Finite Group Databasebeta

← #164083

Subgroups — #164083

Order 512.

940 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35309C211yesmaximal, characteristic
2#22471C211yesmaximal, characteristic
3#22469C211yesmaximal, characteristic
4#7731C211yesmaximal, characteristic, complemented
5#7721C211yesmaximal, characteristic, complemented
6#7707C211yesmaximal, characteristic, complemented
7#7695C211yesmaximal, characteristic, complemented
8#7691C211yesmaximal, characteristic, complemented
9#7681C211yesmaximal, characteristic, complemented
10#7667C211yesmaximal, characteristic, complemented
11#7655C211yesmaximal, characteristic, complemented
12#7600C211yesmaximal, characteristic, complemented
13#7589C211yesmaximal, characteristic, complemented
14#7583C211yesmaximal, characteristic, complemented
15#7566C211yesmaximal, characteristic, complemented
16C2×#541C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#435—21no—
19C2×C8×D4—21no—
20#2857C2211yescharacteristic
21#2855C2211yescharacteristic
22#2855C2211yescharacteristic
23#2851C2211yescharacteristic
24#2629C411yescharacteristic
25#2621C411yescharacteristic
26#2616C411yescharacteristic
27#2616C411yescharacteristic
28#2468—21no—
29#2468—21no—
30#2466—21no—
31#2466—21no—
32#2462—21no—
33#2462—21no—
34#2460—21no—
35#2460—21no—
36#2323C411yescharacteristic
37#2323C411yescharacteristic
38#2314C411yescharacteristic
39C4×C22⋊Q8C2211yescharacteristic
40C4×C22⋊Q8C2211yescharacteristic
41#1985—21no—
42#1984—21no—
43#1870—21no—
44#1869—21no—
45#1869—21no—
46#1869—21no—
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1817C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1749C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55#1728C2211yescharacteristic
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C22×#41C2×C411yescharacteristic
76C22×M16—21no—
77C23×C8—21no—
78C23×C8—21no—
79C23×C8—21no—
80#541—44no—
81C2×C4⋊Q8C2×C411yescharacteristic
82C2×#126—21no—
83C2×#126—21no—
84C2×#125—21no—
85C2×#125—21no—
86C2×#125C2×C411yescharacteristic
87C2×C22⋊Q8C2×C411yescharacteristic
88C2×C22⋊Q8C2×C411yescharacteristic
89C2×C22⋊Q8C2×C411yescharacteristic
90C2×C22⋊Q8C2×C411yescharacteristic
91C2×C22⋊Q8C2×C411yescharacteristic
92C2×C22⋊Q8C2×C411yescharacteristic
93C4×#41—44no—
94C2×C4×Q8C2×C411yescharacteristic
95C2×C4×Q8C2×C411yescharacteristic
96C2×C4×Q8D411yescharacteristic
97C2×C4×Q8D411yescharacteristic
98C2×C4×D4D411yescharacteristic
99C2×C4×D4D411yescharacteristic
100C2×C4×D4D411yescharacteristic
101C2×C4×D4D411yescharacteristic
102C2×C4×D4D411yescharacteristic
103C2×C4×D4D411yescharacteristic
104C2×#117C2×C411yescharacteristic
105C2×#117C2×C411yescharacteristic
106C2×#117C2×C411yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#435—44no—
111C8×D4—44no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
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×#97—21no—
123C2×C4×C8—21no—
124#398—21no—
125#398—21no—
126#398—21no—
127#398—21no—
128#397—21no—
129#397—21no—
130#396—21no—
131#396—21no—
132#395—21no—
133#395—21no—
134#394—21no—
135#394—21no—
136#394—21no—
137#394—21no—
138#394—21no—
139#394—21no—
140#391C2×C411yescharacteristic
141#391C2×C411yescharacteristic
142#391C2311yescharacteristic
143#391C2311yescharacteristic
144#389—21no—
145#389—21no—
146#389—21no—
147#389—21no—
148#389C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#389C2311yescharacteristic
151#389C2311yescharacteristic
152#388—21no—
153#388—21no—
154#388—21no—
155#388—21no—
156#388—21no—
157#388—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#387—21no—
167#387—21no—
168#387—21no—
169#387—21no—
170#385—21no—
171#385—21no—
172#385—21no—
173#385—21no—
174#385—21no—
175#385—21no—
176#384C2×C411yescharacteristic
177#382C2×C411yescharacteristic
178#381—21no—
179#381—21no—
180#380—21no—
181#380—21no—
182#380C2×C411yescharacteristic
183#380C2×C411yescharacteristic
184#380C2×C411yescharacteristic
185#380C2×C411yescharacteristic
186#380C2×C411yescharacteristic
187#380C2×C411yescharacteristic
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4C2×C411yescharacteristic
191C4×C4⋊C4C2×C411yescharacteristic
192C4×C22⋊C4—21no—
193C4×C22⋊C4—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—42no—
212C2×#41—42no—
213C22×Q8C22⋊C411yescharacteristic
214C22×Q8C22×C411yescharacteristic
215C22×Q8C22×C411yescharacteristic
216C22×D4C22⋊C411yescharacteristic
217C22×D4C22⋊C411yescharacteristic
218C22×D4C22⋊C411yescharacteristic
219C23×C4C22×C411yescharacteristic
