Finite Group Databasebeta

← #163763

Subgroups — #163763

Order 512.

948 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35257C211yesmaximal, characteristic
2#22471C211yesmaximal, characteristic
3#22471C211yesmaximal, characteristic
4#7731C211yesmaximal, characteristic, complemented
5#7731C211yesmaximal, characteristic, complemented
6#7708C211yesmaximal, characteristic, complemented
7#7705C211yesmaximal, characteristic, complemented
8#7691C211yesmaximal, characteristic, complemented
9#7691C211yesmaximal, characteristic, complemented
10#7668C211yesmaximal, characteristic, complemented
11#7665C211yesmaximal, characteristic, complemented
12#7646C211yesmaximal, characteristic, complemented
13#7632C211yesmaximal, characteristic, complemented
14#7610C211yesmaximal, characteristic, complemented
15#7607C211yesmaximal, characteristic, complemented
16C2×#544C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#435—21no—
19C2×#435—21no—
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22#2851C2211yescharacteristic
23#2851C2211yescharacteristic
24#2629C411yescharacteristic
25#2629C411yescharacteristic
26#2628C411yescharacteristic
27#2628C411yescharacteristic
28#2454—21no—
29#2453—21no—
30#2452—21no—
31#2451—21no—
32#2450—21no—
33#2450—21no—
34#2449—21no—
35#2449—21no—
36#2317C411yescharacteristic
37#2311C411yescharacteristic
38#2311C411yescharacteristic
39#2296C2211yescharacteristic
40C2×#389C2211yescharacteristic
41#1985—21no—
42#1985—21no—
43#1870—21no—
44#1870—21no—
45#1869—21no—
46#1869—21no—
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1817C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1749C2211yescharacteristic
54#1749C2211yescharacteristic
55#1728C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1271C2211yescharacteristic
60#1271C2211yescharacteristic
61#1271C2211yescharacteristic
62#1271C2211yescharacteristic
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×#41C2×C411yescharacteristic
76C22×M16—21no—
77C22×M16—21no—
78C23×C8—21no—
79C23×C8—21no—
80#544—44no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2×C411yescharacteristic
83C2×C4⋊Q8C2×C411yescharacteristic
84C2×#125—21no—
85C2×#125C2×C411yescharacteristic
86C2×#125C2×C411yescharacteristic
87C2×C22⋊Q8—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C2×C22⋊Q8—21no—
91C4×#41—44no—
92C2×C4×Q8D411yescharacteristic
93C2×C4×Q8D411yescharacteristic
94C2×C4×D4D411yescharacteristic
95C2×C4×D4D411yescharacteristic
96C2×C4×D4D411yescharacteristic
97C2×C4×D4D411yescharacteristic
98C2×C4×D4D411yescharacteristic
99C2×C4×D4D411yescharacteristic
100C2×#117C2×C411yescharacteristic
101C2×#117C2×C411yescharacteristic
102C2×#117C2×C411yescharacteristic
103C2×#117C2×C411yescharacteristic
104C2×#117C2×C411yescharacteristic
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2×C411yescharacteristic
107C22×C4⋊C4C2×C411yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111#435—44no—
112#435—44no—
113C2×C4⋊C8—21no—
114C2×C4⋊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×C22⋊C8—21no—
123C2×#97—21no—
124C2×#97—21no—
125#400—21no—
126#400—21no—
127#399—21no—
128#399—21no—
129#398—21no—
130#398—21no—
131#398—21no—
132#398—21no—
133#396—21no—
134#396—21no—
135#396—21no—
136#396—21no—
137#395—21no—
138#395—21no—
139#393—21no—
140#393—21no—
141#391—21no—
142#391—21no—
143#391—21no—
144#391—21no—
145#390—21no—
146#390—21no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389—21no—
151#389—21no—
152#389—21no—
153#389C2×C411yescharacteristic
154#389C2×C411yescharacteristic
155#389C2×C411yescharacteristic
156#389C2×C411yescharacteristic
157#389C2311yescharacteristic
158#389C2311yescharacteristic
159#389C2311yescharacteristic
160#389C2311yescharacteristic
161#387—21no—
162#387—21no—
163#387—21no—
164#387—21no—
165#386—21no—
166#386—21no—
167#386—21no—
168#386—21no—
169#385—21no—
170#385—21no—
171#385—21no—
172#385—21no—
173#385—21no—
174#385—21no—
175#385—21no—
176#385—21no—
177#384C2×C411yescharacteristic
178#384C2×C411yescharacteristic
179#382C2×C411yescharacteristic
180#382C2×C411yescharacteristic
181#380—21no—
182#380—21no—
183#380C2×C411yescharacteristic
184#380C2×C411yescharacteristic
185#380C2×C411yescharacteristic
186#380C2×C411yescharacteristic
187#380C2×C411yescharacteristic
188#380C2×C411yescharacteristic
189#380C2×C411yescharacteristic
190#380C2×C411yescharacteristic
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C2×#95—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×Q8—21no—
214C22×Q8—21no—
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—81no—
235C2×M16—81no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—42no—
245C22×C8—42no—
