Finite Group Databasebeta

← #25938

Subgroups — #25938

Order 256.

552 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554C211yesmaximal, characteristic, complemented
2C2×#488C211yesmaximal, characteristic, complemented
3C2×#463C211yesmaximal, characteristic
4#2592C211yesmaximal, characteristic, complemented
5#2026C211yesmaximal, characteristic, complemented
6#2025C211yesmaximal, characteristic, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1984C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic
12#1879C211yesmaximal, characteristic
13#1877C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18#554—24no—
19#554—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#125C2211yescharacteristic
22C2×#125C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×C22⋊Q8—21no—
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8—21no—
27C2×C4×Q8C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C22×C4⋊C4—21no—
32#488—24no—
33#463—24no—
34C2×C4⋊C8C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C4×C8C2211yescharacteristic
44#397—21no—
45#393—21no—
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#388—21no—
51#388—21no—
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8D411yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×SD16—22no—
74C2×D8—22no—
75C2×D8—22no—
76C2×D8—22no—
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C4⋊Q8—22no—
82C4⋊Q8—22no—
83#125—22no—
84#125—22no—
85#125—22no—
86#125D414yes—
87#124—22no—
88#124—22no—
89C22⋊Q8—28no—
90C22⋊Q8—28no—
91C22⋊Q8—28no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4D414yes—
100C2×C4⋊C4—28no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4D411yescharacteristic
107C2×C4⋊C4D411yescharacteristic
108C2×C4⋊C4D411yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
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×C42—21no—
124C2×C42C2311yescharacteristic
125C2×C42C2311yescharacteristic
126C2×C42C2311yescharacteristic
127C4⋊C8—22no—
128#103—22no—
129#103—22no—
130#103—22no—
131#102—22no—
132#102—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136C4×C8—22no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142C24—21no—
143C24—21no—
144C2×Q8—24no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—41no—
154C2×Q8—81no—
155C2×Q8—81no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4C2×D412yes—
166C2×D4#4112yes—
167C2×D4#4112yes—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2411yescharacteristic
190SD16—44no—
191D8—44no—
192C2×C8—21no—
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—21no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—22no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4C2×D412yes—
220C4⋊C4C2×D412yes—
221C4⋊C4C2×D412yes—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—81no—
236C42C2×D412yes—
237C42C2×D412yes—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23C22×D411yescharacteristic
253Q8—44no—
254Q8—41no—
255Q8—41no—
256Q8—41no—
257Q8—41no—
258Q8—84no—
259D4—22no—
260D4—22no—
261D4—22no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4#12414yes—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—48no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4#12012yes—
304C2×C4#12012yes—
305C2×C4#13512yes—
306C2×C4C22×D411yescharacteristic
307C2×C4C22×D411yescharacteristic
308C2×C4C2×#4111yescharacteristic
309C2×C4C2×#4111yescharacteristic
310C2×C42−1+411yescharacteristic
311C2×C42−1+411yescharacteristic
312C8—41no—
313C8—41no—
314C8—41no—
315C8—41no—
316C22—24no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22C2×#12011yescharacteristic
331C22#54811yescharacteristic
332C22#54811yescharacteristic
333C22C2×#13511yescharacteristic
334C22#57511yescharacteristic
335C22#57611yescharacteristic
336C22#57711yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—41no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—42no—
345C4—42no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4C2×#12411yescharacteristic
350C4#53611yescharacteristic
351C4#54811yescharacteristic
352C4#54811yescharacteristic
353C2—44no—
354C2—161no—
355C2#243611yescharacteristic
356C2#300211yescharacteristic
357C2#301211yescharacteristic
358C2#336011yescharacteristic
359C2#336111yescharacteristic
360C2#337011yescharacteristic
361C2#337611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#554C2semidirect
2. C2×#488C2semidirect
4. #2592C2semidirect
5. #2026C2semidirect
6. #2025C2semidirect
7. #2024C2semidirect
8. #2023C2semidirect
9. #1989C2semidirect
10. #1984C2semidirect
13. #1877C2semidirect
14. #1877C2semidirect
15. C4×#102C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4882
C2×#4632
#25922
#20262
#20252
#20242
#20232
#19892
#19842
#18792
#18772
C4×#1022