Finite Group Databasebeta

← #7501447

Subgroups — #7501447

Order 512.

2116 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic, complemented
2C2×#3097C211yesmaximal, characteristic, complemented
3#35677C211yesmaximal, characteristic
4#35574C211yesmaximal, characteristic, complemented
5#30660C214yesmaximal, complemented
6#30659C214yesmaximal, complemented
7#30658C214yesmaximal, complemented
8#30657C214yesmaximal, complemented
9#23735C211yesmaximal, characteristic, complemented
10#23614C211yesmaximal, characteristic, complemented
11#23552C211yesmaximal, characteristic, complemented
12#23506C211yesmaximal, characteristic, complemented
13#23486C211yesmaximal, characteristic
14#23484C211yesmaximal, characteristic, complemented
15#21154C211yesmaximal, characteristic, complemented
16#21047C211yesmaximal, characteristic, complemented
17#20914C211yesmaximal, characteristic
18#20880C211yesmaximal, characteristic, complemented
19#20651C211yesmaximal, characteristic, complemented
20C2×#536C2211yescharacteristic
21C2×#535C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#520C2211yescharacteristic
25C2×#519C2211yescharacteristic
26C2×#489—28no—
27C2×#489C2214yes—
28C2×#489C2214yes—
29#3097—28no—
30C2×#484C2212yescomplemented
31C2×#485C2212yescomplemented
32C2×#485C2212yescomplemented
33C2×#483C2212yescomplemented
34C2×#428C2211yescharacteristic
35C2×#428C2211yescharacteristic
36C2×#427C2211yescharacteristic
37C2×#427C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#419C2211yescharacteristic
40C2×#419C2211yescharacteristic
41C2×#418C2211yescharacteristic
42C2×#418C2211yescharacteristic
43C22×#103C2211yescharacteristic
44C22×#103C2211yescharacteristic
45C22×#102C2211yescharacteristic
46C22×#102C2211yescharacteristic
47C2×#408C2211yescharacteristic
48C2×#408C2211yescharacteristic
49C22×#97C2211yescharacteristic
50#2579—22no—
51#2548—22no—
52#2545—22no—
53#2452—22no—
54#2451—22no—
55#2440C2214yescomplemented
56#2439C2214yescomplemented
57#2426C2214yes—
58#2426C2214yes—
59#2411—22no—
60#2411—22no—
61#2409—22no—
62#2317C2211yescharacteristic
63#2316C2211yescharacteristic
64#2311C2211yescharacteristic
65#2309C2211yescharacteristic
66#2297C2211yescharacteristic
67#2296C2211yescharacteristic
68#2296C2211yescharacteristic
69#2094C2212yes—
70#2093C2212yes—
71#2092C2212yes—
72#2092C2212yes—
73#2067C2214yes—
74#2065C2214yes—
75#2058C2214yescomplemented
76#2057C2214yescomplemented
77#2042C2214yes—
78#2042C2214yes—
79#2036C2214yescomplemented
80#2035C2214yescomplemented
81#2034C2214yes—
82#2033C2214yes—
83#1944C2212yescomplemented
84#1944C2212yescomplemented
85#1943C2212yescomplemented
86#1943C2212yescomplemented
87#1924C2214yescomplemented
88#1924C2214yescomplemented
89#1875C2214yes—
90#1874C2214yes—
91#1865C2214yescomplemented
92#1863C2214yescomplemented
93#1805C2214yescomplemented
94#1803C2214yescomplemented
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×Q8—21no—
103C23×Q8C2311yescharacteristic
104C23×D4C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#536—24no—
110#536—42no—
111#535—24no—
112#535—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—24no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8—42no—
117C2×C4⋊Q8C2314yes—
118C2×#127—21no—
119C2×#124—21no—
120C2×#124—21no—
121C2×#124—24no—
122C2×#124—22no—
123C2×#124—42no—
124C2×#124C2314yes—
125C2×#123—22no—
126C2×C22⋊Q8—24no—
127C2×C22⋊Q8C2312yes—
128C2×C22⋊Q8C2312yes—
129C2×#121C2314yes—
130C2×#120—22no—
131#520—24no—
132#520—42no—
133#519—24no—
134#519—42no—
135C2×C4×Q8—21no—
136C2×C4×Q8—22no—
137C2×C4×Q8C2314yes—
138C2×C4×D4—22no—
