Finite Group Databasebeta

← #25675

Subgroups — #25675

Order 256.

541 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#529C211yesmaximal, characteristic
2C2×#499C211yesmaximal, characteristic, complemented
3C2×#488C211yesmaximal, characteristic, complemented
4#2851C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic
8#2033C211yesmaximal, characteristic
9#1991C211yesmaximal, characteristic, complemented
10#1958C211yesmaximal, characteristic, complemented
11#1865C211yesmaximal, characteristic, complemented
12#1864C211yesmaximal, characteristic, complemented
13#1863C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15C4×C22⋊C8C211yesmaximal, characteristic, complemented
16#529—24no—
17#529—24no—
18C2×#126—21no—
19C2×#125C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#123—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×D4—21no—
27C2×C4×D4C2211yescharacteristic
28C2×#117—21no—
29C22×C42C2211yescharacteristic
30#499—24no—
31#488—24no—
32C2×#107C2211yescharacteristic
33C2×#107C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C4×C8C2211yescharacteristic
47C2×C4×C8C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#387—21no—
51#387—21no—
52#385—21no—
53#385—21no—
54#382C2211yescharacteristic
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4D411yescharacteristic
68C23×C4D411yescharacteristic
69C23×C4C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74#126—28no—
75#125—22no—
76#125—22no—
77#125—22no—
78#125D414yes—
79#124—22no—
80#124D414yes—
81#123—28no—
82C22⋊Q8—22no—
83C22⋊Q8—22no—
84#121—22no—
85#121—22no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×D4—28no—
89C4×D4—22no—
90C4×D4—22no—
91#117—28no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C42—21no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42D411yescharacteristic
117C2×C42D411yescharacteristic
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120#107—22no—
121#107—22no—
122#106—22no—
123#106—22no—
124#103—22no—
125#103—22no—
126#103—22no—
127#103—22no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102—22no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C4×C8—22no—
135C4×C8—22no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142C24—21no—
143C24C2×D411yescharacteristic
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8C2×D412yes—
150C2×D4—24no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4C2×D412yes—
159C22×C4—24no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2411yescharacteristic
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8#4112yes—
198C2×C8#4112yes—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—24no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4C2×D412yes—
218C4⋊C4C2×D412yes—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C42—24no—
227C42—22no—
228C42—22no—
229C42—21no—
230C42—21no—
231C42C2×D412yes—
232C42C2×D412yes—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23#13512yes—
247C23C22×D411yescharacteristic
248Q8—22no—
249Q8—22no—
250Q8—41no—
251Q8—41no—
252D4—22no—
253D4—22no—
254D4—48no—
255D4—41no—
256D4—41no—
257C2×C4—21no—
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—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—48no—
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—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4#12012yes—
298C2×C4#12012yes—
299C2×C4#13512yes—
300C2×C4C22×D411yescharacteristic
301C2×C4C22×D411yescharacteristic
302C2×C4C2×#4111yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C2×C42−1+411yescharacteristic
305C2×C42−1+411yescharacteristic
306C8—22no—
307C8—22no—
308C8—41no—
309C8—41no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22C2×#12011yescharacteristic
325C22#53211yescharacteristic
326C22#53611yescharacteristic
327C22C2×#13511yescharacteristic
328C22C2×#13511yescharacteristic
329C22#57511yescharacteristic
330C22#57511yescharacteristic
331C4—22no—
332C4—22no—
333C4—22no—
334C4—41no—
335C4—42no—
336C4—41no—
337C4—42no—
338C4—42no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4#54711yescharacteristic
344C4#54711yescharacteristic
345C4#54811yescharacteristic
346C4#54811yescharacteristic
347C2—42no—
348C2—82no—
349C2#243211yescharacteristic
350C2#300211yescharacteristic
351C2#300211yescharacteristic
352C2#316811yescharacteristic
353C2#318311yescharacteristic
354C2#323211yescharacteristic
355C2#323311yescharacteristic
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
2. C2×#499C2semidirect
3. C2×#488C2semidirect
4. #2851C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
9. #1991C2semidirect
10. #1958C2semidirect
11. #1865C2semidirect
12. #1864C2semidirect
13. #1863C2semidirect
14. #1862C2semidirect
15. C4×C22⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4992
C2×#4882
#28512
#20362
#20352
#20342
#20332
#19912
#19582
#18652
#18642
#18632
#18622
C4×C22⋊C82