Finite Group Databasebeta

← #190957

Subgroups — #190957

Order 512.

903 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#34084C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#21430C211yesmaximal, characteristic, complemented
4#8710C211yesmaximal, characteristic, complemented
5#8709C211yesmaximal, characteristic, complemented
6#8566C211yesmaximal, characteristic, complemented
7#8565C211yesmaximal, characteristic, complemented
8#8564C211yesmaximal, characteristic, complemented
9#8563C211yesmaximal, characteristic, complemented
10#8315C211yesmaximal, characteristic
11#8315C211yesmaximal, characteristic
12#8179C211yesmaximal, characteristic, complemented
13#8178C211yesmaximal, characteristic, complemented
14#8177C211yesmaximal, characteristic, complemented
15#8176C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2335—21no—
18#2335—21no—
19#2334—21no—
20#2334—21no—
21#2330C411yescharacteristic
22#2330C411yescharacteristic
23#2326C411yescharacteristic
24#2326C411yescharacteristic
25#2324C411yescharacteristic
26#2324C411yescharacteristic
27#2323C411yescharacteristic
28#2323C411yescharacteristic
29#2298—21no—
30#2296—21no—
31#2295C2211yescharacteristic
32#2293—21no—
33#2293C2211yescharacteristic
34#2290—21no—
35#2285C2211yescharacteristic
36C42×Q8C2211yescharacteristic
37C42×D4C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1924—21no—
40#1924—21no—
41#1923—21no—
42#1923—21no—
43#1917—21no—
44#1917—21no—
45#1916—21no—
46#1916—21no—
47#1842C2211yescharacteristic
48#1831C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C4×C22⋊C8C2211yescharacteristic
56C4×C22⋊C8C2211yescharacteristic
57#1744C2211yescharacteristic
58C42×C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×D4⋊C8C2211yescharacteristic
66C2×D4⋊C8C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75#529—22no—
76#529—22no—
77#529—22no—
78#529—22no—
79C2×#126—21no—
80C2×#125C2×C411yescharacteristic
81C2×#124C2×C411yescharacteristic
82C2×#123—21no—
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2311yescharacteristic
88C2×C4×D4—21no—
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4C2×C411yescharacteristic
91C2×C4×D4C2311yescharacteristic
92C2×#117—21no—
93C2×#117—21no—
94C2×#117—21no—
95C2×#117C2×C411yescharacteristic
96C2×#117C2×C411yescharacteristic
97C22×C42—21no—
98C22×C42C2311yescharacteristic
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#390—21no—
122#390—21no—
123#390C2×C411yescharacteristic
124#390C2×C411yescharacteristic
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#388—21no—
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#388—21no—
141#388—21no—
142#388—21no—
143#388—21no—
144#388—21no—
145#388—21no—
146#388—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#386—21no—
156#386—21no—
157#385—21no—
158#385—21no—
159#385—21no—
160#385—21no—
161#385—21no—
162#385—21no—
163#384C2×C411yescharacteristic
164#384C2×C411yescharacteristic
165#382—21no—
166#380—21no—
167#380C2×C411yescharacteristic
168#380C2×C411yescharacteristic
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⋊C4D411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C4⋊C4C2311yescharacteristic
183C4×C4⋊C4C2311yescharacteristic
184C4×C22⋊C4—21no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4D411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191C4×C22⋊C4D411yescharacteristic
192#376—21no—
193#376C2×C411yescharacteristic
194#376C2×C411yescharacteristic
195C43—21no—
196C43C2311yescharacteristic
197C43C2311yescharacteristic
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—
210Q8⋊C8—22no—
211Q8⋊C8—22no—
212Q8⋊C8—22no—
213Q8⋊C8—22no—
214D4⋊C8—22no—
215D4⋊C8—22no—
216D4⋊C8—22no—
217D4⋊C8—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—22no—
238#125—22no—
239#124—22no—
240#124—22no—
241#123—24no—
242#123—24no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245#121—22no—
246#121—22no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—24no—
250C4×Q8—42no—
251C4×Q8#4112yes—
252C4×Q8#4112yes—
253C4×D4—24no—
254C4×D4—24no—
255C4×D4—24no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—42no—
259C4×D4#4112yes—
260C4×D4#4112yes—
261#117—24no—
262#117—24no—
263#117—24no—
264#117—42no—
265#117—81no—
266#117—81no—
267#117—81no—
268#117—81no—
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—21no—
