Finite Group Databasebeta

← #28576

Subgroups — #28576

Order 256.

471 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×#467C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic
3#2850C211yesmaximal, characteristic
4C4×C22⋊Q8C211yesmaximal, characteristic, complemented
5#2085C211yesmaximal, characteristic, complemented
6#2074C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2015C211yesmaximal, characteristic, complemented
9#1927C211yesmaximal, characteristic, complemented
10#1925C211yesmaximal, characteristic, complemented
11#1893C211yesmaximal, characteristic
12#1892C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1878C211yesmaximal, characteristic, complemented
15C2×#336C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C22×C42C2211yescharacteristic
26#467—24no—
27#465—24no—
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#390C2211yescharacteristic
41#389C2211yescharacteristic
42#388—21no—
43#388—21no—
44#387—21no—
45#386—21no—
46#386—21no—
47#385—21no—
48#382C2211yescharacteristic
49#380C2211yescharacteristic
50C4×C4⋊C4—21no—
51C4×C4⋊C4C2211yescharacteristic
52C4×C22⋊C4—21no—
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4D411yescharacteristic
67C23×C4C2311yescharacteristic
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—41no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16—41no—
74C22×C8—22no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84#124—41no—
85#124—41no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8D414yes—
90#121—22no—
91#121—22no—
92C4×Q8—41no—
93C4×Q8—41no—
94C2×C4⋊C4—21no—
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×C22⋊C4—21no—
104C2×C22⋊C4—21no—
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×C42—21no—
112C2×C42—21no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42D411yescharacteristic
116C2×C42D411yescharacteristic
117C2×C42C2311yescharacteristic
118C2×C42C2311yescharacteristic
119#107—22no—
120#106—22no—
121#103—22no—
122#103—22no—
123#103—22no—
124#103—22no—
125#103—41no—
126#102—22no—
127#102—22no—
128#102—41no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24C2×D411yescharacteristic
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8C2×D412yes—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—42no—
159C2×D4—42no—
160C2×D4—42no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D411yescharacteristic
182C22×C4C2×D412yes—
183C22×C4C2×D411yescharacteristic
184C22×C4#4112yes—
185C22×C4C2411yescharacteristic
186Q16—44no—
187SD16—44no—
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—44no—
199C2×C8—44no—
200C2×C8#4112yes—
201C2×C8#4112yes—
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—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4C2×D412yes—
216C22⋊C4—24no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—41no—
229C42—41no—
230C42—42no—
231C42—42no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23#12112yes—
244C23#13512yes—
245C23#13712yes—
246C23C22×D411yescharacteristic
247Q8—22no—
248Q8—22no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255D4—41no—
256D4—41no—
257D4—41no—
258D4—41no—
259D4—84no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—44no—
280C2×C4—44no—
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—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4#12112yes—
301C2×C4#13512yes—
302C2×C4C22×D411yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4H(5,𝔽2)11yescharacteristic
307C2×C4H(5,𝔽2)11yescharacteristic
308C8—22no—
309C8—22no—
310C8—41no—
311C8—41no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22#46414yes—
326C22C2×#12111yescharacteristic
327C22#53211yescharacteristic
328C22#53411yescharacteristic
329C22C2×#13511yescharacteristic
330C22C2×#13511yescharacteristic
331C22C2×#13711yescharacteristic
332C22#57511yescharacteristic
333C4—22no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—42no—
339C4—42no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4#54611yescharacteristic
346C4#54611yescharacteristic
347C4#55311yescharacteristic
348C4#55311yescharacteristic
349C2—24no—
350C2—161no—
351C2#254311yescharacteristic
352C2C2×#46411yescharacteristic
353C2#303511yescharacteristic
354C2#316811yescharacteristic
355C2#318111yescharacteristic
356C2#319011yescharacteristic
357C2#319111yescharacteristic
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
1. C2×#467C2semidirect
4. C4×C22⋊Q8C2semidirect
5. #2085C2semidirect
6. #2074C2semidirect
7. #2042C2semidirect
8. #2015C2semidirect
9. #1927C2semidirect
10. #1925C2semidirect
12. #1892C2semidirect
13. #1880C2semidirect
14. #1878C2semidirect
15. C2×#336C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
C2×#4652
#28502
C4×C22⋊Q82
#20852
#20742
#20422
#20152
#19272
#19252
#18932
#18922
#18802
#18782
C2×#3362