Finite Group Databasebeta

← #6496328

Subgroups — #6496328

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3112C211yesmaximal, characteristic
2C2×#3078C211yesmaximal, characteristic, complemented
3#35298C211yesmaximal, characteristic
4Q8×#117C211yesmaximal, characteristic, complemented
5#33202C211yesmaximal, characteristic, complemented
6#33198C211yesmaximal, characteristic, complemented
7#33031C211yesmaximal, characteristic, complemented
8#33025C211yesmaximal, characteristic, complemented
9#29869C211yesmaximal, characteristic, complemented
10#29867C211yesmaximal, characteristic, complemented
11#29827C211yesmaximal, characteristic, complemented
12#29826C211yesmaximal, characteristic, complemented
13#24824C214yesmaximal, complemented
14#24689C214yesmaximal, complemented
15#24591C214yesmaximal, complemented
16#24541C214yesmaximal, complemented
17#21978C211yesmaximal, characteristic, complemented
18#21893C211yesmaximal, characteristic
19#21839C211yesmaximal, characteristic, complemented
20C2×#533C2211yescharacteristic
21C2×#532C2211yescharacteristic
22C22×C4×Q8C2211yescharacteristic
23#3112—28no—
24C2×#480C2212yes—
25#3078—28no—
26C2×#479C2212yes—
27C2×#479C2212yes—
28C2×#477C2211yescharacteristic
29C2×#477C2211yescharacteristic
30C2×#477C2211yescharacteristic
31C2×#477C2211yescharacteristic
32C2×#475C2211yescharacteristic
33C2×#475C2211yescharacteristic
34C2×#460C2211yescharacteristic
35C2×#460C2211yescharacteristic
36C2×#448C2212yes—
37C2×#440C2212yes—
38C2×#440C2212yes—
39C2×#432C2211yescharacteristic
40C2×#428C2212yes—
41C2×#423C2211yescharacteristic
42C2×#419C2212yes—
43#2868C2211yescharacteristic
44#2862C2211yescharacteristic
45#2858C2211yescharacteristic
46#2858C2211yescharacteristic
47#2620—21no—
48#2620—21no—
49#2566C2214yes—
50#2545—22no—
51#2500C2214yes—
52#2432—22no—
53#2355—22no—
54#2355C2212yes—
55Q8×C4⋊C4C2212yes—
56#2345—22no—
57#2345C2212yes—
58Q8×C22⋊C4C2212yes—
59#2326—21no—
60#2326—21no—
61#2295—21no—
62#2295—21no—
63#2295C2211yescharacteristic
64#2295C2211yescharacteristic
65#2285—21no—
66#2285C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68C42×Q8C2211yescharacteristic
69C4×#117—21no—
70C4×#117C2211yescharacteristic
71#2085C2214yes—
72#2083C2214yes—
73#2074C2214yes—
74#2072C2214yes—
75#2067C2214yes—
76#2065C2214yes—
77#2055C2214yes—
78#2055C2214yes—
79#1992C2214yes—
80#1988C2211yescharacteristic
81#1988C2211yescharacteristic
82#1987C2211yescharacteristic
83#1987C2211yescharacteristic
84#1987C2211yescharacteristic
85#1987C2211yescharacteristic
86#1982C2211yescharacteristic
87#1982C2211yescharacteristic
88#1949C2214yes—
89#1947C2214yes—
90#1940C2214yes—
91#1926C2214yes—
92#1925C2214yes—
93#1895C2214yes—
94#1880C2214yes—
95#1880C2214yes—
96#1851C2214yes—
97#1842C2211yescharacteristic
98#1842C2211yescharacteristic
99#1831C2211yescharacteristic
100#1831C2211yescharacteristic
101#1825C2214yes—
102#1805C2214yes—
103#1803C2214yes—
104#1802C2214yes—
105C22×#41—21no—
106C23×Q8C2311yescharacteristic
107C22×SD16C2312yes—
108C22×SD16C2312yes—
109C22×M16C2312yes—
110#533—24no—
111#533—24no—
112#532—24no—
113#532—24no—
114C2×C4⋊Q8—22no—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×#127C2311yescharacteristic
119C2×#125C2312yes—
120C2×#124—21no—
121C2×#124—21no—
122C2×#124—21no—
123C2×#124—22no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×#121—21no—
127C2×#121C2312yes—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8—24no—
133C2×C4×Q8D418yes—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2312yes—
136C2×C4×Q8C2311yescharacteristic
137C2×C4×Q8C2311yescharacteristic
138C2×C4×Q8C2311yescharacteristic
139C2×C4×Q8C2311yescharacteristic
140C2×C4×D4—21no—
141C2×C4×D4C2312yes—
142C2×#117—21no—
143C2×#117D411yescharacteristic
144C2×#117D411yescharacteristic
145C2×#117C2312yes—
146C22×C4⋊C4—21no—
147C22×C4⋊C4C2311yescharacteristic
148C22×C42—21no—
149C22×C42C2311yescharacteristic
150#480—28no—
151#479—28no—
152#479—28no—
153#477—24no—
154#477—24no—
