C4—22no—
267C4—41no—
268C4—41no—
269C4—41no—
270C4—42no—
271C4—41no—
272C4—42no—
273C4—42no—
274C4—81no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4#52911yescharacteristic
279C4#54111yescharacteristic
280C4#55611yescharacteristic
281C4#55611yescharacteristic
282C2—161no—
283C2#246911yescharacteristic
284C2#311211yescharacteristic
285C2#311811yescharacteristic
286C2#334611yescharacteristic
287C2#335411yescharacteristic
288C2#336611yescharacteristic
289C2#336911yescharacteristic
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
2. #2369C2semidirect
3. #2100C2semidirect
5. #2058C2semidirect
7. #1947C2semidirect
12. #1919C2semidirect
13. #1918C2semidirect
14. #1917C2semidirect
15. #1776C2semidirect

Maximal subgroups

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

subgroupindex
#25122
#23692
#21002
#20652
#20582
#20232
#19472
#19362
#19262
#19252
#19192
#19182
#19172
#17762