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×C42—81no—
324C2×C42—81no—
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C22×C411yescharacteristic
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×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2411yescharacteristic
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8—21no—
349C4⋊C8—42no—
350C4⋊C8—42no—
351C4⋊C8—81no—
352C4⋊C8—81no—
353C22⋊C8—42no—
354C22⋊C8—42no—
355C22⋊C8—81no—
356C22⋊C8—81no—
357#97—81no—
358#97—81no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—21no—
365C4×C8—21no—
366C4×C8—21no—
367C4×C8—42no—
368C4×C8—42no—
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#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95C22⋊C411yescharacteristic
386#95C22⋊C411yescharacteristic
387#95C22⋊C411yescharacteristic
388#95C22⋊C411yescharacteristic
389#95C22⋊C411yescharacteristic
390#95C22⋊C411yescharacteristic
391#95C22⋊C411yescharacteristic
392#95C22⋊C411yescharacteristic
393C24—21no—
394C24C2×C22⋊C411yescharacteristic
395C2×Q8—21no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8#11712yes—
399C2×D4—24no—
400C2×D4—21no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—81no—
404C2×D4—81no—
405C2×D4#11712yes—
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—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—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C2×C8—22no—
454C2×C8—22no—
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—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4#11712yes—
503C4⋊C4#11712yes—
504C22⋊C4—24no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—24no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—22no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—42no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—42no—
551C42—81no—
552C42—81no—
553C42—81no—
554C42—81no—
555C42#13114yes—
556C42C2×#4111yescharacteristic
557C42C2×#4111yescharacteristic
558C42C2×#4111yescharacteristic
559C42C2×#4111yescharacteristic
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—22no—
567C23—21no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23C22×C22⋊C411yescharacteristic
575Q8—24no—
576Q8—42no—
577D4—24no—
578D4—42no—
579D4—84no—
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—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
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—84no—
647C2×C4—84no—
648C2×C4#40812yes—
649C2×C4#40812yes—
650C2×C4C2×#11711yescharacteristic
651C2×C4C2×#11711yescharacteristic
652C2×C4#52511yescharacteristic
653C2×C4#52511yescharacteristic
654C2×C4#52511yescharacteristic
655C2×C4#52511yescharacteristic
656C2×C4C2×#13112yes—
657C2×C4#56812yes—
658C2×C4#57611yescharacteristic
659C2×C4#57711yescharacteristic
660C2×C4#57711yescharacteristic
661C2×C4#57811yescharacteristic
662C8—42no—
663C8—42no—
664C8—81no—
665C8—81no—
666C8—81no—
667C8—81no—
668C8—81no—
669C8—81no—
670C22—41no—
671C22—42no—
672C22—41no—
673C22—42no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—42no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22#228011yescharacteristic
685C22C2×#40811yescharacteristic
686C22#287911yescharacteristic
687C22#289711yescharacteristic
688C22#289811yescharacteristic
689C22#290111yescharacteristic
690C22#290111yescharacteristic
691C4—21no—
692C4—21no—
693C4—22no—
694C4—22no—
695C4—41no—
696C4—42no—
697C4—41no—
698C4—42no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—42no—
707C4—81no—
708C4—81no—
709C4—81no—
710C4—81no—
711C4#292512yes—
712C4#292512yes—
713C4#311911yescharacteristic
714C4#312011yescharacteristic
715C4#312411yescharacteristic
716C4#312511yescharacteristic
717C2—81no—
718C2—82no—
719C2#1748311yescharacteristic
720C2#1748411yescharacteristic
721C2#1750511yescharacteristic
722C2#1750611yescharacteristic
723C2#2072111yescharacteristic
724C2#2074611yescharacteristic
725C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21926C2semidirect
3. #21430C2semidirect
4. #8710C2semidirect
5. #8709C2semidirect
6. #8566C2semidirect
7. #8565C2semidirect
8. #8564C2semidirect
9. #8563C2semidirect
12. #8179C2semidirect
13. #8178C2semidirect
14. #8177C2semidirect
15. #8176C2semidirect

Maximal subgroups

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

subgroupindex
#340842
#219262
#214302
#87102
#87092
#85662
#85652
#85642
#85632
#83152
#81792
#81782
#81772
#81762