Finite Group Databasebeta

← #6354169

Subgroups — #6354169

Order 512.

1564 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60966C212yesmaximal
2C2×#3160C211yesmaximal, characteristic, complemented
3#60204C211yesmaximal, characteristic
4#19361C214yesmaximal, complemented
5#19360C214yesmaximal, complemented
6#19298C214yesmaximal, complemented
7#19273C214yesmaximal, complemented
8#19209C211yesmaximal, characteristic, complemented
9#19208C211yesmaximal, characteristic, complemented
10#19207C212yesmaximal, complemented
11#19193C211yesmaximal, characteristic, complemented
12#19192C211yesmaximal, characteristic, complemented
13#19191C212yesmaximal, complemented
14#17793C211yesmaximal, characteristic, complemented
15#17747C212yesmaximal, complemented
16#3444C2211yescharacteristic
17C2×#533C2212yes—
18#3252C2218yes—
19#3231—22no—
20#3230C2212yes—
21#3227C2212yes—
22#3210—22no—
23#3209C2212yes—
24#3206C2212yes—
25#3196—24no—
26#3185—22no—
27#3182—22no—
28#3160—21no—
29#3160—21no—
30#3160—21no—
31#3160—21no—
32#3160C2212yes—
33#3160C2212yes—
34#3160C2212yes—
35#3160C2212yes—
36C2×#500C2211yescharacteristic
37C2×#500C2211yescharacteristic
38C2×#499—21no—
39C2×#499—21no—
40C2×#498C2212yes—
41#3117—24no—
42#3116C2218yes—
43#2966—21no—
44#2966—21no—
45#2966C2212yes—
46#2966C2212yes—
47#2965—21no—
48#2965C2212yes—
49C8×#41—21no—
50C8×#41C2212yes—
51#2960C2211yescharacteristic
52#2960C2211yescharacteristic
53#2931C2211yescharacteristic
54#2931C2211yescharacteristic
55#2929—21no—
56#2929—21no—
57#2922—21no—
58#2920C2212yes—
59#2920C2211yescharacteristic
60C2×#427—21no—
61C2×#427—21no—
62C4×#131—21no—
63C22×C4×C8C2211yescharacteristic
64#1696C2214yescomplemented
65#1695C2214yescomplemented
66#1694C2214yescomplemented
67#1693C2214yescomplemented
68#1692C2218yescomplemented
69#1691C2218yescomplemented
70#1676C2214yescomplemented
71#1675C2214yescomplemented
72#1674C2214yescomplemented
73#1673C2214yescomplemented
74#1672C2218yescomplemented
75#1671C2218yescomplemented
76#1568C2212yes—
77#1568C2212yes—
78#1567C2212yes—
79#1566C2212yes—
80#1564C2212yes—
81#1564C2211yescharacteristic
82#1564C2211yescharacteristic
83#1559C2218yes—
84#1554C2218yes—
85#1549C2212yes—
86#1549C2212yes—
87#1548C2212yes—
88#1548C2212yes—
89C2×#131—21no—
90C2×#131—21no—
91C23×C8D411yescharacteristic
92C23×C8D411yescharacteristic
93#533—22no—
94#533—22no—
95#533—22no—
96#533—22no—
97#533—22no—
98#533—22no—
99#533D411yescharacteristic
100#533D411yescharacteristic
101#533C2314yes—
102#533C2314yes—
103#533C2311yescharacteristic
104#532D411yescharacteristic
105#532D411yescharacteristic
106#532C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109#529—21no—
110#529—21no—
111#529—21no—
112#529—21no—
113#529D412yes—
114#529C2312yes—
115C2×#125—22no—
116#525—21no—
117#525—21no—
118#525D412yes—
119C2×C22⋊Q8—22no—
120C2×C22⋊Q8—22no—
121C4×#41—21no—
122C4×#41—21no—
123C4×#41C2312yes—
124C2×C4×Q8—22no—
125C2×C4×Q8—22no—
126C2×#117—22no—
127C22×C42C2311yescharacteristic
128#500—22no—
129#500—21no—
130#500—21no—
131#500—21no—
132#500—21no—
133#500C2312yes—
134#500C2312yes—
135#500C2312yes—
136#500C2312yes—
137#499—22no—
138#499—22no—
139#499—22no—
140#499—22no—
141#499—22no—
142#499—22no—
143#499—22no—
144#499—22no—
145#498—22no—
146#498—22no—
147#498—22no—
148#498C2314yes—
149#498C2314yes—
150#494—22no—
151#492—22no—
152#487—22no—
153#487—22no—
154#487—22no—
155#486—22no—
