Finite Group Databasebeta

← #6434688

Subgroups — #6434688

Order 512.

3059 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic, complemented
2C2×#3092C211yesmaximal, characteristic
3C2×#3091C211yesmaximal, characteristic
4#35263C211yesmaximal, characteristic
5#29649C214yesmaximal, complemented
6#29647C214yesmaximal, complemented
7#29643C212yesmaximal, complemented
8#29641C212yesmaximal, complemented
9#29640C212yesmaximal, complemented
10#29638C212yesmaximal, complemented
11#23557C214yesmaximal, complemented
12#23458C214yesmaximal, complemented
13#20702C211yesmaximal, characteristic, complemented
14C2×#1823C211yesmaximal, characteristic, complemented
15#20685C211yesmaximal, characteristic, complemented
16C2×#544—22no—
17C2×#544—22no—
18C2×#544—28no—
19C2×#544—28no—
20C2×#544C2214yes—
21C2×#544C2214yes—
22C22×C4⋊Q8—21no—
23C22×C4⋊Q8C2212yes—
24C22×#125—21no—
25C22×#125C2212yes—
26C2×#525C2211yescharacteristic
27C22×C22⋊Q8—22no—
28C22×#117C2211yescharacteristic
29C22×#117C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31#3092—28no—
32#3091—28no—
33C2×#484C2211yescharacteristic
34C2×#484C2212yes—
35C2×#481C2211yescharacteristic
36C2×#481C2212yes—
37C2×#485C2212yes—
38C2×#482C2212yes—
39C2×#483C2211yescharacteristic
40C2×#483C2212yes—
41C2×#480C2211yescharacteristic
42C2×#480C2212yes—
43C2×#428C2212yes—
44C2×#427C2212yes—
45C2×#417C2212yes—
46C2×#416C2212yes—
47C2×#407C2211yescharacteristic
48C22×C22⋊C8C2211yescharacteristic
49#2855C2212yes—
50#2854C2212yes—
51#2625C2212yes—
52#2624C2212yes—
53#2498—22no—
54#2487—21no—
55#2486—21no—
56#2485—22no—
57#2412—22no—
58#2316C2212yes—
59#2311C2212yes—
60C2×#380C2211yescharacteristic
61#2059C2218yescomplemented
62#2059C2218yescomplemented
63#2058C2218yescomplemented
64#2057C2218yescomplemented
65#2026C2218yescomplemented
66#2025C2218yescomplemented
67#2024C2218yescomplemented
68#2023C2218yescomplemented
69#1825C2214yes—
70#1824C2212yes—
71#1823C2212yes—
72#1823C2212yes—
73#1823C2212yes—
74#1823C2214yes—
75#1818C2214yes—
76#1817C2212yes—
77#1817C2212yes—
78#1736C2214yes—
79C2×#340C2214yes—
80#1729C2214yes—
81C22×#41C2311yescharacteristic
82C23×Q8—22no—
83C24×C4—21no—
84C24×C4C2311yescharacteristic
85C22×M16C2312yes—
86C23×C8C2312yes—
87#544—232no—
88#544—232no—
89#544—216no—
90#544—216no—
91#544—216no—
92#544—216no—
93C2×C4⋊Q8—216no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8C2318yes—
100C2×#125—216no—
101C2×#125—24no—
102C2×#125—24no—
103C2×#125—28no—
104C2×#125—28no—
105C2×#125—24no—
106C2×#125C2318yes—
107#525—24no—
108#525—24no—
109C2×#123—21no—
110C2×#123—22no—
111C2×C22⋊Q8—232no—
112C2×C22⋊Q8—28no—
113C2×C22⋊Q8—28no—
114C2×C22⋊Q8C2311yescharacteristic
115C2×C22⋊Q8C2311yescharacteristic
116C2×#121C2311yescharacteristic
117C2×#121C2311yescharacteristic
118C2×#120—21no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×#117—24no—
122C2×#117—21no—
123C2×#117—21no—
124C2×#117—24no—
125C2×#117—24no—
126C2×#117D418yes—
127C2×#117D412yes—
128C2×#117D412yes—
129C2×#117C2311yescharacteristic
130C2×#117C2312yes—
131C2×#117C2312yes—
132C2×#117C2312yes—
133C2×#117C2312yes—
134C22×C4⋊C4—21no—
135C22×C4⋊C4—24no—
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4—22no—
139C22×C4⋊C4—22no—
140C22×C4⋊C4—22no—
141C22×C4⋊C4D411yescharacteristic
142C22×C4⋊C4D411yescharacteristic
143C22×C4⋊C4D418yes—
144C22×C4⋊C4C2312yes—
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2312yes—
150C22×C22⋊C4—21no—
151C22×C22⋊C4—21no—
152C22×C22⋊C4D412yes—
153C22×C42—21no—
154C22×C42C2312yes—
155#485—28no—
156#484—24no—
157#484—28no—
158#483—24no—
159#483—28no—
160#482—28no—
161#481—24no—
162#481—28no—
163#480—24no—
164#480—28no—
165#428—28no—
