Finite Group Databasebeta

← #60953

Subgroups — #60953

Order 256.

912 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3444C211yesmaximal, characteristic, complemented
2#3444C211yesmaximal, characteristic, complemented
3#3252C212yesmaximal, complemented
4#3251C212yesmaximal, complemented
5#3240C212yesmaximal, complemented
6#3237C211yesmaximal, characteristic, complemented
7#3235C211yesmaximal, characteristic, complemented
8#3219C212yesmaximal, complemented
9#3216C211yesmaximal, characteristic, complemented
10#3214C211yesmaximal, characteristic, complemented
11#3197C214yesmaximal, complemented
12#3172C212yesmaximal, complemented
13#3171C212yesmaximal, complemented
14#3107C212yesmaximal, complemented
15#3106C212yesmaximal, complemented
16#3102C214yesmaximal, complemented
17#2920C211yesmaximal, characteristic, complemented
18#533C2211yescharacteristic
19#533C2211yescharacteristic
20#533C2211yescharacteristic
21#533C2211yescharacteristic
22#533C2211yescharacteristic
23#533C2211yescharacteristic
24#532C2211yescharacteristic
25#532C2211yescharacteristic
26#532C2211yescharacteristic
27#532C2211yescharacteristic
28#532C2211yescharacteristic
29#532C2211yescharacteristic
30#529—21no—
31#529—21no—
32#529—21no—
33#529—21no—
34#529—21no—
35#529—21no—
36#529—21no—
37#529—21no—
38#529C2212yes—
39#529C2212yes—
40#529C2212yes—
41#529C2212yes—
42#525—21no—
43#525—21no—
44#525—21no—
45#525—21no—
46#525C2212yes—
47#525C2212yes—
48C4×#41—21no—
49C4×#41—21no—
50C4×#41—21no—
51C4×#41—21no—
52C4×#41C2212yes—
53C4×#41C2212yes—
54C22×C42C2211yescharacteristic
55#501C2212yescomplemented
56#501C2212yescomplemented
57#497C2212yescomplemented
58#496C2212yescomplemented
59#491C2212yescomplemented
60#490C2212yescomplemented
61#489C2212yescomplemented
62#489C2212yescomplemented
63#485C2214yes—
64#485C2214yes—
65#482C2214yes—
66#482C2214yes—
67#479C2214yescomplemented
68#478C2214yescomplemented
69#477C2212yescomplemented
70#476C2212yescomplemented
71#475C2212yescomplemented
72#474C2212yescomplemented
73#470C2214yes—
74#469C2214yes—
75#468C2214yes—
76#468C2214yes—
77#464C2214yescomplemented
78#463C2214yescomplemented
79#462C2212yescomplemented
80#461C2212yescomplemented
81#460C2212yescomplemented
82#459C2212yescomplemented
83#452C2214yes—
84#452C2214yes—
85#449C2214yes—
86#449C2214yes—
87C4×Q16C2214yescomplemented
88C4×SD16C2214yescomplemented
89C4×SD16C2214yescomplemented
90C4×D8C2214yescomplemented
91#432C2211yescharacteristic
92#432C2211yescharacteristic
93#432C2212yes—
94#431C2212yes—
95#431C2212yes—
96#423C2212yes—
97#407C2212yes—
98C4×M16C2212yes—
99C2×#41—21no—
100C2×#41—21no—
101C2×#41—21no—
102C2×#41—21no—
103C2×#41C2312yes—
104C2×#41C2312yes—
105C23×C4D411yescharacteristic
106C23×C4D411yescharacteristic
107C23×C4C2311yescharacteristic
108C2×Q16C2314yes—
109C2×SD16C2314yes—
110C2×SD16C2314yes—
111C2×D8C2314yes—
112C2×M16C2314yes—
113C4⋊Q8C2311yescharacteristic
114C4⋊Q8C2311yescharacteristic
115C4⋊Q8C2311yescharacteristic
116C4⋊Q8C2311yescharacteristic
117C4⋊Q8C2311yescharacteristic
118C4⋊Q8C2311yescharacteristic
119#127C2311yescharacteristic
120#127C2311yescharacteristic
121#126—22no—
122#126—22no—
123#126—22no—
124#126—22no—
125#125—21no—
126#125—21no—
127#125—21no—
128#125—21no—
129#125C2312yes—
130#125C2312yes—
131#124—21no—
132#124—21no—
133#124—21no—
134#124—21no—
135#124C2312yes—
136#124C2312yes—
137#123—22no—
138#123—22no—
139#123—22no—
140#123—22no—
141#123—22no—
142#123—22no—
143C22⋊Q8—21no—
144C22⋊Q8—21no—
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#121—21no—
162#121—21no—
163#121C2312yes—
164#121C2312yes—
165#121C2312yes—
166#121C2312yes—
167#120—22no—
