Finite Group Databasebeta

← #42967

Subgroups — #42967

Order 256.

1325 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
1C2×#547C211yesmaximal, characteristic, complemented
2C2×#535C211yesmaximal, characteristic, complemented
3#2813C212yesmaximal, complemented
4#2697C214yesmaximal, complemented
5#2678C214yesmaximal, complemented
6#2661C212yesmaximal, complemented
7#2625C211yesmaximal, characteristic, complemented
8#2618C212yesmaximal, complemented
9#2591C212yesmaximal, complemented
10#2545C212yesmaximal, complemented
11#2445C212yesmaximal, complemented
12#2435C211yesmaximal, characteristic, complemented
13#2431C212yesmaximal, complemented
14#2426C212yesmaximal, complemented
15#2408C211yesmaximal, characteristic, complemented
16#2395C211yesmaximal, characteristic, complemented
17#2341C211yesmaximal, characteristic, complemented
18C22×#41C2211yescharacteristic
19C22×#41C2211yescharacteristic, complemented
20C23×D4C2211yescharacteristic
21#547—28no—
22#535—28no—
23C2×C4⋊Q8C2211yescharacteristic, complemented
24C2×#127C2212yes—
25C2×#124C2212yes—
26C2×#124C2212yes—
27C2×#123C2212yes—
28C2×#123C2212yescomplemented
29C2×#123C2214yescomplemented
30C2×C22⋊Q8C2211yescharacteristic
31C2×C22⋊Q8C2211yescharacteristic, complemented
32C2×#121C2212yes—
33C2×#121C2212yes—
34C2×#121C2212yes—
35C2×#121C2212yes—
36C2×#120C2214yes—
37C2×#120C2212yes—
38C2×C4×Q8C2212yescomplemented
39C2×C4×D4C2211yescharacteristic
40C2×C4×D4C2211yescharacteristic, complemented
41C2×#117C2211yescharacteristic
42C22×C4⋊C4C2212yescomplemented
43#400C2211yescharacteristic, complemented
44#400C2212yescomplemented
45#399C2211yescharacteristic, complemented
46#398C2211yescharacteristic, complemented
47#398C2212yescomplemented
48#397C2214yes—
49#397C2212yes—
50#397C2211yescharacteristic, complemented
51#396C2214yescomplemented
52#396C2214yescomplemented
53#396C2214yescomplemented
54#394C2212yes—
55#394C2212yes—
56#394C2212yes—
57#394C2212yes—
58#394C2214yescomplemented
59#394C2214yescomplemented
60#394C2214yescomplemented
61#392C2212yes—
62#392C2214yescomplemented
63#392C2214yescomplemented
64#391C2212yescomplemented
65#390C2214yes—
66#390C2212yes—
67#390C2212yes—
68#390C2212yescomplemented
69#390C2214yescomplemented
70#389C2212yescomplemented
71#388C2214yescomplemented
72#388C2214yescomplemented
73#388C2214yescomplemented
74#386C2214yescomplemented
75#386C2214yescomplemented
76#386C2214yescomplemented
77#380C2211yescharacteristic
78#380C2212yescomplemented
79#379C2212yes—
80#379C2214yes—
81C4×C4⋊C4C2212yescomplemented
82C4×C22⋊C4C2211yescharacteristic
83C4×C22⋊C4C2212yescomplemented
84C4×C22⋊C4C2212yescomplemented
85C25C2312yes—
86C2×#41—24no—
87C2×#41—44no—
88C2×#41—44no—
89C2×#41D418yescomplemented
90C22×Q8C2311yescharacteristic
91C22×Q8C2311yescharacteristic
92C22×D4—24no—
93C22×D4—44no—
94C22×D4C2312yes—
95C22×D4C2311yescharacteristic
96C22×D4C2311yescharacteristic
97C22×D4C2314yes—
98C22×D4C2314yes—
