Finite Group Databasebeta

← #188570

Subgroups — #188570

Order 512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34229C211yesmaximal, characteristic
2#21355C211yesmaximal, characteristic, complemented
3#20427C211yesmaximal, characteristic, complemented
4C2×#1532C211yesmaximal, characteristic
5C2×#1524C211yesmaximal, characteristic
6#8584C211yesmaximal, characteristic, complemented
7#8583C211yesmaximal, characteristic, complemented
8#8582C211yesmaximal, characteristic, complemented
9#8582C211yesmaximal, characteristic, complemented
10#8581C211yesmaximal, characteristic, complemented
11#8581C211yesmaximal, characteristic, complemented
12#8580C211yesmaximal, characteristic, complemented
13#8579C211yesmaximal, characteristic, complemented
14#8033C211yesmaximal, characteristic, complemented
15C2×#1295C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#424—21no—
19C2×C4×M16C2211yescharacteristic
20C22×#97C2211yescharacteristic
21C22×#97C2211yescharacteristic
22#2361C411yescharacteristic
23#2361C411yescharacteristic
24#2360C411yescharacteristic
25#2360C411yescharacteristic
26#2359—21no—
27#2359—21no—
28#2357C411yescharacteristic
29#2356C411yescharacteristic
30#2351C411yescharacteristic
31#2349C411yescharacteristic
32#2346—21no—
33#2343—21no—
34#2334—21no—
35#2334—21no—
36#2318—21no—
37C4×#125C2211yescharacteristic
38C4×#125C2211yescharacteristic
39C4×#124C2211yescharacteristic
40C4×#124C2211yescharacteristic
41C2×#382C2211yescharacteristic
42#2275—21no—
43C4×#117C2211yescharacteristic
44#1940—21no—
45#1940—21no—
46#1850—21no—
47#1850—21no—
48#1768—21no—
49#1768—21no—
50#1767C2211yescharacteristic
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53#1767C2211yescharacteristic
54#1767C2211yescharacteristic
55#1767C2211yescharacteristic
56#1767C2211yescharacteristic
57#1767C2211yescharacteristic
58#1729—21no—
59#1532—24no—
60#1524—24no—
61C2×#330C2211yescharacteristic
62C2×#330C2211yescharacteristic
63C2×#330C2211yescharacteristic
64C2×#330C2211yescharacteristic
65C2×#329C2211yescharacteristic
66C2×#329C2211yescharacteristic
67C2×#329C2211yescharacteristic
68C2×#329C2211yescharacteristic
69#1295C2211yescharacteristic
70#1295C2211yescharacteristic
71#1295C2211yescharacteristic
72#1295C2211yescharacteristic
73#1295C2211yescharacteristic
74#1295C2211yescharacteristic
75#1295C2211yescharacteristic
76#1295C2211yescharacteristic
77C22×M16—21no—
78C23×C8—21no—
79C23×C8—21no—
80#529—22no—
81#529—22no—
82#529D414yes—
83#529Q814yes—
84C2×#126—21no—
85C2×#125C2311yescharacteristic
86C2×#124C2311yescharacteristic
87C2×#123—21no—
88C2×C22⋊Q8C2×C411yescharacteristic
89C2×#121C2×C411yescharacteristic
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2×C411yescharacteristic
92C2×C4×Q8C2×C411yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4C2×C411yescharacteristic
97C2×C4×D4C2×C411yescharacteristic
98C2×C4×D4C2×C411yescharacteristic
99C2×#117—21no—
100C2×#117D411yescharacteristic
101C2×#117D411yescharacteristic
102C22×C4⋊C4C2×C411yescharacteristic
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C42C2311yescharacteristic
105#432—22no—
106#432—22no—
107#424—28no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C22⋊C8—21no—
118C4×M16—22no—
119C4×M16—22no—
120C2×#97—22no—
121C2×#97—22no—
122C2×#97—22no—
123C2×#97—22no—
124C2×#97—21no—
125C2×#97—21no—
126C2×#97C2311yescharacteristic
127C2×#97C2311yescharacteristic
128C2×#97C2311yescharacteristic
129C2×#97C2311yescharacteristic
130C2×#97C2311yescharacteristic
131C2×#97C2311yescharacteristic
132C2×#97C2311yescharacteristic
133C2×#97C2311yescharacteristic
134C2×C4×C8—21no—
135#391C2×C411yescharacteristic
136#391C2×C411yescharacteristic
137#390C2×C411yescharacteristic
138#390C2×C411yescharacteristic
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#388—21no—
148#388—21no—
149#388—21no—
150#388—21no—
151#388—21no—