168#120—22no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8C2312yes—
178C4×Q8C2312yes—
179C4×Q8C2312yes—
180C4×Q8C2312yes—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—22no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4C2312yes—
202C4×D4C2312yes—
203C4×D4C2312yes—
204C4×D4C2312yes—
205#117—21no—
206#117—21no—
207#117—21no—
208#117—21no—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—22no—
213#117—22no—
214#117—22no—
215#117C2312yes—
216#117C2312yes—
217C2×C42—22no—
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⋊C8C2312yes—
237C4⋊C8C2314yes—
238#103C2314yes—
239#103C2314yes—
240#103C2314yes—
241#103C2314yes—
242#102C2314yes—
243#102C2314yes—
244#102C2314yes—
245#102C2314yes—
246C22⋊C8C2312yes—
247C22⋊C8C2314yes—
248C22⋊C8C2312yes—
249#97C2312yes—
250#97C2312yes—
251C4×C8C2314yes—
252C24C2×D411yescharacteristic
253#41—42no—
254#41—44no—
255#41—42no—
256#41—44no—
257#41—44no—
258#41—44no—
259#41—44no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8C2412yes—
265C2×Q8C2412yes—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4C2412yes—
277C2×D4C2412yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
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—
309M16—44no—
310C2×C8C2414yes—
311C2×C8C2414yes—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
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—
354C42—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—41no—
379C23—41no—
380C23—41no—
381C23C22×D411yescharacteristic
382C23C22×D411yescharacteristic
383C23C22×D411yescharacteristic
384Q8—22no—
385Q8—24no—
386Q8—22no—
387Q8—42no—
388Q8—42no—
389Q8—42no—
390Q8—42no—
391D4—22no—
392D4—24no—
393D4—22no—
394D4—42no—
395D4—42no—
396D4—42no—
397D4—44no—
398D4—44no—
399D4—44no—
400D4—44no—
401D4—44no—
402D4—44no—
403D4—42no—
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—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
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—21no—
456C22—21no—
457C22—44no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22C23×D411yescharacteristic
465C4—21no—
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—42no—
475C4—42no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4#57512yes—
481C4#57512yes—
482C4#58512yes—
483C2—41no—
484C2—41no—
485C2—41no—
486C2—82no—
487C2—82no—
488C2#344911yescharacteristic
489C2C2×#57511yescharacteristic
490C2C2×#57511yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutator, Frattini, cosocleC2×C4C25
socleC22C23×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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. #3444C2semidirect
3. #3252C2semidirect
4. #3251C2semidirect
5. #3240C2semidirect
6. #3237C2semidirect
7. #3235C2semidirect
8. #3219C2semidirect
9. #3216C2semidirect
10. #3214C2semidirect
11. #3197C2semidirect
12. #3172C2semidirect
13. #3171C2semidirect
14. #3107C2semidirect
15. #3106C2semidirect
16. #3102C2semidirect
17. #2920C2semidirect
55. #501C22semidirect
56. #501C22semidirect
57. #497C22semidirect
58. #496C22semidirect
59. #491C22semidirect
60. #490C22semidirect
61. #489C22semidirect
62. #489C22semidirect
67. #479C22semidirect
68. #478C22semidirect
69. #477C22semidirect
70. #476C22semidirect
71. #475C22semidirect
72. #474C22semidirect
77. #464C22semidirect
78. #463C22semidirect
79. #462C22semidirect
80. #461C22semidirect
81. #460C22semidirect
82. #459C22semidirect
87. C4×Q16C22semidirect
88. C4×SD16C22semidirect
89. C4×SD16C22semidirect
90. C4×D8C22semidirect

Maximal subgroups

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

subgroupindex
#34442
#32522
#32512
#32402
#32372
#32352
#32192
#32162
#32142
#31972
#31722
#31712
#31072
#31062
#31022
#29202