220C23×C4C22×C411yescharacteristic
221C23×C4C22×C411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C2×M16—42no—
229C2×M16—42no—
230C2×M16—81no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—42no—
240C22×C8—42no—
241C22×C8—42no—
242C22×C8—42no—
243C22×C8—42no—
244C22×C8—42no—
245C22×C8—81no—
246C22×C8—81no—
247C22×C8—81no—
248C4⋊Q8—42no—
249#126—44no—
250#126—44no—
251#125—42no—
252#125—44no—
253#125—44no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—42no—
257C22⋊Q8—42no—
258C22⋊Q8—42no—
259C22⋊Q8—42no—
260C4×Q8—42no—
261C4×Q8—42no—
262C4×Q8—42no—
263C4×Q8—42no—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—42no—
267C4×D4—42no—
268C4×D4—42no—
269C4×D4—42no—
270#117—24no—
271#117—42no—
272#117—42no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
292C2×C4⋊C4C22⋊C411yescharacteristic
293C2×C4⋊C4C22⋊C411yescharacteristic
294C2×C4⋊C4C22⋊C411yescharacteristic
295C2×C4⋊C4C22⋊C411yescharacteristic
296C2×C4⋊C4C22⋊C411yescharacteristic
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×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4C22⋊C411yescharacteristic
316C2×C22⋊C4C22⋊C411yescharacteristic
317C2×C22⋊C4C22⋊C411yescharacteristic
318C2×C22⋊C4C22⋊C411yescharacteristic
319C2×C22⋊C4C22⋊C411yescharacteristic
320C2×C22⋊C4C22⋊C411yescharacteristic
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—42no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42C22×C411yescharacteristic
329C2×C42C22×C411yescharacteristic
330C2×C42C22×C411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2411yescharacteristic
338C4⋊C8—42no—
339C4⋊C8—42no—
340C22⋊C8—42no—
341C22⋊C8—42no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—42no—
346C22⋊C8—81no—
347C22⋊C8—81no—
348#97—42no—
349C4×C8—42no—
350#95—21no—
351#95—21no—
352#95—21no—
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—
372C24C2×C22⋊C411yescharacteristic
373C24C2×C22⋊C411yescharacteristic
374C24C2×C22⋊C411yescharacteristic
375#41—416no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—41no—
393C2×D4—41no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
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—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4#13114yes—
444C22×C4C23×C411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447M16—84no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
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—84no—
473C2×C8—84no—
474C2×C8—84no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4—81no—
506C4⋊C4—81no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C22⋊C4—81no—
523C22⋊C4—81no—
524C22⋊C4#9914yes—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—42no—
536C42—42no—
537C42—81no—
538C42—81no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23#40812yes—
558C23C2×#9912yes—
559C23#41312yes—
560C23C22×C22⋊C411yescharacteristic
561Q8—44no—
562Q8—82no—
563Q8—82no—
564D4—28no—
565D4—44no—
566D4—44no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
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—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4#40812yes—
637C2×C4C2×#9912yes—
638C2×C4C2×#9912yes—
639C2×C4C2×#13112yes—
640C2×C4#57711yescharacteristic
641C2×C4#57711yescharacteristic
642C2×C4#57811yescharacteristic
643C2×C4#57811yescharacteristic
644C8—42no—
645C8—42no—
646C8—81no—
647C8—81no—
648C8—81no—
649C8—81no—
650C8—81no—
651C8—81no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22C2×#40811yescharacteristic
674C22C22×#9911yescharacteristic
675C22C2×#41311yescharacteristic
676C22#289811yescharacteristic
677C22#289811yescharacteristic
678C22#290111yescharacteristic
679C22#290111yescharacteristic
680C4—21no—
681C4—21no—
682C4—21no—
683C4—21no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4—81no—
699C4—81no—
700C4—81no—
701C4—81no—
702C2—42no—
703C2—42no—
704C2—42no—
705C2#1738111yescharacteristic
706C2#1746911yescharacteristic
707C2#1748411yescharacteristic
708C2#1760811yescharacteristic
709C2#1760811yescharacteristic
710C2#1769011yescharacteristic
711C2#1778011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
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. #7707C2semidirect
7. #7695C2semidirect
8. #7691C2semidirect
9. #7681C2semidirect
10. #7667C2semidirect
11. #7655C2semidirect
12. #7600C2semidirect
13. #7589C2semidirect
14. #7583C2semidirect
15. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#353092
#224712
#224692
#77312
#77212
#77072
#76952
#76912
#76812
#76672
#76552
#76002
#75892
#75832
#75662