Finite Group Databasebeta

← #25511

Subgroups — #25511

Order 256.

535 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#541C211yesmaximal, characteristic, complemented
2C2×#468C211yesmaximal, characteristic
3C2×#465C211yesmaximal, characteristic
4#2624C211yesmaximal, characteristic
5#2059C211yesmaximal, characteristic, complemented
6#2043C211yesmaximal, characteristic, complemented
7#2025C211yesmaximal, characteristic, complemented
8#2013C211yesmaximal, characteristic, complemented
9#1894C211yesmaximal, characteristic, complemented
10#1892C211yesmaximal, characteristic, complemented
11#1880C211yesmaximal, characteristic, complemented
12#1880C211yesmaximal, characteristic, complemented
13#1865C211yesmaximal, characteristic, complemented
14#1863C211yesmaximal, characteristic, complemented
15#1729C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#541—24no—
21#541—24no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×#126—21no—
24C2×#126—21no—
25C2×#125—21no—
26C2×#125—21no—
27C2×#125C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×C22⋊Q8C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×C4×Q8C2211yescharacteristic
32C2×C4×Q8C2211yescharacteristic
33C2×#117C2211yescharacteristic
34#468—24no—
35#465—24no—
36C2×#107C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46#399—21no—
47#397—21no—
48#394—21no—
49#394—21no—
50#390C2211yescharacteristic
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#389C2211yescharacteristic
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C23×C4C2311yescharacteristic
68C2×SD16—22no—
69C2×SD16—22no—
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—41no—
73C2×SD16—41no—
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—41no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C4⋊Q8—22no—
85C4⋊Q8—22no—
86#126—28no—
87#126—28no—
88#125—28no—
89#125—28no—
90#125—22no—
91#125—22no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8D414yes—
95C22⋊Q8D414yes—
96#121—22no—
97#121—22no—
98C4×Q8—22no—
99C4×Q8—22no—
100C4×Q8—22no—
101C4×Q8—22no—
102#117—22no—
103#117—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4D411yescharacteristic
115C2×C4⋊C4D411yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4D411yescharacteristic
130C2×C22⋊C4D411yescharacteristic
131C2×C42—21no—
132C2×C42—21no—
133C2×C42C2311yescharacteristic
134C2×C42C2311yescharacteristic
135#107—22no—
136#106—22no—
137#103—22no—
138#103—22no—
139#103—22no—
140#103—22no—
141#103—41no—
142#103—41no—
143#102—22no—
144#102—22no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24—21no—
149C24C2×D411yescharacteristic
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8C2×D412yes—
161C2×Q8C2×D412yes—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—42no—
169C2×D4—81no—
170C2×D4—81no—
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×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D412yes—
188C22×C4C2×D412yes—
189C22×C4C2411yescharacteristic
190SD16—44no—
191SD16—44no—
192M16—44no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—44no—
204C2×C8#4112yes—
205C2×C8#4112yes—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—22no—
215C4⋊C4—22no—
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—81no—
225C4⋊C4—81no—
226C4⋊C4C2×D412yes—
227C4⋊C4C2×D412yes—
228C22⋊C4—24no—
229C22⋊C4—24no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—42no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C42—24no—
239C42—24no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#12012yes—
258C23C22×D411yescharacteristic
259Q8—22no—
260Q8—22no—
261Q8—22no—
262Q8—22no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267D4—41no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—84no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
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#12012yes—
312C2×C4C22×D411yescharacteristic
313C2×C4C22×D411yescharacteristic
314C2×C4C2×#4111yescharacteristic
315C2×C4C2×#4111yescharacteristic
316C2×C4H(5,𝔽2)11yescharacteristic
317C2×C42−1+411yescharacteristic
318C8—22no—
319C8—22no—
320C8—41no—
321C8—41no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22C2×#12011yescharacteristic
337C22#53211yescharacteristic
338C22#53711yescharacteristic
339C22#57711yescharacteristic
340C22#57711yescharacteristic
341C22#57711yescharacteristic
342C22#57811yescharacteristic
343C4—22no—
344C4—42no—
345C4—42no—
346C4—41no—
347C4—41no—
348C4—42no—
349C4—41no—
350C4—41no—
351C4—42no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4#54611yescharacteristic
356C4#54611yescharacteristic
357C4#54711yescharacteristic
358C4#54711yescharacteristic
359C2—42no—
360C2—161no—
361C2#243011yescharacteristic
362C2#301011yescharacteristic
363C2#301311yescharacteristic
364C2#317411yescharacteristic
365C2#317811yescharacteristic
366C2#325811yescharacteristic
367C2#326011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#541C2semidirect
5. #2059C2semidirect
6. #2043C2semidirect
7. #2025C2semidirect
8. #2013C2semidirect
9. #1894C2semidirect
10. #1892C2semidirect
11. #1880C2semidirect
12. #1880C2semidirect
13. #1865C2semidirect
14. #1863C2semidirect
15. #1729C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5412
C2×#4682
C2×#4652
#26242
#20592
#20432
#20252
#20132
#18942
#18922
#18802
#18652
#18632
#17292