Finite Group Databasebeta

← #6345312

Subgroups — #6345312

Order 512.

1573 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2#60250C211yesmaximal, characteristic
3C2×#2984C211yesmaximal, characteristic
4#18062C214yesmaximal, complemented
5#18058C214yesmaximal, complemented
6#18032C218yesmaximal, complemented
7#17934C212yesmaximal, complemented
8#17931C212yesmaximal, complemented
9#17906C212yesmaximal, complemented
10#17903C212yesmaximal, complemented
11#17816C211yesmaximal, characteristic
12#17496C211yesmaximal, characteristic
13#17449C211yesmaximal, characteristic
14#17326C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C22×C4×Q8C2211yescharacteristic
17#3104C412yes—
18#3104C412yes—
19#3104C412yes—
20#3104C412yes—
21#3104C2211yescharacteristic
22#3104C2211yescharacteristic
23#3104C2214yes—
24#3104C2211yescharacteristic
25#3104C2211yescharacteristic
26C2×#484C412yes—
27C2×#477C412yes—
28C2×#462C412yes—
29C2×#451C412yes—
30#2986C2214yes—
31#2985—22no—
32#2984—21no—
33#2984—21no—
34#2984—22no—
35#2984C2214yes—
36#2984C2212yes—
37#2984C2212yes—
38#2984C2212yes—
39#2984C2212yes—
40#2983—21no—
41#2982—21no—
42#2962—22no—
43#2960C2212yes—
44#2960C2212yes—
45C2×#446—22no—
46C2×#446C2212yes—
47C2×C8×Q8C2212yes—
48C2×C4×Q16C2214yes—
49#2924—21no—
50#2924—21no—
51#2920C2211yescharacteristic
52#2920C2211yescharacteristic
53C2×#432—21no—
54C2×#432C2211yescharacteristic
55C2×#431—21no—
56C2×#431C2211yescharacteristic
57C2×#431C2211yescharacteristic
58#1593C2218yescomplemented
59#1591C2218yescomplemented
60#1587C2218yescomplemented
61#1587C2218yescomplemented
62#1584C2218yescomplemented
63#1584C2218yescomplemented
64#1579C2218yescomplemented
65C8×Q16C2218yescomplemented
66#1572C2214yes—
67#1571C2212yes—
68#1570C2212yes—
69#1568C2214yes—
70#1497C2214yes—
71#1491C2214yes—
72#1491C2214yes—
73#1487C2212yes—
74#1487C2212yes—
75#1481C2212yes—
76#1481C2212yes—
77#1479C2214yes—
78#1454C2214yes—
79#1452C2212yes—
80#1451C2212yes—
81#1450C2212yes—
82C8×M16C2212yes—
83C23×Q8C2×C411yescharacteristic
84C22×Q16C2×C414yes—
85#533—21no—
86#533—21no—
87#533—21no—
88#533—21no—
89#533C2×C412yes—
90#533C2×C411yescharacteristic
91#533C2×C411yescharacteristic
92#533C2311yescharacteristic
93#533C2311yescharacteristic
94#533C2312yes—
95C2×C4⋊Q8C2×C411yescharacteristic
96C2×C4⋊Q8C2×C411yescharacteristic
97C2×#125—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2×C412yes—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8C2×C412yes—
108C2×C4×Q8C2×C411yescharacteristic
109C2×C4×Q8C2×C412yes—
110C2×C4×Q8C2×C411yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×#117—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2×C411yescharacteristic
119C22×C42—21no—
120C22×C42C2311yescharacteristic
121#484C2×C412yes—
122#484C2×C412yes—
123#484C2×C414yes—
124#484C2×C414yes—
125#484C2×C412yes—
126#484C2×C412yes—
127#477C2×C412yes—
128#477C2×C412yes—
129#477C2×C414yes—
130#477C2×C414yes—
131#477C2×C412yes—
132#477C2×C412yes—
133#462C2×C414yes—
134#462C2×C412yes—
135#462C2×C412yes—
136#462C2×C412yes—
137#462C2×C412yes—
138#462C2×C414yes—
139#451C2×C412yes—
140#451C2×C412yes—
141#451C2×C414yes—
142#451C2×C414yes—
143#451C2×C412yes—
144#451C2×C412yes—
145#446—24no—
