Finite Group Databasebeta

← #24308

Subgroups — #24308

Order 256.

534 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544C211yesmaximal, characteristic, complemented
2C2×#478C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic, complemented
4#2324C211yesmaximal, characteristic
5#2059C211yesmaximal, characteristic, complemented
6#2057C211yesmaximal, characteristic, complemented
7#2025C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1989C211yesmaximal, characteristic, complemented
10#1987C211yesmaximal, characteristic, complemented
11#1875C211yesmaximal, characteristic
12#1874C211yesmaximal, characteristic
13#1854C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1798C211yesmaximal, characteristic, complemented
16#544—24no—
17#544—24no—
18C2×C4⋊Q8—21no—
19C2×C4⋊Q8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×#125—21no—
22C2×#125C2211yescharacteristic
23C2×#125C2211yescharacteristic
24C2×C22⋊Q8—21no—
25C2×C22⋊Q8—21no—
26C2×#121C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31#478—24no—
32#474—24no—
33C2×#107C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#107C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×C4⋊C8C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45#390—21no—
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#388—21no—
49#386—21no—
50#385—21no—
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C4⋊Q8—28no—
75C4⋊Q8—22no—
76C4⋊Q8—22no—
77C4⋊Q8—22no—
78C4⋊Q8D414yescomplemented
79#125—28no—
80#125—22no—
81#125—22no—
82#125—22no—
83#125D414yescomplemented
84C22⋊Q8—28no—
85C22⋊Q8—28no—
86#121—22no—
87#121Q814yescomplemented
88C4×D4—22no—
89C4×D4—22no—
90#117—22no—
91#117—22no—
92#117—22no—
93#117—22no—
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—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4D411yescharacteristic
105C2×C4⋊C4D411yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4D411yescharacteristic
120C2×C22⋊C4D411yescharacteristic
121C2×C42—21no—
122C2×C42—21no—
123C2×C42C2311yescharacteristic
124C2×C42C2311yescharacteristic
125#107—22no—
126#107—22no—
127#107—22no—
128#106—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131#102—22no—
132#102—22no—
133#102—22no—
134#102—22no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137#95—21no—
138#95—21no—
139#95—21no—
140C24—21no—
141C24C2×D411yescharacteristic
142C2×Q8—24no—
143C2×Q8—24no—
144C2×Q8—81no—
145C2×Q8—81no—
146C2×Q8—81no—
147C2×Q8—81no—
148C2×D4—24no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—81no—
155C2×D4—81no—
156C2×D4C2×Q812yes—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—24no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—42no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×Q812yes—
180C22×C4#4112yes—
181C22×C4C2411yescharacteristic
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—
193C2×C8—21no—
194C4⋊C4—24no—
195C4⋊C4—24no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—81no—
213C4⋊C4C2×D412yes—
214C4⋊C4C2×D412yes—
215C4⋊C4C2×D412yes—
216C4⋊C4C2×D412yes—
217C4⋊C4C2×Q812yes—
218C22⋊C4—24no—
219C22⋊C4—22no—
220C22⋊C4—22no—
221C22⋊C4—24no—
222C22⋊C4—42no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C42—24no—
227C42—21no—
228C42—21no—
229C42—81no—
230C42C2×D412yes—
231C42C2×D412yes—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23C22⋊Q812yes—
246C23C22×D411yescharacteristic
247Q8—84no—
248Q8—84no—
249D4—22no—
250D4—22no—
251D4—41no—
252D4—41no—
253D4—84no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
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—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—84no—
292C2×C4—84no—
293C2×C4#12012yes—
294C2×C4#12012yes—
295C2×C4C22⋊Q812yes—
296C2×C4C22×D411yescharacteristic
297C2×C4C22×D411yescharacteristic
298C2×C4C22×Q811yescharacteristic
299C2×C4C2×#4111yescharacteristic
300C2×C42−1+411yescharacteristic
301C2×C42−1+411yescharacteristic
302C8—41no—
303C8—41no—
304C8—41no—
305C8—41no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22C2×#12011yescharacteristic
321C22C2×C22⋊Q811yescharacteristic
322C22#53611yescharacteristic
323C22#57611yescharacteristic
324C22#57611yescharacteristic
325C22#57711yescharacteristic
326C22#57811yescharacteristic
327C4—22no—
328C4—22no—
329C4—22no—
330C4—42no—
331C4—41no—
332C4—42no—
333C4—42no—
334C4—42no—
335C4—41no—
336C4—42no—
337C4—81no—
338C4—81no—
339C4#54811yescharacteristic
340C4#54811yescharacteristic
341C4D4×Q811yescharacteristic
342C4D4×Q811yescharacteristic
343C2—42no—
344C2—82no—
345C2#240911yescharacteristic
346C2#300811yescharacteristic
347C2#301311yescharacteristic
348C2#308111yescharacteristic
349C2#308311yescharacteristic
350C2#324111yescharacteristic
351C2#324311yescharacteristic
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×#544C2semidirect
2. C2×#478C2semidirect
3. C2×#474C2semidirect
5. #2059C2semidirect
6. #2057C2semidirect
7. #2025C2semidirect
8. #2023C2semidirect
9. #1989C2semidirect
10. #1987C2semidirect
13. #1854C2semidirect
14. #1852C2semidirect
15. #1798C2semidirect
78. C4⋊Q8D4semidirect
83. #125D4semidirect
87. #121Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#4782
C2×#4742
#23242
#20592
#20572
#20252
#20232
#19892
#19872
#18752
#18742
#18542
#18522
#17982