Finite Group Databasebeta

← #24839

Subgroups — #24839

Order 256.

472 conjugacy classes of subgroups.

The subgroup lattice has 337 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478C211yesmaximal, characteristic, complemented
2C22×C4⋊C8C211yesmaximal, characteristic, complemented
3#2851C211yesmaximal, characteristic, complemented
4C4×#121C211yesmaximal, characteristic
5#2088C211yesmaximal, characteristic, complemented
6#2087C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1949C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic
12#1893C211yesmaximal, characteristic
13#1808C211yesmaximal, characteristic, complemented
14#1807C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C22×C42C2211yescharacteristic
23#478—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8—24no—
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
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×C8—22no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77#125—22no—
78#125—22no—
79#121—22no—
80#121Q814yes—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—44no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C42—22no—
104C2×C42—22no—
105C2×C42—21no—
106C2×C42D411yescharacteristic
107C2×C42D411yescharacteristic
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110#107—22no—
111#107—41no—
112#106—22no—
113#106—41no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—22no—
117C4⋊C8D414yescomplemented
118#102—22no—
119#102—22no—
120#102—41no—
121#102—41no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131C24—21no—
132C24C2×D411yescharacteristic
133C2×D4—24no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4C2×Q812yes—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—41no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×Q812yes—
168C22×C4#4112yes—
169C22×C4C2411yescharacteristic
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—44no—
181C2×C8—44no—
182C2×C8C2×D412yes—
183C2×C8C2×D412yes—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4C2×Q812yes—
203C22⋊C4—24no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C42—22no—
211C42—22no—
212C42—21no—
213C42—21no—
214C42—42no—
215C42C2×D412yes—
216C42#4112yes—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23C22⋊Q812yes—
229C23#13512yes—
230C23AGL(1,ℤ/8)12yes—
231C23C22×D411yescharacteristic
232D4—22no—
233D4—22no—
234D4—48no—
235D4—41no—
236D4—41no—
237C2×C4—28no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—44no—
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—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—84no—
276C2×C4#12112yes—
277C2×C4#12112yes—
278C2×C4C22⋊Q812yes—
279C2×C4#13512yes—
280C2×C4C22×D411yescharacteristic
281C2×C4C22×Q811yescharacteristic
282C2×C4C2×#4111yescharacteristic
283C2×C4C2×#4111yescharacteristic
284C2×C4H(5,𝔽2)11yescharacteristic
285C2×C42−1+411yescharacteristic
286C8—22no—
287C8—22no—
288C8—41no—
289C8—41no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22#47814yescomplemented
304C22C2×#12111yescharacteristic
305C22C2×C22⋊Q811yescharacteristic
306C22#53711yescharacteristic
307C22C2×#13511yescharacteristic
308C22C2×#13511yescharacteristic
309C22C2×AGL(1,ℤ/8)11yescharacteristic
310C22#57511yescharacteristic
311C4—22no—
312C4—22no—
313C4—22no—
314C4—22no—
315C4—42no—
316C4—41no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4#54611yescharacteristic
324C4D4×Q811yescharacteristic
325C4#55211yescharacteristic
326C4#55411yescharacteristic
327C2—24no—
328C2—82no—
329C2#241711yescharacteristic
330C2#305011yescharacteristic
331C2#305511yescharacteristic
332C2C2×#47811yescharacteristic
333C2#307611yescharacteristic
334C2#324711yescharacteristic
335C2#325011yescharacteristic
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. C22×C4⋊C8C2semidirect
3. #2851C2semidirect
5. #2088C2semidirect
6. #2087C2semidirect
7. #2067C2semidirect
8. #2065C2semidirect
9. #1989C2semidirect
10. #1949C2semidirect
13. #1808C2semidirect
14. #1807C2semidirect
15. #1804C2semidirect
117. C4⋊C8D4semidirect
303. C22#478semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C22×C4⋊C82
#28512
C4×#1212
#20882
#20872
#20672
#20652
#19892
#19492
#18932
#18082
#18072
#18042