139C2×C4×D4—21no—
140C2×#117—22no—
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C2×#117C2312yes—
145C2×#117C2311yescharacteristic
146C2×#117C2311yescharacteristic
147C22×C4⋊C4—21no—
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2312yes—
150C22×C4⋊C4C2311yescharacteristic
151C22×C22⋊C4—21no—
152C22×C22⋊C4—21no—
153C22×C42C2311yescharacteristic
154#489—216no—
155#489—216no—
156#485—28no—
157#485—28no—
158#484—28no—
159#483—28no—
160#428—24no—
161#428—42no—
162#427—24no—
163#427—24no—
164C2×#107C2312yes—
165C2×#107C2311yescharacteristic
166C2×#107C2311yescharacteristic
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#106C2312yes—
170C2×#106C2311yescharacteristic
171C2×#106C2311yescharacteristic
172C2×#106C2312yes—
173C2×#106C2312yes—
174#424—42no—
175C2×C4⋊C8C2314yes—
176#419—24no—
177#419—42no—
178#418—42no—
179#418—42no—
180C2×#103—24no—
181C2×#103D418yescomplemented
182C2×#103C2312yes—
183C2×#103C2314yes—
184C2×#103C2312yes—
185C2×#103C2314yes—
186C2×#103C2312yes—
187C2×#103C2312yes—
188C2×#102—24no—
189C2×#102—24no—
190C2×#102C2314yes—
191C2×#102C2312yes—
192C2×#102C2312yes—
193C2×#102C2314yes—
194C2×#102C2312yes—
195C2×#102C2312yes—
196#408—24no—
197#408—42no—
198C2×C22⋊C8C2312yes—
199C2×C22⋊C8C2312yes—
200C2×C22⋊C8C2314yes—
201C2×#97—24no—
202C2×#97C2312yes—
203C2×#97C2312yes—
204#398—24no—
205#396—24no—
206#396—24no—
207#396—24no—
208#395—24no—
209#394—24no—
210#394—24no—
211#393—24no—
212#391—21no—
213#391—24no—
214#391D412yes—
215#390—21no—
216#390—22no—
217#390—24no—
218#390—24no—
219#390D412yes—
220#390C2312yes—
221#390C2312yes—
222#389—22no—
223#389—22no—
224#389C2312yes—
225#389C2312yes—
226#389C2314yes—
227#389C2312yes—
228#389C2312yes—
229#388—24no—
230#388—24no—
231#387—24no—
232#386—22no—
233#386D412yes—
234#386D412yes—
235#385—21no—
236#385—21no—
237#385—21no—
238#385—21no—
239#385—22no—
240#385—24no—
241#384C2312yes—
242#382C2312yes—
243#380—22no—
244#380C2314yes—
245#379—22no—
246C4×C4⋊C4—22no—
247C4×C4⋊C4—22no—
248C4×C22⋊C4C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2312yes—
252#340C2312yes—
253#340C2312yes—
254#340C2312yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C25—21no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C22×Q8—21no—
266C22×Q8—24no—
267C22×Q8—22no—
268C22×Q8—22no—
269C22×Q8—22no—
270C22×Q8—22no—
271C22×Q8—22no—
272C22×Q8—44no—
273C22×Q8—84no—
274C22×Q8C2×D414yes—
275C22×Q8C2×D411yescharacteristic
276C22×Q8C2412yes—
277C22×D4—24no—
278C22×D4—22no—
279C22×D4—21no—
280C22×D4—22no—
281C22×D4—22no—
282C22×D4—22no—
283C22×D4—22no—
284C22×D4—22no—
285C22×D4C2×D412yes—
286C22×D4C2×D411yescharacteristic
287C22×D4C2412yes—
288C23×C4—21no—
289C23×C4—21no—
290C23×C4—21no—
291C23×C4—22no—
292C23×C4C2×D411yescharacteristic
293C23×C4C2×D411yescharacteristic
294C23×C4C2412yes—
295C23×C4C2411yescharacteristic
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C2×M16—22no—
300C2×M16—22no—
301C2×M16—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8—22no—
307C22×C8C2×D414yes—
308C22×C8C2412yes—
309C22×C8C2412yes—
310C22×C8C2412yes—
311C22×C8C2411yescharacteristic
312C22×C8C2411yescharacteristic
313C4⋊Q8—28no—
314C4⋊Q8—44no—
315C4⋊Q8—48no—
316C4⋊Q8—44no—
317#127—44no—
318#124—28no—
319#124—44no—
320#124—48no—
321#124—44no—
322#124—44no—
323#123—48no—
324C22⋊Q8—24no—
325C22⋊Q8—24no—
326C22⋊Q8—42no—
327C22⋊Q8—42no—
328C22⋊Q8—88no—
329#121—28no—
330#121—44no—
331#120—48no—
332C4×Q8—28no—
333C4×Q8—44no—
334C4×Q8—48no—
335C4×Q8—44no—
336C4×D4—44no—
337C4×D4—48no—
338#117—22no—
339#117—22no—
340#117—22no—
341#117—22no—
342#117—22no—
343#117—22no—
344#117—22no—
