Finite Group Databasebeta

← #36013

Subgroups — #36013

Order 256.

855 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2652C212yesmaximal, complemented
2#2626C211yesmaximal, characteristic
3#2575C212yesmaximal, complemented
4#2573C212yesmaximal, complemented
5#2569C211yesmaximal, characteristic, complemented
6#2527C212yesmaximal, complemented
7#2523C212yesmaximal, complemented
8#2517C212yesmaximal, complemented
9#2515C212yesmaximal, complemented
10#2503C211yesmaximal, characteristic, complemented
11#2493C212yesmaximal, complemented
12#2483C212yesmaximal
13#2401C212yesmaximal, complemented
14C2×#397C211yesmaximal, characteristic
15#2368C211yesmaximal, characteristic, complemented
16#2344C211yesmaximal, characteristic, complemented
17#2343C212yesmaximal, complemented
18#2342C212yesmaximal
19#2273C211yesmaximal, characteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#117C2212yes—
22C22×C4⋊C4C2211yescharacteristic
23C22×C22⋊C4C2211yescharacteristic
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27#401C2212yescomplemented
28#400C2212yescomplemented
29#400C2212yescomplemented
30#400C2212yescomplemented
31#400C2212yescomplemented
32#400C2212yescomplemented
33#400C2212yescomplemented
34#399C2211yescharacteristic
35#399C2211yescharacteristic
36#398C2212yescomplemented
37#397C2211yescharacteristic
38#397C2212yes—
39#397C2211yescharacteristic
40#397C2211yescharacteristic
41#397C2211yescharacteristic
42#397C2212yes—
43#397C2212yes—
44#397C2212yes—
45#397C2212yes—
46#397C2212yes—
47#396C2212yes—
48#396C2212yes—
49#393C2211yescharacteristic
50#393C2211yescharacteristic
51#391C2212yescomplemented
52#391C2212yescomplemented
53#389C2212yescomplemented
54#388C2212yes—
55#388C2212yes—
56#388C2212yes—
57#388C2212yes—
58#387C2212yescomplemented
59#387C2212yescomplemented
60#387C2212yescomplemented
61#387C2212yescomplemented
62#387C2212yescomplemented
63#387C2212yescomplemented
64#387C2212yescomplemented
65#387C2212yescomplemented
66#387C2212yescomplemented
67#387C2212yescomplemented
68#387C2212yescomplemented
69#387C2212yescomplemented
70#387C2212yescomplemented
71#385C2212yes—
72#385C2212yes—
73#385C2212yes—
74#385C2212yes—
75#385C2212yes—
76#385C2212yes—
77#385C2212yes—
78#383C2212yescomplemented
79#383C2212yescomplemented
80#382C2212yescomplemented
81#382C2212yescomplemented
82#381C2212yes—
83#381C2212yes—
84#381C2212yes—
85#381C2212yes—
86#381C2212yes—
87#380C2212yes—
88#380C2211yescharacteristic
89#379C2212yes—
90C4×C4⋊C4C2212yescomplemented
91C4×C4⋊C4C2212yescomplemented
92C4×C4⋊C4C2212yescomplemented
93C4×C22⋊C4C2211yescharacteristic
94C4×C22⋊C4C2212yes—
95C4×C22⋊C4C2212yes—
96C4×C22⋊C4C2212yes—
97C4×C22⋊C4C2212yes—
98C4×C22⋊C4C2212yes—
99C4×C22⋊C4C2211yescharacteristic
100C4×C22⋊C4C2211yescharacteristic
101C4×C22⋊C4C2212yes—
102C4×C22⋊C4C2211yescharacteristic
103C4×C22⋊C4C2212yes—
104#376C2212yescomplemented
105C2×#95C2211yescharacteristic
106C2×#95C2212yes—
107C25C2311yescharacteristic
108C22×Q8C2311yescharacteristic
109C23×C4C2312yes—
110C23×C4C2311yescharacteristic
111C23×C4C2312yes—
112C23×C4C2312yes—
113C23×C4C2312yes—
114C23×C4C2311yescharacteristic
115C22⋊Q8—24no—
116#117—28no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2312yes—
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2312yes—
122C2×C4⋊C4C2312yes—
123C2×C4⋊C4C2312yes—
124C2×C4⋊C4C2311yescharacteristic
125C2×C4⋊C4C2312yes—
126C2×C4⋊C4C2312yes—
127C2×C4⋊C4C2312yes—
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2312yes—
134C2×C22⋊C4—28no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4C2311yescharacteristic
139C2×C22⋊C4C2311yescharacteristic
140C2×C22⋊C4C2311yescharacteristic
141C2×C22⋊C4C2312yes—
142C2×C22⋊C4C2311yescharacteristic
143C2×C22⋊C4C2311yescharacteristic
144C2×C22⋊C4C2311yescharacteristic
145C2×C22⋊C4C2311yescharacteristic
146C2×C22⋊C4C2312yes—
147C2×C22⋊C4C2312yes—
148C2×C22⋊C4C2312yes—
149C2×C22⋊C4C2312yes—
150C2×C22⋊C4C2312yes—
151C2×C22⋊C4C2312yes—
152C2×C22⋊C4C2312yes—
153C2×C22⋊C4C2312yes—
