Finite Group Databasebeta

← #25937

Subgroups — #25937

Order 256.

514 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#557C211yesmaximal, characteristic, complemented
2C2×#488C211yesmaximal, characteristic
3C2×#464C211yesmaximal, characteristic
4#2592C211yesmaximal, characteristic
5#2026C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic
7#2024C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic
9#1989C211yesmaximal, characteristic, complemented
10#1984C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic
12#1880C211yesmaximal, characteristic
13#1878C211yesmaximal, characteristic, complemented
14#1878C211yesmaximal, characteristic, complemented
15C4×#103C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18#557—24no—
19#557—24no—
20C2×C4⋊Q8—21no—
21C2×C4⋊Q8C2211yescharacteristic
22C2×#125—21no—
23C2×#125—21no—
24C2×#125C2211yescharacteristic
25C2×#125C2211yescharacteristic
26C2×#124C2211yescharacteristic
27C2×C4×Q8—21no—
28C2×C4×Q8C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31C2×C4×Q8C2211yescharacteristic
32#488—24no—
33#464—24no—
34C2×C4⋊C8C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#397—21no—
45#393—21no—
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#388—21no—
51#388—21no—
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8D411yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×Q8C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—22no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16—22no—
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C4⋊Q8—28no—
79C4⋊Q8—22no—
80C4⋊Q8—22no—
81#125—28no—
82#125—28no—
83#125—22no—
84#125—22no—
85#125—22no—
86#125D414yes—
87#124—22no—
88#124—22no—
89C4×Q8—28no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8D414yes—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4D411yescharacteristic
107C2×C4⋊C4D411yescharacteristic
108C2×C4⋊C4D411yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42C2311yescharacteristic
125C2×C42C2311yescharacteristic
126C2×C42C2311yescharacteristic
127C4⋊C8—22no—
128#103—22no—
129#103—22no—
130#103—22no—
131#103—22no—
132#103—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136C4×C8—22no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142C24—21no—
143C2×Q8—24no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—41no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8C2×D412yes—
159C2×Q8#4112yes—
160C2×Q8#4112yes—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2411yescharacteristic
181Q16—44no—
182SD16—44no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4C2×D412yes—
214C4⋊C4C2×D412yes—
215C4⋊C4C2×D412yes—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—81no—
221C22⋊C4—81no—
222C42—24no—
223C42—24no—
224C42—24no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—81no—
230C42C2×D412yes—
231C42C2×D412yes—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23C22×D411yescharacteristic
240Q8—22no—
241Q8—22no—
242Q8—22no—
243Q8—44no—
244Q8—48no—
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—41no—
249Q8#12414yes—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—41no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4#12012yes—
291C2×C4#12012yes—
292C2×C4#13512yes—
293C2×C4C22×D411yescharacteristic
294C2×C4C22×D411yescharacteristic
295C2×C4C2×#4111yescharacteristic
296C2×C4C2×#4111yescharacteristic
297C2×C42−1+411yescharacteristic
298C2×C42−1+411yescharacteristic
299C8—41no—
300C8—41no—
301C8—41no—
302C8—41no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22C2×#12011yescharacteristic
311C22#54811yescharacteristic
312C22#54811yescharacteristic
313C22C2×#13511yescharacteristic
314C22#57511yescharacteristic
315C22#57711yescharacteristic
316C22#57811yescharacteristic
317C4—24no—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—41no—
322C4—42no—
323C4—42no—
324C4—41no—
325C4—41no—
326C4—42no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4C2×#12411yescharacteristic
331C4#53611yescharacteristic
332C4#54811yescharacteristic
333C4#54811yescharacteristic
334C2—161no—
335C2#243611yescharacteristic
336C2#300211yescharacteristic
337C2#301311yescharacteristic
338C2#336411yescharacteristic
339C2#336511yescharacteristic
340C2#337111yescharacteristic
341C2#337711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#557C2semidirect
5. #2026C2semidirect
7. #2024C2semidirect
9. #1989C2semidirect
10. #1984C2semidirect
13. #1878C2semidirect
14. #1878C2semidirect
15. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#4882
C2×#4642
#25922
#20262
#20252
#20242
#20232
#19892
#19842
#18802
#18782
C4×#1032