345#117—22no—
346#117—22no—
347#117—24no—
348#117—48no—
349#117—42no—
350#117#4114yes—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—42no—
369C2×C4⋊C4—44no—
370C2×C4⋊C4C2×D412yes—
371C2×C4⋊C4C2×D414yes—
372C2×C4⋊C4C2411yescharacteristic
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2411yescharacteristic
375C2×C4⋊C4C2411yescharacteristic
376C2×C4⋊C4C2411yescharacteristic
377C2×C4⋊C4C2412yes—
378C2×C4⋊C4C2412yes—
379C2×C4⋊C4C2412yes—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—24no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—24no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—44no—
394C2×C22⋊C4—44no—
395C2×C22⋊C4C2×D412yes—
396C2×C22⋊C4C2×D414yes—
397C2×C22⋊C4C2×D414yes—
398C2×C42—21no—
399C2×C42—22no—
400C2×C42—22no—
401C2×C42—24no—
402C2×C42—21no—
403C2×C42—22no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42C2×D412yes—
407C2×C42C2412yes—
408C2×C42C2412yes—
409#107—22no—
410#107—22no—
411#107—24no—
412#107—24no—
413#107—42no—
414#106—22no—
415#106—22no—
416#106—24no—
417#106—24no—
418#106—42no—
419C4⋊C8—44no—
420#103—28no—
421#103—24no—
422#103—24no—
423#103—44no—
424#103C2×D418yes—
425#103C2×D418yes—
426#102—24no—
427#102—24no—
428#102—28no—
429#102—44no—
430#102—42no—
431#102—42no—
432C22⋊C8—24no—
433C22⋊C8—24no—
434C22⋊C8—44no—
435#97—24no—
436#97—24no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—24no—
442#95—24no—
443#95—24no—
444#95C2×D414yes—
445#95C2×D414yes—
446C24—22no—
447C24—42no—
448C24—42no—
449C24—42no—
450C24—82no—
451C24—82no—
452C24C22×D412yes—
453C24C22×D411yescharacteristic
454#41—44no—
455#41—44no—
456#41—44no—
457#41—44no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—24no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—28no—
464C2×Q8—28no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—48no—
470C2×Q8—48no—
471C2×Q8—42no—
472C2×Q8—44no—
473C2×Q8—42no—
474C2×Q8—44no—
475C2×Q8—84no—
476C2×Q8—816no—
477C2×Q8C22×D414yes—
478C2×D4—24no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—22no—
484C2×D4—22no—
485C2×D4—22no—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—48no—
491C2×D4—42no—
492C2×D4—44no—
493C2×D4—44no—
494C2×D4—42no—
495C2×D4—42no—
496C2×D4—44no—
497C2×D4—44no—
498C2×D4—84no—
499C2×D4—84no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—24no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—42no—
520C22×C4—42no—
521C22×C4—44no—
522C22×C4—42no—
523C22×C4—42no—
524C22×C4—42no—
525C22×C4—82no—
526C22×C4—82no—
527C22×C4—84no—
528C22×C4C22×D412yes—
529C22×C4C22×D414yes—
530C22×C4C22×D412yes—
531C22×C4C22×D412yes—
532C22×C4C22×D411yescharacteristic
533C22×C4C22×D412yes—
534C22×C4C2×#4112yes—
535C22×C4H(5,𝔽2)12yes—
536C22×C4H(5,𝔽2)12yes—
537C22×C42−1+412yes—
538C22×C42−1+412yes—
539C22×C4C2511yescharacteristic
540M16—44no—
541M16—44no—
542C2×C8—28no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—22no—
550C2×C8—22no—
551C2×C8—22no—
552C2×C8—22no—
553C2×C8—22no—
554C2×C8—22no—
555C2×C8—22no—
556C2×C8—44no—
557C2×C8C22×D412yes—
558C2×C8C22×D412yes—
559C4⋊C4—24no—
560C4⋊C4—24no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—21no—
565C4⋊C4—21no—
566C4⋊C4—21no—
567C4⋊C4—21no—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—21no—
573C4⋊C4—22no—
574C4⋊C4—21no—
575C4⋊C4—44no—
576C4⋊C4—48no—
577C4⋊C4—48no—
578C4⋊C4—44no—
579C4⋊C4—44no—
580C4⋊C4—44no—
581C4⋊C4—44no—
582C4⋊C4—42no—
583C4⋊C4—82no—
584C4⋊C4—82no—
585C4⋊C4—84no—
586C4⋊C4—84no—
587C4⋊C4C22×D414yes—
588C4⋊C4C2×#4114yes—
589C22⋊C4—24no—
590C22⋊C4—24no—
591C22⋊C4—44no—
592C22⋊C4—44no—
593C22⋊C4—48no—
594C22⋊C4—48no—
595C22⋊C4—48no—
