Finite Group Databasebeta

← #190649

Subgroups — #190649

Order 512.

863 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34242C211yesmaximal, characteristic
2#21435C211yesmaximal, characteristic, complemented
3#20783C211yesmaximal, characteristic, complemented
4#8722C211yesmaximal, characteristic, complemented
5#8721C211yesmaximal, characteristic, complemented
6#8584C211yesmaximal, characteristic, complemented
7#8582C211yesmaximal, characteristic, complemented
8#8581C211yesmaximal, characteristic, complemented
9#8579C211yesmaximal, characteristic, complemented
10#8336C211yesmaximal, characteristic
11#8336C211yesmaximal, characteristic
12#8227C211yesmaximal, characteristic, complemented
13#8226C211yesmaximal, characteristic, complemented
14#8225C211yesmaximal, characteristic, complemented
15#8224C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2373—21no—
18#2373—21no—
19#2363C411yescharacteristic
20#2363C411yescharacteristic
21#2362C411yescharacteristic
22#2362C411yescharacteristic
23#2355C411yescharacteristic
24#2354—21no—
25#2354C411yescharacteristic
26#2352—21no—
27Q8×C4⋊C4C411yescharacteristic
28D4×C4⋊C4C411yescharacteristic
29#2335—21no—
30#2335—21no—
31#2314—21no—
32C4×#125C2211yescharacteristic
33C4×#125C2211yescharacteristic
34C4×#124C2211yescharacteristic
35C4×#124C2211yescharacteristic
36#2285C2211yescharacteristic
37#2274—21no—
38C4×#117C2211yescharacteristic
39#1924—21no—
40#1923—21no—
41#1917—21no—
42#1916—21no—
43#1840—21no—
44#1832—21no—
45#1818—21no—
46#1798—21no—
47#1767C2211yescharacteristic
48#1767C2211yescharacteristic
49#1767C2211yescharacteristic
50#1767C2211yescharacteristic
51#1749C2211yescharacteristic
52#1749C2211yescharacteristic
53#1749C2211yescharacteristic
54#1749C2211yescharacteristic
55#1745C2211yescharacteristic
56#1745C2211yescharacteristic
57#1745C2211yescharacteristic
58C4×#97C2211yescharacteristic
59C2×#330C2211yescharacteristic
60C2×#330C2211yescharacteristic
61C2×#330C2211yescharacteristic
62C2×#330C2211yescharacteristic
63C2×#329C2211yescharacteristic
64C2×#329C2211yescharacteristic
65C2×#329C2211yescharacteristic
66C2×#329C2211yescharacteristic
67#1295C2211yescharacteristic
68#1295C2211yescharacteristic
69#1295C2211yescharacteristic
70#1295C2211yescharacteristic
71#1295C2211yescharacteristic
72#1295C2211yescharacteristic
73#1295C2211yescharacteristic
74#1295C2211yescharacteristic
75#529—22no—
76#529—22no—
77#529—22no—
78#529—22no—
79C2×#126—21no—
80C2×#125C2311yescharacteristic
81C2×#124C2311yescharacteristic
82C2×#123—21no—
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×D4—21no—
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4C2×C411yescharacteristic
91C2×C4×D4C2×C411yescharacteristic
92C2×#117—21no—
93C2×#117—21no—
94C2×#117C2×C411yescharacteristic
95C2×#117C2×C411yescharacteristic
96C2×#117D411yescharacteristic
97C2×#117D411yescharacteristic
98C22×C4⋊C4—21no—
99C22×C42C2311yescharacteristic
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×#97—21no—
109C2×#97C2311yescharacteristic
110C2×#97C2311yescharacteristic
111C2×#97C2311yescharacteristic
112C2×#97C2311yescharacteristic
113C2×#97C2311yescharacteristic
114C2×#97C2311yescharacteristic
115C2×#97C2311yescharacteristic
116C2×#97C2311yescharacteristic
117C2×C4×C8—21no—
118#391C2×C411yescharacteristic
119#391C2×C411yescharacteristic
120#390—21no—
121#390—21no—
122#390C2×C411yescharacteristic
123#390C2×C411yescharacteristic
124#389—21no—
125#389—21no—
126#389C2×C411yescharacteristic
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#388—21no—
131#388—21no—
132#388—21no—
133#388—21no—
134#388—21no—
135#388—21no—
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#388—21no—
141#388—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#386—21no—
151#386—21no—
152#385—21no—
153#385—21no—
154#384—21no—
155#384C2×C411yescharacteristic
156#384C2×C411yescharacteristic
157#382—21no—
158#382C2311yescharacteristic
159#382C2311yescharacteristic
160#380—21no—
161#380—21no—
162#380—21no—
163#380—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4—21no—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4C2×C411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4C2×C411yescharacteristic
189C4×C22⋊C4C2×C411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191C4×C22⋊C4D411yescharacteristic
192#376D411yescharacteristic
193#376D411yescharacteristic
194C43C2311yescharacteristic
195C43C2311yescharacteristic
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336—21no—
201#336—21no—
202#336—21no—
203#336—21no—
204#336—21no—
205#336—21no—
206#336—21no—
207#336—21no—
208#336—21no—
209#336—21no—
210#330—22no—
211#330—22no—
212#330—22no—
213#330—22no—
214#329—22no—
215#329—22no—
216#329—22no—
217#329—22no—
218C22×Q8C22×C411yescharacteristic
219C22×D4—21no—
220C22×D4C22×C411yescharacteristic
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4C22×C411yescharacteristic
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235#126—24no—
236#126—24no—
237#125—21no—
238#125—21no—
239#125#4112yes—
240#125#4112yes—
241#124—21no—
242#124—21no—
