Finite Group Databasebeta

← #24068

Subgroups — #24068

Order 256.

552 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#529C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#482C211yesmaximal, characteristic
4#2296C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2059C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2042C211yesmaximal, characteristic, complemented
9#1875C211yesmaximal, characteristic
10#1873C211yesmaximal, characteristic, complemented
11#1865C211yesmaximal, characteristic, complemented
12#1864C211yesmaximal, characteristic, complemented
13#1863C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15#1817C211yesmaximal, characteristic, complemented
16#529—24no—
17#529—24no—
18C2×#126—21no—
19C2×#125C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#123—21no—
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4C2211yescharacteristic
27C2×#117—21no—
28C2×#117C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C22×C42C2211yescharacteristic
31#485—24no—
32#482—24no—
33C2×#107C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#385—21no—
52#385—21no—
53C4×C4⋊C4—21no—
54C4×C4⋊C4—21no—
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
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#124—22no—
78#124—22no—
79#123—28no—
80C22⋊Q8—22no—
81C22⋊Q8D414yes—
82#121—22no—
83#121D414yes—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×D4—28no—
87C4×D4—22no—
88C4×D4—22no—
89#117—28no—
90#117—22no—
91#117—22no—
92#117—22no—
93#117—22no—
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⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—22no—
118C2×C42—22no—
119C2×C42—21no—
120C2×C42D411yescharacteristic
121C2×C42D411yescharacteristic
122C2×C42C2311yescharacteristic
123C2×C42C2311yescharacteristic
124#107—22no—
125#107—22no—
126#106—22no—
127#106—22no—
128#103—22no—
129#103—22no—
130#103—22no—
131#103—22no—
132#102—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
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—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D412yes—
184C22×C4C2×D412yes—
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—21no—
198C2×C8—21no—
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—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4C2×D412yes—
216C4⋊C4C2×D412yes—
217C4⋊C4#4112yes—
218C4⋊C4#4112yes—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C22⋊C4—24no—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—42no—
226C42—24no—
227C42—24no—
228C42—22no—
229C42—22no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—42no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23#12012yes—
249C23C22×D411yescharacteristic
250Q8—22no—
251Q8—22no—
252Q8—41no—
253Q8—41no—
254D4—22no—
255D4—22no—
256D4—48no—
257D4—41no—
258D4—41no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—44no—
282C2×C4—48no—
283C2×C4—44no—
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—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4#12012yes—
301C2×C4#13512yes—
302C2×C4C22×D411yescharacteristic
303C2×C4C22×D411yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C42−1+411yescharacteristic
307C2×C42−1+411yescharacteristic
308C8—41no—
309C8—41no—
310C8—41no—
311C8—41no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22C2×#12011yescharacteristic
327C22#52511yescharacteristic
328C22#53611yescharacteristic
329C22C2×#13511yescharacteristic
330C22#57511yescharacteristic
331C22#57511yescharacteristic
332C22#57511yescharacteristic
333C4—22no—
334C4—22no—
335C4—42no—
336C4—42no—
337C4—42no—
338C4—41no—
339C4—42no—
340C4—41no—
341C4—41no—
342C4—42no—
343C4—81no—
344C4—81no—
345C4#54711yescharacteristic
346C4#54711yescharacteristic
347C4#54711yescharacteristic
348C4#54711yescharacteristic
349C2—42no—
350C2—82no—
351C2#240611yescharacteristic
352C2#300211yescharacteristic
353C2#300511yescharacteristic
354C2#310211yescharacteristic
355C2#311111yescharacteristic
356C2#323311yescharacteristic
357C2#324011yescharacteristic
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×#485C2semidirect
4. #2296C2semidirect
5. #2059C2semidirect
6. #2059C2semidirect
7. #2042C2semidirect
8. #2042C2semidirect
10. #1873C2semidirect
11. #1865C2semidirect
12. #1864C2semidirect
13. #1863C2semidirect
14. #1862C2semidirect
15. #1817C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4852
C2×#4822
#22962
#20592
#20422
#18752
#18732
#18652
#18642
#18632
#18622
#18172