Finite Group Databasebeta

← #33028

Subgroups — #33028

Order 256.

492 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#478C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic
3#2861C211yesmaximal, characteristic, complemented
4C42×D4C211yesmaximal, characteristic, complemented
5#1989C211yesmaximal, characteristic, complemented
6#1983C211yesmaximal, characteristic, complemented
7#1928C211yesmaximal, characteristic, complemented
8#1927C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1925C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic
12#1879C211yesmaximal, characteristic
13#1877C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1831C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C22×C42—21no—
24#478—24no—
25#463—24no—
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#390C2211yescharacteristic
41#390C2211yescharacteristic
42#388—21no—
43#388—21no—
44#388—21no—
45#388—21no—
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50C4×C22⋊C4—21no—
51C4×C22⋊C4—21no—
52C43C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C2×SD16—22no—
68C2×SD16—22no—
69C2×SD16—22no—
70C2×D8—22no—
71C2×D8—22no—
72C2×D8—22no—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77#125—22no—
78#125—22no—
79#124—22no—
80#124—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—22no—
86C4×D4D414yes—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—44no—
107C2×C42D411yescharacteristic
108C2×C42D411yescharacteristic
109C2×C42D411yescharacteristic
110C2×C42D411yescharacteristic
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#107—22no—
115#106—22no—
116C4⋊C8—22no—
117C4⋊C8—22no—
118#103—22no—
119#103—22no—
120#103—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24—21no—
134C24—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4C2×D412yes—
151C2×D4#4112yes—
152C2×D4#4112yes—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—24no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2411yescharacteristic
175SD16—44no—
176D8—44no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4C2×D412yes—
199C4⋊C4#4112yes—
200C4⋊C4#4112yes—
201C22⋊C4—24no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C42—22no—
211C42—22no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—41no—
218C42—42no—
219C42—42no—
220C42—42no—
221C42C2×D412yes—
222C42#4112yes—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23C22×D411yescharacteristic
238Q8—41no—
239Q8—41no—
240Q8—41no—
241Q8—41no—
242D4—22no—
243D4—22no—
244D4—22no—
245D4—41no—
246D4—41no—
247D4—41no—
248D4—41no—
249D4#12414yes—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—48no—
268C2×C4—48no—
269C2×C4—48no—
270C2×C4—41no—
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#12112yes—
286C2×C4#12112yes—
287C2×C4#13512yes—
288C2×C4AGL(1,ℤ/8)12yes—
289C2×C4C22×D411yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C4C2×#4111yescharacteristic
292C2×C4C2×#4111yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C8—41no—
296C8—41no—
297C8—41no—
298C8—41no—
299C22—24no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22C2×#12111yescharacteristic
314C22#52511yescharacteristic
315C22#52511yescharacteristic
316C22C2×#13511yescharacteristic
317C22C2×AGL(1,ℤ/8)11yescharacteristic
318C22#57511yescharacteristic
319C22#57511yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—42no—
326C4—41no—
327C4—42no—
328C4—42no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4C2×#12411yescharacteristic
333C4#52911yescharacteristic
334C4#52911yescharacteristic
335C4#53211yescharacteristic
336C2—44no—
337C2—161no—
338C2#285011yescharacteristic
339C2#303511yescharacteristic
340C2#303611yescharacteristic
341C2#310211yescharacteristic
342C2#310511yescharacteristic
343C2#311111yescharacteristic
344C2#311411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#478C2semidirect
3. #2861C2semidirect
4. C42×D4C2semidirect
5. #1989C2semidirect
6. #1983C2semidirect
7. #1928C2semidirect
8. #1927C2semidirect
9. #1926C2semidirect
10. #1925C2semidirect
13. #1877C2semidirect
14. #1877C2semidirect
15. #1831C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C2×#4632
#28612
C42×D42
#19892
#19832
#19282
#19272
#19262
#19252
#18792
#18772
#18312