Finite Group Databasebeta

← #6446106

Subgroups — #6446106

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3166C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic
3#35172C211yesmaximal, characteristic
4#34217C211yesmaximal, characteristic, complemented
5#25466C214yesmaximal, complemented
6#25465C214yesmaximal, complemented
7#25458C214yesmaximal, complemented
8#25457C214yesmaximal, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23660C211yesmaximal, characteristic, complemented
11#23614C211yesmaximal, characteristic
12#23614C211yesmaximal, characteristic
13#21311C211yesmaximal, characteristic, complemented
14#21174C211yesmaximal, characteristic, complemented
15#21173C211yesmaximal, characteristic, complemented
16#21145C211yesmaximal, characteristic, complemented
17#21063C211yesmaximal, characteristic
18#21063C211yesmaximal, characteristic
19C4×#418C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#3166—28no—
24C2×#501C2212yescomplemented
25C2×#501C2212yescomplemented
26C2×#499C2212yescomplemented
27C2×#499C2212yescomplemented
28C2×#478—28no—
29C2×#478C2214yes—
30C2×#478C2214yes—
31C2×#428C2211yescharacteristic
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C22×#107C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C2×#424C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C22×C4⋊C8C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C2×#418C2211yescharacteristic
42C22×#102C2211yescharacteristic
43C22×#102C2211yescharacteristic
44C2×C4×M16C2211yescharacteristic
45#2848—22no—
46#2626—22no—
47#2580—22no—
48#2566C2212yes—
49#2566C2212yes—
50#2563C2212yes—
51#2542C2212yes—
52#2538—22no—
53#2370C2214yescomplemented
54#2370C2214yescomplemented
55#2345—22no—
56#2345—22no—
57#2330—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2316C2211yescharacteristic
61#2316C2211yescharacteristic
62#2309C2211yescharacteristic
63#2309C2211yescharacteristic
64C4×#125—22no—
65#2285C2211yescharacteristic
66#2274C2211yescharacteristic
67#2274C2211yescharacteristic
68C4×#117C2211yescharacteristic
69#2085C2214yescomplemented
70#2085C2214yescomplemented
71#2084C2214yescomplemented
72#2083C2214yescomplemented
73#2067C2214yes—
74#2067C2214yes—
75#2065C2214yes—
76#2065C2214yes—
77#1989C2212yescomplemented
78#1989C2212yescomplemented
79#1988C2212yescomplemented
80#1987C2212yescomplemented
81#1949C2214yescomplemented
82#1949C2214yescomplemented
83#1948C2214yescomplemented
84#1947C2214yescomplemented
85#1940C2214yescomplemented
86#1939C2214yescomplemented
87#1926C2214yes—
88#1926C2214yes—
89#1925C2214yes—
90#1925C2214yes—
91#1762C2214yes—
92C4×#102C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1735C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C23×D4C2311yescharacteristic
102C24×C4—21no—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#533—24no—
108#533—24no—
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×#125—21no—
112C2×#125—21no—
113C2×#125—24no—
114C2×#125D418yes—
115C2×#125C2311yescharacteristic
116C2×#125C2311yescharacteristic
117C2×#125C2312yes—
118C2×#125C2312yes—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—24no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—24no—
127C2×C4×D4—24no—
128C2×C4×D4C2312yes—
129C2×C4×D4C2312yes—
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C22×C4⋊C4—21no—
137C22×C4⋊C4—21no—
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2311yescharacteristic
141C22×C22⋊C4—21no—
142C22×C42C2311yescharacteristic
143#501—28no—
144#501—28no—
145#499—28no—
146#499—28no—
147#478—216no—
148#478—216no—
149#428—24no—
150#428—24no—
151#428—24no—
152#428—24no—
153C2×#107—24no—
154C2×#107C2314yes—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#106—24no—
160C2×#106C2314yes—
161C2×#106C2312yes—
162C2×#106C2312yes—
163C2×#106C2312yes—
164C2×#106C2312yes—
165#424—24no—
166#424—24no—
167C2×C4⋊C8—24no—