155#477—24no—
156#477—24no—
157#475—24no—
158#475—24no—
159#460—24no—
160#460—24no—
161#448—28no—
162#440—28no—
163#440—28no—
164#432—24no—
165#428—28no—
166C2×#107C2312yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#106C2312yes—
171C2×#106C2312yes—
172#423—24no—
173C2×C4⋊C8C2311yescharacteristic
174C2×C4⋊C8C2311yescharacteristic
175C2×C4⋊C8C2311yescharacteristic
176C2×C4⋊C8C2311yescharacteristic
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2314yes—
179C2×C4⋊C8C2314yes—
180#419—28no—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2314yes—
190C2×#102C2314yes—
191C2×C22⋊C8C2312yes—
192C2×#97C2312yes—
193#396—24no—
194#396—24no—
195#394—24no—
196#394—24no—
197#391—21no—
198#391—21no—
199#391—22no—
200#391—22no—
201#391—22no—
202#391C2312yes—
203#391C2312yes—
204#390—21no—
205#390—21no—
206#390—22no—
207#390C2314yes—
208#390C2314yes—
209#389—22no—
210#389—22no—
211#389C2312yes—
212#389C2312yes—
213#388—24no—
214#388—24no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#385—24no—
222#385—24no—
223#384—22no—
224#384—21no—
225#384—21no—
226#384C2314yes—
227#384C2312yes—
228#382—22no—
229#382—21no—
230#382C2314yes—
231#380—21no—
232#380—22no—
233#380—22no—
234#380C2312yes—
235C4×C4⋊C4—22no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4D412yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2311yescharacteristic
248C4×C4⋊C4C2311yescharacteristic
249C4×C4⋊C4C2312yes—
250C4×C22⋊C4—22no—
251C4×C22⋊C4—22no—
252C4×C22⋊C4D412yes—
253C4×C22⋊C4C2312yes—
254#376—21no—
255#376—21no—
256#376D411yescharacteristic
257#376D411yescharacteristic
258C43—21no—
259C43—21no—
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2314yes—
269#336C2314yes—
270C2×#41—44no—
271C2×#41—84no—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×Q8—24no—
278C22×Q8—42no—
279C22×Q8C2×D414yes—
280C22×Q8C2411yescharacteristic
281C22×Q8C2411yescharacteristic
282C22×D4—22no—
283C22×D4—21no—
284C22×D4—21no—
285C22×D4C2412yes—
286C23×C4—21no—
287C23×C4—21no—
288C23×C4—22no—
289C23×C4C2×D411yescharacteristic
290C23×C4C2×D411yescharacteristic
291C23×C4C2411yescharacteristic
292C2×SD16—24no—
293C2×SD16—24no—
294C2×SD16—28no—
295C2×SD16—44no—
296C2×M16—24no—
297C2×M16—44no—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—84no—
309#127—22no—
310#127—22no—
311#125—24no—
312#125—24no—
313#124—44no—
314#124—44no—
315#124—44no—
316#124—84no—
317C22⋊Q8—44no—
318C22⋊Q8—44no—
319#121—24no—
320#121—24no—
321#121—44no—
322C4×Q8—28no—
323C4×Q8—28no—
324C4×Q8—22no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×Q8—24no—
329C4×Q8—24no—
330C4×Q8—42no—
331C4×Q8—42no—
332C4×Q8—44no—
333C4×Q8—44no—
334C4×Q8C2×D414yes—
335C4×Q8C2×D414yes—
336C4×Q8C2×D414yes—
337C4×Q8C2×D414yes—
338C4×D4—24no—
339C4×D4—24no—
340C4×D4—44no—
341#117—24no—
342#117—24no—
343#117—24no—
344#117—42no—
345#117—44no—
346C2×C4⋊C4—28no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4C2×D412yes—
367C2×C4⋊C4C2×D414yes—
368C2×C4⋊C4C2×D412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2411yescharacteristic
375C2×C4⋊C4C2411yescharacteristic
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D412yes—
386C2×C42—28no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—21no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—22no—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D412yes—
402C2×C42C2×D412yes—
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D414yes—
406C2×C42C2×D411yescharacteristic
407C2×C42#4114yes—
408C2×C42C2412yes—
409C2×C42C2412yes—
410C2×C42C2411yescharacteristic
411C2×C42C2411yescharacteristic
412#107—24no—
413#107—24no—
414#107—24no—
415#107—24no—
416#106—24no—
417#106—24no—
418C4⋊C8—22no—
419C4⋊C8—22no—
420C4⋊C8—24no—
421C4⋊C8—22no—
422C4⋊C8—22no—
423C4⋊C8—44no—
424C4⋊C8—44no—
425#103—24no—
426#103—24no—
427#103—24no—
428#103—24no—
429#103—24no—
430#103—24no—
431#102—24no—
432#102—24no—
433#102—44no—
434#102—44no—
435C22⋊C8—24no—
436#97—24no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—21no—