146#446—24no—
147#446—24no—
148#446—24no—
149#446—24no—
150#446—24no—
151#446—24no—
152#446—22no—
153#446—22no—
154#446—22no—
155#446—22no—
156#446C2314yes—
157#446C2314yes—
158#446C2314yes—
159#446C2314yes—
160C8×Q8—24no—
161C8×Q8—22no—
162C8×Q8—22no—
163C8×Q8—22no—
164C8×Q8—22no—
165C8×Q8C2314yes—
166C8×Q8C2314yes—
167C8×Q8C2314yes—
168C8×Q8C2314yes—
169C4×Q16C2×C414yes—
170C4×Q16C2×C414yes—
171C4×Q16C2×C414yes—
172C4×Q16C2×C414yes—
173C4×Q16C2314yes—
174C4×Q16C2314yes—
175C4×Q16C2314yes—
176C4×Q16C2314yes—
177#433—24no—
178#433—24no—
179#432—21no—
180#432—21no—
181#432—21no—
182#432—21no—
183#432—21no—
184#432—22no—
185#432—22no—
186#432—24no—
187#432—22no—
188#432—22no—
189#432C2312yes—
190#432C2312yes—
191#432C2312yes—
192#431—21no—
193#431—22no—
194#431—22no—
195#431—24no—
196#431—22no—
197#431—22no—
198#431C2×C412yes—
199#431C2×C412yes—
200#431D411yescharacteristic
201#431D411yescharacteristic
202#431D412yes—
203#431C2312yes—
204#431C2312yes—
205#431C2312yes—
206#431C2311yescharacteristic
207#431C2311yescharacteristic
208#431C2312yes—
209#431C2311yescharacteristic
210#431C2311yescharacteristic
211#431C2312yes—
212C2×#107C2×C414yes—
213#424—22no—
214#424—22no—
215#423—21no—
216#423C2312yes—
217C2×C4⋊C8—22no—
218C2×C4⋊C8—22no—
219C2×C4⋊C8C2×C412yes—
220C2×C4⋊C8D412yes—
221C2×C4⋊C8C2312yes—
222C2×#103C2×C414yes—
223C2×#103C2×C414yes—
224#407—21no—
225#407—21no—
226C2×C22⋊C8—21no—
227C2×C22⋊C8C2×C412yes—
228C2×C22⋊C8D411yescharacteristic
229C2×C22⋊C8D411yescharacteristic
230#405—22no—
231#405—22no—
232C4×M16—21no—
233C4×M16C2312yes—
234C2×#97—22no—
235C2×#97—21no—
236C2×C4×C8—22no—
237C2×C4×C8C2312yes—
238C2×C4×C8C2312yes—
239#335C2318yes—
240#334C2318yes—
241Q8⋊C8C2318yes—
242Q8⋊C8C2318yes—
243Q8⋊C8C2318yes—
244Q8⋊C8C2318yes—
245#322C2314yes—
246#322C2314yes—
247#322C2314yes—
248C82C2314yes—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—44no—
254C22×Q8C22×C411yescharacteristic
255C22×Q8C22×C412yes—
256C22×Q8C22×C412yes—
257C22×Q8C22×C411yescharacteristic
258C22×Q8C22×C412yes—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4C22×C411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4#4111yescharacteristic
264C2×Q16—48no—
265C2×Q16C22×C414yes—
266C2×Q16C22×C414yes—
267C2×Q16C22×C414yes—
268C2×Q16C22×C414yes—
269C2×M16—22no—
270C2×M16—22no—
271C22×C8—22no—
272C22×C8C22×C414yes—
273C22×C8C2×D412yes—
274C22×C8C2×D412yes—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8C22×C411yescharacteristic
280C4⋊Q8C22×C411yescharacteristic
281C4⋊Q8C22×C412yes—
282C4⋊Q8C22×C411yescharacteristic
283C4⋊Q8C22×C411yescharacteristic
284C4⋊Q8C22×C412yes—
285#125—21no—
286#125—21no—
287#125—22no—
288#125—21no—
289#125—21no—
290#125—22no—
291C22⋊Q8—22no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—22no—
299C22⋊Q8C22×C412yes—
300C22⋊Q8C22×C412yes—
301C22⋊Q8C22×C412yes—
302C22⋊Q8C22×C412yes—
303C4×Q8—21no—
304C4×Q8—22no—
305C4×Q8—21no—
306C4×Q8—21no—
307C4×Q8—22no—
308C4×Q8—21no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—22no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—21no—
318C4×Q8—22no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—21no—
324C4×Q8—21no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8C22×C412yes—
