Finite Group Databasebeta

← #24706

Subgroups — #24706

Order 256.

434 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478C211yesmaximal, characteristic, complemented
2C2×#432C211yesmaximal, characteristic, complemented
3#2851C211yesmaximal, characteristic, complemented
4C4×#121C211yesmaximal, characteristic
5#2077C211yesmaximal, characteristic, complemented
6#2075C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic
8#2033C211yesmaximal, characteristic
9#1989C211yesmaximal, characteristic, complemented
10#1949C211yesmaximal, characteristic, complemented
11#1944C211yesmaximal, characteristic, complemented
12#1943C211yesmaximal, characteristic, complemented
13#1936C211yesmaximal, characteristic, complemented
14#1936C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C2×#125C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×C4×D4—21no—
19C2×C4×D4C2211yescharacteristic
20C22×C42C2211yescharacteristic
21#478—24no—
22#432—24no—
23C2×#107C2211yescharacteristic
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×#97C2211yescharacteristic
36C2×#97C2211yescharacteristic
37#390C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#389C2211yescharacteristic
41#388—21no—
42#387—21no—
43#387—21no—
44#386—21no—
45#385—21no—
46#385—21no—
47#382C2211yescharacteristic
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4C2311yescharacteristic
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71#125—22no—
72#125—22no—
73#121—22no—
74#121—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—44no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
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×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42—21no—
100C2×C42D411yescharacteristic
101C2×C42D411yescharacteristic
102C2×C42C2311yescharacteristic
103C2×C42C2311yescharacteristic
104#107—22no—
105#107—22no—
106#106—22no—
107#106—22no—
108C4⋊C8—22no—
109C4⋊C8D414yescomplemented
110#102—22no—
111#102—22no—
112#102—22no—
113#102—22no—
114C22⋊C8—22no—
115C22⋊C8—22no—
116#97—22no—
117#97—22no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127C24—21no—
128C24C2×D411yescharacteristic
129C2×D4—24no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4#4112yes—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2411yescharacteristic
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8C2×D412yes—
177C2×C8C2×D412yes—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4#4112yes—
197C22⋊C4—24no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C42—22no—
205C42—22no—
206C42—21no—
207C42—21no—
208C42—42no—
209C42C2×D412yes—
210C42#4112yes—
211C23—21no—
212C23—21no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—22no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23AGL(1,ℤ/8)12yes—
225C23C22×D411yescharacteristic
226D4—22no—
227D4—22no—
228D4—48no—
229D4—41no—
230D4—41no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—44no—
246C2×C4—44no—
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—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—84no—
272C2×C4#12112yes—
273C2×C4#12112yes—
274C2×C4C22×D411yescharacteristic
275C2×C4C2×#4111yescharacteristic
276C2×C4C2×#4111yescharacteristic
277C2×C4C2×#4111yescharacteristic
278C2×C42−1+411yescharacteristic
279C2×C42−1+411yescharacteristic
280C8—22no—
281C8—22no—
282C8—41no—
283C8—41no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—81no—
295C22—81no—
296C22—81no—
297C22—81no—
298C22C2×#12111yescharacteristic
299C22#52511yescharacteristic
300C22#53611yescharacteristic
301C22C2×AGL(1,ℤ/8)11yescharacteristic
302C22#57511yescharacteristic
303C22#57511yescharacteristic
304C22#57511yescharacteristic
305C4—22no—
306C4—22no—
307C4—42no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4—81no—
317C4#54711yescharacteristic
318C4#54811yescharacteristic
319C4#55211yescharacteristic
320C4#55611yescharacteristic
321C2—42no—
322C2—82no—
323C2#241511yescharacteristic
324C2#305411yescharacteristic
325C2#305511yescharacteristic
326C2#311111yescharacteristic
327C2#311411yescharacteristic
328C2#323411yescharacteristic
329C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#478C2semidirect
2. C2×#432C2semidirect
3. #2851C2semidirect
5. #2077C2semidirect
6. #2075C2semidirect
9. #1989C2semidirect
10. #1949C2semidirect
11. #1944C2semidirect
12. #1943C2semidirect
13. #1936C2semidirect
14. #1936C2semidirect
15. #1804C2semidirect
109. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C2×#4322
#28512
C4×#1212
#20772
#20752
#20342
#20332
#19892
#19492
#19442
#19432
#19362
#18042