Finite Group Databasebeta

← #164101

Subgroups — #164101

Order 512.

971 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#22469C211yesmaximal, characteristic
4#7731C211yesmaximal, characteristic, complemented
5#7728C211yesmaximal, characteristic, complemented
6#7711C211yesmaximal, characteristic, complemented
7#7708C211yesmaximal, characteristic, complemented
8#7691C211yesmaximal, characteristic, complemented
9#7688C211yesmaximal, characteristic, complemented
10#7671C211yesmaximal, characteristic, complemented
11#7668C211yesmaximal, characteristic, complemented
12#7593C211yesmaximal, characteristic, complemented
13#7579C211yesmaximal, characteristic, complemented
14#7579C211yesmaximal, characteristic, complemented
15C2×#1271C211yesmaximal, characteristic, complemented
16C2×#544C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#435—21no—
19C2×C8×D4—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#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#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—
77C23×C8—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—
112C8×D4—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×C4×C8—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—24no—
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—81no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—42no—
242C22×C8—42no—
243C22×C8—42no—
244C22×C8—42no—
245C22×C8—42no—
246C22×C8—42no—
247C22×C8—81no—
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—24no—
272#117—42no—
273#117—42no—
274#117—42no—
275C2×C4⋊C4—24no—
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—21no—
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—
355C4×C8—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—22no—
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—22no—
394C2×D4—22no—
395C2×D4—22no—
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—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
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—41no—
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—42no—
439C22×C4—42no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—81no—
444C22×C4C2×C22⋊C411yescharacteristic
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×C4#13114yes—
454C22×C4C23×C411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457M16—84no—
458C2×C8—22no—
459C2×C8—22no—
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—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—84no—
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—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—42no—
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#40812yes—
569C23C2×#9912yes—
570C23C2×#10112yes—
571C23C2×#10112yes—
572C23C22×C22⋊C411yescharacteristic
573Q8—44no—
574Q8—162no—
575Q8—162no—
576D4—28no—
577D4—44no—
578D4—44no—
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—28no—
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—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
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—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—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—84no—
648C2×C4—162no—
649C2×C4#40812yes—
650C2×C4C2×#9912yes—
651C2×C4C2×#9912yes—
652C2×C4C2×#10112yes—
653C2×C4C2×#13112yes—
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#147214yes—
687C22C2×#40811yescharacteristic
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—22no—
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—24no—
717C2—42no—
718C2—81no—
719C2C2×#147211yescharacteristic
720C2#1746811yescharacteristic
721C2#1748411yescharacteristic
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. #7728C2semidirect
6. #7711C2semidirect
7. #7708C2semidirect
8. #7691C2semidirect
9. #7688C2semidirect
10. #7671C2semidirect
11. #7668C2semidirect
12. #7593C2semidirect
13. #7579C2semidirect
14. #7579C2semidirect
15. C2×#1271C2semidirect

Maximal subgroups

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

subgroupindex
#352572
#224712
#224692
#77312
#77282
#77112
#77082
#76912
#76882
#76712
#76682
#75932
#75792
C2×#12712