156#485—28no—
157#484—24no—
158#481—24no—
159#479—28no—
160#477C2318yes—
161#476C2318yes—
162#470—28no—
163#468—28no—
164#464—28no—
165#462C2318yes—
166#461C2318yes—
167#452—28no—
168#451—24no—
169#448—24no—
170#446—21no—
171#446—21no—
172#446C2312yes—
173#446C2312yes—
174C8×Q8—21no—
175C8×Q8—21no—
176C8×Q8C2312yes—
177C8×Q8C2312yes—
178#441—28no—
179#440—28no—
180#436—21no—
181#436—21no—
182#436C2312yes—
183#436C2312yes—
184#435—22no—
185#435—22no—
186#435—22no—
187#435—22no—
188C8×D4—22no—
189C8×D4—22no—
190C8×D4—22no—
191C8×D4—22no—
192C8×D4—21no—
193C8×D4—21no—
194C8×D4C2312yes—
195C8×D4C2312yes—
196#433—22no—
197#433—22no—
198#432—22no—
199#432—21no—
200#432C2314yes—
201#432C2312yes—
202#431—22no—
203#431—21no—
204#431—22no—
205#431—22no—
206#431C2312yes—
207#431C2312yes—
208#431C2312yes—
209#427—22no—
210#427—22no—
211#427—22no—
212#427—22no—
213#427—22no—
214#427—22no—
215#427—22no—
216#427—22no—
217C2×#107—21no—
218C2×#107—21no—
219C2×#107—21no—
220C2×#107—21no—
221C2×#106—22no—
222C2×#106—22no—
223#424—21no—
224#424—21no—
225#423C2311yescharacteristic
226#423C2311yescharacteristic
227#423C2314yes—
228#408—21no—
229#408—21no—
230#407—21no—
231#407—22no—
232#405—21no—
233#405—21no—
234#405—22no—
235#405—22no—
236C4×M16—21no—
237C4×M16—22no—
238C4×M16—22no—
239C2×C4×C8—22no—
240C2×C4×C8D411yescharacteristic
241C2×C4×C8D411yescharacteristic
242C2×C4×C8D411yescharacteristic
243C2×C4×C8D411yescharacteristic
244C2×C4×C8D411yescharacteristic
245C2×C4×C8D411yescharacteristic
246C2×C4×C8C2311yescharacteristic
247C2×C4×C8C2311yescharacteristic
248C2×C4×C8C2311yescharacteristic
249C2×C4×C8C2311yescharacteristic
250C2×C4×C8C2311yescharacteristic
251C2×C4×C8C2311yescharacteristic
252#335C2312yes—
253#335C2312yes—
254#335C2314yes—
255#334C2314yes—
256#334C2312yes—
257#334C2312yes—
258#333C2318yes—
259#333C2318yes—
260#332C2318yes—
261#332C2318yes—
262C2×#41—21no—
263C2×#41—21no—
264C2×#41C2×D412yes—
265C22×Q8—22no—
266C22×Q8—22no—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D411yescharacteristic
269C23×C4#4111yescharacteristic
270C2×Q16—28no—
271C2×SD16—28no—
272#131—44no—
273#131—44no—
274C2×M16—21no—
275C2×M16—21no—
276C2×M16—24no—
277C2×M16—22no—
278C2×M16—22no—
279C22×C8—24no—
280C22×C8—24no—
281C22×C8—22no—
282C22×C8—22no—
283C22×C8C2×D411yescharacteristic
284C22×C8C2×D411yescharacteristic
285C22×C8C2×D411yescharacteristic
286C22×C8C2×D411yescharacteristic
287C22×C8C2×D411yescharacteristic
288C22×C8C2×D411yescharacteristic
289C22×C8C2×D411yescharacteristic
290C22×C8C2×D411yescharacteristic
291C22×C8C2×D411yescharacteristic
292C22×C8C2×D411yescharacteristic
293C22×C8C2×D411yescharacteristic
294C22×C8C2×D411yescharacteristic
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8C2×D411yescharacteristic
302C4⋊Q8C2×D411yescharacteristic
303C4⋊Q8C2×D411yescharacteristic
304C4⋊Q8C2414yes—
305C4⋊Q8C2414yes—
306#127C2×D411yescharacteristic
307#126—22no—
308#126—22no—
309#125—21no—
310#125—24no—
311#125—24no—
312#125—21no—
313#125—42no—
314#125—42no—
315#125C2×D412yes—
316#124—21no—
317#124—21no—
318#124C2×D412yes—
319#123—22no—
320#123—22no—
321#123—22no—
322C22⋊Q8—21no—
323C22⋊Q8—21no—
324C22⋊Q8—21no—
325C22⋊Q8—21no—
326C22⋊Q8—24no—
327C22⋊Q8—24no—
328C22⋊Q8—24no—
329C22⋊Q8—24no—
330C22⋊Q8—42no—
331C22⋊Q8—42no—
332C22⋊Q8—42no—
333C22⋊Q8—42no—
334C22⋊Q8C2×D412yes—
335C22⋊Q8C2×D412yes—
336#121—21no—
337#121—21no—
338#121—21no—
