Finite Group Databasebeta

← #7505372

Subgroups — #7505372

Order 512.

2098 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35554C211yesmaximal, characteristic, complemented
2#35532C211yesmaximal, characteristic, complemented
3#31641C214yesmaximal, complemented
4#31638C214yesmaximal, complemented
5#31637C214yesmaximal, complemented
6#31634C214yesmaximal, complemented
7#23745C211yesmaximal, characteristic, complemented
8#23632C211yesmaximal, characteristic, complemented
9#23544C211yesmaximal, characteristic, complemented
10#23456C211yesmaximal, characteristic, complemented
11#23456C211yesmaximal, characteristic, complemented
12#23430C211yesmaximal, characteristic
13C2×#1957C211yesmaximal, characteristic, complemented
14#21083C211yesmaximal, characteristic, complemented
15#21020C211yesmaximal, characteristic, complemented
16#20932C211yesmaximal, characteristic, complemented
17#20932C211yesmaximal, characteristic, complemented
18#20929C211yesmaximal, characteristic, complemented
19#20650C211yesmaximal, characteristic, complemented
20C22×AGL(1,ℤ/8)C2211yescharacteristic
21C22×AGL(1,ℤ/8)C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C22×#127C2211yescharacteristic
24C2×#520C2211yescharacteristic
25C2×#518C2211yescharacteristic
26C2×#424C2211yescharacteristic
27C2×#418C2211yescharacteristic
28C2×#418C2211yescharacteristic
29C2×#416C2211yescharacteristic
30C2×#416C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×C4×C8C2211yescharacteristic
36#2567C2212yescomplemented
37#2566C2212yescomplemented
38#2565C2212yescomplemented
39#2542C2212yescomplemented
40#2449—24no—
41#2445—22no—
42#2445—22no—
43#2445C2214yes—
44#2445C2214yes—
45#2441—22no—
46#2440—22no—
47#2405—24no—
48#2317C2211yescharacteristic
49#2311C2211yescharacteristic
50#2309C2211yescharacteristic
51#2309C2211yescharacteristic
52#2309C2211yescharacteristic
53#2297C2211yescharacteristic
54#2296C2211yescharacteristic
55#2296C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2100C2214yescomplemented
58#2099C2214yescomplemented
59#2073C2214yescomplemented
60#2071C2214yescomplemented
61#2055C2214yescomplemented
62#2054C2214yescomplemented
63#2025C2214yescomplemented
64#2025C2214yescomplemented
65#2023C2214yescomplemented
66#2023C2214yescomplemented
67#2014C2212yes—
68#2014C2212yes—
69#2011C2212yes—
70#2011C2212yes—
71#1957C2214yes—
72#1957C2214yes—
73#1934C2212yescomplemented
74#1934C2212yescomplemented
75#1933C2212yescomplemented
76#1933C2212yescomplemented
77#1916C2214yescomplemented
78#1916C2214yescomplemented
79#1880C2214yescomplemented
80#1880C2214yescomplemented
81#1879C2214yescomplemented
82#1879C2214yescomplemented
83#1879C2214yescomplemented
84#1877C2214yescomplemented
85#1804C2214yescomplemented
86#1802C2214yescomplemented
87#1736C2211yescharacteristic
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1735C2211yescharacteristic
92#1735C2211yescharacteristic
93#1729C2211yescharacteristic
94#1729C2211yescharacteristic
95#1729C2211yescharacteristic
96#1728C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C23×D4—22no—
99C23×D4C2311yescharacteristic
100C23×D4C2311yescharacteristic
101C2×AGL(1,ℤ/8)—24no—
102C2×AGL(1,ℤ/8)—24no—
103C2×AGL(1,ℤ/8)—24no—
104C2×AGL(1,ℤ/8)—42no—
105C2×AGL(1,ℤ/8)—42no—
106C2×AGL(1,ℤ/8)—42no—
107C22×SD16C2312yes—
108C22×SD16C2312yes—
109C22×D8C2312yes—
110C22×D8C2312yes—
111C22×M16C2311yescharacteristic
112C22×M16C2311yescharacteristic
113C23×C8C2311yescharacteristic
114C23×C8C2311yescharacteristic
115#544—42no—
116#544—42no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C2312yes—