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2314yes—
171C2×C4⋊C8C2314yes—
172#418—24no—
173#418Q818yescomplemented
174C2×#102—24no—
175C2×#102—24no—
176C2×#102C2314yes—
177C2×#102C2314yes—
178C2×#102C2314yes—
179C2×#102C2314yes—
180C4×M16—24no—
181C2×#97C2312yes—
182C2×C4×C8C2312yes—
183#399—24no—
184#396—24no—
185#396—24no—
186#393—24no—
187#390—22no—
188#390—22no—
189#390C2312yes—
190#390C2312yes—
191#390C2312yes—
192#390C2312yes—
193#389—22no—
194#389—22no—
195#389—22no—
196#389—22no—
197#389—24no—
198#389—24no—
199#389C2314yes—
200#389C2314yes—
201#387—24no—
202#387—24no—
203#387—24no—
204#387—24no—
205#385—22no—
206#385—22no—
207#385—24no—
208#385—24no—
209#384D412yes—
210#384C2312yes—
211#384C2312yes—
212#382D412yes—
213#382C2312yes—
214#382C2312yes—
215#380—24no—
216#380—22no—
217#380—22no—
218#380D412yes—
219#380D412yes—
220#379—22no—
221#379—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—24no—
225C4×C4⋊C4—24no—
226C4×C4⋊C4C2314yes—
227C4×C4⋊C4C2314yes—
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C4⋊C4C2312yes—
231C4×C4⋊C4C2312yes—
232C4×C22⋊C4—21no—
233C4×C22⋊C4—21no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236#376—21no—
237C43—21no—
238#340C2312yes—
239#340C2312yes—
240#340C2312yes—
241#340C2312yes—
242#340C2314yes—
243#340C2312yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#340C2314yes—
248#336C2314yes—
249#336C2314yes—
250C25—21no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×D4—22no—
254C22×D4—21no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4C2×Q814yes—
260C22×D4C2412yes—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4—44no—
267C23×C4—84no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2411yescharacteristic
271C2×M16—22no—
272C2×M16—22no—
273C2×M16—22no—
274C2×M16—22no—
275C2×M16C2×Q814yes—
276C2×M16#4114yes—
277C22×C8—22no—
278C22×C8—22no—
279C22×C8—22no—
280C22×C8—22no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291#125—28no—
292#125—28no—
293#125—22no—
294#125—22no—
295#125—24no—
296#125—24no—
297#125—22no—
298#125—22no—
299#125C2×D418yes—
300#125C2×D418yes—
301C22⋊Q8—44no—
302C22⋊Q8—44no—
303C22⋊Q8—88no—
304C4×Q8—44no—
305C4×Q8—44no—
306C4×D4—24no—
307C4×D4—24no—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4—44no—
311C4×D4—44no—
312#117—22no—
313#117—22no—
314#117—22no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117C2×Q814yes—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D412yes—
349C2×C4⋊C4C2×D414yes—
350C2×C4⋊C4C2×D414yes—
351C2×C4⋊C4C2×D414yes—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2411yescharacteristic
355C2×C4⋊C4C2411yescharacteristic
356C2×C4⋊C4C2411yescharacteristic
357C2×C4⋊C4C2411yescharacteristic
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—24no—
370C2×C22⋊C4—24no—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—44no—
373C2×C22⋊C4C2×D412yes—
374C2×C22⋊C4C2×D412yes—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—21no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—24no—
382C2×C42—24no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D414yes—
386C2×C42#4114yes—
387C2×C42C2412yes—
388C2×C42C2412yes—
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391#107—28no—
392#107—24no—
393#107—24no—
394#107—24no—
395#107—24no—
396#106—28no—
397#106—24no—
398#106—24no—
399#106—24no—
400#106—24no—
401C4⋊C8—28no—
402C4⋊C8—28no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405#102—28no—
406#102—28no—
407#102—28no—
408#102C2×Q8116yes—
409#97—24no—
410C4×C8—24no—
411#95—22no—
412#95—22no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95C2×D414yes—
420#95C2×D414yes—
421C24—22no—
422C24—24no—
423C24—42no—
424C24—42no—
425C24—82no—
426C24—82no—
427C24C22×D411yescharacteristic
428C2×Q8—44no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—82no—
432C2×D4—28no—
433C2×D4—28no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—42no—
