Finite Group Databasebeta

← #61710

Subgroups — #61710

Order 256.

913 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3505C211yesmaximal, characteristic, complemented
2C2×#555C211yesmaximal, characteristic, complemented
3#3348C214yesmaximal, complemented
4#3347C214yesmaximal, complemented
5#3321C214yesmaximal, complemented
6#3319C214yesmaximal, complemented
7#3239C212yesmaximal, complemented
8#3193C211yesmaximal, characteristic, complemented
9#3192C211yesmaximal, characteristic, complemented
10#3188C212yesmaximal, complemented
11#3114C212yesmaximal, complemented
12#3095C211yesmaximal, characteristic, complemented
13#3094C211yesmaximal, characteristic, complemented
14#3091C212yesmaximal, complemented
15#2973C211yesmaximal, characteristic, complemented
16#555—22no—
17#555—22no—
18#555C2214yes—
19#555C2214yescomplemented
20#553—22no—
21#553C2212yescomplemented
22#552C2212yescomplemented
23#551—22no—
24#548C2212yescomplemented
25#546C2212yescomplemented
26#539—21no—
27#538—21no—
28#534—21no—
29C2×C4⋊Q8C2211yescharacteristic, complemented
30#529C2212yes—
31C2×#124—21no—
32C2×#124—21no—
33#525C2212yes—
34C2×C22⋊Q8C2211yescharacteristic
35C2×#121—21no—
36C2×#121C2211yescharacteristic
37C2×#121C2211yescharacteristic
38C2×#121C2211yescharacteristic, complemented
39#518—21no—
40C2×C4×D4C2211yescharacteristic
41C2×C4×D4C2211yescharacteristic, complemented
42C2×#117—21no—
43C2×#117C2211yescharacteristic
44C2×#117C2211yescharacteristic, complemented
45#501C2214yescomplemented
46#489C2214yescomplemented
47#485C2214yescomplemented
48#485C2214yescomplemented
49#484C2212yescomplemented
50#484C2212yescomplemented
51#483C2212yescomplemented
52#482C2214yescomplemented
53#482C2214yescomplemented
54#482C2214yescomplemented
55#481C2212yes—
56#481C2212yes—
57#481C2212yescomplemented
58#480C2212yes—
59#480C2212yescomplemented
60#478C2214yescomplemented
61#475C2214yescomplemented
62#474C2214yescomplemented
63#469C2214yescomplemented
64#468C2214yescomplemented
65#466C2214yescomplemented
66#465C2214yescomplemented
67#463C2214yescomplemented
68#450C2212yescomplemented
69#449C2214yescomplemented
70#448C2212yescomplemented
71#447C2212yes—
72#447C2212yescomplemented
73#442C2214yescomplemented
74#440C2214yescomplemented
75#435C2214yescomplemented
76#435C2214yescomplemented
77#432C2212yescomplemented
78#428C2212yescomplemented
79#428C2212yescomplemented
80#427C2214yescomplemented
81#419C2212yescomplemented
82#418C2212yescomplemented
83#416C2214yescomplemented
84#407C2211yescharacteristic
85#407C2211yescharacteristic
86C2×C22⋊C8C2212yescomplemented
87C2×#41C2312yes—
88C22×Q8—21no—
89C22×Q8C2311yescharacteristic
90C22×D4—21no—
91C22×D4—21no—
92C22×D4—21no—
93C22×D4D411yescharacteristic
94C22×D4C2311yescharacteristic
95C22×D4C2311yescharacteristic
96C23×C4—21no—
97C23×C4C2311yescharacteristic
98C23×C4C2311yescharacteristic
99C2×SD16C2314yes—
100C2×D8C2314yes—
101C2×M16C2312yes—
102C2×M16C2312yes—
103C22×C8C2314yes—
104C4⋊Q8—22no—
105C4⋊Q8C2314yes—
106#126—22no—
107#126—22no—
108#126—22no—
109#125—22no—
110#125C2312yes—
111#124—24no—
112#124—24no—
113#124—24no—
114#124—24no—
115#124—22no—
116#124C2312yes—
117#123—22no—
118#123—22no—
119#123—22no—
120#123—22no—
121#123—22no—
122C22⋊Q8—21no—
123C22⋊Q8—21no—
124C22⋊Q8—22no—
125C22⋊Q8C2312yes—
126C22⋊Q8C2312yes—
127C22⋊Q8C2312yes—
128#121—24no—
129#121—24no—
130#121—22no—
131#121—22no—
132#121—21no—
133#121—21no—
134#121—21no—
135#121—21no—
136#121—22no—
137#121—22no—
138#121C2312yes—
139#121C2312yes—
140#121C2312yes—
141#121C2312yes—
142#121C2314yes—
143#121C2312yes—
144#120—22no—
145#120—22no—
146C4×Q8C2312yes—
147C4×D4—22no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4D412yes—
154C4×D4D412yes—
155C4×D4C2314yes—
156C4×D4C2312yes—
157C4×D4C2312yes—
158C4×D4C2312yes—
159#117—21no—
160#117—21no—
161#117—22no—
162#117—24no—
163#117—24no—
164#117—22no—
165#117—22no—
166#117C2312yes—
167#117C2312yes—
168#117C2314yes—
169#117C2312yes—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4D411yescharacteristic
173C2×C4⋊C4C2311yescharacteristic
174C2×C4⋊C4C2311yescharacteristic
175C2×C4⋊C4C2311yescharacteristic
176C2×C4⋊C4C2311yescharacteristic
177C2×C4⋊C4C2311yescharacteristic
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4D411yescharacteristic
187C2×C22⋊C4D411yescharacteristic
188C2×C42—21no—
189C2×C42—21no—
190C2×C42C2311yescharacteristic
191#107C2314yes—
192#107C2314yes—
193#107C2314yes—
194#106C2314yes—
195#106C2314yes—
196#106C2314yes—
