Finite Group Databasebeta

← #6466014

Subgroups — #6466014

Order 512.

2078 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2961C211yesmaximal, characteristic, complemented
2C2×#2949C211yesmaximal, characteristic
3#35662C211yesmaximal, characteristic, complemented
4#35515C211yesmaximal, characteristic
5#28277C214yesmaximal, complemented
6#28275C214yesmaximal, complemented
7#28272C214yesmaximal, complemented
8#28270C214yesmaximal, complemented
9#23619C211yesmaximal, characteristic, complemented
10#23618C211yesmaximal, characteristic
11#23615C211yesmaximal, characteristic, complemented
12#23614C211yesmaximal, characteristic
13#21231C211yesmaximal, characteristic
14C2×#1973C211yesmaximal, characteristic
15#21158C211yesmaximal, characteristic, complemented
16#21154C211yesmaximal, characteristic, complemented
17#20803C211yesmaximal, characteristic, complemented
18#20651C211yesmaximal, characteristic, complemented
19#20650C211yesmaximal, characteristic
20C23×SD16C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C2×#519C2211yescharacteristic
23C2×#518C2211yescharacteristic
24#2961—28no—
25C2×#446C2214yescomplemented
26C2×#446C2214yescomplemented
27#2949—28no—
28C2×#440C2214yes—
29C2×C4×SD16C2214yes—
30C2×#428C2211yescharacteristic
31C2×#428C2211yescharacteristic
32C2×#427C2211yescharacteristic
33C2×#427C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C2×#424C2211yescharacteristic
36C2×#424C2211yescharacteristic
37C2×#423C2211yescharacteristic
38C2×#419C2211yescharacteristic
39C2×#419C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C2×#418C2211yescharacteristic
42C22×#103C2211yescharacteristic
43C22×#102C2211yescharacteristic
44C2×#405C2211yescharacteristic
45C2×#405C2211yescharacteristic
46C22×#97C2211yescharacteristic
47#2582C2214yescomplemented
48#2581C2214yescomplemented
49#2580C2214yes—
50#2538C2214yes—
51#2454—22no—
52#2453—22no—
53#2452—22no—
54#2451—22no—
55#2438—24no—
56#2437—24no—
57#2317C2211yescharacteristic
58#2317C2211yescharacteristic
59#2316C2211yescharacteristic
60#2316C2211yescharacteristic
61#2310C2211yescharacteristic
62#2309C2211yescharacteristic
63#2296C2211yescharacteristic
64#2296C2211yescharacteristic
65C2×#391C2211yescharacteristic
66C2×#390C2211yescharacteristic
67#2068C2214yescomplemented
68#2068C2214yescomplemented
69#2067C2214yes—
70#2067C2214yes—
71#2066C2214yescomplemented
72#2066C2214yescomplemented
73#2065C2214yes—
74#2065C2214yes—
75#1973C2212yes—
76#1973C2214yes—
77#1973C2212yes—
78#1973C2214yes—
79#1968C2212yes—
80#1968C2212yes—
81#1944C2212yescomplemented
82#1944C2212yescomplemented
83#1943C2212yescomplemented
84#1943C2214yescomplemented
85#1943C2212yescomplemented
86#1942C2214yescomplemented
87#1847C2212yescomplemented
88#1847C2212yescomplemented
89#1846C2212yescomplemented
90#1845C2212yescomplemented
91#1805C2214yescomplemented
92#1804C2214yes—
93#1803C2214yescomplemented
94#1802C2214yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99C23×Q8C2311yescharacteristic
100C23×D4C2311yescharacteristic
101C22×SD16—24no—
102C22×SD16—24no—
103C22×SD16Q818yescomplemented
104C22×SD16C2312yes—
105C22×SD16C2312yes—
106C22×M16C2311yescharacteristic
107C22×M16C2311yescharacteristic
108C23×C8C2311yescharacteristic
109C23×C8C2311yescharacteristic
110#533—24no—
111#533—24no—
112C2×C4⋊Q8D412yes—
113C2×#125D412yes—
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8D414yes—
116#519—24no—
117#519—24no—
118#518—24no—
119#518—24no—
120C2×C4×Q8—22no—
121C2×C4×Q8C2314yes—
122C2×C4×Q8C2314yes—
123C2×C4×D4—22no—
124C2×C4×D4C2314yes—
125C2×#117—22no—
126C2×#117C2311yescharacteristic
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C22⋊C4—22no—
135C22×C22⋊C4—22no—
136C22×C42C2311yescharacteristic
137#446—216no—
138#446—216no—
139#440—216no—
140C4×SD16—216no—
141#428—42no—
142#428—42no—
143#427—24no—
144#427—24no—
145C2×#107—24no—
146C2×#107C2311yescharacteristic
147C2×#107C2312yes—
148C2×#107C2311yescharacteristic
149C2×#107C2312yes—
150C2×#107C2312yes—