154C2×C22⋊C4C2312yes—
155C2×C22⋊C4C2312yes—
156C2×C22⋊C4C2312yes—
157C2×C22⋊C4C2311yescharacteristic
158C2×C42C2312yes—
159C2×C42C2312yes—
160C2×C42C2312yes—
161C2×C42C2312yes—
162C2×C42C2312yes—
163C2×C42C2312yes—
164C2×C42C2312yes—
165C2×C42C2312yes—
166#95C2312yes—
167#95C2311yescharacteristic
168#95C2312yes—
169#95C2312yes—
170#95C2312yes—
171#95C2311yescharacteristic
172#95C2312yes—
173#95C2312yes—
174#95C2312yes—
175#95C2312yes—
176#95C2312yes—
177#95C2312yes—
178#95C2312yes—
179#95C2312yes—
180#95C2312yes—
181#95C2312yes—
182#95C2312yes—
183#95C2312yes—
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2311yescharacteristic
191#95C2311yescharacteristic
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201C24—22no—
202C24—22no—
203C24—42no—
204C24—42no—
205C24#4114yes—
206C24C2411yescharacteristic
207C24C2411yescharacteristic
208C24C2411yescharacteristic
209C2×Q8—22no—
210C2×Q8—42no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4#4114yes—
227C22×C4C2412yes—
228C22×C4C2411yescharacteristic
229C22×C4C2412yes—
230C22×C4C2411yescharacteristic
231C22×C4C2412yes—
232C22×C4C2412yes—
233C22×C4C2412yes—
234C22×C4C2412yes—
235C22×C4C2412yes—
236C22×C4C2412yes—
237C22×C4C2412yes—
238C22×C4C2412yes—
239C22×C4C2412yes—
240C22×C4C2411yescharacteristic
241C22×C4C2412yes—
242C22×C4C2411yescharacteristic
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—24no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—22no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4#4114yes—
279C42—24no—
280C42—24no—
281C42—42no—
282C42—42no—
283C42—42no—
284C42—42no—
285C42—42no—
286C42—42no—
287C23—22no—
288C23—22no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—28no—
298C23—48no—
299C23—82no—
300C23—82no—
301C23—84no—
302C23C2×#4112yes—
303C23C2×#4112yes—
304C23C2×#4112yes—
305C23C2×#4112yes—
306C23C2×#4112yes—
307C23C2511yescharacteristic
308Q8—82no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4—82no—
356C2×C4C2×#4114yes—
357C2×C4C2×#4112yes—
358C2×C4C2×#4112yes—
359C22—22no—
360C22—24no—
361C22—22no—
362C22—24no—
363C22—22no—
364C22—22no—
365C22—24no—
366C22—24no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—88no—
374C22#52914yes—
375C22C22×#4111yescharacteristic
376C22C22×#4111yescharacteristic
377C22C22×#4111yescharacteristic
378C22#58512yes—
379C22#58512yes—
380C4—22no—
381C4—22no—
382C4—24no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—41no—
392C4—41no—
393C4—42no—
394C4—42no—
395C4—42no—
396C2—24no—
397C2—42no—
398C2—42no—
399C2C2×#52911yescharacteristic
400C2#349111yescharacteristic
401C2#349112yes—
402C2#351011yescharacteristic
403C2#352112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
prime radicalC25C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2652C2semidirect
3. #2575C2semidirect
4. #2573C2semidirect
5. #2569C2semidirect
6. #2527C2semidirect
7. #2523C2semidirect
8. #2517C2semidirect
9. #2515C2semidirect
10. #2503C2semidirect
11. #2493C2semidirect
13. #2401C2semidirect
15. #2368C2semidirect
16. #2344C2semidirect
17. #2343C2semidirect
27. #401C22semidirect
28. #400C22semidirect
29. #400C22semidirect
30. #400C22semidirect
31. #400C22semidirect
32. #400C22semidirect
33. #400C22semidirect
36. #398C22semidirect
51. #391C22semidirect
52. #391C22semidirect
53. #389C22semidirect
58. #387C22semidirect
59. #387C22semidirect
60. #387C22semidirect
61. #387C22semidirect
62. #387C22semidirect
63. #387C22semidirect
64. #387C22semidirect
65. #387C22semidirect
66. #387C22semidirect
67. #387C22semidirect
68. #387C22semidirect
69. #387C22semidirect
70. #387C22semidirect
78. #383C22semidirect
79. #383C22semidirect
80. #382C22semidirect
81. #382C22semidirect
90. C4×C4⋊C4C22semidirect
91. C4×C4⋊C4C22semidirect
92. C4×C4⋊C4C22semidirect
104. #376C22semidirect

Maximal subgroups

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

subgroupindex
#26522
#26262
#25752
#25732
#25692
#25272
#25232
#25172
#25152
#25032
#24932
#24832
#24012
C2×#3972
#23682
#23442
#23432
#23422
#22732