Finite Group Databasebeta

← #6434686

Subgroups — #6434686

Order 512.

2988 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3458C211yesmaximal, characteristic, complemented
2C2×#3094C211yesmaximal, characteristic
3C22×#482C211yesmaximal, characteristic
4#35262C211yesmaximal, characteristic
5#29647C214yesmaximal, complemented
6#29646C214yesmaximal, complemented
7#29644C218yesmaximal, complemented
8#23556C212yesmaximal, complemented
9#23555C212yesmaximal, complemented
10#23456C212yesmaximal, complemented
11#23456C212yesmaximal, complemented
12#20701C211yesmaximal, characteristic, complemented
13#20699C211yesmaximal, characteristic, complemented
14#20685C211yesmaximal, characteristic, complemented
15#3458—28no—
16#3458—28no—
17C2×#544—21no—
18C2×#544—21no—
19C2×#544C2212yes—
20C2×#544C2212yes—
21C2×#544C2212yescomplemented
22C2×#544C2212yescomplemented
23C2×#543—22no—
24C2×#533—22no—
25C2×#533—22no—
26C2×#533C2214yescomplemented
27C22×#121C2211yescharacteristic
28C22×#117C2211yescharacteristic
29C22×#117C2211yescharacteristic
30C22×#117C2211yescharacteristic
31#3094—28no—
32C2×#481C2212yes—
33C2×#482—28no—
34C2×#482C2214yes—
35C2×#482C2214yes—
36C2×#482C2214yes—
37C2×#480C2212yes—
38C2×#428C2212yes—
39C22×#107C2211yescharacteristic
40C22×#106C2211yescharacteristic
41C2×#418C2212yes—
42C22×#102C2211yescharacteristic
43C22×#102C2211yescharacteristic
44C2×#407C2211yescharacteristic
45C22×C22⋊C8C2211yescharacteristic
46#2854C2214yes—
47#2624C2214yes—
48#2504—21no—
49#2501—21no—
50#2500—22no—
51#2482—22no—
52#2405—22no—
53#2317C2212yes—
54#2309C2212yes—
55#2280C2211yescharacteristic
56#2059C2218yescomplemented
57#2059C2218yescomplemented
58#2058C2218yescomplemented
59#2057C2218yescomplemented
60#2025C2218yescomplemented
61#2025C2218yescomplemented
62#2023C2218yescomplemented
63#2023C2218yescomplemented
64#1825C2214yescomplemented
65#1825C2214yescomplemented
66#1825C2214yescomplemented
67#1824C2212yescomplemented
68#1823C2212yescomplemented
69#1818C2214yescomplemented
70#1817C2214yescomplemented
71#1736C2212yes—
72#1736C2212yes—
73#1735C2212yes—
74#1735C2212yes—
75#1729C2212yes—
76#1729C2212yes—
77C23×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C22×M16C2312yes—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83#544—216no—
84#544—216no—
85#544—28no—
86#544—28no—
87#544—28no—
88#544—28no—
89#544—28no—
90#544—28no—
91#544—28no—
92#544—28no—
93#543—232no—
94#533—232no—
95#533—232no—
96#533—216no—
97#533—216no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—22no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8C2314yes—
103C2×C4⋊Q8C2314yes—
104C2×#125—22no—
105C2×#125—22no—
106C2×#125—22no—
107C2×#125—22no—
108C2×#125C2314yes—
109C2×#125C2314yes—
110C2×#123—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C2×#121—21no—
118C2×#121—21no—
119C2×#121—24no—
120C2×#121—24no—
121C2×#121C2312yes—
122C2×#121C2312yes—
123C2×C4×Q8—24no—
124C2×C4×Q8—24no—
125C2×#117—21no—
126C2×#117—21no—
127C2×#117—21no—
128C2×#117—21no—
129C2×#117—24no—
130C2×#117—24no—
131C2×#117—24no—
132C2×#117—24no—
133C2×#117D418yes—
134C2×#117D418yes—
135C2×#117D412yes—
136C2×#117D412yes—
137C2×#117C2312yes—
138C2×#117C2312yes—
139C2×#117C2312yes—
140C2×#117C2312yes—
141C2×#117C2312yes—
142C2×#117C2312yes—
143C2×#117C2312yes—
144C22×C4⋊C4D411yescharacteristic
145C22×C4⋊C4D411yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2311yescharacteristic