151C2×#107C2312yes—
152C2×#107C2312yes—
153C2×#106C2311yescharacteristic
154C2×#106C2311yescharacteristic
155C2×#106C2312yes—
156C2×#106C2312yes—
157C2×#106C2312yes—
158#424—24no—
159#424—24no—
160#423—24no—
161C2×C4⋊C8C2314yes—
162C2×C4⋊C8C2314yes—
163C2×C4⋊C8C2314yes—
164#419—24no—
165#419—42no—
166#418—24no—
167#418—42no—
168C2×#103—42no—
169C2×#103C2314yes—
170C2×#103C2312yes—
171C2×#103C2314yes—
172C2×#103C2312yes—
173C2×#102—42no—
174C2×#102C2314yes—
175C2×#102C2312yes—
176C2×#102C2314yes—
177C2×#102C2312yes—
178#405—24no—
179#405D418yes—
180C2×#97—24no—
181C2×#97C2312yes—
182C2×#97C2312yes—
183C2×#97C2312yes—
184C2×#97C2312yes—
185C2×C4×C8C2312yes—
186C2×C4×C8C2312yes—
187#400—24no—
188#399—24no—
189#398—24no—
190#398—24no—
191#396—24no—
192#396—24no—
193#395—24no—
194#393—24no—
195#391—24no—
196#391—22no—
197#391—21no—
198#391—21no—
199#391C2312yes—
200#391C2312yes—
201#391C2314yes—
202#390—22no—
203#390—21no—
204#390—21no—
205#390C2312yes—
206#390C2312yes—
207#390C2314yes—
208#389—22no—
209#389—22no—
210#389—22no—
211#389—22no—
212#389—24no—
213#389C2314yes—
214#389C2314yes—
215#389C2312yes—
216#389C2312yes—
217#389C2312yes—
218#389C2312yes—
219#387—24no—
220#387—24no—
221#385—22no—
222#385—22no—
223#385—24no—
224#385—24no—
225#385—22no—
226#385—22no—
227#384C2312yes—
228#384C2312yes—
229#382C2312yes—
230#382C2312yes—
231#381—22no—
232#380—24no—
233#380—22no—
234#380—22no—
235#379—22no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C2×#95—22no—
241#340C2312yes—
242#340C2314yes—
243#340C2312yes—
244#340C2314yes—
245#340C2312yes—
246#340C2312yes—
247C25—21no—
248C22×Q8—21no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8C2×D414yes—
255C22×Q8C2×Q814yes—
256C22×Q8C2412yes—
257C22×D4—22no—
258C22×D4—21no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4C2×Q814yes—
264C22×D4C2412yes—
265C23×C4—21no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—24no—
269C23×C4C2×D412yes—
270C23×C4C2411yescharacteristic
271C2×SD16—232no—
272C2×SD16—24no—
273C2×SD16—24no—
274C2×SD16—24no—
275C2×SD16—24no—
276C2×SD16C2×Q818yes—
277C2×SD16C2×Q818yes—
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16C2×D414yes—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8C2×D414yes—
288C22×C8C2×Q814yes—
289C22×C8#4114yes—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2411yescharacteristic
294C22×C8C2411yescharacteristic
295C4⋊Q8—28no—
296#125—28no—
297C22⋊Q8—216no—
298C22⋊Q8—88no—
299C4×Q8—28no—
300C4×Q8—28no—
301C4×Q8—28no—
302C4×Q8—28no—
303C4×Q8—48no—
304C4×D4—28no—
305C4×D4—28no—
306C4×D4—48no—
307#117—22no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—48no—
318#117C2×D414yes—
319#117#4114yes—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—24no—
330C2×C4⋊C4—24no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D414yes—
337C2×C4⋊C4C2×D414yes—
338C2×C4⋊C4C2×D414yes—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—48no—
358C2×C22⋊C4—84no—
359C2×C22⋊C4C2×D412yes—
360C2×C22⋊C4C2×D412yes—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42—21no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2412yes—
371C2×C42C2412yes—
372C2×C42C2412yes—
373#107—24no—
374#107—24no—
375#107—24no—
376#107—22no—
377#107—22no—
378#107—42no—
379#107—42no—
380#106—24no—
381#106—22no—
382#106—22no—
383#106—42no—
384#106—42no—
385C4⋊C8—28no—
386C4⋊C8—28no—
387C4⋊C8—28no—
388#103—28no—
389#103—44no—
390#103—42no—
391#103—42no—
392#102—28no—
393#102—44no—
394#102—42no—
395#102—42no—
396#97—24no—
397#97—24no—
398#97—24no—
399#97C2×D418yes—
400C4×C8—24no—
401C4×C8C2×D418yes—
402#95—24no—
403#95—24no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—24no—
409#95—24no—
410#95—24no—
411#95—24no—
412#95—24no—
413#95—24no—
414C24—22no—
415C24—24no—
416C24—42no—
417C24—42no—
418C24—82no—
419C24—82no—
420C24C22×D411yescharacteristic
421C2×Q8—28no—
422C2×Q8—24no—
423C2×Q8—22no—
424C2×Q8—28no—
425C2×Q8—28no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—44no—
431C2×Q8—44no—
