Finite Group Databasebeta

← #227628

Subgroups — #227628

Order 512.

631 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#18485C211yesmaximal, characteristic
2#18024C211yesmaximal, characteristic, complemented
3#18021C211yesmaximal, characteristic, complemented
4#17522C211yesmaximal, characteristic
5#17517C211yesmaximal, characteristic
6#9858C211yesmaximal, characteristic, complemented
7#9588C211yesmaximal, characteristic, complemented
8#9588C211yesmaximal, characteristic, complemented
9#9214C211yesmaximal, characteristic, complemented
10#9212C211yesmaximal, characteristic, complemented
11#9070C211yesmaximal, characteristic, complemented
12#8899C211yesmaximal, characteristic, complemented
13#8898C211yesmaximal, characteristic, complemented
14#8188C211yesmaximal, characteristic, complemented
15#8186C211yesmaximal, characteristic, complemented
16#3481C2211yescharacteristic
17#3131C2211yescharacteristic
18#3107C2211yescharacteristic
19#3107C2211yescharacteristic
20#2991C2211yescharacteristic
21#2990C2211yescharacteristic
22#2986C2211yescharacteristic, complemented
23#2922C2211yescharacteristic
24C4×#131C2211yescharacteristic
25#1848C2211yescharacteristic, complemented
26#1644C411yescharacteristic
27#1644C411yescharacteristic
28#1643C411yescharacteristic
29#1643C411yescharacteristic
30#1640C411yescharacteristic
31#1640C411yescharacteristic
32#1639C411yescharacteristic
33#1639C411yescharacteristic
34#1593—21no—
35#1592—21no—
36#1584—21no—
37#1584—21no—
38#1583—21no—
39#1582—21no—
40#1579—21no—
41SD16⋊C8—21no—
42#1574C2211yescharacteristic, complemented
43#1574C2211yescharacteristic, complemented
44#1543C2211yescharacteristic
45#1526C2211yescharacteristic, complemented
46#1495—21no—
47#1494—21no—
48#1493C2211yescharacteristic, complemented
49#1493C2211yescharacteristic, complemented
50#1492C2211yescharacteristic
51#1492C2211yescharacteristic
52#1490—21no—
53#1489C2211yescharacteristic
54#1489C2211yescharacteristic
55#1486—21no—
56#1484C2211yescharacteristic
57#1484C2211yescharacteristic
58#1483—21no—
59#1482—21no—
60#1478—21no—
61#1476C2211yescharacteristic
62#1476C2211yescharacteristic
63#1476C2211yescharacteristic
64#1476C2211yescharacteristic
65C2×D4⋊C8—21no—
66#1452C2211yescharacteristic, complemented
67#1452C2211yescharacteristic, complemented
68C2×#322C2211yescharacteristic, complemented
69#1299C2211yescharacteristic, complemented
70#1299C2211yescharacteristic, complemented
71#1278C2211yescharacteristic, complemented
72#1278C2211yescharacteristic, complemented
73#1278C2211yescharacteristic, complemented
74#1278C2211yescharacteristic, complemented
75C2×#131—21no—
76#559C2×C411yescharacteristic
77#554C2×C411yescharacteristic
78#554C2×C411yescharacteristic
79#553—21no—
80#553—21no—
81#551—41no—
82#546—21no—
83#546—21no—
84D42C2×C411yescharacteristic
85#533C2311yescharacteristic
86#532C2311yescharacteristic
87#529—21no—
88#529—21no—
89#525—41no—
90C4×#41C2311yescharacteristic
91C4×#41C2311yescharacteristic
92C2×C4×D4—21no—
93C2×C4×D4—21no—
94#488—41no—
95#487C2×C411yescharacteristic
96#487C2×C411yescharacteristic
97#486C2×C411yescharacteristic
98#486C2×C411yescharacteristic
99#485—21no—
100#485—21no—
101#475C2×C411yescharacteristic
102#475C2×C411yescharacteristic
103#474C2×C411yescharacteristic
104#474C2×C411yescharacteristic
105#462C2×C411yescharacteristic
106#462C2×C411yescharacteristic
107#461C2×C411yescharacteristic
108#461C2×C411yescharacteristic
109#452—21no—
110#452—21no—
111#446—21no—
112#446—21no—
113#446—21no—
114C8×Q8—21no—
115C8×Q8—21no—
116C8×Q8—21no—
117C4×Q16—21no—
118C4×Q16—21no—
119C4×SD16—21no—
120C4×SD16—21no—
121#436—21no—
122#436—41no—
123#435—41no—
124C8×D4—21no—
125#433—41no—
126#433—41no—
127#432—21no—
128#431—21no—
129#431—21no—
130#431—21no—
131#431C2311yescharacteristic
132#431C2311yescharacteristic
