Finite Group Databasebeta

← #60959

Subgroups — #60959

Order 256.

918 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3444C211yesmaximal, characteristic, complemented
2C2×#529C211yesmaximal, characteristic, complemented
3#3250C214yesmaximal, complemented
4#3240C212yesmaximal, complemented
5#3236C211yesmaximal, characteristic, complemented
6#3234C211yesmaximal, characteristic, complemented
7#3219C212yesmaximal, complemented
8#3215C211yesmaximal, characteristic, complemented
9#3213C211yesmaximal, characteristic, complemented
10#3190C212yesmaximal, complemented
11#3188C212yesmaximal, complemented
12#3181C212yesmaximal, complemented
13#3180C212yesmaximal, complemented
14#3117C212yesmaximal, complemented
15#3114C212yesmaximal, complemented
16#3111C214yesmaximal, complemented
17C2×#432C211yesmaximal, characteristic, complemented
18#533C2211yescharacteristic
19#533C2211yescharacteristic, complemented
20#533C2211yescharacteristic, complemented
21#532C2211yescharacteristic
22#532C2211yescharacteristic, complemented
23#532C2211yescharacteristic, complemented
24#529—21no—
25#529—21no—
26#529—21no—
27#529—21no—
28#529—21no—
29#529—21no—
30#529—21no—
31#529—21no—
32#529C2212yes—
33#529C2212yes—
34#529C2212yes—
35#529C2212yes—
36#529C2212yes—
37#529C2212yescomplemented
38C2×#126—21no—
39C2×#125C2211yescharacteristic
40C2×#124C2211yescharacteristic
41#525—21no—
42#525—21no—
43#525C2212yes—
44C2×#123—21no—
45C2×C22⋊Q8C2211yescharacteristic
46C2×#121C2211yescharacteristic
47C4×#41—21no—
48C4×#41—21no—
49C4×#41C2212yescomplemented
50C2×C4×Q8C2211yescharacteristic
51C2×C4×D4—21no—
52C2×C4×D4C2211yescharacteristic
53C2×#117—21no—
54C22×C42C2211yescharacteristic
55#501C2212yescomplemented
56#501C2212yescomplemented
57#497C2212yescomplemented
58#496C2212yescomplemented
59#491C2212yescomplemented
60#490C2212yescomplemented
61#489C2212yescomplemented
62#489C2212yescomplemented
63#485C2214yescomplemented
64#484C2212yes—
65#483C2212yes—
66#482C2214yes—
67#481C2212yes—
68#480C2212yes—
69#479C2214yescomplemented
70#479C2214yescomplemented
71#478C2214yescomplemented
72#478C2214yescomplemented
73#467C2214yescomplemented
74#466C2214yes—
75#465C2214yes—
76#465C2214yescomplemented
77#464C2214yescomplemented
78#464C2214yescomplemented
79#463C2214yescomplemented
80#463C2214yescomplemented
81#452C2214yescomplemented
82#451C2212yes—
83#450C2212yes—
84#449C2214yes—
85#448C2212yes—
86#447C2212yes—
87#442C2214yescomplemented
88#441C2214yescomplemented
89#440C2214yescomplemented
90#440C2214yescomplemented
91#432C2212yes—
92#432C2211yescharacteristic
93#432C2211yescharacteristic
94#432C2212yescomplemented
95#432C2212yescomplemented
96C2×C4⋊C8C2212yescomplemented
97C2×C22⋊C8C2212yes—
98C2×#97C2212yescomplemented
99C2×#41—21no—
100C2×#41—21no—
101C2×#41C2312yes—
102C22×Q8C2311yescharacteristic
103C22×D4—21no—
104C22×D4C2311yescharacteristic
105C23×C4—21no—
106C23×C4D411yescharacteristic
107C23×C4D411yescharacteristic
108C23×C4C2311yescharacteristic
109C2×Q16C2314yes—
110C2×SD16C2314yes—
111C2×SD16C2314yes—
112C2×D8C2314yes—
113C22×C8C2314yes—
114C4⋊Q8C2311yescharacteristic
115C4⋊Q8C2311yescharacteristic
116C4⋊Q8C2311yescharacteristic
