Finite Group Databasebeta

← #60917

Subgroups — #60917

Order 256.

960 conjugacy classes of subgroups.

The subgroup lattice has 511 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#3256C214yesmaximal, complemented
4#3252C212yesmaximal, complemented
5#3251C212yesmaximal, complemented
6#3250C214yesmaximal, complemented
7#3248C212yesmaximal, complemented
8#3247C212yesmaximal, complemented
9#3080C212yesmaximal, complemented
10#3078C211yesmaximal, characteristic, complemented
11#3077C211yesmaximal, characteristic, complemented
12#3056C212yesmaximal, complemented
13#3055C212yesmaximal, complemented
14#3040C212yesmaximal, complemented
15#3037C211yesmaximal, characteristic, complemented
16#3036C211yesmaximal, characteristic, complemented
17C2×#423C211yesmaximal, characteristic, complemented
18C2×#137C2212yes—
19C2×AGL(1,ℤ/8)C2212yes—
20C22×M16C2212yes—
21#533C2211yescharacteristic
22#533C2211yescharacteristic, complemented
23#533C2211yescharacteristic, complemented
24#532C2211yescharacteristic
25#532C2211yescharacteristic, complemented
26#532C2211yescharacteristic, complemented
27#529—21no—
28#529—21no—
29#529—21no—
30#529—21no—
31#529—21no—
32#529—21no—
33#529—21no—
34#529—21no—
35#529C2212yes—
36#529C2212yes—
37#529C2212yes—
38#529C2212yes—
39#529C2212yescomplemented
40#529C2212yescomplemented
41C2×#126—21no—
42C2×#125C2211yescharacteristic
43C2×#124C2211yescharacteristic
44#525—21no—
45#525—21no—
46#525C2212yes—
47C2×#123—21no—
48C2×C22⋊Q8C2211yescharacteristic
49C2×#121C2211yescharacteristic
50C4×#41—21no—
51C4×#41—21no—
52C4×#41C2212yes—
53C2×C4×Q8C2211yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4C2211yescharacteristic
56C2×#117—21no—
57C22×C42C2211yescharacteristic
58#479C2214yescomplemented
59#479C2214yescomplemented
60#479C2214yescomplemented
61#478C2214yescomplemented
62#478C2214yescomplemented
63#478C2214yescomplemented
64#477C2212yescomplemented
65#476C2212yescomplemented
66#475C2212yescomplemented
67#474C2212yescomplemented
68#470C2214yescomplemented
69#469C2214yes—
70#468C2214yes—
71#468C2214yescomplemented
72#467C2214yescomplemented
73#466C2214yes—
74#465C2214yes—
75#465C2214yescomplemented
76#464C2214yescomplemented
77#464C2214yescomplemented
78#464C2214yescomplemented
79#463C2214yescomplemented
80#463C2214yescomplemented
81#463C2214yescomplemented
82#462C2212yescomplemented
83#461C2212yescomplemented
84#460C2212yescomplemented
85#459C2212yescomplemented
86#428C2212yes—
87#428C2212yes—
88#423C2211yescharacteristic
89#423C2211yescharacteristic
90#423C2212yes—
91#423C2212yescomplemented
92#423C2212yescomplemented
93C2×C4⋊C8C2212yescomplemented
94C2×C4⋊C8C2212yescomplemented
95#419C2212yes—
96#418C2212yes—
97#417C2212yes—
98#417C2212yes—
99C2×#41—21no—
100C2×#41—21no—
101C2×#41C2312yes—
102C22×Q8C2311yescharacteristic
103C22×D4—21no—
104C22×D4C2311yescharacteristic
105C23×C4—21no—
106C23×C4D411yescharacteristic
107C23×C4D411yescharacteristic
108C23×C4C2311yescharacteristic
109#137—44no—
110AGL(1,ℤ/8)—44no—
111C2×Q16C2314yes—
112C2×SD16C2314yes—
113C2×SD16C2314yes—
114C2×D8C2314yes—
115C2×M16—44no—
116C2×M16C2312yes—
117C2×M16C2312yes—
118C2×M16C2314yes—
119C22×C8C2314yes—
120C4⋊Q8C2311yescharacteristic