166#427—28no—
167C2×#107C2312yes—
168C2×#107C2314yes—
169C2×#107C2312yes—
170C2×#106C2312yes—
171C2×#106C2314yes—
172C2×#106C2312yes—
173#417—28no—
174#416—28no—
175C2×#103C2312yes—
176C2×#103C2314yes—
177C2×#103C2312yes—
178C2×#102C2312yes—
179C2×#102C2314yes—
180C2×#102C2312yes—
181#407—24no—
182C2×C22⋊C8—24no—
183C2×C22⋊C8C2312yes—
184C2×C22⋊C8C2311yescharacteristic
185C2×C22⋊C8C2311yescharacteristic
186C2×C22⋊C8C2312yes—
187C2×C22⋊C8C2312yes—
188#400—22no—
189#399—22no—
190#398—22no—
191#397—22no—
192#397—24no—
193#394—24no—
194#391C2314yes—
195#390C2314yes—
196#389—22no—
197#389—22no—
198#389C2314yes—
199#389C2314yes—
200#388—24no—
201#387—24no—
202#386—22no—
203#386—24no—
204#385—22no—
205#381—22no—
206#380—21no—
207#380—21no—
208#380—22no—
209#380—22no—
210#380—24no—
211#380C2312yes—
212#380C2312yes—
213C2×#95—21no—
214#340C2318yes—
215#340C2318yes—
216#340C2318yes—
217#340C2318yes—
218#336C2318yes—
219#336C2318yes—
220C25C2×D411yescharacteristic
221C2×#41—22no—
222C2×#41—22no—
223C2×#41—28no—
224C22×Q8—216no—
225C22×Q8—28no—
226C22×Q8—84no—
227C22×Q8—84no—
228C22×Q8C2411yescharacteristic
229C22×D4—21no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2411yescharacteristic
233C23×C4—28no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4—28no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—22no—
241C23×C4—21no—
242C23×C4—82no—
243C23×C4—82no—
244C23×C4C2×D412yes—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D414yes—
249C23×C4C2×D414yes—
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C2×M16—24no—
255C2×M16—28no—
256C22×C8—24no—
257C22×C8—28no—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C4⋊Q8—232no—
262C4⋊Q8—232no—
263C4⋊Q8—232no—
264C4⋊Q8—216no—
265C4⋊Q8—216no—
266#125—232no—
267#125—232no—
268#125—232no—
269#125—216no—
270#125—216no—
271#123—44no—
272#123—84no—
273C22⋊Q8—264no—
274C22⋊Q8—264no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277C22⋊Q8—22no—
278C22⋊Q8—22no—
279#121—22no—
280#121—22no—
281#121—22no—
282#121—22no—
283#120—44no—
284C4×D4—44no—
285C4×D4—44no—
286#117—28no—
287#117—28no—
288#117—28no—
289#117—28no—
290#117—24no—
291#117—24no—
292#117—24no—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—22no—
297#117—22no—
298#117C2×D418yes—
299#117C2×D418yes—
300C2×C4⋊C4—216no—
301C2×C4⋊C4—216no—
302C2×C4⋊C4—216no—
303C2×C4⋊C4—28no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—28no—
316C2×C4⋊C4—28no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—28no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4C2×D412yes—
328C2×C4⋊C4C2×D412yes—
329C2×C4⋊C4C2×D412yes—
330C2×C4⋊C4C2×D412yes—
331C2×C4⋊C4C2×D418yes—
332C2×C4⋊C4C2×D418yes—
333C2×C4⋊C4C2×D418yes—
334C2×C4⋊C4C2×D414yes—
335C2×C4⋊C4C2×D414yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2411yescharacteristic
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—28no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—82no—
356C2×C22⋊C4C2×D414yes—
357C2×C22⋊C4C2×D414yes—
358C2×C42—28no—
359C2×C42—24no—
360C2×C42—24no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—21no—
364C2×C42C2×D414yes—
365C2×C42C2414yes—
366#107—24no—
367#107—24no—
368#107—28no—
369#106—24no—
370#106—24no—
371#106—28no—
372#103—24no—
373#103—24no—
374#103—28no—
375#102—24no—
376#102—24no—
377#102—28no—
378C22⋊C8—22no—
379C22⋊C8—22no—
380C22⋊C8—24no—
381C22⋊C8—24no—
382C22⋊C8—24no—
383#95—22no—
384#95—22no—
385#95—24no—
386#95—24no—
387#95—24no—
388#95—24no—
389C24—24no—
390C24—22no—
391C24—22no—
392C24—21no—
393C24—42no—
394C24#12014yes—
395C24C22×D411yescharacteristic
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398#41—48no—
399#41—48no—
400C2×Q8—232no—
401C2×Q8—21no—
402C2×Q8—21no—