117#127C2311yescharacteristic
118#126—22no—
119#126—22no—
120#126—22no—
121#126—22no—
122#126—22no—
123#126—22no—
124#125—21no—
125#125—21no—
126#125—21no—
127#125—21no—
128#125C2312yes—
129#125C2312yes—
130#125C2312yes—
131#124—21no—
132#124—21no—
133#124—21no—
134#124—21no—
135#124C2312yes—
136#124C2312yes—
137#124C2312yes—
138#123—22no—
139#123—22no—
140#123—22no—
141#123—22no—
142#123—22no—
143#123—22no—
144#123—22no—
145C22⋊Q8—21no—
146C22⋊Q8—21no—
147C22⋊Q8—21no—
148C22⋊Q8—21no—
149C22⋊Q8—21no—
150C22⋊Q8—21no—
151C22⋊Q8C2312yes—
152C22⋊Q8C2312yes—
153C22⋊Q8C2312yes—
154C22⋊Q8C2312yes—
155#121—21no—
156#121—21no—
157#121—21no—
158#121—21no—
159#121—21no—
160#121—21no—
161#121C2312yes—
162#121C2312yes—
163#121C2312yes—
164#121C2312yes—
165#120—22no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8C2312yes—
173C4×Q8C2312yes—
174C4×Q8C2312yes—
175C4×Q8C2312yes—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4C2312yes—
193C4×D4C2312yes—
194C4×D4C2312yes—
195C4×D4C2312yes—
196#117—21no—
197#117—21no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—22no—
202#117—22no—
203#117—22no—
204#117—22no—
205#117C2312yes—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C2311yescharacteristic
211C2×C4⋊C4C2311yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42D411yescharacteristic
220C2×C42D411yescharacteristic
221C2×C42D411yescharacteristic
222C2×C42D411yescharacteristic
223C2×C42D411yescharacteristic
224C2×C42D411yescharacteristic
225C2×C42C2311yescharacteristic
226C2×C42C2311yescharacteristic
227C2×C42C2311yescharacteristic
228C2×C42C2311yescharacteristic
229C2×C42C2311yescharacteristic
230C2×C42C2311yescharacteristic
231#107C2314yes—
232#107C2314yes—
233#106C2314yes—
234#106C2314yes—
235C4⋊C8C2312yes—
236C4⋊C8C2314yes—
237C4⋊C8C2312yes—
238#103C2314yes—
239#103C2314yes—
240#103C2314yes—
241#103C2314yes—
242#102C2314yes—
243#102C2314yes—
244#102C2314yes—
245#102C2314yes—
246C22⋊C8C2312yes—
247C22⋊C8C2312yes—
248C22⋊C8C2314yes—
249#97C2314yes—
250#97C2312yes—
251#97C2312yes—
252C24—21no—
253C24C2×D411yescharacteristic
254#41—44no—
255#41—44no—
256#41—44no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—42no—
261C2×Q8C2412yes—
262C2×Q8C2412yes—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4C2412yes—
274C2×D4C2412yes—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4C2×D411yescharacteristic
288C22×C4C2×D411yescharacteristic
289C22×C4C2×D411yescharacteristic
290C22×C4C2×D411yescharacteristic
291C22×C4C2×D411yescharacteristic
292C22×C4C2×D411yescharacteristic
293C22×C4C2×D411yescharacteristic
294C22×C4C2×D411yescharacteristic
295C22×C4C2×D411yescharacteristic
296C22×C4C2×D411yescharacteristic
297C22×C4C2×D411yescharacteristic
298C22×C4C2×D411yescharacteristic
299C22×C4C2×D411yescharacteristic
300C22×C4C2×D411yescharacteristic
301C22×C4C2×D411yescharacteristic
302C22×C4C2411yescharacteristic
303C22×C4C2411yescharacteristic
304C22×C4C2411yescharacteristic
305Q16—44no—
306SD16—44no—
307SD16—44no—
308D8—44no—
309C2×C8—44no—
310C2×C8C2414yes—
311C2×C8C2414yes—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4C2412yes—
333C4⋊C4C2412yes—
334C4⋊C4C2412yes—
335C4⋊C4C2412yes—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42C2×D411yescharacteristic
