Finite Group Databasebeta

← #59457

Subgroups — #59457

Order 256.

990 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2780C214yesmaximal, complemented
2#2580C212yesmaximal, complemented
3#2571C211yesmaximal, characteristic, complemented
4#2566C212yesmaximal, complemented
5#2566C212yesmaximal, complemented
6#2552C211yesmaximal, characteristic, complemented
7#2538C212yesmaximal, complemented
8#2454C212yesmaximal, complemented
9#2452C212yesmaximal, complemented
10#2450C214yesmaximal, complemented
11#2366C211yesmaximal, characteristic
12#2366C211yesmaximal, characteristic
13#2349C212yesmaximal, complemented
14#2349C212yesmaximal, complemented
15C4×C4⋊Q8C211yesmaximal, characteristic, complemented
16C4×#127C211yesmaximal, characteristic
17C4×#125C211yesmaximal, characteristic, complemented
18C2×C4⋊Q8C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#125C2211yescharacteristic
21C2×#125C2212yes—
22C2×C22⋊Q8C2214yes—
23C2×C4×Q8C2212yes—
24C2×C4×D4C2212yes—
25C2×C4×D4C2212yes—
26C2×C4×D4C2212yes—
27C2×C4×D4C2212yes—
28C2×C4×D4C2212yes—
29C2×C4×D4C2212yes—
30C2×C4×D4C2212yes—
31#400C2214yes—
32#399C2214yes—
33#398C2214yes—
34#396C2214yes—
35#396C2214yes—
36#396C2214yes—
37#396C2214yes—
38#393C2214yes—
39#391C2212yes—
40#390C2212yes—
41#390C2212yes—
42#390C2212yes—
43#390C2212yes—
44#390C2212yes—
45#390C2212yes—
46#390C2212yes—
47#389C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#389C2212yes—
52#389C2212yes—
53#389C2212yes—
54#389C2212yes—
55#389C2212yes—
56#387C2212yes—
57#387C2212yes—
58#386C2214yes—
59#386C2214yes—
60#385C2214yes—
61#385C2214yes—
62#385C2214yes—
63#385C2214yes—
64#385C2214yes—
65#385C2214yes—
66#384C2211yescharacteristic
67#384C2211yescharacteristic
68#384C2212yes—
69#382C2211yescharacteristic
70#382C2212yes—
71#382C2212yes—
72#380C2214yes—
73#380C2214yes—
74C4×C4⋊C4C2212yes—
75C4×C4⋊C4C2211yescharacteristic
76C4×C4⋊C4C2212yes—
77C4×C4⋊C4C2211yescharacteristic
78C4×C4⋊C4C2212yes—
79C4×C4⋊C4C2211yescharacteristic
80C4×C4⋊C4C2211yescharacteristic
81C2×#95C2214yes—
82C43C2211yescharacteristic
83C22×Q8C2312yes—
84C22×D4C2312yes—
85C22×D4C2312yes—
86C22×D4C2312yes—
87C23×C4C2314yes—
88C23×C4C2314yes—
89C23×C4C2314yes—
90C4⋊Q8—24no—
91#127Q818yescomplemented
92#125—24no—
93#125—44no—
94C22⋊Q8—48no—
95C4×Q8—44no—
96C4×D4—28no—
97C4×D4—28no—
98C4×D4—44no—
99C4×D4—44no—
100C4×D4—44no—
101C4×D4—44no—
102C4×D4—44no—
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2312yes—
107C2×C4⋊C4C2312yes—
108C2×C4⋊C4C2312yes—
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2312yes—
113C2×C4⋊C4C2312yes—
114C2×C4⋊C4C2312yes—
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2312yes—
117C2×C4⋊C4C2312yes—
118C2×C4⋊C4C2312yes—
119C2×C4⋊C4C2312yes—
120C2×C4⋊C4C2314yes—
121C2×C4⋊C4C2314yes—
122C2×C4⋊C4C2314yes—
123C2×C4⋊C4C2314yes—
124C2×C4⋊C4C2312yes—
125C2×C4⋊C4C2312yes—
126C2×C4⋊C4C2311yescharacteristic
127C2×C22⋊C4C2314yes—
128C2×C22⋊C4C2314yes—
129C2×C22⋊C4C2314yes—
130C2×C22⋊C4C2314yes—
131C2×C22⋊C4C2314yes—
132C2×C22⋊C4C2314yes—
133C2×C22⋊C4C2314yes—
134C2×C22⋊C4C2314yes—
135C2×C22⋊C4C2314yes—
136C2×C42C2311yescharacteristic
137C2×C42C2311yescharacteristic
138C2×C42C2312yes—
139C2×C42C2312yes—
140C2×C42C2312yes—
141C2×C42C2312yes—
142C2×C42C2311yescharacteristic
143C2×C42C2311yescharacteristic
144C2×C42C2311yescharacteristic
