Finite Group Databasebeta

← #44786

Subgroups — #44786

Order 256.

881 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2798C212yesmaximal, complemented
2#2764C211yesmaximal, characteristic, complemented
3#2763C211yesmaximal, characteristic, complemented
4#2759C212yesmaximal, complemented
5#2754C212yesmaximal, complemented
6#2607C211yesmaximal, characteristic, complemented
7#2589C212yesmaximal, complemented
8#2561C211yesmaximal, characteristic
9#2532C212yesmaximal, complemented
10#2527C212yesmaximal, complemented
11#2523C211yesmaximal, characteristic, complemented
12#2493C212yesmaximal, complemented
13#2483C211yesmaximal, characteristic
14#2482C212yesmaximal
15#2438C212yesmaximal, complemented
16C2×#393C211yesmaximal, characteristic
17#2343C212yesmaximal, complemented
18#2342C212yesmaximal
19#2310C212yesmaximal, complemented
20C23×Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#117C2212yes—
23C2×#117C2212yes—
24C22×C22⋊C4C2212yes—
25C22×C22⋊C4C2211yescharacteristic
26C22×C22⋊C4C2211yescharacteristic
27#401C2212yescomplemented
28#400C2212yescomplemented
29#400C2212yescomplemented
30#400C2212yescomplemented
31#400C2212yescomplemented
32#399C2212yes—
33#399C2212yes—
34#398C2212yescomplemented
35#398C2212yescomplemented
36#397C2211yescharacteristic
37#397C2211yescharacteristic
38#397C2212yes—
39#397C2212yes—
40#397C2212yes—
41#397C2212yes—
42#395C2212yescomplemented
43#393C2211yescharacteristic
44#393C2211yescharacteristic
45#393C2212yes—
46#393C2211yescharacteristic
47#393C2212yes—
48#393C2211yescharacteristic
49#393C2211yescharacteristic
50#393C2211yescharacteristic
51#391C2212yescomplemented
52#391C2212yescomplemented
53#391C2212yescomplemented
54#391C2212yescomplemented
55#391C2212yescomplemented
56#391C2212yescomplemented
57#391C2212yescomplemented
58#391C2212yescomplemented
59#391C2212yescomplemented
60#389C2212yescomplemented
61#388C2212yes—
62#388C2212yes—
63#388C2212yes—
64#388C2212yes—
65#388C2212yes—
66#388C2212yes—
67#388C2212yes—
68#388C2212yes—
69#388C2212yes—
70#388C2212yes—
71#387C2212yescomplemented
72#387C2212yescomplemented
73#387C2212yescomplemented
74#387C2212yescomplemented
75#387C2212yescomplemented
76#387C2212yescomplemented
77#387C2212yescomplemented
78#387C2212yescomplemented
79#387C2212yescomplemented
80#385C2211yescharacteristic
81#385C2212yes—
82#385C2212yes—
83#385C2212yes—
84#385C2212yes—
85#385C2212yes—
86#385C2211yescharacteristic
87#383C2212yescomplemented
88#383C2212yescomplemented
89#382C2212yescomplemented
90#381C2212yes—
91#381C2212yes—
92#381C2212yes—
93#381C2212yes—
94#381C2211yescharacteristic
95#381C2211yescharacteristic
96#380C2212yes—
97#379C2212yes—
98#379C2211yescharacteristic
99C4×C4⋊C4C2212yescomplemented
100C4×C22⋊C4C2211yescharacteristic
101C4×C22⋊C4C2212yes—
102C4×C22⋊C4C2212yes—
103C4×C22⋊C4C2212yes—
104#376C2212yescomplemented
105C2×#95C2212yes—
106C2×#95C2211yescharacteristic
107C25C2311yescharacteristic
108C22×Q8—28no—
109C22×Q8—42no—
110C22×Q8C2312yes—
111C22×Q8C2311yescharacteristic
112C22×Q8C2311yescharacteristic
113C22×Q8C2311yescharacteristic
114C23×C4C2311yescharacteristic
115C23×C4C2312yes—
116C23×C4C2311yescharacteristic
117C23×C4C2312yes—
118C23×C4C2311yescharacteristic
119C23×C4C2311yescharacteristic
120C22⋊Q8—42no—
121#117—28no—
122#117—44no—
123C2×C4⋊C4C2312yes—
124C2×C4⋊C4C2312yes—
125C2×C4⋊C4C2312yes—
126C2×C4⋊C4C2312yes—
127C2×C4⋊C4C2311yescharacteristic
128C2×C4⋊C4C2312yes—
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2312yes—
133C2×C4⋊C4C2312yes—
134C2×C4⋊C4C2312yes—
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C22⋊C4—28no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4C2312yes—
141C2×C22⋊C4C2312yes—
142C2×C22⋊C4C2312yes—
143C2×C22⋊C4C2312yes—
144C2×C22⋊C4C2312yes—
145C2×C22⋊C4C2312yes—
146C2×C22⋊C4C2312yes—
147C2×C22⋊C4C2312yes—
148C2×C22⋊C4C2312yes—
149C2×C22⋊C4C2311yescharacteristic
150C2×C22⋊C4C2311yescharacteristic
151C2×C22⋊C4C2311yescharacteristic
152C2×C22⋊C4C2312yes—
153C2×C22⋊C4C2311yescharacteristic
154C2×C22⋊C4C2311yescharacteristic
155C2×C22⋊C4C2312yes—
156C2×C22⋊C4C2311yescharacteristic
157C2×C22⋊C4C2312yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2312yes—
160C2×C42C2312yes—
161C2×C42C2312yes—
162C2×C42C2312yes—
163C2×C42C2312yes—
164C2×C42C2312yes—