197C4⋊C8C2314yes—
198#103C2314yes—
199#103C2314yes—
200#103C2314yes—
201#102C2314yes—
202#102C2314yes—
203#102C2314yes—
204#102C2314yes—
205#102C2314yes—
206C22⋊C8C2312yes—
207C22⋊C8C2312yes—
208C22⋊C8C2314yes—
209C22⋊C8C2312yes—
210C22⋊C8C2312yes—
211#97C2314yes—
212C24—21no—
213C24—21no—
214C24C2×D411yescharacteristic
215C24C2×D411yescharacteristic
216#41—44no—
217C2×Q8—21no—
218C2×Q8—24no—
219C2×Q8—42no—
220C2×Q8—82no—
221C2×Q8C2412yes—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—24no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—44no—
235C2×D4—82no—
236C2×D4—82no—
237C2×D4C2×D412yes—
238C2×D4C2×D412yes—
239C2×D4C2412yes—
240C2×D4C2412yes—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4C2×D411yescharacteristic
254C22×C4C2×D411yescharacteristic
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D411yescharacteristic
258C22×C4C2×D411yescharacteristic
259C22×C4C2×D412yes—
260C22×C4C2411yescharacteristic
261C22×C4C2411yescharacteristic
262C22×C4C2411yescharacteristic
263C22×C4C2412yes—
264SD16—44no—
265D8—44no—
266M16—42no—
267M16—42no—
268C2×C8—44no—
269C2×C8C2414yes—
270C2×C8C2414yes—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4C2×D412yes—
284C4⋊C4C2×D412yes—
285C4⋊C4C2412yes—
286C4⋊C4C2412yes—
287C4⋊C4C2412yes—
288C4⋊C4C2412yes—
289C4⋊C4C2412yes—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—24no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—24no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—22no—
303C22⋊C4—24no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4C2×D412yes—
308C22⋊C4C2×D412yes—
309C22⋊C4C2×D412yes—
310C22⋊C4C2×D412yes—
311C42—24no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42C2×D412yes—
317C42C2412yes—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—81no—
331C23—81no—
332C23C22×D411yescharacteristic
333C23C22×D412yes—
334C23C22×D411yescharacteristic
335C23C22×D411yescharacteristic
336Q8—24no—
337Q8—81no—
338Q8—84no—
339D4—24no—
340D4—24no—
341D4—42no—
342D4—42no—
343D4—42no—
344D4—44no—
345D4—44no—
346D4—81no—
347D4—82no—
348D4—82no—
349D4—84no—
350D4—82no—
351D4—82no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—48no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4C22×D412yes—
379C2×C4C22×D411yescharacteristic
380C2×C4C22×D411yescharacteristic
381C2×C4C22×D412yes—
382C2×C4C22×D411yescharacteristic
383C2×C4C22×D412yes—
384C2×C4C2511yescharacteristic
385C8—24no—
386C8—24no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—22no—
398C22—42no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—82no—
405C22—82no—
406C22—84no—
407C22—162no—
408C22#57512yes—
409C22C23×D411yescharacteristic
410C4—22no—
411C4—21no—
412C4—22no—
413C4—21no—
414C4—21no—
415C4—22no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—42no—
421C4—42no—
422C4#58512yes—
423C2—22no—
424C2—41no—
425C2—42no—
426C2—41no—
427C2—82no—
428C2—82no—
429C2#348511yescharacteristic
430C2C2×#57511yescharacteristic
431C2#358511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3505C2semidirect
2. C2×#555C2semidirect
3. #3348C2semidirect
4. #3347C2semidirect
5. #3321C2semidirect
6. #3319C2semidirect
7. #3239C2semidirect
8. #3193C2semidirect
9. #3192C2semidirect
10. #3188C2semidirect
11. #3114C2semidirect
12. #3095C2semidirect
13. #3094C2semidirect
14. #3091C2semidirect
15. #2973C2semidirect
19. #555C22semidirect
21. #553C22semidirect
22. #552C22semidirect
24. #548C22semidirect
25. #546C22semidirect
29. C2×C4⋊Q8C22semidirect
38. C2×#121C22semidirect
41. C2×C4×D4C22semidirect
44. C2×#117C22semidirect
45. #501C22semidirect
46. #489C22semidirect
47. #485C22semidirect
48. #485C22semidirect
49. #484C22semidirect
50. #484C22semidirect
51. #483C22semidirect
52. #482C22semidirect
53. #482C22semidirect
54. #482C22semidirect
57. #481C22semidirect
59. #480C22semidirect
60. #478C22semidirect
61. #475C22semidirect
62. #474C22semidirect
63. #469C22semidirect
64. #468C22semidirect
65. #466C22semidirect
66. #465C22semidirect
67. #463C22semidirect
68. #450C22semidirect
69. #449C22semidirect
70. #448C22semidirect
72. #447C22semidirect
73. #442C22semidirect
74. #440C22semidirect
75. #435C22semidirect
76. #435C22semidirect
77. #432C22semidirect
78. #428C22semidirect
79. #428C22semidirect
80. #427C22semidirect
81. #419C22semidirect
82. #418C22semidirect
83. #416C22semidirect
86. C2×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35052
C2×#5552
#33482
#33472
#33212
#33192
#32392
#31932
#31922
#31882
#31142
#30952
#30942
#30912
#29732