Finite Group Databasebeta

← #24561

Subgroups — #24561

Order 256.

542 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#485C211yesmaximal, characteristic, complemented
2C2×#407C211yesmaximal, characteristic
3#2848C211yesmaximal, characteristic
4#2280C211yesmaximal, characteristic
5#2077C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#1894C211yesmaximal, characteristic, complemented
10#1894C211yesmaximal, characteristic, complemented
11#1873C211yesmaximal, characteristic, complemented
12#1854C211yesmaximal, characteristic, complemented
13#1825C211yesmaximal, characteristic, complemented
14#1809C211yesmaximal, characteristic, complemented
15#1809C211yesmaximal, characteristic, complemented
16C24×C4C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#485—24no—
25C2×#107C2211yescharacteristic, complemented
26C2×#107C2211yescharacteristic, complemented
27C2×#106C2211yescharacteristic, complemented
28C2×#106C2211yescharacteristic, complemented
29C2×#103C2211yescharacteristic, complemented
30C2×#103C2211yescharacteristic, complemented
31C2×#103C2211yescharacteristic, complemented
32C2×#103C2211yescharacteristic, complemented
33#407—24no—
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38#385—21no—
39#385—21no—
40#385—21no—
41#381—21no—
42#380C2211yescharacteristic
43#380C2211yescharacteristic
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49#379—21no—
50#379—21no—
51C4×C22⋊C4—21no—
52C4×C22⋊C4—21no—
53#340C2211yescharacteristic, complemented
54#340C2211yescharacteristic, complemented
55#340C2211yescharacteristic, complemented
56#340C2211yescharacteristic, complemented
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61C25D411yescharacteristic
62C22×Q8C2311yescharacteristic
63C23×C4—22no—
64C23×C4—22no—
65C23×C4—22no—
66C23×C4—22no—
67C23×C4—41no—
68C23×C4—41no—
69C23×C4D411yescharacteristic
70C23×C4D411yescharacteristic
71C23×C4D411yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—41no—
80C2×M16—41no—
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91#117—22no—
92#117—22no—
93#117—41no—
94#117—41no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C22⋊C4—21no—
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×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42—21no—
118C2×C42—21no—
119#107—22no—
120#107—41no—
121#106—22no—
122#106—41no—
123#103—22no—
124#103—22no—
125#103—41no—
126#103—41no—
127C22⋊C8—22no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130C22⋊C8D414yes—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—22no—
139C24—22no—
140C24—22no—
141C24—41no—
142C24—41no—
143C24—41no—
144C24—41no—
145C24C2×D411yescharacteristic
146C24C2×D411yescharacteristic
147C24C2×D411yescharacteristic
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—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—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
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—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D412yes—
192C22×C4C2×D411yescharacteristic
193C22×C4#4112yes—
194C22×C4#4112yes—
195C22×C4#4112yes—
196C22×C4C2411yescharacteristic
197M16—44no—
198M16—44no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8C2×D412yes—
210C2×C8C2×D412yes—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C42—42no—
241C42—42no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—44no—
261C23—44no—
262C23—44no—
263C23—44no—
264C23—44no—
265C23—44no—
266C23—44no—
267C23#12112yes—
268C23#12312yes—
269C23#13712yes—
270C23C22×D411yescharacteristic
271Q8—41no—
272Q8—41no—
273Q8—41no—
274Q8—41no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4#12112yes—
322C2×C4#12312yes—
323C2×C4C22×D411yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4C2×#4111yescharacteristic
326C2×C4C2×#4111yescharacteristic
327C2×C4H(5,𝔽2)11yescharacteristic
328C2×C4H(5,𝔽2)11yescharacteristic
329C8—22no—
330C8—22no—
331C8—41no—
332C8—41no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—416no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22C2×#12111yescharacteristic
356C22C2×#12311yescharacteristic
357C22#53411yescharacteristic
358C22C2×#13711yescharacteristic
359C22#57511yescharacteristic
360C22#57511yescharacteristic
361C22#57511yescharacteristic
362C4—22no—
363C4—22no—
364C4—41no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4#54611yescharacteristic
374C4#54611yescharacteristic
375C4#55111yescharacteristic
376C4#55111yescharacteristic
377C2—42no—
378C2—42no—
379C2—42no—
380C2#241211yescharacteristic
381C2#305411yescharacteristic
382C2#305611yescharacteristic
383C2#309311yescharacteristic
384C2#309511yescharacteristic
385C2#319611yescharacteristic
386C2#319711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC25D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#485C2semidirect
5. #2077C2semidirect
6. #2076C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #1894C2semidirect
10. #1894C2semidirect
11. #1873C2semidirect
12. #1854C2semidirect
13. #1825C2semidirect
14. #1809C2semidirect
15. #1809C2semidirect
25. C2×#107C22semidirect
26. C2×#107C22semidirect
27. C2×#106C22semidirect
28. C2×#106C22semidirect
29. C2×#103C22semidirect
30. C2×#103C22semidirect
31. C2×#103C22semidirect
32. C2×#103C22semidirect
53. #340C22semidirect
54. #340C22semidirect
55. #340C22semidirect
56. #340C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4072
#28482
#22802
#20772
#20762
#20362
#20352
#18942
#18732
#18542
#18252
#18092