Finite Group Databasebeta

← #6345401

Subgroups — #6345401

Order 512.

1584 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60560C211yesmaximal, characteristic
2#60285C211yesmaximal, characteristic
3#60248C211yesmaximal, characteristic, complemented
4#18049C214yesmaximal, complemented
5#18047C214yesmaximal, complemented
6#18023C214yesmaximal, complemented
7#18017C214yesmaximal, complemented
8#17953C212yesmaximal, complemented
9#17951C212yesmaximal, complemented
10#17925C212yesmaximal, complemented
11#17923C212yesmaximal, complemented
12#17821C211yesmaximal, characteristic, complemented
13#17503C211yesmaximal, characteristic
14C2×#1491C211yesmaximal, characteristic, complemented
15#17334C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18#3117C412yes—
19#3117C412yes—
20#3113C412yes—
21#3112C412yes—
22#3108C2214yes—
23#3104C2211yescharacteristic
24#3104C2211yescharacteristic
25#3103C2211yescharacteristic
26#3103C2211yescharacteristic
27#3092C412yes—
28#3078C412yes—
29#3037C412yes—
30#3004C412yes—
31#2995—22no—
32#2994C2214yes—
33#2985C2212yes—
34#2984—22no—
35#2984C2214yes—
36#2983—22no—
37#2983C2212yes—
38#2982C2212yes—
39#2981C2212yes—
40#2976—22no—
41#2971C2212yes—
42#2970—22no—
43#2970C2212yes—
44Q8×M16—22no—
45Q8×M16C2212yes—
46Q8×M16C2212yes—
47C4×#137C2214yes—
48#2923—22no—
49#2920—22no—
50#2920C2211yescharacteristic
51#2920C2211yescharacteristic
52#2920C2211yescharacteristic
53C2×#432C2211yescharacteristic
54C2×C4×M16C2211yescharacteristic
55#1593C2214yescomplemented
56#1592C2214yescomplemented
57#1591C2214yescomplemented
58#1590C2214yescomplemented
59#1588C2214yescomplemented
60#1587C2214yescomplemented
61#1587C2214yescomplemented
62#1585C2214yescomplemented
63#1584C2214yescomplemented
64#1584C2214yescomplemented
65#1583C2214yescomplemented
66#1582C2214yescomplemented
67#1579C2214yescomplemented
68SD16⋊C8C2214yescomplemented
69C8×Q16C2214yescomplemented
70C8×SD16C2214yescomplemented
71#1573C2214yes—
72#1572C2212yes—
73#1572C2212yes—
74#1569C2214yes—
75#1494C2212yes—
76#1494C2212yes—
77#1491C2212yes—
78#1491C2212yes—
79#1491C2214yes—
80#1489C2214yes—
81#1485C2212yes—
82#1484C2212yes—
83C2×Q8⋊C8C2212yes—
84C2×Q8⋊C8C2212yes—
85#1455C2212yes—
86#1455C2212yes—
87#1453C2212yes—
88#1453C2212yes—
89#1452C2212yes—
90#1450C2212yes—
91C23×Q8C2×C411yescharacteristic
92C2×#137C2×C414yes—
93C22×M16—21no—
94#533C2×C411yescharacteristic
95#533C2×C411yescharacteristic
96#533C2311yescharacteristic
97#532C2×C411yescharacteristic
98#532C2×C411yescharacteristic
99#532C2311yescharacteristic
100#529—21no—
101#529—21no—
102#529—21no—
103#529—21no—
104#529C2×C412yes—
105#529C2312yes—
106#525—21no—
107#525—21no—
108#525C2×C412yes—
109C4×#41—21no—
110C4×#41—21no—
111C4×#41C2312yes—
112C2×C4×Q8—21no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8—21no—
118C2×C4×Q8C2×C412yes—
119C2×C4×Q8C2×C412yes—
120C2×C4×Q8C2×C411yescharacteristic
121C2×C4×Q8C2×C411yescharacteristic
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4C2×C411yescharacteristic
128C22×C42—21no—
129C22×C42C2311yescharacteristic
130#484C2×C412yes—
131#484C2×C412yes—
132#483C2×C414yes—
133#481C2×C412yes—
134#481C2×C412yes—
135#480C2×C414yes—
136#479C2×C414yes—
137#479C2×C414yes—
138#477C2×C412yes—
139#477C2×C412yes—
140#475C2×C412yes—
141#475C2×C412yes—
142#464C2×C414yes—
143#464C2×C414yes—
144#462C2×C412yes—
145#462C2×C412yes—
146#460C2×C412yes—
147#460C2×C412yes—
148#451C2×C412yes—
149#451C2×C412yes—
