Finite Group Databasebeta

← #27744

Subgroups — #27744

Order 256.

359 conjugacy classes of subgroups.

The subgroup lattice has 295 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2570C211yesmaximal, characteristic, complemented
2#2542C211yesmaximal, characteristic
3#2100C211yesmaximal, characteristic, complemented
4#2099C211yesmaximal, characteristic, complemented
5#2073C211yesmaximal, characteristic
6#2071C211yesmaximal, characteristic
7#1993C211yesmaximal, characteristic, complemented
8#1987C211yesmaximal, characteristic, complemented
9#1936C211yesmaximal, characteristic
10#1934C211yesmaximal, characteristic
11#1836C211yesmaximal, characteristic, complemented
12#1834C211yesmaximal, characteristic, complemented
13#1804C211yesmaximal, characteristic
14#1771C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—21no—
19C2×C4×D4C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×#102C2211yescharacteristic
22C2×#102C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#97C2211yescharacteristic
26C2×#97C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#400—21no—
29#399—21no—
30#399—21no—
31#398—21no—
32#390C2211yescharacteristic
33#390C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2211yescharacteristic
36#387—21no—
37#387—21no—
38#384C2211yescharacteristic
39#384C2211yescharacteristic
40#384C2211yescharacteristic
41#382C2211yescharacteristic
42#382C2211yescharacteristic
43C4×C4⋊C4C2211yescharacteristic
44C4×C4⋊C4C2211yescharacteristic
45#340C2211yescharacteristic
46#340C2211yescharacteristic
47#340C2211yescharacteristic
48#340C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×D4C2311yescharacteristic
61C23×C4—21no—
62C22×C8C2311yescharacteristic
63C22×C8C2311yescharacteristic
64C22×C8C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C4⋊Q8—22no—
67C4⋊Q8—22no—
68C22⋊Q8—44no—
69C22⋊Q8—44no—
70C4×D4—22no—
71C4×D4—22no—
72C2×C4⋊C4—21no—
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×C4⋊C4C2311yescharacteristic
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98C2×C42C2311yescharacteristic
99C4⋊C8—22no—
100#102—22no—
101#102—22no—
102#102—22no—
103#102—22no—
104#97—22no—
105#97—22no—
106C4×C8—22no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95D411yescharacteristic
115#95D411yescharacteristic
116C24—21no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4#4112yes—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2411yescharacteristic
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8#4112yes—
155C4⋊C4—22no—
156C4⋊C4—22no—
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—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—42no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—81no—
190C42#4112yes—
191C42#4112yes—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23C22×D411yescharacteristic
200Q8—84no—
201D4—22no—
202D4—22no—
203D4—41no—
204D4—41no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
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×C4—41no—
240C2×C4—84no—
241C2×C4#12412yes—
242C2×C4#12412yes—
243C2×C4AGL(1,ℤ/8)12yes—
244C2×C4AGL(1,ℤ/8)12yes—
245C2×C4C2×#4111yescharacteristic
246C2×C4C2×#4111yescharacteristic
247C2×C4C2×#4111yescharacteristic
248C2×C4C2×#4111yescharacteristic
249C2×C42−1+411yescharacteristic
250C2×C42−1+411yescharacteristic
251C8—22no—
252C8—41no—
253C8—41no—
254C8—41no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22C2×#12411yescharacteristic
263C22#54711yescharacteristic
264C22#54811yescharacteristic
265C22C2×AGL(1,ℤ/8)11yescharacteristic
266C22C2×AGL(1,ℤ/8)11yescharacteristic
267C22#57711yescharacteristic
268C22#57811yescharacteristic
269C4—22no—
270C4—22no—
271C4—22no—
272C4—22no—
273C4—41no—
274C4—41no—
275C4—81no—
276C4—81no—
277C4—81no—
278C4—81no—
279C4—81no—
280C4—81no—
281C4—81no—
282C4#52911yescharacteristic
283C4#54111yescharacteristic
284C4#55211yescharacteristic
285C4#55211yescharacteristic
286C2—82no—
287C2#248911yescharacteristic
288C2#313711yescharacteristic
289C2#313811yescharacteristic
290C2#334311yescharacteristic
291C2#334511yescharacteristic
292C2#336711yescharacteristic
293C2#337311yescharacteristic
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. #2100C2semidirect
4. #2099C2semidirect
7. #1993C2semidirect
8. #1987C2semidirect
11. #1836C2semidirect
12. #1834C2semidirect
14. #1771C2semidirect

Maximal subgroups

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

subgroupindex
#25702
#25422
#21002
#20992
#20732
#20712
#19932
#19872
#19362
#19342
#18362
#18342
#18042
#17712
#17622