361C42C2×D411yescharacteristic
362C42C2×D411yescharacteristic
363C42C2×D411yescharacteristic
364C42C2×D411yescharacteristic
365C42C2×D411yescharacteristic
366C42C2×D411yescharacteristic
367C42C2×D411yescharacteristic
368C42C2×D411yescharacteristic
369C42C2×D411yescharacteristic
370C42C2×D411yescharacteristic
371C42C2×D411yescharacteristic
372C42C2411yescharacteristic
373C42C2411yescharacteristic
374C42C2411yescharacteristic
375C42C2411yescharacteristic
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—41no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23C22×D411yescharacteristic
385C23C22×D411yescharacteristic
386C23C22×D411yescharacteristic
387Q8—24no—
388Q8—24no—
389Q8—42no—
390Q8—42no—
391Q8—42no—
392D4—24no—
393D4—24no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—44no—
398D4—44no—
399D4—44no—
400D4—44no—
401D4—44no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4C22×D411yescharacteristic
435C2×C4C22×D411yescharacteristic
436C2×C4C22×D411yescharacteristic
437C2×C4C22×D411yescharacteristic
438C2×C4C22×D411yescharacteristic
439C2×C4C22×D411yescharacteristic
440C2×C4C22×D411yescharacteristic
441C2×C4C22×D411yescharacteristic
442C2×C4C22×D411yescharacteristic
443C2×C4C22×D411yescharacteristic
444C2×C4C22×D411yescharacteristic
445C2×C4C2511yescharacteristic
446C8—24no—
447C8—24no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—44no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—84no—
463C22#57512yes—
464C22#57512yes—
465C22C23×D411yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4#58512yes—
484C2—22no—
485C2—22no—
486C2—41no—
487C2—82no—
488C2—82no—
489C2#344911yescharacteristic
490C2C2×#57511yescharacteristic
491C2C2×#57511yescharacteristic
subgroupG/H
center, socleC22C23×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3444C2semidirect
2. C2×#529C2semidirect
3. #3250C2semidirect
4. #3240C2semidirect
5. #3236C2semidirect
6. #3234C2semidirect
7. #3219C2semidirect
8. #3215C2semidirect
9. #3213C2semidirect
10. #3190C2semidirect
11. #3188C2semidirect
12. #3181C2semidirect
13. #3180C2semidirect
14. #3117C2semidirect
15. #3114C2semidirect
16. #3111C2semidirect
17. C2×#432C2semidirect
19. #533C22semidirect
20. #533C22semidirect
22. #532C22semidirect
23. #532C22semidirect
37. #529C22semidirect
49. C4×#41C22semidirect
55. #501C22semidirect
56. #501C22semidirect
57. #497C22semidirect
58. #496C22semidirect
59. #491C22semidirect
60. #490C22semidirect
61. #489C22semidirect
62. #489C22semidirect
63. #485C22semidirect
69. #479C22semidirect
70. #479C22semidirect
71. #478C22semidirect
72. #478C22semidirect
73. #467C22semidirect
76. #465C22semidirect
77. #464C22semidirect
78. #464C22semidirect
79. #463C22semidirect
80. #463C22semidirect
81. #452C22semidirect
87. #442C22semidirect
88. #441C22semidirect
89. #440C22semidirect
90. #440C22semidirect
94. #432C22semidirect
95. #432C22semidirect
96. C2×C4⋊C8C22semidirect
98. C2×#97C22semidirect

Maximal subgroups

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

subgroupindex
#34442
C2×#5292
#32502
#32402
#32362
#32342
#32192
#32152
#32132
#31902
#31882
#31812
#31802
#31172
#31142
#31112
C2×#4322