596C22⋊C4—44no—
597C22⋊C4—44no—
598C22⋊C4—44no—
599C22⋊C4—42no—
600C22⋊C4—42no—
601C22⋊C4—42no—
602C22⋊C4—84no—
603C22⋊C4—84no—
604C22⋊C4—84no—
605C42—24no—
606C42—24no—
607C42—22no—
608C42—22no—
609C42—22no—
610C42—22no—
611C42—44no—
612C42—48no—
613C42—42no—
614C42—42no—
615C42—42no—
616C23—22no—
617C23—22no—
618C23—24no—
619C23—24no—
620C23—22no—
621C23—22no—
622C23—22no—
623C23—44no—
624C23—44no—
625C23—42no—
626C23—42no—
627C23—42no—
628C23—84no—
629C23—84no—
630C23—84no—
631C23—164no—
632C23—164no—
633C23#53612yes—
634C23D4212yes—
635C23#54612yes—
636C23C23×D411yescharacteristic
637Q8—24no—
638Q8—24no—
639Q8—48no—
640Q8—44no—
641Q8—42no—
642Q8—42no—
643Q8—816no—
644D4—48no—
645D4—44no—
646D4—42no—
647D4—42no—
648D4—42no—
649D4—42no—
650D4—84no—
651D4—168no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—24no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—24no—
663C2×C4—24no—
664C2×C4—24no—
665C2×C4—24no—
666C2×C4—22no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—48no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—41no—
687C2×C4—41no—
688C2×C4—41no—
689C2×C4—41no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—84no—
693C2×C4—84no—
694C2×C4—164no—
695C2×C4#53612yes—
696C2×C4#53714yes—
697C2×C4#54214yes—
698C2×C4D4212yes—
699C2×C4#54612yes—
700C2×C4#54714yes—
701C2×C4#56214yes—
702C2×C4C23×D411yescharacteristic
703C2×C4C22×#4111yescharacteristic
704C2×C4C2×H(5,𝔽2)11yescharacteristic
705C2×C4C2×H(5,𝔽2)11yescharacteristic
706C2×C4C2×2−1+411yescharacteristic
707C2×C4C2×2−1+411yescharacteristic
708C8—22no—
709C8—22no—
710C8—42no—
711C8—42no—
712C8—42no—
713C22—24no—
714C22—22no—
715C22—22no—
716C22—24no—
717C22—44no—
718C22—42no—
719C22—42no—
720C22—42no—
721C22—44no—
722C22—84no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—84no—
727C22—168no—
728C22#267214yes—
729C22C2×#53611yescharacteristic
730C22C2×D4211yescharacteristic
731C22C2×#54611yescharacteristic
732C22#358512yes—
733C22#358612yes—
734C4—44no—
735C4—44no—
736C4—42no—
737C4—44no—
738C4—42no—
739C4—44no—
740C4—42no—
741C4—82no—
742C4—81no—
743C4—82no—
744C4—81no—
745C4—81no—
746C4—81no—
747C4#267214yes—
748C4C2×#53711yescharacteristic
749C4C2×#54211yescharacteristic
750C4C2×#54711yescharacteristic
751C4C2×#56211yescharacteristic
752C2—24no—
753C2—82no—
754C2—162no—
755C2C2×#267211yescharacteristic
756C2#6088512yes—
757C2#6105911yescharacteristic
758C2#6106111yescharacteristic
759C2#6122212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#489C2semidirect
2. C2×#3097C2semidirect
4. #35574C2semidirect
5. #30660C2semidirect
6. #30659C2semidirect
7. #30658C2semidirect
8. #30657C2semidirect
9. #23735C2semidirect
10. #23614C2semidirect
11. #23552C2semidirect
12. #23506C2semidirect
14. #23484C2semidirect
15. #21154C2semidirect
16. #21047C2semidirect
18. #20880C2semidirect
19. #20651C2semidirect
30. C2×#484C22semidirect
31. C2×#485C22semidirect
32. C2×#485C22semidirect
33. C2×#483C22semidirect
55. #2440C22semidirect
56. #2439C22semidirect
75. #2058C22semidirect
76. #2057C22semidirect
79. #2036C22semidirect
80. #2035C22semidirect
83. #1944C22semidirect
84. #1944C22semidirect
85. #1943C22semidirect
86. #1943C22semidirect
87. #1924C22semidirect
88. #1924C22semidirect
91. #1865C22semidirect
92. #1863C22semidirect
93. #1805C22semidirect
94. #1803C22semidirect
181. C2×#103D4semidirect

Maximal subgroups

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

subgroupindex
C22×#4892
C2×#30972
#356772
#355742
#306602
#306592
#306582
#306572
#237352
#236142
#235522
#235062
#234862
#234842
#211542
#210472
#209142
#208802
#206512