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#386—21no—
164#386—21no—
165#385—21no—
166#385—21no—
167#385—21no—
168#385—21no—
169#385—21no—
170#385—21no—
171#383—21no—
172#382—21no—
173#382—21no—
174#382—21no—
175#382C2×C411yescharacteristic
176#382C2×C411yescharacteristic
177#382C2311yescharacteristic
178#382C2311yescharacteristic
179#381—21no—
180#380—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4C2×C411yescharacteristic
187C4×C4⋊C4C2×C411yescharacteristic
188C4×C4⋊C4C2×C411yescharacteristic
189C4×C4⋊C4C2×C411yescharacteristic
190C4×C4⋊C4C2×C411yescharacteristic
191C4×C4⋊C4C2×C411yescharacteristic
192C4×C4⋊C4D411yescharacteristic
193C4×C4⋊C4D411yescharacteristic
194C4×C22⋊C4—21no—
195C4×C22⋊C4—21no—
196C4×C22⋊C4—21no—
197C4×C22⋊C4—21no—
198C4×C22⋊C4—21no—
199C4×C22⋊C4—21no—
200C4×C22⋊C4—21no—
201C4×C22⋊C4C2×C411yescharacteristic
202C4×C22⋊C4C2×C411yescharacteristic
203C4×C22⋊C4D411yescharacteristic
204C4×C22⋊C4D411yescharacteristic
205#376—21no—
206#376D411yescharacteristic
207#376D411yescharacteristic
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C43C2311yescharacteristic
212C43C2311yescharacteristic
213#336—21no—
214#336—21no—
215#336—21no—
216#336—21no—
217#336—21no—
218#336—21no—
219#336—21no—
220#336—21no—
221#330—22no—
222#330—22no—
223#330—22no—
224#330—22no—
225#329—22no—
226#329—22no—
227#329—22no—
228#329—22no—
229C22×Q8C22×C411yescharacteristic
230C22×D4—21no—
231C22×D4C22×C411yescharacteristic
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4C22×C411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C2×M16—24no—
241C2×M16—42no—
242C2×M16—42no—
243C22×C8—24no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—42no—
253C22×C8—42no—
254C22×C8—42no—
255C22×C8—42no—
256C22×C8—42no—
257#126—24no—
258#126—24no—
259#125—21no—
260#125—21no—
261#125C2×D412yes—
262#125C2×Q812yes—
263#124—21no—
264#124—21no—
265#124C2×D412yes—
266#124C2×Q812yes—
267#123—24no—
268#123—24no—
269C22⋊Q8—22no—
270C22⋊Q8C4⋊C414yes—
271#121—22no—
272#121C4⋊C414yes—
273C4×Q8—22no—
274C4×Q8—81no—
275C4×Q8—81no—
276C4×Q8C4⋊C414yes—
277C4×D4—24no—
278C4×D4—24no—
279C4×D4—22no—
280C4×D4—44no—
281C4×D4—44no—
282C4×D4—81no—
283C4×D4—81no—
284C4×D4C4⋊C414yes—
285#117—24no—
286#117—24no—
287#117—24no—
288#117—42no—
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—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—81no—
302C2×C4⋊C4—81no—
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⋊C4—21no—
324C2×C22⋊C4C22⋊C411yescharacteristic
325C2×C22⋊C4C22⋊C411yescharacteristic
326C2×C22⋊C4C22⋊C411yescharacteristic
327C2×C22⋊C4C22⋊C411yescharacteristic
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42C22×C411yescharacteristic
336C2×C42C22×C411yescharacteristic
337C2×C42C22×C411yescharacteristic
338C2×C42C22×C411yescharacteristic
339C2×C42C22×C411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D412yes—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×Q812yes—
352C2×C42#4112yes—
353C2×C42#4112yes—
354C2×C42C2411yescharacteristic
355C4⋊C8—24no—
356C4⋊C8—24no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—42no—
360C4⋊C8—42no—
361C4⋊C8—42no—
362C4⋊C8—42no—
363C4⋊C8—42no—
364C22⋊C8—42no—
365#97—21no—
366#97—21no—
367#97—21no—
368#97—21no—
369#97—21no—
370#97—21no—
371#97—21no—
372#97—21no—
373#97—21no—
374#97—21no—
375#97—21no—
376#97—21no—
377#97—21no—
378#97—21no—
379#97—21no—
380#97—21no—
381#97—42no—
382#97—42no—
383C4×C8—42no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95C22⋊C411yescharacteristic
405#95C22⋊C411yescharacteristic
406#95C22⋊C411yescharacteristic
407#95C22⋊C411yescharacteristic
408#95C22⋊C411yescharacteristic
409#95C22⋊C411yescharacteristic
410#95C22⋊C411yescharacteristic
411#95C22⋊C411yescharacteristic
412C24—21no—
413C24C2×C22⋊C411yescharacteristic
414C2×Q8—21no—
415C2×Q8—41no—
416C2×Q8—41no—
417C2×Q8C2×C4⋊C412yes—
418C2×D4—24no—
419C2×D4—21no—
420C2×D4—41no—
