Finite Group Databasebeta

← #165343

Subgroups — #165343

Order 512.

946 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36375C211yesmaximal, characteristic
2#22473C211yesmaximal, characteristic
3#20684C211yesmaximal, characteristic
4#7731C211yesmaximal, characteristic, complemented
5#7721C211yesmaximal, characteristic, complemented
6#7695C211yesmaximal, characteristic, complemented
7#7695C211yesmaximal, characteristic, complemented
8#7681C211yesmaximal, characteristic, complemented
9#7671C211yesmaximal, characteristic, complemented
10#7648C211yesmaximal, characteristic, complemented
11#7632C211yesmaximal, characteristic, complemented
12#7607C211yesmaximal, characteristic, complemented
13#7593C211yesmaximal, characteristic, complemented
14#7583C211yesmaximal, characteristic, complemented
15#7579C211yesmaximal, characteristic, complemented
16C2×#552C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#436—21no—
19C2×#408—21no—
20#2855C2211yescharacteristic
21#2855C2211yescharacteristic
22#2851C2211yescharacteristic
23#2584C411yescharacteristic
24#2581C411yescharacteristic
25#2486C411yescharacteristic
26#2485C411yescharacteristic
27#2478—21no—
28#2475—21no—
29#2466—21no—
30#2462—21no—
31#2419—21no—
32#2417C411yescharacteristic
33#2415C411yescharacteristic
34#2406—21no—
35#2359—21no—
36#2352C411yescharacteristic
37#2344—21no—
38C4×C22⋊Q8C2211yescharacteristic
39#2296C2211yescharacteristic
40C2×#389C2211yescharacteristic
41#1984—21no—
42#1870—21no—
43#1870—21no—
44#1818—21no—
45#1818—21no—
46#1817C2211yescharacteristic
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1817C2211yescharacteristic
51#1817C2211yescharacteristic
52#1797C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54#1729—21no—
55#1728C2211yescharacteristic
56#1728C2211yescharacteristic
57#1728C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C22×#41D411yescharacteristic
76C22×M16—21no—
77C22×M16—21no—
78C22×M16—21no—
79C23×C8—21no—
80#552—44no—
81C2×#126—21no—
82C2×#125—21no—
83C2×#125C2×C411yescharacteristic
84C2×#123—21no—
85C2×#123—21no—
86C2×C22⋊Q8C2×C411yescharacteristic
87C2×C22⋊Q8C2×C411yescharacteristic
88C2×C22⋊Q8C2×C411yescharacteristic
89C4×#41—44no—
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8D411yescharacteristic
93C2×C4×D4C2×C411yescharacteristic
94C2×C4×D4C2×C411yescharacteristic
95C2×C4×D4C2×C411yescharacteristic
96C2×C4×D4D411yescharacteristic
97C2×C4×D4D411yescharacteristic
98C2×C4×D4D411yescharacteristic
99C2×#117—21no—
100C2×#117C2×C411yescharacteristic
101C2×#117C2×C411yescharacteristic
102C2×#117D411yescharacteristic
103C2×#117D411yescharacteristic
104C2×#117D411yescharacteristic
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2×C411yescharacteristic
107C22×C4⋊C4C2×C411yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111#436—44no—
112C2×C4⋊C8—21no—
113#408—44no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C4×C8—21no—
123#400—21no—
124#398—21no—
125#398—21no—
126#398—21no—
127#397—21no—
128#397—21no—
129#397—21no—
130#396—21no—
131#395—21no—
132#395—21no—
133#394—21no—
134#394—21no—
135#391C2×C411yescharacteristic
136#391C2×C411yescharacteristic
137#391C2311yescharacteristic
138#389—21no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—21no—
143#389C2×C411yescharacteristic
144#389C2×C411yescharacteristic
145#389C2×C411yescharacteristic
146#389C2×C411yescharacteristic
147#389C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#389C2311yescharacteristic
150#389C2311yescharacteristic
151#389C2311yescharacteristic
152#388—21no—
153#388—21no—
154#388—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#387—21no—
161#387—21no—
162#387—21no—
163#387—21no—
164#387—21no—
165#387—21no—
166#386—21no—
167#386—21no—
168#385—21no—
169#385—21no—
170#385—21no—
171#385—21no—
172#385—21no—
173#385—21no—
174#385—21no—
175#382C2×C411yescharacteristic
176#381—21no—
177#381—21no—
178#381—21no—
179#380—21no—
180#380C2×C411yescharacteristic
181#380C2×C411yescharacteristic
182#380C2×C411yescharacteristic
183#380C2×C411yescharacteristic
184#380C2×C411yescharacteristic
185C4×C4⋊C4—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4C2×C411yescharacteristic
189C4×C22⋊C4—21no—
190C4×C22⋊C4—21no—
191C2×#95—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336—21no—
201#336—21no—
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208#336C2311yescharacteristic
209#336C2311yescharacteristic
210C2×#41—24no—
211C2×#41—41no—
212C2×#41—41no—
213C2×#41—42no—
214C22×Q8C22⋊C411yescharacteristic
215C22×Q8C22×C411yescharacteristic
216C22×D4C22⋊C411yescharacteristic
217C22×D4C22⋊C411yescharacteristic
218C22×D4C22⋊C411yescharacteristic
219C23×C4—21no—