444#95—21no—
445#95—22no—
446#95—22no—
447#95—22no—
448#95—24no—
449#95—24no—
450#95C2×D412yes—
451#95C2×D412yes—
452#95C2×D412yes—
453#95C2×D412yes—
454C24—22no—
455C24C22×D411yescharacteristic
456#41—816no—
457C2×Q8—24no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—28no—
462C2×Q8—22no—
463C2×Q8—22no—
464C2×Q8—44no—
465C2×Q8—44no—
466C2×Q8—48no—
467C2×Q8—44no—
468C2×Q8—44no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—42no—
473C2×Q8—42no—
474C2×Q8C22×D412yes—
475C2×Q8C22×D412yes—
476C2×Q82−1+414yes—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—42no—
486C2×D4—42no—
487C2×D4—84no—
488C22×C4—21no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—42no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—84no—
507C22×C4—82no—
508C22×C4#12114yes—
509C22×C4#12114yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D411yescharacteristic
520C22×C4C2×#4112yes—
521C22×C4C2511yescharacteristic
522SD16—48no—
523SD16—48no—
524M16—48no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—24no—
530C2×C8—24no—
531C2×C8—24no—
532C2×C8—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—42no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—84no—
564C4⋊C4—84no—
565C4⋊C4C22×D412yes—
566C4⋊C4C22×D412yes—
567C22⋊C4—24no—
568C22⋊C4—42no—
569C22⋊C4—48no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—48no—
575C22⋊C4—84no—
576C22⋊C4—84no—
577C42—24no—
578C42—24no—
579C42—24no—
580C42—24no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—42no—
589C42—42no—
590C42—44no—
591C42—44no—
592C42—44no—
593C42—42no—
594C42—42no—
595C42—82no—
596C42—82no—
597C42#12114yes—
598C42#12114yes—
599C42#13714yes—
600C42C22×D412yes—
601C42C22×D412yes—
602C42C2×#4112yes—
603C42C2×#4112yes—
604C23—22no—
605C23—22no—
606C23—21no—
607C23—21no—
608C23—44no—
609C23—42no—
610C23—44no—
611C23—42no—
612C23—42no—
613C23C2×#12112yes—
614C23C23×D411yescharacteristic
615Q8—22no—
616Q8—24no—
617Q8—22no—
618Q8—22no—
619Q8—48no—
620Q8—42no—
621Q8—84no—
622Q8—84no—
623Q8—84no—
624Q8#53614yes—
625D4—42no—
626D4—42no—
627D4—44no—
628D4—84no—
629D4—88no—
630D4—84no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—28no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—44no—
657C2×C4—48no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—88no—
675C2×C4—84no—
676C2×C4C2×#12112yes—
677C2×C4C2×#12112yes—
678C2×C4C2×#12112yes—
679C2×C4C2×#12112yes—
680C2×C4C2×#12112yes—
681C2×C4C2×#13714yes—
682C2×C4C2×#13712yes—
683C2×C4#57512yes—
684C2×C4C23×D411yescharacteristic
685C2×C4C22×#4111yescharacteristic
686C2×C4C2×2−1+412yes—
687C2×C4#58512yes—
688C8—42no—
689C8—42no—
690C8—42no—
691C8—42no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—42no—
696C22—42no—
697C22—82no—
698C22—84no—
699C22—84no—
700C22—82no—
701C22—82no—
702C22C22×#12111yescharacteristic
703C22#348912yes—
704C22C22×#13711yescharacteristic
705C22C2×#57511yescharacteristic
706C22#358512yes—
707C4—22no—
708C4—22no—
709C4—24no—
710C4—22no—
711C4—24no—
712C4—22no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—42no—
717C4—42no—
718C4—42no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4#303714yes—
725C4C2×#53611yescharacteristic
726C4#344911yescharacteristic
727C4#349512yes—
728C2—42no—
729C2—162no—
730C2#3530611yescharacteristic
731C2C2×#303711yescharacteristic
732C2#6042611yescharacteristic
733C2#6211412yes—
734C2#6211712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3078C2semidirect
4. Q8×#117C2semidirect
5. #33202C2semidirect
6. #33198C2semidirect
7. #33031C2semidirect
8. #33025C2semidirect
9. #29869C2semidirect
10. #29867C2semidirect
11. #29827C2semidirect
12. #29826C2semidirect
13. #24824C2semidirect
14. #24689C2semidirect
15. #24591C2semidirect
16. #24541C2semidirect
17. #21978C2semidirect
19. #21839C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31122
C2×#30782
#352982
Q8×#1172
#332022
#331982
#330312
#330252
#298692
#298672
#298272
#298262
#248242
#246892
#245912
#245412
#219782
#218932
#218392