Finite Group Databasebeta

← #25654

Subgroups — #25654

Order 256.

509 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556C211yesmaximal, characteristic
2C2×#488C211yesmaximal, characteristic
3C2×#479C211yesmaximal, characteristic, complemented
4#2584C211yesmaximal, characteristic, complemented
5#2043C211yesmaximal, characteristic, complemented
6#2043C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic
8#2042C211yesmaximal, characteristic
9#1984C211yesmaximal, characteristic, complemented
10#1983C211yesmaximal, characteristic
11#1865C211yesmaximal, characteristic, complemented
12#1864C211yesmaximal, characteristic
13#1863C211yesmaximal, characteristic, complemented
14#1862C211yesmaximal, characteristic
15C4×#103C211yesmaximal, characteristic, complemented
16#556—24no—
17#556—24no—
18C2×C4⋊Q8C2211yescharacteristic
19C2×#126—21no—
20C2×#126—21no—
21C2×#125C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×C22⋊Q8—21no—
25C2×C4×Q8C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×#117—21no—
30#488—24no—
31#479—24no—
32C2×#107C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
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×Q8C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C23×C4—21no—
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C4⋊Q8—22no—
73C4⋊Q8—22no—
74#126—28no—
75#126—28no—
76#125—22no—
77#125—22no—
78#124—22no—
79#124—22no—
80#124—22no—
81#124D414yes—
82C22⋊Q8—28no—
83C4×Q8—22no—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8D414yes—
89C4×D4—22no—
90C4×D4—22no—
91#117—28no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
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×C42—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#103—22no—
129#103—22no—
130#103—22no—
131#103—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—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—41no—
156C2×Q8C2×D412yes—
157C2×Q8C2×D412yes—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4C2×D412yes—
164C22×C4—24no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2411yescharacteristic
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4C2×D412yes—
214C4⋊C4#4112yes—
215C4⋊C4#4112yes—
216C22⋊C4—24no—
217C22⋊C4—24no—
218C22⋊C4—24no—
219C22⋊C4—24no—
220C22⋊C4—24no—
221C42—24no—
222C42—24no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—21no—
227C42—81no—
228C42C2×D412yes—
229C42C2×D412yes—
230C23—22no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23C22×D411yescharacteristic
238Q8—22no—
239Q8—22no—
240Q8—22no—
241Q8—22no—
242Q8—44no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246Q8—41no—
247D4—22no—
248D4—22no—
249D4—41no—
250D4—41no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—48no—
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—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4#12012yes—
289C2×C4#12012yes—
290C2×C4#13512yes—
291C2×C4C22×D411yescharacteristic
292C2×C4C22×D411yescharacteristic
293C2×C4C2×#4111yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C42−1+411yescharacteristic
296C2×C42−1+411yescharacteristic
297C8—41no—
298C8—41no—
299C8—41no—
300C8—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—81no—
305C22—81no—
306C22—81no—
307C22—81no—
308C22C2×#12011yescharacteristic
309C22#54811yescharacteristic
310C22#54811yescharacteristic
311C22C2×#13511yescharacteristic
312C22#57511yescharacteristic
313C22#57711yescharacteristic
314C22#57811yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—42no—
319C4—42no—
320C4—42no—
321C4—42no—
322C4—41no—
323C4—42no—
324C4—41no—
325C4—41no—
326C4—81no—
327C4—81no—
328C4#53211yescharacteristic
329C4#53611yescharacteristic
330C4#54711yescharacteristic
331C4#54711yescharacteristic
332C2—82no—
333C2#243211yescharacteristic
334C2#300211yescharacteristic
335C2#301311yescharacteristic
336C2#336011yescharacteristic
337C2#336411yescharacteristic
338C2#337611yescharacteristic
339C2#337711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
3. C2×#479C2semidirect
4. #2584C2semidirect
5. #2043C2semidirect
6. #2043C2semidirect
9. #1984C2semidirect
11. #1865C2semidirect
13. #1863C2semidirect
15. C4×#103C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4882
C2×#4792
#25842
#20432
#20422
#19842
#19832
#18652
#18642
#18632
#18622
C4×#1032