150#451C2×C414yes—
151#448C2×C412yes—
152#448C2×C412yes—
153#448C2×C414yes—
154#446—24no—
155#446—24no—
156#446—24no—
157#446—24no—
158#446—24no—
159#446—24no—
160#446C2314yes—
161#446C2314yes—
162#446C2314yes—
163C8×Q8—24no—
164C8×Q8C2314yes—
165C8×Q8C2314yes—
166C8×Q8C2314yes—
167#441C2×C414yes—
168#441C2×C414yes—
169#440C2×C414yes—
170#440C2×C414yes—
171C4×Q16C2314yes—
172C4×Q16C2314yes—
173C4×SD16C2314yes—
174C4×SD16C2314yes—
175#436—24no—
176#436C2314yes—
177#435—24no—
178#435—24no—
179#435—24no—
180C8×D4—24no—
181C8×D4C2314yes—
182#433—24no—
183#433—24no—
184#432—21no—
185#432—21no—
186#432—21no—
187#432—24no—
188#432—24no—
189#432—22no—
190#432—22no—
191#432C2×C412yes—
192#432C2×C412yes—
193#432D412yes—
194#432D412yes—
195#432C2312yes—
196#432C2312yes—
197#432C2312yes—
198#431—21no—
199#431—24no—
200#431—24no—
201#431C2312yes—
202#431C2311yescharacteristic
203#431C2311yescharacteristic
204#431C2312yes—
205#431C2312yes—
206#431C2312yes—
207#428C2×C414yes—
208#424—24no—
209#423—22no—
210#423—22no—
211#423C2×C412yes—
212#423D412yes—
213#423C2312yes—
214C2×C4⋊C8C2312yes—
215#419C2×C414yes—
216#417C2×C414yes—
217#408—24no—
218#407—21no—
219#407C2×C412yes—
220#407D412yes—
221C2×C22⋊C8—21no—
222C4×M16—22no—
223C4×M16—22no—
224C4×M16—21no—
225C4×M16—21no—
226C4×M16—21no—
227C4×M16—22no—
228C4×M16C2312yes—
229C4×M16C2312yes—
230C4×M16C2312yes—
231C4×M16C2311yescharacteristic
232C4×M16C2311yescharacteristic
233C2×#97—21no—
234C2×#97—21no—
235C2×#97—22no—
236C2×C4×C8C2311yescharacteristic
237C2×C4×C8C2311yescharacteristic
238#335C2314yes—
239#335C2314yes—
240#334C2314yes—
241#334C2314yes—
242Q8⋊C8C2314yes—
243Q8⋊C8C2314yes—
244Q8⋊C8C2314yes—
245Q8⋊C8C2314yes—
246Q8⋊C8C2314yes—
247Q8⋊C8C2314yes—
248D4⋊C8C2314yes—
249D4⋊C8C2314yes—
250#322C2312yes—
251#322C2312yes—
252#322C2312yes—
253#322C2312yes—
254#322C2312yes—
255#322C2312yes—
256C82C2312yes—
257C82C2312yes—
258C2×#41—21no—
259C2×#41—21no—
260C2×#41C22×C412yes—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—44no—
265C22×Q8C22×C411yescharacteristic
266C22×Q8C22×C412yes—
267C22×Q8C22×C412yes—
268C22×Q8C22×C411yescharacteristic
269C23×C4—21no—
270C23×C4—21no—
271C23×C4C22×C411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4#4111yescharacteristic
274#137—48no—
275C2×Q16C22×C414yes—
276C2×Q16C22×C414yes—
277C2×SD16C22×C414yes—
278C2×SD16C22×C414yes—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—24no—
283C2×M16—44no—
284C2×M16C22×C414yes—
285C2×M16C2×D412yes—
286C2×M16C2×D412yes—
287C22×C8—21no—
288C22×C8—21no—
289C22×C8—22no—
290C22×C8—24no—
291C4⋊Q8C22×C411yescharacteristic
292C4⋊Q8C22×C411yescharacteristic
293C4⋊Q8C22×C411yescharacteristic
294#127C22×C411yescharacteristic
295#126—22no—
296#126—22no—
297#125—21no—
298#125—21no—
299#125C22×C412yes—
300#124—21no—
301#124—21no—
302#124C22×C412yes—
303#123—22no—
304#123—22no—
305#123—22no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310C22⋊Q8C22×C412yes—
311C22⋊Q8C22×C412yes—
312#121—21no—
313#121—21no—
314#121—21no—
315#121—21no—
316#121C22×C412yes—
317#121C22×C412yes—
318#120—22no—
319C4×Q8—21no—
320C4×Q8—21no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8—22no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—21no—
329C4×Q8—21no—
330C4×Q8—21no—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×Q8—21no—
334C4×Q8—21no—
335C4×Q8—22no—
336C4×Q8—22no—
337C4×Q8—21no—
338C4×Q8—21no—
339C4×Q8C22×C412yes—
340C4×Q8C22×C412yes—
341C4×Q8C22×C412yes—