99C22×D4C2312yes—
100C22×D4C2312yes—
101C22×D4C2314yes—
102C22×D4C2312yes—
103C23×C4C2311yescharacteristic
104C23×C4C2312yes—
105C23×C4C2311yescharacteristic
106C23×C4C2312yes—
107C23×C4C2312yes—
108C4⋊Q8—24no—
109#127—28no—
110#124—28no—
111#124—44no—
112#123—28no—
113#123—28no—
114#123—48no—
115C22⋊Q8—24no—
116C22⋊Q8—24no—
117#121—28no—
118#121—28no—
119#121—28no—
120#121—44no—
121#120—216no—
122#120—44no—
123C4×Q8—44no—
124C4×D4—24no—
125C4×D4—24no—
126#117—24no—
127C2×C4⋊C4—28no—
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2312yes—
130C2×C4⋊C4C2312yes—
131C2×C4⋊C4C2312yes—
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2312yes—
136C2×C4⋊C4C2312yes—
137C2×C4⋊C4C2312yes—
138C2×C4⋊C4C2314yes—
139C2×C4⋊C4C2314yes—
140C2×C4⋊C4C2312yes—
141C2×C22⋊C4C2312yes—
142C2×C22⋊C4C2311yescharacteristic
143C2×C22⋊C4C2311yescharacteristic
144C2×C22⋊C4C2312yes—
145C2×C22⋊C4C2314yes—
146C2×C22⋊C4C2314yes—
147C2×C22⋊C4C2314yes—
148C2×C22⋊C4C2312yes—
149C2×C22⋊C4C2312yes—
150C2×C22⋊C4C2312yes—
151C2×C22⋊C4C2314yes—
152C2×C22⋊C4C2314yes—
153C2×C22⋊C4C2314yes—
154C2×C22⋊C4C2314yes—
155C2×C22⋊C4C2314yes—
156C2×C22⋊C4C2312yes—
157C2×C22⋊C4C2314yes—
158C2×C22⋊C4C2312yes—
159C2×C22⋊C4C2312yes—
160C2×C22⋊C4C2314yes—
161C2×C22⋊C4C2314yes—
162C2×C42C2312yes—
163C2×C42C2311yescharacteristic
164C2×C42C2312yes—
165C2×C42C2314yes—
166#95C2311yescharacteristic
167#95C2311yescharacteristic
168#95C2312yes—
169#95C2312yes—
170#95C2312yes—
171#95C2312yes—
172#95C2312yes—
173#95C2314yes—
174#95C2314yes—
175#95C2314yes—
176C24—24no—
177C24—44no—
178C24—42no—
179C24—44no—
180C24—42no—
181C24C2414yes—
182C24C2411yescharacteristic
183C24C2411yescharacteristic
184C24C2412yes—
185#41—416no—
186#41—416no—
187C2×Q8—22no—
188C2×Q8—24no—
189C2×Q8—24no—
190C2×Q8C2×D414yes—
191C2×D4—22no—
192C2×D4—24no—
193C2×D4—28no—
194C2×D4—28no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—44no—
202C2×D4—41no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—416no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—44no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4C2×D414yes—
214C2×D4C2×D418yes—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4C2×D414yes—
225C22×C4C2×D418yes—
226C22×C4C2412yes—
227C22×C4C2412yes—
228C22×C4C2414yes—
229C22×C4C2412yes—
230C22×C4C2411yescharacteristic
231C22×C4C2412yes—
232C22×C4C2411yescharacteristic
233C22×C4C2412yes—
234C22×C4C2412yes—
235C22×C4C2411yescharacteristic
236C22×C4C2412yes—
237C22×C4C2411yescharacteristic
238C22×C4C2411yescharacteristic
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—42no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—22no—
255C22⋊C4—24no—
256C22⋊C4—22no—
257C22⋊C4—28no—
258C22⋊C4—28no—
259C22⋊C4—24no—
260C22⋊C4—28no—
261C22⋊C4—42no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—42no—
265C22⋊C4—44no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C42—24no—
274C42—22no—
275C42—24no—
276C42—44no—
277C23—22no—