145C2×C42C2311yescharacteristic
146C2×C42C2311yescharacteristic
147#95C2312yes—
148#95C2312yes—
149#95C2314yes—
150#95C2314yes—
151#95C2312yes—
152#95C2312yes—
153#95C2312yes—
154#95C2312yes—
155#95C2312yes—
156#95C2312yes—
157#95C2312yes—
158#95C2312yes—
159#95C2314yes—
160#95C2314yes—
161C24C2414yes—
162C2×Q8—24no—
163C2×Q8—24no—
164C2×Q8—42no—
165C2×D4—24no—
166C2×D4—24no—
167C2×D4—24no—
168C2×D4—24no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4C2×Q818yes—
172C2×D4C2×Q818yes—
173C2×D4C2×Q818yes—
174C22×C4—28no—
175C22×C4—28no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4—44no—
183C22×C4C2412yes—
184C22×C4C2412yes—
185C22×C4C2412yes—
186C22×C4C2412yes—
187C22×C4C2412yes—
188C22×C4C2414yes—
189C22×C4C2411yescharacteristic
190C22×C4C2412yes—
191C22×C4C2412yes—
192C22×C4C2411yescharacteristic
193C22×C4C2411yescharacteristic
194C22×C4C2412yes—
195C22×C4C2411yescharacteristic
196C22×C4C2411yescharacteristic
197C22×C4C2411yescharacteristic
198C22×C4C2411yescharacteristic
199C4⋊C4—24no—
200C4⋊C4—24no—
201C4⋊C4—24no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—24no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4#4114yes—
223C22⋊C4—28no—
224C22⋊C4—28no—
225C22⋊C4—28no—
226C22⋊C4—44no—
227C22⋊C4—44no—
228C22⋊C4—44no—
229C22⋊C4—44no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C42—24no—
233C42—22no—
234C42—22no—
235C42—22no—
236C42—24no—
237C42—24no—
238C42—24no—
239C42—22no—
240C42—41no—
241C42C2×Q814yes—
242C42#4114yes—
243C23—24no—
244C23—24no—
245C23—24no—
246C23—24no—
247C23—24no—
248C23—24no—
249C23C22×Q818yes—
250C23C2511yescharacteristic
251Q8—48no—
252D4—216no—
253D4—216no—
254D4—48no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—88no—
293C2×C4—88no—
294C2×C4—88no—
295C2×C4C22×Q812yes—
296C2×C4C22×Q812yes—
297C2×C4C22×Q812yes—
298C2×C4C2×#4112yes—
299C2×C4C2×#4112yes—
300C2×C4C2×#4114yes—
301C2×C4C2×#4112yes—
302C2×C4C2×#4112yes—
303C2×C4H(5,𝔽2)12yes—
304C2×C4H(5,𝔽2)12yes—
305C2×C4H(5,𝔽2)12yes—
306C22—28no—
307C22—28no—
308C22—28no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—44no—
313C22C23×Q811yescharacteristic
314C22C22×#4111yescharacteristic
315C22C22×#4111yescharacteristic
316C22C2×H(5,𝔽2)11yescharacteristic
317C22C2×H(5,𝔽2)11yescharacteristic
318C22C2×H(5,𝔽2)11yescharacteristic
319C22#58511yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—24no—
324C4—22no—
325C4—42no—
326C4—42no—
327C4—44no—
328C4—42no—
329C4—42no—
330C4—41no—
331C4—42no—
332C4—42no—
333C4—42no—
334C4#55414yes—
335C4#55414yes—
336C2—48no—
337C2C2×#55411yescharacteristic
338C2C2×#55411yescharacteristic
339C2#348011yescharacteristic
340C2#350111yescharacteristic
341C2#350811yescharacteristic
342C2#351911yescharacteristic
343C2#354611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC2×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2780C2semidirect
2. #2580C2semidirect
3. #2571C2semidirect
4. #2566C2semidirect
5. #2566C2semidirect
6. #2552C2semidirect
7. #2538C2semidirect
8. #2454C2semidirect
9. #2452C2semidirect
10. #2450C2semidirect
13. #2349C2semidirect
14. #2349C2semidirect
15. C4×C4⋊Q8C2semidirect
17. C4×#125C2semidirect
91. #127Q8semidirect

Maximal subgroups

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

subgroupindex
#27802
#25802
#25712
#25662
#25522
#25382
#24542
#24522
#24502
#23662
#23492
C4×C4⋊Q82
C4×#1272
C4×#1252