339#121—21no—
340#121C2×D412yes—
341#121C2×D412yes—
342#120—22no—
343C4×Q8—21no—
344C4×Q8—21no—
345C4×Q8—21no—
346C4×Q8—21no—
347C4×Q8—24no—
348C4×Q8—24no—
349C4×Q8—24no—
350C4×Q8—24no—
351C4×Q8—42no—
352C4×Q8—42no—
353C4×Q8—42no—
354C4×Q8—42no—
355C4×Q8C2×D412yes—
356C4×Q8C2412yes—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—21no—
364C4×D4—21no—
365C4×D4—21no—
366C4×D4—21no—
367C4×D4C2×D412yes—
368C4×D4C2412yes—
369#117—21no—
370#117—21no—
371#117—22no—
372#117—22no—
373#117—22no—
374#117—24no—
375#117—24no—
376#117—42no—
377#117—42no—
378#117C2×D412yes—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—42no—
384C2×C4⋊C4—42no—
385C2×C22⋊C4—42no—
386C2×C42—22no—
387C2×C42—42no—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42#4111yescharacteristic
395C2×C42#4111yescharacteristic
396C2×C42#4111yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400#107—22no—
401#107—22no—
402#107—22no—
403#107—22no—
404#107—22no—
405#107—22no—
406#107—22no—
407#107—22no—
408#107—28no—
409#106—24no—
410#106—24no—
411#106—24no—
412#106—24no—
413#106—28no—
414C4⋊C8—22no—
415C4⋊C8—22no—
416C4⋊C8—22no—
417C4⋊C8—22no—
418C4⋊C8—21no—
419C4⋊C8—21no—
420C4⋊C8—24no—
421C4⋊C8C2412yes—
422C4⋊C8C2412yes—
423C4⋊C8C2414yes—
424C4⋊C8C2414yes—
425#103—28no—
426#103—28no—
427#103—28no—
428#102—28no—
429C22⋊C8—22no—
430C22⋊C8—24no—
431C22⋊C8—22no—
432C22⋊C8—22no—
433C22⋊C8—22no—
434C22⋊C8—22no—
435C22⋊C8—22no—
436C22⋊C8—22no—
437C22⋊C8—22no—
438#97—22no—
439#97—22no—
440#97—22no—
441#97—24no—
442C4×C8—22no—
443C4×C8—22no—
444C4×C8—22no—
445C4×C8—22no—
446C4×C8—22no—
447C4×C8C2×D411yescharacteristic
448C4×C8C2×D411yescharacteristic
449C4×C8C2×D411yescharacteristic
450C4×C8C2×D411yescharacteristic
451C4×C8C2×D411yescharacteristic
452C4×C8C2×D411yescharacteristic
453C4×C8C2×D411yescharacteristic
454C4×C8C2×D411yescharacteristic
455C4×C8C2×D411yescharacteristic
456C4×C8C2×D411yescharacteristic
457C4×C8C2×D411yescharacteristic
458C4×C8C2×D411yescharacteristic
459C4×C8C2411yescharacteristic
460C4×C8C2411yescharacteristic
461C4×C8C2411yescharacteristic
462C4×C8C2411yescharacteristic
463C24#12111yescharacteristic
464#41—44no—
465#41—44no—
466#41—42no—
467#41—42no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—24no—
471C2×Q8—24no—
472C2×Q8—42no—
473C2×Q8—42no—
474C2×Q8—84no—
475C2×Q8—84no—
476C2×Q8C22×D412yes—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4C22×D412yes—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4#12111yescharacteristic
493C22×C4#12111yescharacteristic
494C22×C4#12111yescharacteristic
495C22×C4#12111yescharacteristic
496C22×C4#12111yescharacteristic
497C22×C4#12111yescharacteristic
498C22×C4#12111yescharacteristic
499C22×C4#12111yescharacteristic
500C22×C4#12111yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C2×#4111yescharacteristic
508C22×C4C2×#4111yescharacteristic
509C22×C4C2×#4111yescharacteristic
510Q16—88no—
511SD16—88no—
512M16—42no—
513M16—42no—
514M16—44no—
515M16—44no—
516M16—84no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—24no—
530C2×C8—24no—
531C2×C8—44no—
532C2×C8—44no—
533C2×C8#13714yes—
534C2×C8#13714yes—
535C2×C8C22×D411yescharacteristic
536C2×C8C22×D411yescharacteristic
537C2×C8C22×D411yescharacteristic
538C2×C8C22×D411yescharacteristic
539C2×C8C22×D411yescharacteristic
540C2×C8C22×D411yescharacteristic
541C2×C8C22×D411yescharacteristic
542C2×C8C22×D411yescharacteristic
