Finite Group Databasebeta

← #25236

Subgroups — #25236

Order 256.

459 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#483C211yesmaximal, characteristic, complemented
2C2×C8×D4C211yesmaximal, characteristic
3#2486C211yesmaximal, characteristic
4#2417C211yesmaximal, characteristic
5#2088C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#1941C211yesmaximal, characteristic, complemented
10#1933C211yesmaximal, characteristic, complemented
11#1892C211yesmaximal, characteristic, complemented
12#1892C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic, complemented
14#1807C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C23×C8C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C2×#123—21no—
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24#483—24no—
25C8×D4—24no—
26C2×#106C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#398—21no—
39#398—21no—
40#397—21no—
41#396—21no—
42#391C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#386—21no—
47#381—21no—
48#380—21no—
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×D4D411yescharacteristic
63C23×C4—21no—
64C23×C4—21no—
65C23×C4C2311yescharacteristic
66C23×C4C2311yescharacteristic
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77#123—44no—
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C22⋊Q8—22no—
81C22⋊Q8—41no—
82C4×D4—22no—
83C4×D4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
110C2×C22⋊C4D411yescharacteristic
111C2×C42C2311yescharacteristic
112#106—22no—
113#106—22no—
114#106—22no—
115#106—41no—
116C4⋊C8—22no—
117#103—22no—
118#103—22no—
119#103—22no—
120#103—41no—
121C22⋊C8—22no—
122C22⋊C8D414yes—
123C4×C8—22no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24C2×D411yescharacteristic
134C24C2×D411yescharacteristic
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4SD1614yes—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
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—21no—
161C22×C4—21no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D412yes—
172C22×C4C2×D411yescharacteristic
173C22×C4#4112yes—
174C22×C4C2411yescharacteristic
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—44no—
185C2×C8—44no—
186C2×C8C2×D412yes—
187C2×C8C2×D412yes—
188C2×C8#4112yes—
189C4⋊C4—22no—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—81no—
210C4⋊C4#4112yes—
211C22⋊C4—22no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—81no—
219C42—21no—
220C42—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23#12112yes—
232C23C2×SD1612yes—
233C23C2×SD1612yes—
234C23#13512yes—
235C23C22×D411yescharacteristic
236Q8—41no—
237Q8—41no—
238Q8—41no—
239Q8—41no—
240D4—28no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
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—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—84no—
284C2×C4—84no—
285C2×C4#12112yes—
286C2×C4C2×SD1612yes—
287C2×C4C22×D411yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C2×C4H(5,𝔽2)11yescharacteristic
292C2×C42−1+411yescharacteristic
293C8—22no—
294C8—22no—
295C8—22no—
296C8—41no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22#46514yes—
311C22C2×#12111yescharacteristic
312C22#54611yescharacteristic
313C22#54711yescharacteristic
314C22C22×SD1611yescharacteristic
315C22C2×#13511yescharacteristic
316C22#57711yescharacteristic
317C22#57811yescharacteristic
318C4—22no—
319C4—22no—
320C4—41no—
321C4—42no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—81no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4#52911yescharacteristic
331C4#53911yescharacteristic
332C4#54611yescharacteristic
333C4#54711yescharacteristic
334C2—24no—
335C2—42no—
336C2#242511yescharacteristic
337C2C2×#46511yescharacteristic
338C2#306311yescharacteristic
339C2#329611yescharacteristic
340C2#329811yescharacteristic
341C2#333511yescharacteristic
342C2#333611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#483C2semidirect
5. #2088C2semidirect
6. #2077C2semidirect
7. #2066C2semidirect
8. #2035C2semidirect
9. #1941C2semidirect
10. #1933C2semidirect
11. #1892C2semidirect
12. #1892C2semidirect
13. #1853C2semidirect
14. #1807C2semidirect
15. C2×#340C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
C2×C8×D42
#24862
#24172
#20882
#20772
#20662
#20352
#19412
#19332
#18922
#18532
#18072
C2×#3402