246C22×C8—42no—
247C22×C8—42no—
248C22×C8—81no—
249C22×C8—81no—
250C4⋊Q8—42no—
251C4⋊Q8—42no—
252C4⋊Q8—44no—
253#125—42no—
254#125—42no—
255#125—44no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258C22⋊Q8—81no—
259C22⋊Q8—81no—
260C22⋊Q8—81no—
261C22⋊Q8—81no—
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—
273#117—42no—
274#117—42no—
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⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—81no—
298C2×C4⋊C4—81no—
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×C4⋊C4C22×C411yescharacteristic
311C2×C4⋊C4C22×C411yescharacteristic
312C2×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4C22⋊C411yescharacteristic
324C2×C22⋊C4C22⋊C411yescharacteristic
325C2×C22⋊C4C22⋊C411yescharacteristic
326C2×C22⋊C4C22⋊C411yescharacteristic
327C2×C22⋊C4C22⋊C411yescharacteristic
328C2×C22⋊C4C22⋊C411yescharacteristic
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42C22×C411yescharacteristic
335C2×C42C22×C411yescharacteristic
336C2×C42C22×C411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2411yescharacteristic
344C4⋊C8—42no—
345C4⋊C8—42no—
346C22⋊C8—42no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350C22⋊C8—42no—
351C22⋊C8—42no—
352C22⋊C8—81no—
353C22⋊C8—81no—
354#97—42no—
355#97—42no—
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—416no—
380C2×Q8—22no—
381C2×Q8—41no—
382C2×Q8—41no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×Q8—81no—
388C2×Q8—81no—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4#10114yes—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
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—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
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—41no—
430C22×C4—41no—
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—42no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457M16—84no—
458M16—84no—
459C2×C8—22no—
460C2×C8—22no—
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—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—84no—
484C2×C8—84no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—81no—
516C4⋊C4—81no—
517C4⋊C4—81no—
518C4⋊C4—81no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4—81no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—81no—
531C22⋊C4—81no—
532C22⋊C4—81no—
533C22⋊C4—81no—
534C22⋊C4—81no—
535C22⋊C4—81no—
536C22⋊C4—81no—
537C22⋊C4—81no—
538C22⋊C4#9914yes—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—42no—
550C42—42no—
551C23—21no—
552C23—21no—
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—
569C23C2×#9912yes—
570C23C2×#10112yes—
571C23C2×#10112yes—
572C23C22×C22⋊C411yescharacteristic
573Q8—44no—
574Q8—162no—
575Q8—162no—
576D4—44no—
577D4—44no—
578D4—44no—
579C2×C4—28no—
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—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—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
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—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—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—84no—
649C2×C4—162no—
650C2×C4C2×#9912yes—
651C2×C4C2×#9912yes—
652C2×C4C2×#10112yes—
653C2×C4#56812yes—
654C2×C4#57611yescharacteristic
655C2×C4#57711yescharacteristic
656C2×C4#57711yescharacteristic
657C2×C4#57811yescharacteristic
658C8—42no—
659C8—42no—
660C8—81no—
661C8—81no—
662C8—81no—
663C8—81no—
664C8—81no—
665C8—81no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22#287911yescharacteristic
688C22C22×#9911yescharacteristic
689C22C22×#10111yescharacteristic
690C22#289711yescharacteristic
691C22#289811yescharacteristic
692C22#290111yescharacteristic
693C22#290111yescharacteristic
694C4—21no—
695C4—21no—
696C4—22no—
697C4—21no—
698C4—21no—
699C4—22no—
700C4—22no—
701C4—22no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—81no—
709C4—81no—
710C4—81no—
711C4—81no—
712C4—81no—
713C4—81no—
714C4—81no—
715C4—81no—
716C2—42no—
717C2—42no—
718C2—42no—
719C2#1739711yescharacteristic
720C2#1750511yescharacteristic
721C2#1750611yescharacteristic
722C2#1760311yescharacteristic
723C2#1760811yescharacteristic
724C2#1768911yescharacteristic
725C2#1777411yescharacteristic
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. #7731C2semidirect
6. #7708C2semidirect
7. #7705C2semidirect
8. #7691C2semidirect
9. #7691C2semidirect
10. #7668C2semidirect
11. #7665C2semidirect
12. #7646C2semidirect
13. #7632C2semidirect
14. #7610C2semidirect
15. #7607C2semidirect

Maximal subgroups

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

subgroupindex
#352572
#224712
#77312
#77082
#77052
#76912
#76682
#76652
#76462
#76322
#76102
#76072