150C22×C4⋊C4C2311yescharacteristic
151C22×C22⋊C4—21no—
152C22×C22⋊C4—21no—
153C22×C22⋊C4—21no—
154C22×C22⋊C4D412yes—
155C22×C42—21no—
156C22×C42—21no—
157C22×C42C2312yes—
158#482—216no—
159#482—216no—
160#482—216no—
161#481—28no—
162#480—28no—
163#428—28no—
164C2×#107—24no—
165C2×#107C2314yes—
166C2×#107C2314yes—
167C2×#106—24no—
168C2×#106C2314yes—
169C2×#106C2314yes—
170#418—28no—
171C2×#102—24no—
172C2×#102—24no—
173C2×#102C2314yes—
174C2×#102C2314yes—
175C2×#102C2314yes—
176C2×#102C2314yes—
177#407—24no—
178C2×C22⋊C8—24no—
179C2×C22⋊C8C2312yes—
180C2×C22⋊C8C2312yes—
181C2×C22⋊C8C2312yes—
182C2×C22⋊C8C2312yes—
183#399—24no—
184#397—24no—
185#394—28no—
186#390—22no—
187#390—22no—
188#390C2318yes—
189#389C2318yes—
190#388—28no—
191#386—24no—
192#385—24no—
193#384—22no—
194#382—22no—
195#381—21no—
196#380C2312yes—
197#380C2312yes—
198#379—21no—
199#379—24no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4—21no—
202#340C2314yes—
203#340C2318yes—
204#340C2314yes—
205#340C2318yes—
206#340C2314yes—
207#340C2314yes—
208#336C2318yes—
209#336C2318yes—
210C25—21no—
211C25C2×D411yescharacteristic
212C22×Q8—24no—
213C22×Q8—24no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—21no—
219C22×D4—24no—
220C22×D4—24no—
221C22×D4—44no—
222C22×D4—84no—
223C22×D4C2412yes—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D412yes—
237C23×C4C2×D412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C2×M16—24no—
242C2×M16—24no—
243C2×M16—24no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—24no—
247C22×C8—24no—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C4⋊Q8—216no—
252C4⋊Q8—216no—
253C4⋊Q8—216no—
254C4⋊Q8—216no—
255C4⋊Q8—28no—
256C4⋊Q8—28no—
257C4⋊Q8—28no—
258C4⋊Q8—28no—
259#125—216no—
260#125—216no—
261#125—216no—
262#125—216no—
263#125—28no—
264#125—28no—
265#125—28no—
266#125—28no—
267#123—88no—
268C22⋊Q8—232no—
269C22⋊Q8—232no—
270C22⋊Q8—232no—
271C22⋊Q8—232no—
272C22⋊Q8—232no—
273C22⋊Q8—232no—
274#121—24no—
275#121—24no—
276#121—24no—
277#121—24no—
278#121—44no—
279#121—44no—
280C4×Q8—232no—
281C4×Q8—232no—
282#117—28no—
283#117—28no—
284#117—28no—
285#117—28no—
286#117—28no—
287#117—28no—
288#117—24no—
289#117—24no—
290#117—24no—
291#117—24no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—24no—
297#117—24no—
298#117C2×D418yes—
299#117C2×D418yes—
300#117C2×D418yes—
301#117C2×D418yes—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2×D414yes—
327C2×C4⋊C4C2×D414yes—
328C2×C4⋊C4C2×D414yes—
329C2×C4⋊C4C2×D414yes—
330C2×C4⋊C4C2412yes—
331C2×C4⋊C4C2414yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C22⋊C4—28no—
337C2×C22⋊C4—28no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—28no—
350C2×C22⋊C4—44no—
351C2×C22⋊C4C2×D414yes—
352C2×C22⋊C4C2×D414yes—
353C2×C42—28no—
354C2×C42—28no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42C2×D414yes—
363C2×C42C2414yes—
364#107—28no—
365#107—28no—
366#106—28no—
367#106—28no—
368#102—28no—
369#102—28no—
370#102—28no—
371#102—28no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376#95—22no—
377#95—22no—
378#95—28no—
379C24—24no—
380C24—22no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—44no—
385C24—82no—
386C24—82no—
387C24#12014yes—
388C24C22×D411yescharacteristic
389C24C22×D411yescharacteristic
390C24C22×D411yescharacteristic
391C2×Q8—216no—
392C2×Q8—216no—
393C2×Q8—88no—