119C2×#127—24no—
120C2×#127—22no—
121C2×#127—42no—
122C2×#127C2312yes—
123C2×#127C2312yes—
124C2×#125—21no—
125C2×#125C2312yes—
126C2×#123—24no—
127C2×#123—24no—
128C2×C22⋊Q8—24no—
129C2×C22⋊Q8—22no—
130#520—42no—
131#520—42no—
132#518—24no—
133#518—42no—
134C2×C4×Q8C2312yes—
135C2×C4×D4—22no—
136C2×C4×D4—22no—
137C2×C4×D4C2314yes—
138C2×C4×D4C2312yes—
139C2×#117—21no—
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C2×#117C2311yescharacteristic
145C22×C4⋊C4—22no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2311yescharacteristic
149C22×C22⋊C4—22no—
150C22×C42C2311yescharacteristic
151#424—42no—
152C2×C4⋊C8C2314yes—
153#418—42no—
154#418—42no—
155#416—24no—
156#416—42no—
157C2×#103C2312yes—
158C2×#103C2312yes—
159C2×#102—42no—
160C2×#102—42no—
161C2×#102—42no—
162C2×#102—42no—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#102C2312yes—
166C2×#102C2314yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#102C2314yes—
170C2×#102C2312yes—
171C2×#102C2314yes—
172C2×#102C2312yes—
173C2×C4×C8—42no—
174C2×C4×C8C2312yes—
175C2×C4×C8C2312yes—
176#399—24no—
177#398—24no—
178#396—24no—
179#395—24no—
180#394—24no—
181#394—24no—
182#394—24no—
183#394—24no—
184#391C2311yescharacteristic
185#391C2312yes—
186#391C2311yescharacteristic
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—22no—
192#390C2314yes—
193#390C2314yes—
194#390C2314yes—
195#390C2312yes—
196#390C2311yescharacteristic
197#390C2311yescharacteristic
198#389—24no—
199#389—22no—
200#389D412yes—
201#389D412yes—
202#389C2312yes—
203#389C2312yes—
204#387—24no—
205#386—22no—
206#386—22no—
207#386—24no—
208#386—24no—
209#385—22no—
210#385—22no—
211#385—24no—
212#385D414yes—
213#384—22no—
214#384C2312yes—
215#384C2312yes—
216#384C2312yes—
217#382C2312yes—
218#380—22no—
219#379—22no—
220#379—22no—
221#379—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4C2312yes—
224C4×C4⋊C4C2312yes—
225C4×C22⋊C4—22no—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#336C2314yes—
235#336C2312yes—
236#336C2312yes—
237#336C2314yes—
238#336C2314yes—
239C25—21no—
240C25—21no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41#4114yes—
247C22×Q8—22no—
248C22×Q8C2411yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252C22×D4—24no—
253C22×D4—24no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—22no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4—22no—
261C22×D4—22no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—22no—
265C22×D4—48no—
266C22×D4—84no—
267C22×D4—84no—
268C22×D4#4114yes—
269C22×D4C2412yes—
270C22×D4C2412yes—
271C22×D4C2411yescharacteristic
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—24no—
276C23×C4—22no—
277C23×C4C2×D412yes—
278C23×C4C2411yescharacteristic
279AGL(1,ℤ/8)—416no—
280AGL(1,ℤ/8)—416no—
281C2×SD16—24no—
282C2×SD16—24no—
283C2×SD16—24no—
284C2×SD16—42no—
285C2×SD16—42no—
286C2×SD16—42no—
287C2×D8—24no—
288C2×D8—24no—
289C2×D8—24no—
290C2×D8—42no—
291C2×D8—42no—
292C2×D8—42no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—22no—
297C2×M16—22no—
298C2×M16—22no—
299C22×C8—22no—
300C22×C8—22no—
301C22×C8—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8—22no—
305C22×C8C2412yes—
306C22×C8C2412yes—
307C22×C8C2412yes—
308C22×C8C2412yes—
309C4⋊Q8—42no—
310C4⋊Q8—42no—
311C4⋊Q8—44no—
312#127—24no—
313#127—24no—
314#127—42no—
315#127—42no—
316#127—48no—
317#125—42no—
318#125—42no—
319#125—44no—
320#123—88no—
321#123—88no—
322C22⋊Q8—48no—
323C22⋊Q8—88no—
324C4×Q8—42no—
325C4×Q8—42no—
326C4×D4—28no—
327C4×D4—44no—
328C4×D4—42no—
329C4×D4—42no—
330C4×D4—48no—
331C4×D4—48no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—22no—
