Finite Group Databasebeta

← #25499

Subgroups — #25499

Order 256.

582 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#529C211yesmaximal, characteristic
2C2×#467C211yesmaximal, characteristic, complemented
3C2×#465C211yesmaximal, characteristic
4#2855C211yesmaximal, characteristic, complemented
5#2059C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2026C211yesmaximal, characteristic, complemented
8#2015C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic
10#1892C211yesmaximal, characteristic, complemented
11#1878C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic, complemented
13#1865C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15C2×#336C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#529—24no—
21#529—24no—
22C2×#126—21no—
23C2×#125C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#123—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30C2×C4×D4—21no—
31C2×C4×D4C2211yescharacteristic
32C2×#117—21no—
33C22×C42C2211yescharacteristic
34#467—24no—
35#465—24no—
36C2×#107C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#387—21no—
51#387—21no—
52#385—21no—
53#381—21no—
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×C4—21no—
68C23×C4D411yescharacteristic
69C23×C4D411yescharacteristic
70C23×C4C2311yescharacteristic
71C2×Q16—22no—
72C2×Q16—22no—
73C2×Q16—41no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—41no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87#126—28no—
88#125—22no—
89#125—22no—
90#124—22no—
91#124—22no—
92#123—28no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96C22⋊Q8D414yes—
97#121—22no—
98#121D414yes—
99C4×Q8—22no—
100C4×Q8—22no—
101C4×D4—28no—
102C4×D4—22no—
103C4×D4—22no—
104#117—28no—
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⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C42—21no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42D411yescharacteristic
128C2×C42D411yescharacteristic
129C2×C42C2311yescharacteristic
130C2×C42C2311yescharacteristic
131#107—22no—
132#106—22no—
133#103—22no—
134#103—22no—
135#103—22no—
136#103—22no—
137#103—41no—
138#102—22no—
139#102—22no—
140#102—41no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
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×Q8—21no—
161C2×Q8C2×D412yes—
162C2×D4—24no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4C2×D412yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—24no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D412yes—
196C22×C4C2×D412yes—
197C22×C4C2×D411yescharacteristic
198C22×C4C2411yescharacteristic
199Q16—44no—
200SD16—44no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—44no—
212C2×C8—44no—
213C2×C8#4112yes—
214C2×C8#4112yes—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4C2×D412yes—
231C4⋊C4C2×D412yes—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—24no—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C42—24no—
240C42—22no—
241C42—22no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—22no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#12012yes—
258C23#13512yes—
259C23#13712yes—
260C23C22×D411yescharacteristic
261Q8—22no—
262Q8—22no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269D4—22no—
270D4—22no—
271D4—48no—
272D4—41no—
273D4—41no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—48no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—44no—
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—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4#12012yes—
316C2×C4#13512yes—
317C2×C4C22×D411yescharacteristic
318C2×C4C22×D411yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4H(5,𝔽2)11yescharacteristic
322C2×C42−1+411yescharacteristic
323C8—22no—
324C8—22no—
325C8—41no—
326C8—41no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22#45214yes—
341C22C2×#12011yescharacteristic
342C22#53211yescharacteristic
343C22#53711yescharacteristic
344C22C2×#13511yescharacteristic
345C22C2×#13511yescharacteristic
346C22C2×#13711yescharacteristic
347C22#57511yescharacteristic
348C4—22no—
349C4—22no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—42no—
354C4—41no—
355C4—42no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4#54611yescharacteristic
361C4#54611yescharacteristic
362C4#54711yescharacteristic
363C4#54711yescharacteristic
364C2—24no—
365C2—82no—
366C2#243011yescharacteristic
367C2C2×#45211yescharacteristic
368C2#300211yescharacteristic
369C2#316811yescharacteristic
370C2#318111yescharacteristic
371C2#324811yescharacteristic
372C2#325011yescharacteristic
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
2. C2×#467C2semidirect
4. #2855C2semidirect
5. #2059C2semidirect
6. #2042C2semidirect
7. #2026C2semidirect
8. #2015C2semidirect
10. #1892C2semidirect
11. #1878C2semidirect
12. #1877C2semidirect
13. #1865C2semidirect
14. #1862C2semidirect
15. C2×#336C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4672
C2×#4652
#28552
#20592
#20422
#20262
#20152
#18932
#18922
#18782
#18772
#18652
#18622
C2×#3362