220C23×C4—21no—
221C23×C4C22×C411yescharacteristic
222C23×C4C22×C411yescharacteristic
223C23×C4C22×C411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C2×M16—42no—
231C2×M16—42no—
232C2×M16—42no—
233C2×M16—42no—
234C2×M16—42no—
235C2×M16—42no—
236C2×M16—42no—
237C2×M16—81no—
238C2×M16—81no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—42no—
248C22×C8—42no—
249C22×C8—42no—
250#126—44no—
251#125—42no—
252#125—44no—
253#123—44no—
254#123—44no—
255C22⋊Q8—22no—
256C22⋊Q8—41no—
257C22⋊Q8—42no—
258C22⋊Q8—42no—
259C4×Q8—42no—
260C4×Q8—42no—
261C4×Q8—42no—
262C4×D4—24no—
263C4×D4—24no—
264C4×D4—42no—
265C4×D4—42no—
266C4×D4—42no—
267C4×D4—42no—
268#117—22no—
269#117—42no—
270#117—42no—
271#117—44no—
272#117—41no—
273#117—42no—
274#117—42no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4C22⋊C411yescharacteristic
298C2×C4⋊C4C22⋊C411yescharacteristic
299C2×C4⋊C4C22⋊C411yescharacteristic
300C2×C4⋊C4C22⋊C411yescharacteristic
301C2×C4⋊C4C22⋊C411yescharacteristic
302C2×C4⋊C4C22⋊C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C411yescharacteristic
305C2×C4⋊C4C22×C411yescharacteristic
306C2×C4⋊C4C22×C411yescharacteristic
307C2×C4⋊C4C22×C411yescharacteristic
308C2×C4⋊C4C22×C411yescharacteristic
309C2×C4⋊C4C22×C411yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4C22⋊C411yescharacteristic
321C2×C22⋊C4C22⋊C411yescharacteristic
322C2×C22⋊C4C22⋊C411yescharacteristic
323C2×C22⋊C4C22⋊C411yescharacteristic
324C2×C22⋊C4C22⋊C411yescharacteristic
325C2×C22⋊C4C22⋊C411yescharacteristic
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42C22×C411yescharacteristic
334C2×C42C22×C411yescharacteristic
335C2×C42C22×C411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2411yescharacteristic
343C4⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—42no—
346C22⋊C8—42no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351C22⋊C8—81no—
352C4×C8—42no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376C24C2×C22⋊C411yescharacteristic
377C24C2×C22⋊C411yescharacteristic
378C24C2×C22⋊C411yescharacteristic
379#41—48no—
380#41—48no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—41no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4#9914yes—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—81no—
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C23×C411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459M16—84no—
460M16—84no—
461M16—84no—
462C2×C8—22no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—84no—
487C4⋊C4—21no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4#13112yes—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—81no—
538C22⋊C4—81no—
539C42—22no—
540C42—22no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—42no—
550C42—42no—
551C42—42no—
552C42—81no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23C2×#9912yes—
573C23C2×#9912yes—
574C23C22×C22⋊C411yescharacteristic
575Q8—44no—
576Q8—81no—
577Q8—81no—
578D4—44no—
579D4—44no—
580D4—44no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—84no—
655C2×C4—84no—
656C2×C4C2×#9912yes—
657C2×C4#42112yes—
658C2×C4C2×#13111yescharacteristic
659C2×C4#56811yescharacteristic
660C2×C4#57511yescharacteristic
661C2×C4#57511yescharacteristic
662C2×C4#57711yescharacteristic
663C2×C4#57811yescharacteristic
664C8—42no—
665C8—81no—
666C8—81no—
667C8—81no—
668C8—81no—
669C8—81no—
670C8—81no—
671C8—81no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22#288011yescharacteristic
694C22C22×#9911yescharacteristic
695C22#288911yescharacteristic
696C22#289611yescharacteristic
697C22#289811yescharacteristic
698C22C2×#42111yescharacteristic
699C22#290111yescharacteristic
700C4—21no—
701C4—21no—
702C4—21no—
703C4—21no—
704C4—22no—
705C4—22no—
706C4—21no—
707C4—21no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—42no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—81no—
717C4—81no—
718C4—81no—
719C4—81no—
720C4—81no—
721C4—81no—
722C4—81no—
723C2—42no—
724C2—42no—
725C2—81no—
726C2#1741611yescharacteristic
727C2#1751311yescharacteristic
728C2#1752511yescharacteristic
729C2#1758711yescharacteristic
730C2#1760811yescharacteristic
731C2#1768211yescharacteristic
732C2#1776711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7731C2semidirect
5. #7721C2semidirect
6. #7695C2semidirect
7. #7695C2semidirect
8. #7681C2semidirect
9. #7671C2semidirect
10. #7648C2semidirect
11. #7632C2semidirect
12. #7607C2semidirect
13. #7593C2semidirect
14. #7583C2semidirect
15. #7579C2semidirect

Maximal subgroups

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

subgroupindex
#363752
#224732
#206842
#77312
#77212
#76952
#76812
#76712
#76482
#76322
#76072
#75932
#75832
#75792