339#117—22no—
340#117—22no—
341#117—22no—
342#117—44no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—44no—
359C2×C4⋊C4—84no—
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D412yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2×D414yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2411yescharacteristic
370C2×C4⋊C4C2412yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—24no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—24no—
383C2×C22⋊C4—24no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—48no—
386C2×C22⋊C4C2×D412yes—
387C2×C22⋊C4C2×D412yes—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42C2×D412yes—
395C2×C42C2412yes—
396C2×C42C2412yes—
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C4⋊C8—44no—
400#103—24no—
401#103—42no—
402#102—24no—
403#102—44no—
404#102—44no—
405#102—44no—
406#102—44no—
407#102—42no—
408#102—42no—
409#102—42no—
410#102—42no—
411#102—42no—
412C4×C8—42no—
413C4×C8—42no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—24no—
419#95—24no—
420#95C2×D414yes—
421#95C2×D414yes—
422C24—21no—
423C24—22no—
424C24—24no—
425C24—22no—
426C24—42no—
427C24—42no—
428C24—42no—
429C24—42no—
430C24—42no—
431C24—82no—
432C24—82no—
433C24—82no—
434C24—82no—
435C24C22×D411yescharacteristic
436#41—28no—
437#41—28no—
438#41—44no—
439#41—44no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—44no—
446C2×Q8—84no—
447C2×Q8C2×#4112yes—
448C2×D4—24no—
449C2×D4—28no—
450C2×D4—24no—
451C2×D4—28no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—21no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—22no—
468C2×D4—48no—
469C2×D4—48no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—44no—
475C2×D4—44no—
476C2×D4—44no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—44no—
481C2×D4—44no—
482C2×D4—44no—
483C2×D4—48no—
484C2×D4—48no—
485C2×D4—84no—
486C2×D4—84no—
487C2×D4—832no—
488C2×D4C2×#4112yes—
489C2×D4C2×#4114yes—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—21no—
494C22×C4—24no—
495C22×C4—21no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—21no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—48no—
502C22×C4—44no—
503C22×C4—42no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—42no—
507C22×C4—84no—
508C22×C4—84no—
509C22×C4—84no—
510C22×C4—82no—
511C22×C4—82no—
512C22×C4C22×D414yes—
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C2×#4112yes—
519C22×C4C2×#4112yes—
520C22×C4H(5,𝔽2)12yes—
521C22×C4H(5,𝔽2)12yes—
522C22×C4H(5,𝔽2)12yes—
523C22×C42−1+412yes—
524C22×C4C2511yescharacteristic
525SD16—48no—
526SD16—48no—
527D8—48no—
528D8—48no—
529M16—44no—
530M16—44no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—44no—
544C2×C8—44no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—24no—
548C4⋊C4—28no—
549C4⋊C4—28no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C4⋊C4—44no—
566C4⋊C4—42no—
567C4⋊C4—42no—
568C4⋊C4—48no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—84no—
572C4⋊C4—84no—
573C22⋊C4—24no—
574C22⋊C4—24no—
575C22⋊C4—48no—
576C22⋊C4—44no—
577C22⋊C4—42no—
578C22⋊C4—42no—
579C22⋊C4—42no—
580C22⋊C4—44no—
581C22⋊C4—44no—
582C22⋊C4—84no—
583C22⋊C4—84no—
584C22⋊C4—84no—
585C22⋊C4—84no—
586C22⋊C4—84no—
587C22⋊C4—84no—
588C22⋊C4—84no—
589C42—24no—
590C42—21no—
591C42—21no—
592C42—22no—
593C42—21no—
594C42—21no—
595C42—22no—
596C42—22no—
597C42—22no—
598C42—44no—
599C42—42no—
600C42—42no—
601C42—42no—
602C23—22no—
603C23—22no—
604C23—24no—
605C23—22no—
606C23—41no—
607C23—42no—
608C23—41no—
609C23—42no—
610C23—42no—
611C23—44no—
612C23—42no—
613C23—44no—