133#431C2311yescharacteristic
134#431C2311yescharacteristic
135C2×#108—41no—
136#424—21no—
137#423—21no—
138#423C2311yescharacteristic
139#423C2311yescharacteristic
140C2×C4⋊C8—21no—
141#416—41no—
142#408—21no—
143#405—21no—
144#405—41no—
145C4×M16—21no—
146C4×M16C2311yescharacteristic
147C4×M16C2311yescharacteristic
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152#335—21no—
153#335—21no—
154#334—21no—
155#334—21no—
156#333C2×C411yescharacteristic
157#333C2×C411yescharacteristic
158#331C2×C411yescharacteristic
159#331C2×C411yescharacteristic
160#330—41no—
161Q8⋊C8—21no—
162Q8⋊C8—21no—
163Q8⋊C8—21no—
164Q8⋊C8—21no—
165Q8⋊C8—21no—
166Q8⋊C8—21no—
167Q8⋊C8C2×C411yescharacteristic
168Q8⋊C8C2×C411yescharacteristic
169Q8⋊C8C2×C411yescharacteristic
170Q8⋊C8C2×C411yescharacteristic
171Q8⋊C8D411yescharacteristic
172Q8⋊C8D411yescharacteristic
173Q8⋊C8D411yescharacteristic
174Q8⋊C8D411yescharacteristic
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185D4⋊C8C2×C411yescharacteristic
186D4⋊C8C2×C411yescharacteristic
187D4⋊C8C2×C411yescharacteristic
188D4⋊C8C2×C411yescharacteristic
189D4⋊C8D411yescharacteristic
190D4⋊C8D411yescharacteristic
191D4⋊C8D411yescharacteristic
192D4⋊C8D411yescharacteristic
193#322—21no—
194#322—21no—
195#322—21no—
196#322—21no—
197#322—41no—
198C2×#41—21no—
199C2×#41#4111yescharacteristic
200C2×#41#4111yescharacteristic
201C22×D4—21no—
202C22×D4—21no—
203C23×C4—41no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×SD16—21no—
207C2×SD16—21no—
208#131—42no—
209#131—42no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—41no—
215C2×M16—41no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—41no—
220C22×C8—41no—
221C4⋊Q8C22×C411yescharacteristic
222C4⋊Q8C22×C411yescharacteristic
223C4⋊Q8C22×C411yescharacteristic
224#127C22×C411yescharacteristic
225#126—41no—
226#126—41no—
227#125—21no—
228#125—21no—
229#125—21no—
230#124—21no—
231#124—21no—
232#124—21no—
233#123—41no—
234#123—41no—
235#123—41no—
236#123—41no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—41no—
242C22⋊Q8—41no—
243#121—21no—
244#121—21no—
245#121—21no—
246#121—21no—
247#121—41no—
248#121—41no—
249#120—41no—
250#120—41no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C22×C411yescharacteristic
254C4×Q8C22×C411yescharacteristic
255C4×Q8C2×D411yescharacteristic
256C4×Q8C2×D411yescharacteristic
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—41no—
270C4×D4—41no—
271C4×D4C22×C411yescharacteristic
272C4×D4C22×C411yescharacteristic
273C4×D4C2×D411yescharacteristic
274C4×D4C2×D411yescharacteristic
275#117—21no—
276#117—21no—
277#117—21no—
278#117—41no—
279#117#4111yescharacteristic
280#117#4111yescharacteristic
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—41no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42C2411yescharacteristic
288#108—82no—
289#107—21no—
290#107—21no—
291#106—21no—
292#106—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—41no—
298C4⋊C8—41no—
299C4⋊C8—41no—
300C4⋊C8—41no—
301C4⋊C8—41no—
302C4⋊C8—41no—
303C4⋊C8C22×C411yescharacteristic
304C4⋊C8C22×C411yescharacteristic
305C4⋊C8C22×C411yescharacteristic
306C4⋊C8C22×C411yescharacteristic
307C4⋊C8C2×D411yescharacteristic
308C4⋊C8C2×D411yescharacteristic
309C4⋊C8C2×D411yescharacteristic
310C4⋊C8C2×D411yescharacteristic
311#103—21no—
312#103—21no—
313#103—21no—
314#103—21no—
315#103—21no—
316#103—21no—
317#103—41no—
318#103—41no—
319#102—21no—
320#102—21no—
321#102—41no—
322#102—41no—
323C22⋊C8—21no—
324C22⋊C8—21no—
325C22⋊C8—21no—
326C22⋊C8—21no—
327C22⋊C8—41no—
328C22⋊C8—41no—
329C22⋊C8—41no—
330C22⋊C8—41no—
331#97—21no—
332#97—21no—
333#97—41no—
334#97—41no—
335#97—41no—
336C4×C8—21no—
337C4×C8—21no—
338C4×C8—21no—
339C4×C8—21no—
340C4×C8—21no—
341C4×C8—21no—
342C4×C8—21no—
343C4×C8—21no—
344C4×C8—21no—
345C4×C8—41no—
346C4×C8—41no—
347C4×C8—41no—