432C2×Q8—42no—
433C2×Q8—44no—
434C2×Q8—42no—
435C2×Q8—44no—
436C2×Q8C22×Q814yes—
437C2×D4—28no—
438C2×D4—24no—
439C2×D4—28no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—42no—
446C2×D4—44no—
447C2×D4—42no—
448C2×D4—44no—
449C2×D4C22×Q814yes—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—48no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D412yes—
478C22×C4C22×Q812yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C22×C4C2511yescharacteristic
484SD16—216no—
485SD16—216no—
486M16—28no—
487M16—44no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—44no—
497C2×C8C22⋊Q8116yescomplemented
498C2×C8C22×D414yes—
499C2×C8C22×D412yes—
500C2×C8C22×D412yes—
501C2×C8C22×Q812yes—
502C2×C8C22×Q812yes—
503C2×C8C2×#4112yes—
504C2×C8C2×#4112yes—
505C4⋊C4—24no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—84no—
531C4⋊C4—82no—
532C4⋊C4—84no—
533C4⋊C4—82no—
534C4⋊C4—84no—
535C4⋊C4—84no—
536C4⋊C4#12118yes—
537C22⋊C4—24no—
538C22⋊C4—48no—
539C22⋊C4—42no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—48no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4#12118yes—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—42no—
555C42—42no—
556C42—48no—
557C42—42no—
558C42—42no—
559C42C22×D414yes—
560C42C2×#4114yes—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—48no—
569C23—88no—
570C23—164no—
571C23—162no—
572C23—162no—
573C23C2×#12112yes—
574C23D4×Q812yes—
575C23#55012yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—24no—
579Q8—42no—
580Q8—42no—
581Q8—48no—
582Q8—88no—
583D4—24no—
584D4—24no—
585D4—42no—
586D4—42no—
587D4—48no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—48no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—84no—
622C2×C4—168no—
623C2×C4—162no—
624C2×C4—162no—
625C2×C4C2×#12118yes—
626C2×C4C2×#12112yes—
627C2×C4C2×C22⋊Q814yes—
628C2×C4#53714yes—
629C2×C4D4×Q812yes—
630C2×C4#55012yes—
631C2×C4#55414yes—
632C2×C4#55614yes—
633C2×C4C23×D411yescharacteristic
634C2×C4C23×Q811yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C22×#4111yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C8—24no—
640C8—42no—
641C8—42no—
642C8C2×C22⋊Q814yes—
643C8C2×C22⋊Q814yes—
644C22—24no—
645C22—24no—
646C22—22no—
647C22—22no—
648C22—44no—
649C22—48no—
650C22—84no—
651C22—82no—
652C22—82no—
653C22—168no—
654C22#253814yes—
655C22C22×#12111yescharacteristic
656C22C2×D4×Q811yescharacteristic
657C22C2×#55011yescharacteristic
658C22#358512yes—
659C22#358612yes—
660C4—28no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—82no—
666C4—82no—
667C4—81no—
668C4—81no—
669C4—82no—
670C4—82no—
671C4—81no—
672C4—81no—
673C4#253814yes—
674C4C22×C22⋊Q811yescharacteristic
675C4C2×#53711yescharacteristic
676C4C2×#55411yescharacteristic
677C4C2×#55611yescharacteristic
678C2—24no—
679C2—84no—
680C2C2×#253811yescharacteristic
681C2#6046012yes—
682C2#6137712yes—
683C2#6141711yescharacteristic
684C2#6141811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#533 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2961C2semidirect
3. #35662C2semidirect
5. #28277C2semidirect
6. #28275C2semidirect
7. #28272C2semidirect
8. #28270C2semidirect
9. #23619C2semidirect
11. #23615C2semidirect
15. #21158C2semidirect
16. #21154C2semidirect
17. #20803C2semidirect
18. #20651C2semidirect
25. C2×#446C22semidirect
26. C2×#446C22semidirect
47. #2582C22semidirect
48. #2581C22semidirect
67. #2068C22semidirect
68. #2068C22semidirect
71. #2066C22semidirect
72. #2066C22semidirect
81. #1944C22semidirect
82. #1944C22semidirect
83. #1943C22semidirect
84. #1943C22semidirect
85. #1943C22semidirect
86. #1942C22semidirect
87. #1847C22semidirect
88. #1847C22semidirect
89. #1846C22semidirect
90. #1845C22semidirect
91. #1805C22semidirect
93. #1803C22semidirect
103. C22×SD16Q8semidirect
497. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#29612
C2×#29492
#356622
#355152
#282772
#282752
#282722
#282702
#236192
#236182
#236152
#236142
#212312
C2×#19732
#211582
#211542
#208032
#206512
#206502