441C2×D4—44no—
442C2×D4—42no—
443C2×D4—44no—
444C2×D4C22×Q814yes—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—48no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—84no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—816no—
467C22×C4#12114yes—
468C22×C4#12114yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×Q812yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C42−1+412yes—
481C22×C42−1+412yes—
482C22×C4C2511yescharacteristic
483M16—28no—
484M16—28no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—44no—
496C2×C8—44no—
497C2×C8C22×Q814yes—
498C2×C8C2×#4114yes—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—28no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
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—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4—84no—
535C4⋊C4C22×D414yes—
536C4⋊C4C22×D414yes—
537C4⋊C4C22×Q814yes—
538C22⋊C4—24no—
539C22⋊C4—24no—
540C22⋊C4—24no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C42—24no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—42no—
556C42—42no—
557C42—42no—
558C42—82no—
559C42—82no—
560C42—84no—
561C42—84no—
562C42#12114yes—
563C42#12114yes—
564C42C22×D412yes—
565C42C22×D412yes—
566C42C2×#4112yes—
567C42C2×#4112yes—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—44no—
572C23—44no—
573C23—42no—
574C23—48no—
575C23—42no—
576C23—88no—
577C23—84no—
578C23—84no—
579C23—88no—
580C23C2×#12112yes—
581C23#54712yes—
582C23D4×Q812yes—
583C23C23×D411yescharacteristic
584Q8—84no—
585Q8—84no—
586D4—24no—
587D4—24no—
588D4—48no—
589D4—42no—
590D4—42no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—88no—
624C2×C4—88no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—84no—
628C2×C4C2×#12112yes—
629C2×C4C2×#12112yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#12112yes—
632C2×C4C2×#12112yes—
633C2×C4#53314yes—
634C2×C4#54114yes—
635C2×C4#54712yes—
636C2×C4#54814yes—
637C2×C4D4×Q812yes—
638C2×C4D4×Q814yes—
639C2×C4C23×D411yescharacteristic
640C2×C4C23×Q811yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C2×C4C2×2−1+411yescharacteristic
645C8—24no—
646C8—24no—
647C8—42no—
648C8—42no—
649C22—22no—
650C22—22no—
651C22—24no—
652C22—24no—
653C22—44no—
654C22—48no—
655C22—816no—
656C22—82no—
657C22—82no—
658C22—84no—
659C22#242914yes—
660C22C22×#12111yescharacteristic
661C22C2×#54711yescharacteristic
662C22C2×D4×Q811yescharacteristic
663C22#358412yes—
664C22#358512yes—
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—81no—
675C4—81no—
676C4—82no—
677C4—81no—
678C4—81no—
679C4—82no—
680C4#242914yes—
681C4C2×#53311yescharacteristic
682C4C2×#54111yescharacteristic
683C4C2×#54811yescharacteristic
684C4C2×D4×Q811yescharacteristic
685C2—24no—
686C2—84no—
687C2C2×#242911yescharacteristic
688C2#6042211yescharacteristic
689C2#6042611yescharacteristic
690C2#6128312yes—
691C2#6138512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3166C2semidirect
4. #34217C2semidirect
5. #25466C2semidirect
6. #25465C2semidirect
7. #25458C2semidirect
8. #25457C2semidirect
9. #23661C2semidirect
10. #23660C2semidirect
13. #21311C2semidirect
14. #21174C2semidirect
15. #21173C2semidirect
16. #21145C2semidirect
24. C2×#501C22semidirect
25. C2×#501C22semidirect
26. C2×#499C22semidirect
27. C2×#499C22semidirect
53. #2370C22semidirect
54. #2370C22semidirect
69. #2085C22semidirect
70. #2085C22semidirect
71. #2084C22semidirect
72. #2083C22semidirect
77. #1989C22semidirect
78. #1989C22semidirect
79. #1988C22semidirect
80. #1987C22semidirect
81. #1949C22semidirect
82. #1949C22semidirect
83. #1948C22semidirect
84. #1947C22semidirect
85. #1940C22semidirect
86. #1939C22semidirect
173. #418Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31662
C22×#4782
#351722
#342172
#254662
#254652
#254582
#254572
#236612
#236602
#236142
#213112
#211742
#211732
#211452
#210632
C4×#4182