Finite Group Databasebeta

← #26722

Subgroups — #26722

Order 256.

424 conjugacy classes of subgroups.

The subgroup lattice has 339 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2492C211yesmaximal, characteristic, complemented
2#2361C211yesmaximal, characteristic
3#2055C211yesmaximal, characteristic, complemented
4#2034C211yesmaximal, characteristic
5#2023C211yesmaximal, characteristic, complemented
6#2000C211yesmaximal, characteristic
7#1987C211yesmaximal, characteristic, complemented
8#1933C211yesmaximal, characteristic, complemented
9#1927C211yesmaximal, characteristic, complemented
10#1923C211yesmaximal, characteristic, complemented
11#1919C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic, complemented
13#1872C211yesmaximal, characteristic, complemented
14#1823C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×#103C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#102C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C4×C8C2211yescharacteristic
34#400—21no—
35#393—21no—
36#391—21no—
37#391C2211yescharacteristic
38#391C2211yescharacteristic
39#390C2211yescharacteristic
40#389C2211yescharacteristic
41#388—21no—
42#388—21no—
43#386—21no—
44#385—21no—
45#380C2211yescharacteristic
46C4×C4⋊C4C2211yescharacteristic
47C4×C4⋊C4C2211yescharacteristic
48C4×C22⋊C4C2211yescharacteristic
49#340C2211yescharacteristic
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8—21no—
60C22×Q8C2311yescharacteristic
61C22×D4—21no—
62C22×D4C2311yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C2×SD16—22no—
67C2×SD16—41no—
68C2×SD16—41no—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C4⋊Q8—22no—
74C4⋊Q8—41no—
75C22⋊Q8—22no—
76C22⋊Q8—41no—
77#121—22no—
78#121—22no—
79C4×D4—22no—
80C4×D4—41no—
81C4×D4—44no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
101C2×C22⋊C4D411yescharacteristic
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#106—22no—
107C4⋊C8—41no—
108#103—22no—
109#103—22no—
110#103—41no—
111#102—22no—
112#102—22no—
113#102—41no—
114C22⋊C8—41no—
115C22⋊C8—41no—
116C4×C8—41no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95D411yescharacteristic
126#95D411yescharacteristic
127C24—21no—
128C24C2×D411yescharacteristic
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—81no—
138C2×D4—24no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4#4112yes—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172SD16—44no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4#4112yes—
199C4⋊C4#4112yes—
200C4⋊C4#4112yes—
201C22⋊C4—24no—
202C22⋊C4—41no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—81no—
207C22⋊C4—81no—
208C22⋊C4SD1614yescomplemented
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C42—81no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—41no—
223C23—41no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23C2×SD1612yes—
229C23C22×D411yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—41no—
233Q8—41no—
234Q8—84no—
235D4—22no—
236D4—22no—
237D4—48no—
238D4—41no—
239D4—41no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
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—84no—
279C2×C4—84no—
280C2×C4C2×SD1612yes—
281C2×C4C2×SD1612yes—
282C2×C4C2×#4111yescharacteristic
283C2×C4C2×#4111yescharacteristic
284C2×C4C2×#4111yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C4H(5,𝔽2)11yescharacteristic
287C2×C4H(5,𝔽2)11yescharacteristic
288C8—41no—
289C8—41no—
290C8—41no—
291C8—41no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—81no—
306C22#52511yescharacteristic
307C22#54611yescharacteristic
308C22#54611yescharacteristic
309C22C22×SD1611yescharacteristic
310C22#57511yescharacteristic
311C22#57611yescharacteristic
312C22#57811yescharacteristic
313C4—22no—
314C4—22no—
315C4—42no—
316C4—41no—
317C4—42no—
318C4—41no—
319C4—41no—
320C4—42no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4—81no—
325C4#52911yescharacteristic
326C4#53811yescharacteristic
327C4#55511yescharacteristic
328C4#55911yescharacteristic
329C2—42no—
330C2—82no—
331C2#245911yescharacteristic
332C2#310111yescharacteristic
333C2#312411yescharacteristic
334C2#329511yescharacteristic
335C2#329811yescharacteristic
336C2#331311yescharacteristic
337C2#332211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2492C2semidirect
3. #2055C2semidirect
5. #2023C2semidirect
7. #1987C2semidirect
8. #1933C2semidirect
9. #1927C2semidirect
10. #1923C2semidirect
11. #1919C2semidirect
12. #1877C2semidirect
13. #1872C2semidirect
14. #1823C2semidirect
15. C4×#102C2semidirect
208. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#24922
#23612
#20552
#20342
#20232
#20002
#19872
#19332
#19272
#19232
#19192
#18772
#18722
#18232
C4×#1022