Finite Group Databasebeta

← #24099

Subgroups — #24099

Order 256.

529 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#552C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#427C211yesmaximal, characteristic, complemented
4#2485C211yesmaximal, characteristic
5#2059C211yesmaximal, characteristic, complemented
6#2059C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic, complemented
10#1869C211yesmaximal, characteristic
11#1865C211yesmaximal, characteristic, complemented
12#1863C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1825C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17#552—24no—
18#552—24no—
19C2×#126—21no—
20C2×#125—21no—
21C2×#125C2211yescharacteristic
22C2×#123—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C2×#117—21no—
28C2×#117C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31#485—24no—
32#427—24no—
33C2×#107C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#107C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46#397—21no—
47#394—21no—
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#388—21no—
51#387—21no—
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4D411yescharacteristic
65C23×C4—21no—
66C23×C4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#126—28no—
76#125—28no—
77#125—22no—
78#125—22no—
79#123—28no—
80C22⋊Q8—22no—
81C22⋊Q8—22no—
82C22⋊Q8—22no—
83C22⋊Q8D414yes—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×D4—22no—
87C4×D4—22no—
88#117—28no—
89#117—22no—
90#117—22no—
91#117—22no—
92#117D414yes—
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—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4D411yescharacteristic
119C2×C22⋊C4D411yescharacteristic
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42C2311yescharacteristic
124#107—22no—
125#107—22no—
126#107—22no—
127#106—22no—
128#106—22no—
129#106—22no—
130#103—22no—
131#103—22no—
132#103—22no—
133#103—22no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8C2×D412yes—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—41no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—24no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D412yes—
179C22×C4C2×D412yes—
180C22×C4C2411yescharacteristic
181C2×C8—28no—
182C2×C8—21no—
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#4112yes—
193C2×C8#4112yes—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—21no—
202C4⋊C4—21no—
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—42no—
213C4⋊C4—81no—
214C4⋊C4C2×D412yes—
215C4⋊C4C2×D412yes—
216C4⋊C4C2×D412yes—
217C22⋊C4—24no—
218C22⋊C4—24no—
219C22⋊C4—24no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C42—24no—
227C42—24no—
228C42—21no—
229C42—21no—
230C42—42no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23#12012yes—
244C23#13512yes—
245C23C22×D411yescharacteristic
246Q8—22no—
247Q8—22no—
248Q8—41no—
249Q8—41no—
250D4—44no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—48no—
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—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#12012yes—
296C2×C4C22×D411yescharacteristic
297C2×C4C22×D411yescharacteristic
298C2×C4C2×#4111yescharacteristic
299C2×C4C2×#4111yescharacteristic
300C2×C42−1+411yescharacteristic
301C2×C42−1+411yescharacteristic
302C8—22no—
303C8—22no—
304C8—41no—
305C8—41no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22C2×#12011yescharacteristic
321C22#54711yescharacteristic
322C22#54711yescharacteristic
323C22C2×#13511yescharacteristic
324C22#57511yescharacteristic
325C22#57711yescharacteristic
326C22#57811yescharacteristic
327C4—22no—
328C4—41no—
329C4—42no—
330C4—41no—
331C4—42no—
332C4—42no—
333C4—41no—
334C4—42no—
335C4—81no—
336C4—81no—
337C4—81no—
338C4—81no—
339C4#52511yescharacteristic
340C4#53611yescharacteristic
341C4#54711yescharacteristic
342C4#54711yescharacteristic
343C2—42no—
344C2—42no—
345C2#240611yescharacteristic
346C2#300211yescharacteristic
347C2#301311yescharacteristic
348C2#333611yescharacteristic
349C2#333711yescharacteristic
350C2#334811yescharacteristic
351C2#334911yescharacteristic
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
2. C2×#485C2semidirect
3. C2×#427C2semidirect
5. #2059C2semidirect
6. #2059C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #1892C2semidirect
11. #1865C2semidirect
12. #1863C2semidirect
13. #1853C2semidirect
14. #1852C2semidirect
15. #1825C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5522
C2×#4852
C2×#4272
#24852
#20592
#20362
#20352
#18922
#18692
#18652
#18632
#18532
#18522
#18252