348C4×C8C22×C411yescharacteristic
349C4×C8C22×C411yescharacteristic
350C4×C8C2×D411yescharacteristic
351C4×C8C2×D411yescharacteristic
352C4×C8C2×D411yescharacteristic
353C4×C8C2×D411yescharacteristic
354C24—41no—
355#41—22no—
356#41—41no—
357#41—41no—
358#41—41no—
359#41—81no—
360#41—81no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8C4×D411yescharacteristic
365C2×Q8C4×D411yescharacteristic
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4C4×D411yescharacteristic
379C2×D4C4×D411yescharacteristic
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—161no—
391C22×C4—161no—
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4H(5,𝔽2)11yescharacteristic
395Q16—42no—
396Q16—81no—
397SD16—42no—
398SD16—81no—
399M16—22no—
400M16—41no—
401M16—44no—
402M16—41no—
403M16—41no—
404M16—81no—
405M16—81no—
406M16—161no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—44no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—161no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4C4×D411yescharacteristic
450C4⋊C4C4×D411yescharacteristic
451C4⋊C4C4×D411yescharacteristic
452C4⋊C4C4×D411yescharacteristic
453C22⋊C4—21no—
454C22⋊C4—21no—
455C22⋊C4—21no—
456C22⋊C4—21no—
457C22⋊C4—21no—
458C22⋊C4—21no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—41no—
475C42C23×C411yescharacteristic
476C42C22×D411yescharacteristic
477C42C22×D411yescharacteristic
478C42H(5,𝔽2)11yescharacteristic
479C23—21no—
480C23—21no—
481C23—41no—
482C23—82no—
483C23—82no—
484C23—161no—
485C23#55111yescharacteristic
486Q8—21no—
487Q8—21no—
488Q8—21no—
489Q8—21no—
490Q8—41no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494Q8—81no—
495D4—21no—
496D4—21no—
497D4—21no—
498D4—42no—
499D4—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—81no—
504D4—81no—
505D4—82no—
506D4—82no—
507D4—161no—
508D4#44412yes—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—161no—
537C2×C4—161no—
538C2×C4—161no—
539C2×C4C2×C4×D411yescharacteristic
540C2×C4C2×C4×D411yescharacteristic
541C2×C4#51811yescharacteristic
542C2×C4D4211yescharacteristic
543C2×C4#54611yescharacteristic
544C2×C4#54611yescharacteristic
545C8—41no—
546C8—41no—
547C8—41no—
548C8—41no—
549C8—41no—
550C8—41no—
551C8—42no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8—41no—
558C8—41no—
559C8—41no—
560C8—81no—
561C8—81no—
562C8—81no—
563C22—21no—
564C22—22no—
565C22—21no—
566C22—21no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—164no—
576C22#235811yescharacteristic
577C4—21no—
578C4—21no—
579C4—21no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4C2×#44411yescharacteristic
590C4#295511yescharacteristic
591C4#295611yescharacteristic
592C4#295711yescharacteristic
593C4#328811yescharacteristic
594C4#328911yescharacteristic
595C2—41no—
596C2—42no—
597C2—81no—
598C2—161no—
599C2#2288611yescharacteristic
600C2#2288711yescharacteristic
601C2#2289011yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2358
prime radical#3481C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #18024C2semidirect
3. #18021C2semidirect
6. #9858C2semidirect
7. #9588C2semidirect
8. #9588C2semidirect
9. #9214C2semidirect
10. #9212C2semidirect
11. #9070C2semidirect
12. #8899C2semidirect
13. #8898C2semidirect
14. #8188C2semidirect
15. #8186C2semidirect
22. #2986C22semidirect
25. #1848C22semidirect
42. #1574C22semidirect
43. #1574C22semidirect
45. #1526C22semidirect
48. #1493C22semidirect
49. #1493C22semidirect
66. #1452C22semidirect
67. #1452C22semidirect
68. C2×#322C22semidirect
69. #1299C22semidirect
70. #1299C22semidirect
71. #1278C22semidirect
72. #1278C22semidirect
73. #1278C22semidirect
74. #1278C22semidirect

Maximal subgroups

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

subgroupindex
#184852
#180242
#180212
#175222
#175172
#98582
#95882
#92142
#92122
#90702
#88992
#88982
#81882
#81862