121C4⋊Q8C2311yescharacteristic
122C4⋊Q8C2311yescharacteristic
123#127C2311yescharacteristic
124#126—22no—
125#126—22no—
126#126—22no—
127#126—22no—
128#126—22no—
129#126—22no—
130#125—21no—
131#125—21no—
132#125—21no—
133#125—21no—
134#125C2312yes—
135#125C2312yes—
136#125C2312yes—
137#124—21no—
138#124—21no—
139#124—21no—
140#124—21no—
141#124C2312yes—
142#124C2312yes—
143#124C2312yes—
144#123—22no—
145#123—22no—
146#123—22no—
147#123—22no—
148#123—22no—
149#123—22no—
150#123—22no—
151C22⋊Q8—21no—
152C22⋊Q8—21no—
153C22⋊Q8—21no—
154C22⋊Q8—21no—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157C22⋊Q8C2312yes—
158C22⋊Q8C2312yes—
159C22⋊Q8C2312yes—
160C22⋊Q8C2312yes—
161#121—21no—
162#121—21no—
163#121—21no—
164#121—21no—
165#121—21no—
166#121—21no—
167#121C2312yes—
168#121C2312yes—
169#121C2312yes—
170#121C2312yes—
171#120—22no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8C2312yes—
179C4×Q8C2312yes—
180C4×Q8C2312yes—
181C4×Q8C2312yes—
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—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4C2312yes—
199C4×D4C2312yes—
200C4×D4C2312yes—
201C4×D4C2312yes—
202#117—21no—
203#117—21no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117—22no—
211#117C2312yes—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C2311yescharacteristic
217C2×C4⋊C4C2311yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42D411yescharacteristic
226C2×C42D411yescharacteristic
227C2×C42D411yescharacteristic
228C2×C42D411yescharacteristic
229C2×C42D411yescharacteristic
230C2×C42D411yescharacteristic
231C2×C42C2311yescharacteristic
232C2×C42C2311yescharacteristic
233C2×C42C2311yescharacteristic
234C2×C42C2311yescharacteristic
235C2×C42C2311yescharacteristic
236C2×C42C2311yescharacteristic
237#107C2314yes—
238#107C2314yes—
239#106C2314yes—
240#106C2314yes—
241C4⋊C8C2312yes—
242C4⋊C8C2312yes—
243C4⋊C8C2312yes—
244C4⋊C8C2314yes—
245C4⋊C8C2314yes—
246C4⋊C8C2312yes—
247#103C2314yes—
248#103C2314yes—
249#103C2314yes—
250#103C2314yes—
251#102C2314yes—
252#102C2314yes—
253#102C2314yes—
254#102C2314yes—
255C24—21no—
256C24C2×D411yescharacteristic
257#41—24no—
258#41—24no—
259#41—44no—
260#41—44no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8C2412yes—
267C2×Q8C2412yes—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—44no—
279C2×D4C2412yes—
280C2×D4C2412yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—44no—
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×C4C2×D411yescharacteristic
303C22×C4C2×D411yescharacteristic
304C22×C4C2×D411yescharacteristic
305C22×C4C2×D411yescharacteristic
306C22×C4C2×D411yescharacteristic
307C22×C4C2×D411yescharacteristic
308C22×C4C2×D411yescharacteristic
309C22×C4C2411yescharacteristic
310C22×C4C2411yescharacteristic
311C22×C4C2411yescharacteristic
312Q16—44no—
313SD16—44no—
314SD16—44no—
315D8—44no—
316M16—24no—
317M16—24no—
318M16—44no—
319C2×C8—44no—
320C2×C8C2414yes—
321C2×C8C2414yes—
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—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4C2412yes—
343C4⋊C4C2412yes—
344C4⋊C4C2412yes—
345C4⋊C4C2412yes—
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—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42C2×D411yescharacteristic
