Finite Group Databasebeta

← #25653

Subgroups — #25653

Order 256.

541 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#559C211yesmaximal, characteristic
2C2×#488C211yesmaximal, characteristic, complemented
3C2×#478C211yesmaximal, characteristic, complemented
4#2584C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2041C211yesmaximal, characteristic
8#2041C211yesmaximal, characteristic
9#1984C211yesmaximal, characteristic, complemented
10#1983C211yesmaximal, characteristic, complemented
11#1865C211yesmaximal, characteristic, complemented
12#1864C211yesmaximal, characteristic, complemented
13#1863C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16#559—24no—
17#559—24no—
18C2×#127C2211yescharacteristic
19C2×#125C2211yescharacteristic
20C2×#124—21no—
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×#123—21no—
24C2×#123—21no—
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×D4—21no—
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30#488—24no—
31#478—24no—
32C2×#107C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×C4×C8C2211yescharacteristic
48#398—21no—
49#395—21no—
50#389—21no—
51#389—21no—
52#389C2211yescharacteristic
53#389C2211yescharacteristic
54#389C2211yescharacteristic
55#389C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8D411yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×D4—21no—
66C22×D4—21no—
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#127—22no—
76#127—22no—
77#125—22no—
78#125—22no—
79#124—28no—
80#124—22no—
81#124—22no—
82#124—22no—
83#124D414yes—
84#123—28no—
85#123—28no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×D4—28no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4D414yes—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4D411yescharacteristic
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4D411yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—22no—
122#106—22no—
123C4⋊C8—22no—
124#103—22no—
125#103—22no—
126#103—22no—
127#103—22no—
128#102—22no—
129#102—22no—
130#102—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—22no—
145C2×Q8—22no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—41no—
152C2×Q8C2×D412yes—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—21no—
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×D4—42no—
166C2×D4—42no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4C2×D412yes—
170C2×D4C2×D412yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—21no—
178C22×C4—21no—
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
190C2×C8—21no—
191C2×C8—21no—
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—
202C4⋊C4—24no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4C2×D412yes—
220C4⋊C4#4112yes—
221C4⋊C4#4112yes—
222C22⋊C4—24no—
223C22⋊C4—24no—
224C22⋊C4—24no—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—24no—
228C42—24no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—81no—
234C42C2×D412yes—
235C42C2×D412yes—
236C23—22no—
237C23—22no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23C22×D411yescharacteristic
251Q8—22no—
252Q8—22no—
253Q8—44no—
254Q8—41no—
255Q8—41no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—22no—
260D4—48no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—84no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
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—41no—
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—84no—
300C2×C4—84no—
301C2×C4#12012yes—
302C2×C4#12012yes—
303C2×C4#13512yes—
304C2×C4C22×D411yescharacteristic
305C2×C4C22×D411yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C42−1+411yescharacteristic
309C2×C42−1+411yescharacteristic
310C8—41no—
311C8—41no—
312C8—41no—
313C8—41no—
314C22—42no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22C2×#12011yescharacteristic
329C22#54811yescharacteristic
330C22#54811yescharacteristic
331C22C2×#13511yescharacteristic
332C22#57511yescharacteristic
333C22#57611yescharacteristic
334C22#57711yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—42no—
345C4—81no—
346C4—81no—
347C4#53211yescharacteristic
348C4#53611yescharacteristic
349C4#54711yescharacteristic
350C4#54711yescharacteristic
351C2—82no—
352C2—82no—
353C2#243211yescharacteristic
354C2#300211yescharacteristic
355C2#301211yescharacteristic
356C2#336111yescharacteristic
357C2#336511yescharacteristic
358C2#337011yescharacteristic
359C2#337111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#127C22
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
2. C2×#488C2semidirect
3. C2×#478C2semidirect
4. #2584C2semidirect
5. #2042C2semidirect
6. #2042C2semidirect
9. #1984C2semidirect
10. #1983C2semidirect
11. #1865C2semidirect
12. #1864C2semidirect
13. #1863C2semidirect
14. #1862C2semidirect
15. C4×#102C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5592
C2×#4882
C2×#4782
#25842
#20422
#20412
#19842
#19832
#18652
#18642
#18632
#18622
C4×#1022