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—24no—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4C22×D412yes—
566C4⋊C4C22×D412yes—
567C22⋊C4—22no—
568C22⋊C4—22no—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—22no—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—22no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—44no—
582C42—44no—
583C42#12111yescharacteristic
584C42#12111yescharacteristic
585C42#12111yescharacteristic
586C42#12111yescharacteristic
587C42#12111yescharacteristic
588C42#12111yescharacteristic
589C42C22×D411yescharacteristic
590C42C22×D411yescharacteristic
591C42C22×D411yescharacteristic
592C42C22×D411yescharacteristic
593C42C22×D411yescharacteristic
594C42C22×D411yescharacteristic
595C42C2×#4111yescharacteristic
596C42C2×#4111yescharacteristic
597C42C2×#4111yescharacteristic
598C42C2511yescharacteristic
599C23—22no—
600C23—22no—
601C23—41no—
602C23—41no—
603C23C2×#12111yescharacteristic
604C23C2×#12111yescharacteristic
605C23C2×#12111yescharacteristic
606Q8—24no—
607Q8—42no—
608Q8—42no—
609Q8—48no—
610Q8—84no—
611Q8—84no—
612Q8—84no—
613D4—24no—
614D4—44no—
615D4—44no—
616D4—44no—
617D4—42no—
618D4—42no—
619C2×C4—22no—
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—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—84no—
648C2×C4—84no—
649C2×C4C2×#12111yescharacteristic
650C2×C4C2×#12111yescharacteristic
651C2×C4C2×#12111yescharacteristic
652C2×C4C2×#12111yescharacteristic
653C2×C4C2×#12111yescharacteristic
654C2×C4C2×#12111yescharacteristic
655C2×C4C2×#12111yescharacteristic
656C2×C4C2×#12111yescharacteristic
657C2×C4C2×#12111yescharacteristic
658C2×C4C2×#13712yes—
659C2×C4C2×#13712yes—
660C2×C4C23×D411yescharacteristic
661C2×C4C23×D411yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C8—21no—
664C8—21no—
665C8—21no—
666C8—21no—
667C8—44no—
668C8—44no—
669C8—42no—
670C8—42no—
671C8C2×#13712yes—
672C8C2×#13712yes—
673C8C2×#13712yes—
674C8C2×#13712yes—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—21no—
681C22—21no—
682C22—21no—
683C22—44no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22#303912yes—
688C22C22×#12111yescharacteristic
689C4—21no—
690C4—21no—
691C4—21no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—21no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—84no—
701C4—84no—
702C4#303912yes—
703C4#306712yes—
704C4C22×#13711yescharacteristic
705C4C22×#13711yescharacteristic
706C4#358412yes—
707C2—22no—
708C2—41no—
709C2—41no—
710C2—82no—
711C2C2×#303911yescharacteristic
712C2#6045011yescharacteristic
713C2C2×#306711yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3160C2semidirect
4. #19361C2semidirect
5. #19360C2semidirect
6. #19298C2semidirect
7. #19273C2semidirect
8. #19209C2semidirect
9. #19208C2semidirect
10. #19207C2semidirect
11. #19193C2semidirect
12. #19192C2semidirect
13. #19191C2semidirect
14. #17793C2semidirect
15. #17747C2semidirect
64. #1696C22semidirect
65. #1695C22semidirect
66. #1694C22semidirect
67. #1693C22semidirect
68. #1692C22semidirect
69. #1691C22semidirect
70. #1676C22semidirect
71. #1675C22semidirect
72. #1674C22semidirect
73. #1673C22semidirect
74. #1672C22semidirect
75. #1671C22semidirect

Maximal subgroups

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

subgroupindex
#609662
C2×#31602
#602042
#193612
#193602
#192982
#192732
#192092
#192082
#192072
#191932
#191922
#191912
#177932
#177472