371C42C2×D411yescharacteristic
372C42C2×D411yescharacteristic
373C42C2×D411yescharacteristic
374C42C2×D411yescharacteristic
375C42C2×D411yescharacteristic
376C42C2×D411yescharacteristic
377C42C2×D411yescharacteristic
378C42C2×D411yescharacteristic
379C42C2×D411yescharacteristic
380C42C2×D411yescharacteristic
381C42C2×D411yescharacteristic
382C42C2411yescharacteristic
383C42C2411yescharacteristic
384C42C2411yescharacteristic
385C42C2411yescharacteristic
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—41no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23C22×D411yescharacteristic
395C23C22×D411yescharacteristic
396C23C22×D411yescharacteristic
397Q8—22no—
398Q8—22no—
399Q8—22no—
400Q8—22no—
401Q8—42no—
402Q8—42no—
403Q8—42no—
404D4—22no—
405D4—22no—
406D4—22no—
407D4—22no—
408D4—42no—
409D4—44no—
410D4—44no—
411D4—44no—
412D4—44no—
413D4—44no—
414D4—42no—
415D4—42no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4C22×D411yescharacteristic
450C2×C4C22×D411yescharacteristic
451C2×C4C22×D411yescharacteristic
452C2×C4C22×D411yescharacteristic
453C2×C4C22×D411yescharacteristic
454C2×C4C22×D411yescharacteristic
455C2×C4C22×D411yescharacteristic
456C2×C4C22×D411yescharacteristic
457C2×C4C22×D411yescharacteristic
458C2×C4C22×D411yescharacteristic
459C2×C4C22×D411yescharacteristic
460C2×C4H(5,𝔽2)12yes—
461C2×C42−1+412yes—
462C2×C4C2511yescharacteristic
463C8—24no—
464C8—24no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—44no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—84no—
480C22#53712yes—
481C22#57512yes—
482C22C23×D411yescharacteristic
483C4—21no—
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4#53712yes—
499C4#57512yes—
500C4C2×H(5,𝔽2)11yescharacteristic
501C4C2×2−1+411yescharacteristic
502C2—22no—
503C2—22no—
504C2—41no—
505C2—82no—
506C2—82no—
507C2C2×#53711yescharacteristic
508C2C2×#57511yescharacteristic
509C2C2×#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. #3256C2semidirect
4. #3252C2semidirect
5. #3251C2semidirect
6. #3250C2semidirect
7. #3248C2semidirect
8. #3247C2semidirect
9. #3080C2semidirect
10. #3078C2semidirect
11. #3077C2semidirect
12. #3056C2semidirect
13. #3055C2semidirect
14. #3040C2semidirect
15. #3037C2semidirect
16. #3036C2semidirect
17. C2×#423C2semidirect
22. #533C22semidirect
23. #533C22semidirect
25. #532C22semidirect
26. #532C22semidirect
39. #529C22semidirect
40. #529C22semidirect
58. #479C22semidirect
59. #479C22semidirect
60. #479C22semidirect
61. #478C22semidirect
62. #478C22semidirect
63. #478C22semidirect
64. #477C22semidirect
65. #476C22semidirect
66. #475C22semidirect
67. #474C22semidirect
68. #470C22semidirect
71. #468C22semidirect
72. #467C22semidirect
75. #465C22semidirect
76. #464C22semidirect
77. #464C22semidirect
78. #464C22semidirect
79. #463C22semidirect
80. #463C22semidirect
81. #463C22semidirect
82. #462C22semidirect
83. #461C22semidirect
84. #460C22semidirect
85. #459C22semidirect
91. #423C22semidirect
92. #423C22semidirect
93. C2×C4⋊C8C22semidirect
94. C2×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34442
C2×#5292
#32562
#32522
#32512
#32502
#32482
#32472
#30802
#30782
#30772
#30562
#30552
#30402
#30372
#30362
C2×#4232