Finite Group Databasebeta

← #31376

Subgroups — #31376

Order 256.

382 conjugacy classes of subgroups.

The subgroup lattice has 325 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2440C211yesmaximal, characteristic, complemented
2#2420C211yesmaximal, characteristic
3#2099C211yesmaximal, characteristic, complemented
4#2088C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic
8#2023C211yesmaximal, characteristic, complemented
9#1993C211yesmaximal, characteristic, complemented
10#1936C211yesmaximal, characteristic, complemented
11#1924C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic
13#1862C211yesmaximal, characteristic, complemented
14#1824C211yesmaximal, characteristic, complemented
15#1802C211yesmaximal, characteristic, complemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×C22⋊Q8—21no—
18C2×#121C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C22×C4⋊C4C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#102C2211yescharacteristic
25C2×#102C2211yescharacteristic
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×#97C2211yescharacteristic
31#398—21no—
32#397—21no—
33#395—21no—
34#394—21no—
35#390C2211yescharacteristic
36#389C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#386—21no—
40#385—21no—
41#384C2211yescharacteristic
42#382C2211yescharacteristic
43#381—21no—
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46C4×C22⋊C4C2211yescharacteristic
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×D4—21no—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8—22no—
70C4⋊Q8—41no—
71C22⋊Q8—44no—
72#121—22no—
73#121—41no—
74C4×D4—22no—
75C4×D4—41no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4D411yescharacteristic
99C2×C22⋊C4D411yescharacteristic
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#107—41no—
103#106—41no—
104C4⋊C8—41no—
105#102—22no—
106#102—22no—
107#102—41no—
108#102—41no—
109C22⋊C8—41no—
110C22⋊C8—41no—
111#97—41no—
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—
122C24C2×D411yescharacteristic
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8—81no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—81no—
134C2×D4#4112yes—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—81no—
153C22×C4—81no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2411yescharacteristic
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
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—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4#4112yes—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—42no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C22⋊C4—81no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23—41no—
213C23—41no—
214C23—41no—
215C23—41no—
216C23—41no—
217C23—41no—
218C23#13712yes—
219C23C22×D411yescharacteristic
220Q8—84no—
221D4—22no—
222D4—22no—
223D4—41no—
224D4—41no—
225D4—84no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
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—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4AGL(1,ℤ/8)12yes—
268C2×C4C2×#4111yescharacteristic
269C2×C4C2×#4111yescharacteristic
270C2×C4H(5,𝔽2)11yescharacteristic
271C2×C4H(5,𝔽2)11yescharacteristic
272C2×C42−1+411yescharacteristic
273C2×C42−1+411yescharacteristic
274C8—41no—
275C8—41no—
276C8—41no—
277C8—41no—
278C22—22no—
279C22—22no—
280C22—22no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—81no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22#53611yescharacteristic
293C22#54611yescharacteristic
294C22#54611yescharacteristic
295C22C2×AGL(1,ℤ/8)11yescharacteristic
296C22C2×#13711yescharacteristic
297C22#57611yescharacteristic
298C22#57711yescharacteristic
299C4—22no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—42no—
305C4—81no—
306C4—81no—
307C4—81no—
308C4—81no—
309C4—81no—
310C4—81no—
311C4#53911yescharacteristic
312C4#54211yescharacteristic
313C4#55211yescharacteristic
314C4#56211yescharacteristic
315C2—42no—
316C2—82no—
317C2#268711yescharacteristic
318C2#323911yescharacteristic
319C2#324111yescharacteristic
320C2#330611yescharacteristic
321C2#331411yescharacteristic
322C2#332011yescharacteristic
323C2#332511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2440C2semidirect
3. #2099C2semidirect
4. #2088C2semidirect
5. #2085C2semidirect
6. #2076C2semidirect
8. #2023C2semidirect
9. #1993C2semidirect
10. #1936C2semidirect
11. #1924C2semidirect
13. #1862C2semidirect
14. #1824C2semidirect
15. #1802C2semidirect

Maximal subgroups

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

subgroupindex
#24402
#24202
#20992
#20882
#20852
#20762
#20342
#20232
#19932
#19362
#19242
#18742
#18622
#18242
#18022