Finite Group Databasebeta

← #25256

Subgroups — #25256

Order 256.

432 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484C211yesmaximal, characteristic, complemented
2C2×#436C211yesmaximal, characteristic
3#2486C211yesmaximal, characteristic
4#2415C211yesmaximal, characteristic
5#2086C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#1942C211yesmaximal, characteristic, complemented
10#1933C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic, complemented
12#1894C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1809C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C2×#123—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#484—24no—
25#436—24no—
26C2×#107C2211yescharacteristic
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#398—21no—
39#397—21no—
40#397—21no—
41#394—21no—
42#391C2211yescharacteristic
43#389—21no—
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#389C2211yescharacteristic
47#388—21no—
48#381—21no—
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4D411yescharacteristic
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C23×C4C2311yescharacteristic
66C2×M16—22no—
67C2×M16—22no—
68C2×M16—22no—
69C2×M16—22no—
70C2×M16—41no—
71C2×M16—41no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76#123—44no—
77C22⋊Q8—22no—
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C22⋊Q8—41no—
81C4×D4—22no—
82C4×D4—22no—
83#117—22no—
84#117—41no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4D411yescharacteristic
109C2×C22⋊C4D411yescharacteristic
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112#107—22no—
113#107—22no—
114#107—22no—
115#106—41no—
116C4⋊C8—22no—
117#103—22no—
118#103—22no—
119#103—22no—
120#103—41no—
121C22⋊C8—22no—
122C22⋊C8D414yes—
123C4×C8—22no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24C2×D411yescharacteristic
134C24C2×D411yescharacteristic
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—41no—
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—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D412yes—
169C22×C4C2×D411yescharacteristic
170C22×C4#4112yes—
171C22×C4C2411yescharacteristic
172M16—44no—
173M16—44no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8C2×D412yes—
184C2×C8C2×D412yes—
185C2×C8#4112yes—
186C4⋊C4—22no—
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—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4#4112yes—
207C22⋊C4—22no—
208C22⋊C4—22no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—21no—
217C42—21no—
218C42—42no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23#12112yes—
233C23C22×D411yescharacteristic
234Q8—41no—
235Q8—41no—
236Q8—41no—
237Q8—41no—
238D4—44no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
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—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—84no—
282C2×C4#12112yes—
283C2×C4AGL(1,ℤ/8)12yes—
284C2×C4C22×D411yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C4C2×#4111yescharacteristic
287C2×C4C2×#4111yescharacteristic
288C2×C4H(5,𝔽2)11yescharacteristic
289C2×C42−1+411yescharacteristic
290C8—22no—
291C8—22no—
292C8—22no—
293C8—41no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22C2×#12111yescharacteristic
309C22#54611yescharacteristic
310C22#54711yescharacteristic
311C22C2×AGL(1,ℤ/8)11yescharacteristic
312C22#57511yescharacteristic
313C22#57811yescharacteristic
314C22#57811yescharacteristic
315C4—22no—
316C4—22no—
317C4—42no—
318C4—41no—
319C4—41no—
320C4—81no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4#52911yescharacteristic
328C4#53911yescharacteristic
329C4#54611yescharacteristic
330C4#54711yescharacteristic
331C2—42no—
332C2—81no—
333C2#242511yescharacteristic
334C2#305511yescharacteristic
335C2#305911yescharacteristic
336C2#332711yescharacteristic
337C2#333011yescharacteristic
338C2#335211yescharacteristic
339C2#335311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#484C2semidirect
5. #2086C2semidirect
6. #2076C2semidirect
7. #2066C2semidirect
8. #2036C2semidirect
9. #1942C2semidirect
10. #1933C2semidirect
11. #1894C2semidirect
12. #1894C2semidirect
13. #1854C2semidirect
14. #1809C2semidirect
15. #1736C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4362
#24862
#24152
#20862
#20762
#20662
#20362
#19422
#19332
#18942
#18542
#18092
#17362