614C23—48no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—44no—
619C23—44no—
620C23—88no—
621C23—88no—
622C23—84no—
623C23—162no—
624C23—162no—
625C23—164no—
626C23—164no—
627C23—162no—
628C23—162no—
629C23#53512yes—
630C23#54612yes—
631C23#54812yes—
632C23C23×D411yescharacteristic
633Q8—22no—
634Q8—22no—
635Q8—41no—
636Q8—41no—
637Q8—164no—
638D4—22no—
639D4—24no—
640D4—22no—
641D4—24no—
642D4—48no—
643D4—42no—
644D4—42no—
645D4—41no—
646D4—42no—
647D4—48no—
648D4—42no—
649D4—42no—
650D4—41no—
651D4—42no—
652D4—816no—
653D4—816no—
654D4—84no—
655C2×C4—216no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—22no—
662C2×C4—24no—
663C2×C4—22no—
664C2×C4—24no—
665C2×C4—24no—
666C2×C4—24no—
667C2×C4—24no—
668C2×C4—41no—
669C2×C4—41no—
670C2×C4—48no—
671C2×C4—41no—
672C2×C4—42no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—168no—
686C2×C4—164no—
687C2×C4—162no—
688C2×C4—162no—
689C2×C4#53512yes—
690C2×C4#53814yes—
691C2×C4#53914yes—
692C2×C4#54612yes—
693C2×C4#54812yes—
694C2×C4#55114yes—
695C2×C4#56214yes—
696C2×C4#57614yes—
697C2×C4#57714yes—
698C2×C4C22×#4111yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C2×C4C2×H(5,𝔽2)11yescharacteristic
701C2×C4C2×H(5,𝔽2)11yescharacteristic
702C2×C4C2×H(5,𝔽2)11yescharacteristic
703C2×C4C2×2−1+411yescharacteristic
704C8—42no—
705C8—42no—
706C8—42no—
707C8—42no—
708C22—22no—
709C22—24no—
710C22—22no—
711C22—24no—
712C22—44no—
713C22—42no—
714C22—44no—
715C22—48no—
716C22—81no—
717C22—82no—
718C22—84no—
719C22—81no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—84no—
724C22—82no—
725C22—82no—
726C22—84no—
727C22—168no—
728C22—168no—
729C22#269614yes—
730C22C2×#53511yescharacteristic
731C22C2×#54611yescharacteristic
732C22C2×#54811yescharacteristic
733C22C2×#57612yes—
734C22C2×#57712yes—
735C4—24no—
736C4—24no—
737C4—42no—
738C4—42no—
739C4—42no—
740C4—44no—
741C4—82no—
742C4—82no—
743C4—81no—
744C4—82no—
745C4—81no—
746C4—82no—
747C4#269614yes—
748C4C2×#53811yescharacteristic
749C4C2×#53911yescharacteristic
750C4C2×#55111yescharacteristic
751C4C2×#56211yescharacteristic
752C2—24no—
753C2—82no—
754C2—84no—
755C2—162no—
756C2C2×#269611yescharacteristic
757C2#6081711yescharacteristic
758C2#6082011yescharacteristic
759C2#6116712yes—
760C2#6132912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#23430C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35554C2semidirect
2. #35532C2semidirect
3. #31641C2semidirect
4. #31638C2semidirect
5. #31637C2semidirect
6. #31634C2semidirect
7. #23745C2semidirect
8. #23632C2semidirect
9. #23544C2semidirect
10. #23456C2semidirect
11. #23456C2semidirect
13. C2×#1957C2semidirect
14. #21083C2semidirect
15. #21020C2semidirect
16. #20932C2semidirect
17. #20932C2semidirect
18. #20929C2semidirect
19. #20650C2semidirect
36. #2567C22semidirect
37. #2566C22semidirect
38. #2565C22semidirect
39. #2542C22semidirect
57. #2100C22semidirect
58. #2099C22semidirect
59. #2073C22semidirect
60. #2071C22semidirect
61. #2055C22semidirect
62. #2054C22semidirect
63. #2025C22semidirect
64. #2025C22semidirect
65. #2023C22semidirect
66. #2023C22semidirect
73. #1934C22semidirect
74. #1934C22semidirect
75. #1933C22semidirect
76. #1933C22semidirect
77. #1916C22semidirect
78. #1916C22semidirect
79. #1880C22semidirect
80. #1880C22semidirect
81. #1879C22semidirect
82. #1879C22semidirect
83. #1879C22semidirect
84. #1877C22semidirect
85. #1804C22semidirect
86. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#355542
#355322
#316412
#316382
#316372
#316342
#237452
#236322
#235442
#234562
#234302
C2×#19572
#210832
#210202
#209322
#209292
#206502