342C4×Q8C22×C412yes—
343C4×Q8C22×C412yes—
344C4×Q8C22×C412yes—
345C4×Q8C22×C412yes—
346C4×Q8C2412yes—
347C4×Q8C2412yes—
348C4×Q8C2412yes—
349C4×D4—22no—
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—22no—
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4C22×C412yes—
360C4×D4C2412yes—
361#117—21no—
362#117—21no—
363#117—22no—
364#117—22no—
365#117—22no—
366#117C22×C412yes—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C22×C412yes—
379C2×C4⋊C4C22×C412yes—
380C2×C4⋊C4C22×C411yescharacteristic
381C2×C4⋊C4C22×C411yescharacteristic
382C2×C42—21no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42—22no—
387C2×C42—21no—
388C2×C42—22no—
389C2×C42—21no—
390C2×C42—22no—
391C2×C42C22×C411yescharacteristic
392C2×C42C22×C411yescharacteristic
393C2×C42C22×C411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42#4111yescharacteristic
398C2×C42#4111yescharacteristic
399C2×C42#4111yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403#107C22×C414yes—
404#107C22×C414yes—
405#106C22×C414yes—
406#106C22×C414yes—
407C4⋊C8—24no—
408C4⋊C8—24no—
409C4⋊C8—24no—
410C4⋊C8—24no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413C4⋊C8—22no—
414C4⋊C8—22no—
415C4⋊C8C22×C412yes—
416C4⋊C8C22×C412yes—
417C4⋊C8C22×C414yes—
418C4⋊C8C2×D414yes—
419C4⋊C8C2×D414yes—
420C4⋊C8C2414yes—
421C4⋊C8C2414yes—
422#103C22×C414yes—
423#103C22×C414yes—
424#103C22×C414yes—
425#103C22×C414yes—
426#103C22×C414yes—
427#103C22×C414yes—
428#102C22×C414yes—
429#102C22×C414yes—
430C22⋊C8—22no—
431C22⋊C8—22no—
432C22⋊C8—22no—
433C22⋊C8—22no—
434C22⋊C8—24no—
435C22⋊C8—24no—
436C22⋊C8C22×C414yes—
437C22⋊C8C22×C412yes—
438C22⋊C8C22×C412yes—
439C22⋊C8C2×D414yes—
440C22⋊C8C2×D414yes—
441#97—24no—
442#97—24no—
443#97—24no—
444#97—22no—
445#97—22no—
446#97—22no—
447#97—22no—
448#97—22no—
449#97C22×C414yes—
450#97C2×D412yes—
451#97C2×D412yes—
452C4×C8—24no—
453C4×C8—24no—
454C4×C8—24no—
455C4×C8—22no—
456C4×C8—21no—
457C4×C8—21no—
458C4×C8C2412yes—
459C4×C8C2412yes—
460C4×C8C2412yes—
461C4×C8C2412yes—
462C4×C8C2412yes—
463C4×C8C2412yes—
464C24C4×D411yescharacteristic
465#41—44no—
466#41—82no—
467#41—82no—
468C2×Q8—24no—
469C2×Q8—21no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—24no—
476C2×Q8—24no—
477C2×Q8—42no—
478C2×Q8—42no—
479C2×Q8—42no—
480C2×Q8—44no—
481C2×Q8—44no—
482C2×Q8—44no—
483C2×Q8C23×C412yes—
484C2×Q8C23×C412yes—
485C2×Q8C23×C412yes—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4C23×C412yes—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—44no—
507C22×C4—82no—
508C22×C4C4×D411yescharacteristic
509C22×C4C4×D411yescharacteristic
510C22×C4C4×D411yescharacteristic
511C22×C4C4×D411yescharacteristic
512C22×C4C4×D411yescharacteristic
513C22×C4C4×D411yescharacteristic
514C22×C4C4×D411yescharacteristic
515C22×C4C4×D411yescharacteristic
516C22×C4C4×D411yescharacteristic
517C22×C4C23×C411yescharacteristic
518C22×C4C23×C411yescharacteristic
519C22×C4C23×C411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C2×#4111yescharacteristic
524C22×C4C2×#4111yescharacteristic
525C22×C4C2×#4111yescharacteristic
526Q16—28no—
527Q16—44no—
528SD16—28no—
529SD16—44no—
530M16—44no—
531M16—42no—
532M16—42no—
533M16—44no—
534M16—44no—
535M16—44no—
536M16—84no—
537C2×C8—24no—
538C2×C8—22no—
539C2×C8—24no—
540C2×C8—24no—
541C2×C8—22no—
542C2×C8—42no—
543C2×C8—84no—
544C2×C8—82no—
545C2×C8C23×C414yes—
546C2×C8C23×C414yes—
547C2×C8C22×D414yes—
548C2×C8C22×D414yes—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—22no—