328C4×Q8C22×C412yes—
329C4×Q8C22×C412yes—
330C4×Q8C22×C412yes—
331C4×Q8C22×C412yes—
332C4×Q8C22×C412yes—
333C4×Q8C22×C412yes—
334C4×Q8C22×C412yes—
335C4×Q8C2412yes—
336C4×Q8C2412yes—
337C4×Q8C2412yes—
338C4×Q8C2412yes—
339#117—21no—
340#117—21no—
341#117—22no—
342#117—21no—
343#117—21no—
344#117—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4C22×C412yes—
362C2×C4⋊C4C22×C411yescharacteristic
363C2×C4⋊C4C22×C411yescharacteristic
364C2×C4⋊C4C22×C412yes—
365C2×C4⋊C4C22×C412yes—
366C2×C22⋊C4—22no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—21no—
373C2×C42—21no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42C22×C411yescharacteristic
377C2×C42C22×C411yescharacteristic
378C2×C42C22×C411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42#4111yescharacteristic
383C2×C42#4111yescharacteristic
384C2×C42#4111yescharacteristic
385C2×C42C2411yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388#107C22×C414yes—
389#107C22×C414yes—
390#107C22×C414yes—
391#107C22×C414yes—
392C4⋊C8—24no—
393C4⋊C8—24no—
394C4⋊C8—24no—
395C4⋊C8—24no—
396C4⋊C8—24no—
397C4⋊C8—24no—
398C4⋊C8—24no—
399C4⋊C8—24no—
400C4⋊C8—22no—
401C4⋊C8—22no—
402C4⋊C8—22no—
403C4⋊C8—22no—
404C4⋊C8C22×C412yes—
405C4⋊C8C22×C412yes—
406C4⋊C8C22×C414yes—
407C4⋊C8C2×D412yes—
408C4⋊C8C2×D412yes—
409C4⋊C8C2×D412yes—
410C4⋊C8C2×D412yes—
411C4⋊C8C2414yes—
412C4⋊C8C2414yes—
413#103C22×C414yes—
414#103C22×C414yes—
415#103C22×C414yes—
416#103C22×C414yes—
417#103C22×C414yes—
418#103C22×C414yes—
419#103C22×C414yes—
420#103C22×C414yes—
421C22⋊C8—22no—
422C22⋊C8—22no—
423C22⋊C8—22no—
424C22⋊C8—22no—
425C22⋊C8—22no—
426C22⋊C8—22no—
427C22⋊C8C22×C412yes—
428C22⋊C8C22×C412yes—
429C22⋊C8C22×C414yes—
430C22⋊C8C2×D411yescharacteristic
431C22⋊C8C2×D411yescharacteristic
432C22⋊C8C2×D411yescharacteristic
433C22⋊C8C2×D412yes—
434C22⋊C8C2×D411yescharacteristic
435C22⋊C8C2×D412yes—
436#97—24no—
437#97—24no—
438#97—24no—
439#97—24no—
440#97—22no—
441#97—22no—
442#97—22no—
443#97—22no—
444C4×C8—24no—
445C4×C8—24no—
446C4×C8—24no—
447C4×C8—24no—
448C4×C8—22no—
449C4×C8C22×C414yes—
450C4×C8C2×D412yes—
451C4×C8C2×D412yes—
452C4×C8C2414yes—
453C4×C8C2412yes—
454C4×C8C2412yes—
455C4×C8C2412yes—
456C4×C8C2412yes—
457C24C4×D411yescharacteristic
458C2×Q8—24no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—21no—
463C2×Q8—22no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—22no—
467C2×Q8—24no—
468C2×Q8—24no—
469C2×Q8—42no—
470C2×Q8—42no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—44no—
474C2×Q8—44no—
475C2×Q8—44no—
476C2×Q8—44no—
477C2×Q8C23×C412yes—
478C2×Q8C23×C412yes—
479C2×Q8C23×C412yes—
480C2×Q8C23×C412yes—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4C4×D411yescharacteristic
497C22×C4C4×D411yescharacteristic
498C22×C4C4×D411yescharacteristic
499C22×C4C4×D411yescharacteristic
500C22×C4C4×D411yescharacteristic
501C22×C4C4×D411yescharacteristic
502C22×C4C4×D411yescharacteristic
503C22×C4C4×D411yescharacteristic
504C22×C4C4×D411yescharacteristic
505C22×C4C23×C411yescharacteristic
506C22×C4C23×C411yescharacteristic
507C22×C4C23×C411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C2×#4111yescharacteristic
512C22×C4C2×#4111yescharacteristic