403C2×Q8—24no—
404C2×Q8—88no—
405C2×Q8—88no—
406C2×Q8—832no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—24no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—42no—
415C2×D4—42no—
416C2×D4—82no—
417C22×C4—24no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—28no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—42no—
446C22×C4—44no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—816no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—82no—
455C22×C4—82no—
456C22×C4#12014yes—
457C22×C4#12014yes—
458C22×C4#12014yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C42−1+418yes—
471C22×C4C2511yescharacteristic
472M16—48no—
473C2×C8—22no—
474C2×C8—24no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—28no—
479C2×C8—48no—
480C4⋊C4—232no—
481C4⋊C4—232no—
482C4⋊C4—232no—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—28no—
486C4⋊C4—24no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—28no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4C22×D418yes—
506C4⋊C4C22×D414yes—
507C4⋊C4C22×D414yes—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—44no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—84no—
519C22⋊C4—84no—
520C22⋊C4—82no—
521C42—28no—
522C42—28no—
523C42—28no—
524C42—24no—
525C42—24no—
526C42—42no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—21no—
531C23—21no—
532C23—22no—
533C23—28no—
534C23—22no—
535C23—28no—
536C23—21no—
537C23—21no—
538C23—44no—
539C23—48no—
540C23—48no—
541C23—42no—
542C23—41no—
543C23—42no—
544C23—41no—
545C23—41no—
546C23C2×#12012yes—
547C23C2×#12012yes—
548C23C2×#12012yes—
549C23#53614yes—
550C23#57612yes—
551C23#57712yes—
552C23C23×D411yescharacteristic
553Q8—42no—
554Q8—42no—
555Q8—832no—
556D4—42no—
557D4—42no—
558D4—84no—
559D4—84no—
560D4—84no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—48no—
580C2×C4—48no—
581C2×C4—44no—
582C2×C4—48no—
583C2×C4—416no—
584C2×C4—48no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—88no—
600C2×C4—88no—
601C2×C4—84no—
602C2×C4—164no—
603C2×C4C2×#12012yes—
604C2×C4C2×#12012yes—
605C2×C4C2×#12012yes—
606C2×C4#53614yes—
607C2×C4#53618yes—
608C2×C4#53618yes—
609C2×C4C23×D411yescharacteristic
610C2×C4C23×D411yescharacteristic
611C2×C4C2×2−1+414yes—
612C8—42no—
613C8—42no—
614C8—44no—
615C22—22no—
616C22—24no—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—82no—
631C22—82no—
632C22—81no—
633C22—81no—
634C22#261414yes—
635C22#301214yes—
636C22C22×#12011yescharacteristic
637C22C2×#53612yes—
638C22C2×#57611yescharacteristic
639C22C2×#57711yescharacteristic
640C22#358612yes—
641C4—22no—
642C4—22no—
643C4—44no—
644C4—48no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—81no—
649C4—84no—
650C4—81no—
651C4—81no—
652C4#261414yes—
653C4C2×#53612yes—
654C4C2×#53612yes—
655C2—24no—
656C2—24no—
657C2—42no—
658C2—161no—
659C2C2×#261411yescharacteristic
660C2C2×#301211yescharacteristic
661C2#6036811yescharacteristic
662C2#6087112yes—
663C2#6087212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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. C22×#544C2semidirect
5. #29649C2semidirect
6. #29647C2semidirect
7. #29643C2semidirect
8. #29641C2semidirect
9. #29640C2semidirect
10. #29638C2semidirect
11. #23557C2semidirect
12. #23458C2semidirect
13. #20702C2semidirect
14. C2×#1823C2semidirect
15. #20685C2semidirect
61. #2059C22semidirect
62. #2059C22semidirect
63. #2058C22semidirect
64. #2057C22semidirect
65. #2026C22semidirect
66. #2025C22semidirect
67. #2024C22semidirect
68. #2023C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#30922
C2×#30912
#352632
#296492
#296472
#296432
#296412
#296402
#296382
#235572
#234582
#207022
C2×#18232
#206852