571C4⋊C4C23×C412yes—
572C4⋊C4C23×C412yes—
573C4⋊C4C23×C412yes—
574C4⋊C4C23×C412yes—
575C22⋊C4—22no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—22no—
579C22⋊C4—22no—
580C22⋊C4—22no—
581C22⋊C4—22no—
582C22⋊C4—22no—
583C22⋊C4—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42C4×D411yescharacteristic
596C42C4×D411yescharacteristic
597C42C4×D411yescharacteristic
598C42C4×D411yescharacteristic
599C42C4×D411yescharacteristic
600C42C4×D411yescharacteristic
601C42C23×C411yescharacteristic
602C42C23×C411yescharacteristic
603C42C23×C411yescharacteristic
604C42C22×D411yescharacteristic
605C42C22×D411yescharacteristic
606C42C22×D411yescharacteristic
607C42C2×#4111yescharacteristic
608C42C2×#4111yescharacteristic
609C42C2×#4111yescharacteristic
610C42C2511yescharacteristic
611C23—22no—
612C23—22no—
613C23—41no—
614C23—41no—
615C23C2×C4×D411yescharacteristic
616C23C2×C4×D411yescharacteristic
617C23C2×C4×D411yescharacteristic
618Q8—24no—
619Q8—22no—
620Q8—22no—
621Q8—24no—
622Q8—22no—
623Q8—44no—
624Q8—44no—
625Q8—42no—
626Q8—81no—
627Q8—81no—
628Q8#56814yes—
629D4—24no—
630D4—44no—
631D4—81no—
632D4—81no—
633D4—82no—
634D4—82no—
635C2×C4—24no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—82no—
667C2×C4—82no—
668C2×C4—82no—
669C2×C4—81no—
670C2×C4—81no—
671C2×C4C2×C4×D411yescharacteristic
672C2×C4C2×C4×D411yescharacteristic
673C2×C4C2×C4×D411yescharacteristic
674C2×C4C2×C4×D411yescharacteristic
675C2×C4C2×C4×D411yescharacteristic
676C2×C4C2×C4×D411yescharacteristic
677C2×C4C2×C4×D411yescharacteristic
678C2×C4C2×C4×D411yescharacteristic
679C2×C4C2×C4×D411yescharacteristic
680C2×C4C24×C411yescharacteristic
681C2×C4C23×D411yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C8—24no—
684C8—24no—
685C8—24no—
686C8—24no—
687C8—24no—
688C8—42no—
689C8—44no—
690C8—44no—
691C22—21no—
692C22—21no—
693C22—21no—
694C22—21no—
695C22—21no—
696C22—21no—
697C22—21no—
698C22—21no—
699C22—44no—
700C22—42no—
701C22—42no—
702C22—42no—
703C22#295612yes—
704C22C22×C4×D411yescharacteristic
705C4—21no—
706C4—21no—
707C4—21no—
708C4—21no—
709C4—21no—
710C4—21no—
711C4—21no—
712C4—24no—
713C4—21no—
714C4—21no—
715C4—21no—
716C4—42no—
717C4—42no—
718C4—42no—
719C4—42no—
720C4—42no—
721C4—42no—
722C4C2×#56812yes—
723C4#358612yes—
724C2—22no—
725C2—41no—
726C2—41no—
727C2—82no—
728C2#6014111yescharacteristic
729C2#6016911yescharacteristic
730C2C2×#295611yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#3444C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #2920 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #60248C2semidirect
4. #18049C2semidirect
5. #18047C2semidirect
6. #18023C2semidirect
7. #18017C2semidirect
8. #17953C2semidirect
9. #17951C2semidirect
10. #17925C2semidirect
11. #17923C2semidirect
12. #17821C2semidirect
14. C2×#1491C2semidirect
15. #17334C2semidirect
55. #1593C22semidirect
56. #1592C22semidirect
57. #1591C22semidirect
58. #1590C22semidirect
59. #1588C22semidirect
60. #1587C22semidirect
61. #1587C22semidirect
62. #1585C22semidirect
63. #1584C22semidirect
64. #1584C22semidirect
65. #1583C22semidirect
66. #1582C22semidirect
67. #1579C22semidirect
68. SD16⋊C8C22semidirect
69. C8×Q16C22semidirect
70. C8×SD16C22semidirect

Maximal subgroups

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

subgroupindex
#605602
#602852
#602482
#180492
#180472
#180232
#180172
#179532
#179512
#179252
#179232
#178212
#175032
C2×#14912
#173342