165C2×C42C2312yes—
166C2×C42C2312yes—
167#95C2312yes—
168#95C2312yes—
169#95C2311yescharacteristic
170#95C2312yes—
171#95C2312yes—
172#95C2312yes—
173#95C2311yescharacteristic
174#95C2312yes—
175#95C2312yes—
176#95C2312yes—
177#95C2312yes—
178#95C2312yes—
179#95C2312yes—
180#95C2312yes—
181#95C2311yescharacteristic
182#95C2312yes—
183#95C2311yescharacteristic
184#95C2312yes—
185#95C2312yes—
186#95C2312yes—
187#95C2312yes—
188#95C2312yes—
189#95C2312yes—
190#95C2312yes—
191#95C2312yes—
192#95C2312yes—
193#95C2312yes—
194#95C2312yes—
195#95C2312yes—
196#95C2312yes—
197#95C2312yes—
198#95C2312yes—
199#95C2312yes—
200#95C2312yes—
201#95C2312yes—
202C24—22no—
203C24—24no—
204C24—22no—
205C24—42no—
206C24—41no—
207C24C2411yescharacteristic
208C24C2411yescharacteristic
209C24C2411yescharacteristic
210C2×Q8—24no—
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—41no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—416no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4#4118yes—
235C22×C4C2411yescharacteristic
236C22×C4C2411yescharacteristic
237C22×C4C2412yes—
238C22×C4C2412yes—
239C22×C4C2412yes—
240C22×C4C2412yes—
241C22×C4C2412yes—
242C22×C4C2411yescharacteristic
243C22×C4C2412yes—
244C22×C4C2412yes—
245C22×C4C2411yescharacteristic
246C22×C4C2412yes—
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2412yes—
250C22×C4C2412yes—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—22no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—42no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—42no—
285C42—24no—
286C42—24no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—42no—
292C23—21no—
293C23—22no—
294C23—21no—
295C23—22no—
296C23—21no—
297C23—22no—
298C23—21no—
299C23—22no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—21no—
304C23—48no—
305C23—44no—
306C23—44no—
307C23—84no—
308C23—82no—
309C23C2×#4114yes—
310C23H(5,𝔽2)12yes—
311C232−1+412yes—
312C23C2511yescharacteristic
313Q8—44no—
314Q8—48no—
315Q8—44no—
316Q8—82no—
317C2×C4—216no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
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—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
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—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—44no—
361C2×C4—84no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4C2×#4114yes—
366C2×C4C2×#4114yes—
367C22—24no—
368C22—24no—
369C22—22no—
370C22—22no—
371C22—24no—
372C22—22no—
373C22—41no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—42no—
378C22—42no—
379C22—41no—
380C22—41no—
381C22—88no—
382C22#55114yes—
383C22C22×#4112yes—
384C22C2×H(5,𝔽2)11yescharacteristic
385C22C2×2−1+411yescharacteristic
386C22#58512yes—
387C22#58511yescharacteristic
388C4—24no—
389C4—24no—
390C4—41no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—42no—
400C4—42no—
401C4—41no—
402C4—41no—
403C4—42no—
404C2—24no—
405C2—42no—
406C2—81no—
407C2C2×#55111yescharacteristic
408C2#349512yes—
409C2#351012yes—
410C2#353511yescharacteristic
411C2#354511yescharacteristic
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. #2798C2semidirect
2. #2764C2semidirect
3. #2763C2semidirect
4. #2759C2semidirect
5. #2754C2semidirect
6. #2607C2semidirect
7. #2589C2semidirect
9. #2532C2semidirect
10. #2527C2semidirect
11. #2523C2semidirect
12. #2493C2semidirect
15. #2438C2semidirect
17. #2343C2semidirect
19. #2310C2semidirect
27. #401C22semidirect
28. #400C22semidirect
29. #400C22semidirect
30. #400C22semidirect
31. #400C22semidirect
34. #398C22semidirect
35. #398C22semidirect
42. #395C22semidirect
51. #391C22semidirect
52. #391C22semidirect
53. #391C22semidirect
54. #391C22semidirect
55. #391C22semidirect
56. #391C22semidirect
57. #391C22semidirect
58. #391C22semidirect
59. #391C22semidirect
60. #389C22semidirect
71. #387C22semidirect
72. #387C22semidirect
73. #387C22semidirect
74. #387C22semidirect
75. #387C22semidirect
76. #387C22semidirect
77. #387C22semidirect
78. #387C22semidirect
79. #387C22semidirect
87. #383C22semidirect
88. #383C22semidirect
89. #382C22semidirect
99. 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
#27982
#27642
#27632
#27592
#27542
#26072
#25892
#25612
#25322
#25272
#25232
#24932
#24832
#24822
#24382
C2×#3932
#23432
#23422
#23102