513C22×C4C2×#4111yescharacteristic
514Q16—28no—
515Q16—28no—
516Q16—44no—
517Q16—44no—
518M16—44no—
519M16—44no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—44no—
526C2×C8—44no—
527C2×C8C23×C414yes—
528C2×C8C23×C414yes—
529C2×C8C22×D412yes—
530C2×C8C22×D412yes—
531C2×C8C22×D412yes—
532C2×C8C22×D412yes—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—22no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—22no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—22no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4C23×C412yes—
564C4⋊C4C23×C412yes—
565C4⋊C4C23×C412yes—
566C4⋊C4C23×C412yes—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42C4×D411yescharacteristic
584C42C4×D411yescharacteristic
585C42C4×D411yescharacteristic
586C42C4×D411yescharacteristic
587C42C4×D411yescharacteristic
588C42C4×D411yescharacteristic
589C42C23×C411yescharacteristic
590C42C23×C411yescharacteristic
591C42C23×C411yescharacteristic
592C42C22×D411yescharacteristic
593C42C22×D411yescharacteristic
594C42C22×D411yescharacteristic
595C42C2×#4111yescharacteristic
596C42C2×#4111yescharacteristic
597C42C2×#4111yescharacteristic
598C42C2511yescharacteristic
599C23—22no—
600C23—22no—
601C23—41no—
602C23C2×C4×D411yescharacteristic
603C23C2×C4×D411yescharacteristic
604C23C2×C4×D411yescharacteristic
605Q8—24no—
606Q8—22no—
607Q8—24no—
608Q8—24no—
609Q8—22no—
610Q8—22no—
611Q8—44no—
612Q8—44no—
613Q8—44no—
614Q8—42no—
615Q8—42no—
616Q8—42no—
617Q8—82no—
618Q8#56814yes—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—82no—
654C2×C4C2×C4×D411yescharacteristic
655C2×C4C2×C4×D411yescharacteristic
656C2×C4C2×C4×D411yescharacteristic
657C2×C4C2×C4×D411yescharacteristic
658C2×C4C2×C4×D411yescharacteristic
659C2×C4C2×C4×D411yescharacteristic
660C2×C4C2×C4×D411yescharacteristic
661C2×C4C2×C4×D411yescharacteristic
662C2×C4C2×C4×D411yescharacteristic
663C2×C4#57814yes—
664C2×C4C24×C411yescharacteristic
665C2×C4C23×D411yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C8—24no—
668C8—24no—
669C8—22no—
670C8—22no—
671C8—22no—
672C8—22no—
673C8—44no—
674C8—44no—
675C8—44no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—44no—
684C22#295012yes—
685C22#295512yes—
686C22C22×C4×D411yescharacteristic
687C4—21no—
688C4—21no—
689C4—21no—
690C4—21no—
691C4—21no—
692C4—21no—
693C4—24no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—42no—
704C4#295012yes—
705C4C2×#56812yes—
706C4C2×#57812yes—
707C2—22no—
708C2—22no—
709C2—41no—
710C2#6014111yescharacteristic
711C2C2×#295011yescharacteristic
712C2C2×#295511yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radicalC24C4×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#431 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #18062C2semidirect
5. #18058C2semidirect
6. #18032C2semidirect
7. #17934C2semidirect
8. #17931C2semidirect
9. #17906C2semidirect
10. #17903C2semidirect
58. #1593C22semidirect
59. #1591C22semidirect
60. #1587C22semidirect
61. #1587C22semidirect
62. #1584C22semidirect
63. #1584C22semidirect
64. #1579C22semidirect
65. C8×Q16C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
#602502
C2×#29842
#180622
#180582
#180322
#179342
#179312
#179062
#179032
#178162
#174962
#174492
#173262