278C23—21no—
279C23—22no—
280C23—24no—
281C23—24no—
282C23—22no—
283C23—22no—
284C23—21no—
285C23—21no—
286C23—24no—
287C23—24no—
288C23—24no—
289C23—24no—
290C23—22no—
291C23—21no—
292C23—21no—
293C23—22no—
294C23—24no—
295C23—24no—
296C23—24no—
297C23—48no—
298C23—84no—
299C23—84no—
300C23—88no—
301C23—88no—
302C23C22×D412yes—
303C23C22×D414yes—
304C23C2511yescharacteristic
305Q8—28no—
306Q8—44no—
307D4—28no—
308D4—48no—
309D4—44no—
310D4—48no—
311D4—48no—
312D4—48no—
313D4—88no—
314D4—88no—
315D4—88no—
316C2×C4—21no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—28no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4C22×D414yes—
350C2×C4C22×D412yes—
351C2×C4C22×D412yes—
352C2×C4H(5,𝔽2)12yes—
353C2×C4H(5,𝔽2)14yes—
354C22—22no—
355C22—24no—
356C22—24no—
357C22—22no—
358C22—24no—
359C22—22no—
360C22—44no—
361C22—42no—
362C22—44no—
363C22—42no—
364C22—41no—
365C22—44no—
366C22—41no—
367C22—44no—
368C22—44no—
369C22—41no—
370C22—42no—
371C22—44no—
372C22—44no—
373C22—42no—
374C22—44no—
375C22—44no—
376C22—816no—
377C22C23×D411yescharacteristic
378C22C2×H(5,𝔽2)12yes—
379C22C2×H(5,𝔽2)11yescharacteristic
380C22#58511yescharacteristic
381C22#58512yes—
382C4—22no—
383C4—24no—
384C4—22no—
385C4—41no—
386C4—42no—
387C4—42no—
388C4—42no—
389C4—41no—
390C4—42no—
391C4—42no—
392C4—44no—
393C4—42no—
394C4#53414yes—
395C2—44no—
396C2—42no—
397C2—81no—
398C2—84no—
399C2C2×#53411yescharacteristic
400C2#348811yescharacteristic
401C2#350011yescharacteristic
402C2#354612yes—
403C2#356212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#547C2semidirect
2. C2×#535C2semidirect
3. #2813C2semidirect
4. #2697C2semidirect
5. #2678C2semidirect
6. #2661C2semidirect
7. #2625C2semidirect
8. #2618C2semidirect
9. #2591C2semidirect
10. #2545C2semidirect
11. #2445C2semidirect
12. #2435C2semidirect
13. #2431C2semidirect
14. #2426C2semidirect
15. #2408C2semidirect
16. #2395C2semidirect
17. #2341C2semidirect
19. C22×#41C22semidirect
23. C2×C4⋊Q8C22semidirect
28. C2×#123C22semidirect
29. C2×#123C22semidirect
31. C2×C22⋊Q8C22semidirect
38. C2×C4×Q8C22semidirect
40. C2×C4×D4C22semidirect
42. C22×C4⋊C4C22semidirect
43. #400C22semidirect
44. #400C22semidirect
45. #399C22semidirect
46. #398C22semidirect
47. #398C22semidirect
50. #397C22semidirect
51. #396C22semidirect
52. #396C22semidirect
53. #396C22semidirect
58. #394C22semidirect
59. #394C22semidirect
60. #394C22semidirect
62. #392C22semidirect
63. #392C22semidirect
64. #391C22semidirect
68. #390C22semidirect
69. #390C22semidirect
70. #389C22semidirect
71. #388C22semidirect
72. #388C22semidirect
73. #388C22semidirect
74. #386C22semidirect
75. #386C22semidirect
76. #386C22semidirect
78. #380C22semidirect
81. C4×C4⋊C4C22semidirect
83. C4×C22⋊C4C22semidirect
84. C4×C22⋊C4C22semidirect
89. C2×#41D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#5352
#28132
#26972
#26782
#26612
#26252
#26182
#25912
#25452
#24452
#24352
#24312
#24262
#24082
#23952
#23412