243#124#4112yes—
244#124#4112yes—
245#123—24no—
246#123—24no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249#121—22no—
250#121—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—81no—
254C4×Q8—81no—
255C4×D4—24no—
256C4×D4—24no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—81no—
260C4×D4—81no—
261#117—24no—
262#117—24no—
263#117—42no—
264#117—42no—
265#117—44no—
266#117—81no—
267#117—81no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
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—44no—
283C2×C4⋊C4C22⋊C411yescharacteristic
284C2×C4⋊C4C22⋊C411yescharacteristic
285C2×C4⋊C4C22⋊C411yescharacteristic
286C2×C4⋊C4C22⋊C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4C22⋊C411yescharacteristic
307C2×C22⋊C4C22⋊C411yescharacteristic
308C2×C22⋊C4C22⋊C411yescharacteristic
309C2×C22⋊C4C22⋊C411yescharacteristic
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42C22×C411yescharacteristic
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2411yescharacteristic
339C4⋊C8—42no—
340C4⋊C8—42no—
341C4⋊C8—42no—
342C4⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—42no—
346C22⋊C8—42no—
347#97—21no—
348#97—21no—
349#97—21no—
350#97—21no—
351#97—21no—
352#97—21no—
353#97—21no—
354#97—21no—
355#97—21no—
356#97—21no—
357#97—21no—
358#97—21no—
359#97—21no—
360#97—21no—
361#97—21no—
362#97—21no—
363#97—42no—
364C4×C8—42no—
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—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95C22⋊C411yescharacteristic
382#95C22⋊C411yescharacteristic
383#95C22⋊C411yescharacteristic
384#95C22⋊C411yescharacteristic
385#95C22⋊C411yescharacteristic
386#95C22⋊C411yescharacteristic
387#95C22⋊C411yescharacteristic
388#95C22⋊C411yescharacteristic
389C24—21no—
390C24C2×C22⋊C411yescharacteristic
391C2×Q8—21no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8#11712yes—
395C2×D4—24no—
396C2×D4—21no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—81no—
400C2×D4—81no—
401C2×D4#11712yes—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—81no—
427C22×C4—81no—
428C22×C4—81no—
429C22×C4—81no—
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
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—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4#11712yes—
495C4⋊C4#11712yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—41no—
505C22⋊C4—41no—
506C22⋊C4—41no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—81no—
510C22⋊C4—81no—
511C22⋊C4—81no—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C22⋊C4—81no—
515C42—24no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—22no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—42no—
535C42—42no—
536C42—81no—
537C42—81no—
538C42—81no—
539C42—81no—
540C42—81no—
541C42—81no—
542C42C2×#4111yescharacteristic
543C42C2×#4111yescharacteristic
544C42C2×#4111yescharacteristic
545C42C2×#4111yescharacteristic
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—21no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23C22×C22⋊C411yescharacteristic
561Q8—42no—
562Q8—42no—
563D4—42no—
564D4—42no—
565D4—84no—
566C2×C4—21no—
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—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—48no—
609C2×C4—42no—
610C2×C4—42no—
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—81no—
630C2×C4—81no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4#42112yes—
634C2×C4#42112yes—
635C2×C4C2×#11711yescharacteristic
636C2×C4C2×#11711yescharacteristic
637C2×C4#52511yescharacteristic
638C2×C4#52511yescharacteristic
639C2×C4#52511yescharacteristic
640C2×C4#52511yescharacteristic
641C8—42no—
642C8—42no—
643C8—81no—
644C8—81no—
645C8—81no—
646C8—81no—
647C8—81no—
648C8—81no—
649C22—41no—
650C22—42no—
651C22—41no—
652C22—42no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—42no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22#228011yescharacteristic
664C22#288811yescharacteristic
665C22#288911yescharacteristic
666C22C2×#42111yescharacteristic
667C22C2×#42111yescharacteristic
668C22#290111yescharacteristic
669C22#290111yescharacteristic
670C4—21no—
671C4—21no—
672C4—22no—
673C4—21no—
674C4—21no—
675C4—22no—
676C4—21no—
677C4—21no—
678C4—42no—
679C4—41no—
680C4—42no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—42no—
690C4—81no—
691C4—81no—
692C4—81no—
693C4—81no—
694C2—81no—
695C2—82no—
696C2#1767911yescharacteristic
697C2#1768211yescharacteristic
698C2#1768311yescharacteristic
699C2#1768411yescharacteristic
700C2#2073111yescharacteristic
701C2#2074811yescharacteristic
702C2#2074911yescharacteristic
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. #21435C2semidirect
3. #20783C2semidirect
4. #8722C2semidirect
5. #8721C2semidirect
6. #8584C2semidirect
7. #8582C2semidirect
8. #8581C2semidirect
9. #8579C2semidirect
12. #8227C2semidirect
13. #8226C2semidirect
14. #8225C2semidirect
15. #8224C2semidirect

Maximal subgroups

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

subgroupindex
#342422
#214352
#207832
#87222
#87212
#85842
#85822
#85812
#85792
#83362
#82272
#82262
#82252
#82242