394C2×Q8—88no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—48no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—42no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—48no—
406C2×D4—44no—
407C2×D4—84no—
408C2×D4—816no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—216no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—82no—
435C22×C4—82no—
436C22×C4—82no—
437C22×C4—82no—
438C22×C4—82no—
439C22×C4#12014yes—
440C22×C4#12014yes—
441C22×C4#12014yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C42−1+414yes—
454C22×C42−1+414yes—
455C22×C4C2511yescharacteristic
456M16—48no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—44no—
465C2×C8—44no—
466C4⋊C4—216no—
467C4⋊C4—216no—
468C4⋊C4—216no—
469C4⋊C4—216no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—42no—
489C4⋊C4—88no—
490C4⋊C4C22×D414yes—
491C4⋊C4C22×D414yes—
492C4⋊C4C22×D414yes—
493C4⋊C4C22×D414yes—
494C22⋊C4—28no—
495C22⋊C4—28no—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—82no—
507C42—28no—
508C42—28no—
509C42—28no—
510C42—28no—
511C42—28no—
512C42—24no—
513C42—24no—
514C42—44no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—28no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—44no—
530C23—88no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—168no—
535C23C2×#12012yes—
536C23C2×#12012yes—
537C23C2×#12012yes—
538C23#53614yes—
539C23C23×D411yescharacteristic
540Q8—816no—
541Q8—816no—
542D4—42no—
543D4—42no—
544D4—44no—
545D4—88no—
546D4—88no—
547D4—84no—
548C2×C4—28no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—28no—
552C2×C4—28no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4C2×#12012yes—
584C2×C4C2×#12012yes—
585C2×C4C2×#12012yes—
586C2×C4#53614yes—
587C2×C4#53614yes—
588C2×C4#53614yes—
589C2×C4#53618yes—
590C2×C4C23×D411yescharacteristic
591C2×C4C23×D411yescharacteristic
592C2×C4C2×2−1+412yes—
593C2×C4C2×2−1+412yes—
594C8—42no—
595C8—42no—
596C8—44no—
597C22—28no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—44no—
607C22—44no—
608C22—88no—
609C22—82no—
610C22—88no—
611C22—82no—
612C22—82no—
613C22—168no—
614C22#261414yes—
615C22C22×#12011yescharacteristic
616C22C2×#53612yes—
617C22#358412yes—
618C22#358512yes—
619C4—22no—
620C4—22no—
621C4—44no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—44no—
627C4—84no—
628C4—82no—
629C4#261414yes—
630C4C2×#53611yescharacteristic
631C4C2×#53611yescharacteristic
632C4C2×#53612yes—
633C2—24no—
634C2—42no—
635C2—42no—
636C2—162no—
637C2C2×#261411yescharacteristic
638C2#6036611yescharacteristic
639C2#6036911yescharacteristic
640C2#6087514yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3458C2semidirect
5. #29647C2semidirect
6. #29646C2semidirect
7. #29644C2semidirect
8. #23556C2semidirect
9. #23555C2semidirect
10. #23456C2semidirect
11. #23456C2semidirect
12. #20701C2semidirect
13. #20699C2semidirect
14. #20685C2semidirect
21. C2×#544C22semidirect
22. C2×#544C22semidirect
26. C2×#533C22semidirect
56. #2059C22semidirect
57. #2059C22semidirect
58. #2058C22semidirect
59. #2057C22semidirect
60. #2025C22semidirect
61. #2025C22semidirect
62. #2023C22semidirect
63. #2023C22semidirect
64. #1825C22semidirect
65. #1825C22semidirect
66. #1825C22semidirect
67. #1824C22semidirect
68. #1823C22semidirect
69. #1818C22semidirect
70. #1817C22semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
C2×#30942
C22×#4822
#352622
#296472
#296462
#296442
#235562
#235552
#234562
#207012
#206992
#206852