421C2×D4—41no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4C2×C4⋊C412yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4—81no—
451C22×C4—81no—
452C22×C4—81no—
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C2×C22⋊C411yescharacteristic
461C22×C4C2×C22⋊C411yescharacteristic
462C22×C4C2×C22⋊C411yescharacteristic
463C22×C4C2×C22⋊C411yescharacteristic
464C22×C4C2×C4⋊C412yes—
465C22×C4#11712yes—
466C22×C4#12112yes—
467C22×C4#12112yes—
468C22×C4C22⋊Q812yes—
469C22×C4C22⋊Q812yes—
470C22×C4C23×C411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473M16—48no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—48no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—48no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
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—81no—
519C4⋊C4—81no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4—81no—
523C4⋊C4—81no—
524C4⋊C4C2×C4⋊C412yes—
525C4⋊C4C2×C4⋊C412yes—
526C22⋊C4—24no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—22no—
530C22⋊C4—22no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—81no—
539C22⋊C4—81no—
540C22⋊C4—81no—
541C22⋊C4—81no—
542C22⋊C4#10414yes—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—42no—
557C42—42no—
558C42—81no—
559C42—81no—
560C42—81no—
561C42—81no—
562C42C2×C4⋊C412yes—
563C42#11712yes—
564C42#12112yes—
565C42#12112yes—
566C42C22⋊Q812yes—
567C42C22⋊Q812yes—
568C42C22×D411yescharacteristic
569C42C22×Q811yescharacteristic
570C42C2×#4111yescharacteristic
571C42C2×#4111yescharacteristic
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—22no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23#38012yes—
584C23C2×#10412yes—
585C23#42112yes—
586C23C22×C22⋊C411yescharacteristic
587Q8—42no—
588Q8—42no—
589D4—42no—
590D4—42no—
591D4—48no—
592C2×C4—24no—
593C2×C4—24no—
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—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
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—41no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4—84no—
654C2×C4—162no—
655C2×C4—162no—
656C2×C4—162no—
657C2×C4—162no—
658C2×C4#38012yes—
659C2×C4#38012yes—
660C2×C4#38012yes—
661C2×C4C2×#10412yes—
662C2×C4C2×#10412yes—
663C2×C4C22×C4⋊C411yescharacteristic
664C2×C4C2×#11711yescharacteristic
665C2×C4C2×#12111yescharacteristic
666C2×C4C2×#12111yescharacteristic
667C2×C4C2×C22⋊Q811yescharacteristic
668C2×C4C2×C22⋊Q811yescharacteristic
669C8—42no—
670C8—42no—
671C8—42no—
672C8—42no—
673C8—42no—
674C8—42no—
675C8—81no—
676C8—81no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—81no—
687C22—81no—
688C22—81no—
689C22—81no—
690C22#180614yes—
691C22C2×#38011yescharacteristic
692C22#288811yescharacteristic
693C22#288911yescharacteristic
694C22C22×#10411yescharacteristic
695C22C2×#42111yescharacteristic
696C22#290111yescharacteristic
697C22#290111yescharacteristic
698C4—21no—
699C4—21no—
700C4—22no—
701C4—21no—
702C4—21no—
703C4—22no—
704C4—21no—
705C4—22no—
706C4—21no—
707C4—22no—
708C4—22no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—42no—
714C4—42no—
715C4—42no—
716C4—41no—
717C4—41no—
718C4—81no—
719C4—81no—
720C4—81no—
721C4—81no—
722C2—24no—
723C2—82no—
724C2#1765211yescharacteristic
725C2#1765311yescharacteristic
726C2#1767911yescharacteristic
727C2#1768011yescharacteristic
728C2#2064011yescharacteristic
729C2C2×#180611yescharacteristic
730C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21355C2semidirect
3. #20427C2semidirect
6. #8584C2semidirect
7. #8583C2semidirect
8. #8582C2semidirect
9. #8582C2semidirect
10. #8581C2semidirect
11. #8581C2semidirect
12. #8580C2semidirect
13. #8579C2semidirect
14. #8033C2semidirect
15. C2×#1295C2semidirect

Maximal subgroups

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

subgroupindex
#342292
#213552
#204272
C2×#15322
C2×#15242
#85842
#85832
#85822
#85812
#85802
#85792
#80332
C2×#12952