Finite Group Databasebeta

← #224565

Subgroups — #224565

Order 512.

847 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34902C211yesmaximal, characteristic
2#22823C211yesmaximal, characteristic, complemented
3C4×C8×Q8C211yesmaximal, characteristic, complemented
4#9677C211yesmaximal, characteristic, complemented
5#9676C211yesmaximal, characteristic
6#9675C211yesmaximal, characteristic
7#9639C211yesmaximal, characteristic
8#9639C211yesmaximal, characteristic
9#9637C211yesmaximal, characteristic, complemented
10#8570C211yesmaximal, characteristic, complemented
11#8567C211yesmaximal, characteristic
12#8566C211yesmaximal, characteristic, complemented
13#8177C211yesmaximal, characteristic
14#8176C211yesmaximal, characteristic, complemented
15C4×Q8⋊C8C211yesmaximal, characteristic, complemented
16C2×C8×Q8—21no—
17#2596—21no—
18#2594—21no—
19#2592C411yescharacteristic
20#2581C411yescharacteristic
21#2570C411yescharacteristic
22#2545C411yescharacteristic
23#2525—21no—
24#2514C411yescharacteristic
25#2512C411yescharacteristic
26#2469—21no—
27#2448C411yescharacteristic
28#2446C411yescharacteristic
29#2401—21no—
30#2396—21no—
31C4×C4⋊Q8C2211yescharacteristic
32C4×#126—21no—
33C4×#124C2211yescharacteristic
34C4×#124C2211yescharacteristic
35#2298—21no—
36#2295C2211yescharacteristic
37#2293C2211yescharacteristic
38C42×Q8C2211yescharacteristic
39C42×Q8C2211yescharacteristic
40#1985—21no—
41#1917—21no—
42#1916—21no—
43#1842C2211yescharacteristic
44#1831C2211yescharacteristic
45#1831C2211yescharacteristic
46#1771—21no—
47C8×C4⋊C4—21no—
48C8×C4⋊C4—21no—
49C8×C4⋊C4—21no—
50#1767C2211yescharacteristic
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C4×C4⋊C8C2211yescharacteristic
56C42×C8C2211yescharacteristic
57C42×C8C2211yescharacteristic
58C42×C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×Q8⋊C8C2211yescharacteristic
64C2×Q8⋊C8C2211yescharacteristic
65C2×Q8⋊C8C2211yescharacteristic
66C2×Q8⋊C8C2211yescharacteristic
67C2×D4⋊C8C2211yescharacteristic
68C2×D4⋊C8C2211yescharacteristic
69C2×D4⋊C8C2211yescharacteristic
70C2×D4⋊C8C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#126—21no—
77C2×#124C2×C411yescharacteristic
78C2×#124C2×C411yescharacteristic
79C2×C4×Q8C2×C411yescharacteristic
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8D411yescharacteristic
83C2×C4×Q8D411yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×Q8C2311yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×#117—21no—
88C8×Q8—28no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#97—21no—
99C2×#97—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#401—21no—
112#400—21no—
113#400—21no—
114#400—21no—
115#400—21no—
116#399—21no—
117#398—21no—
118#398—21no—
119#397—21no—
120#397—21no—
121#397—21no—
122#396—21no—
123#396—21no—
124#396—21no—
125#395—21no—
126#393—21no—
127#391C2×C411yescharacteristic
128#391C2×C411yescharacteristic
129#391C2×C411yescharacteristic
130#391C2×C411yescharacteristic
131#391C2×C411yescharacteristic
132#391C2×C411yescharacteristic
133#390C2×C411yescharacteristic
134#390C2×C411yescharacteristic
135#390C2×C411yescharacteristic
136#390C2×C411yescharacteristic
137#389—21no—
138#389—21no—
139#389C2×C411yescharacteristic
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#388—21no—
144#388—21no—
145#388—21no—
146#388—21no—
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#388—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#387—21no—
161#387—21no—
162#387—21no—
163#387—21no—
164#387—21no—
165#384C2×C411yescharacteristic
166#383—21no—
167#382C2×C411yescharacteristic
168#382C2×C411yescharacteristic
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
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×C4⋊C4D411yescharacteristic
183C4×C4⋊C4D411yescharacteristic
184C4×C4⋊C4D411yescharacteristic
185C4×C4⋊C4D411yescharacteristic
186C4×C4⋊C4C2311yescharacteristic
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—21no—
191C4×C22⋊C4—21no—
192#376—21no—
193C43—21no—
194C43C2311yescharacteristic
195C43C2311yescharacteristic
196C43C2311yescharacteristic
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—
207Q8⋊C8—22no—
208Q8⋊C8—22no—
209Q8⋊C8—22no—
210Q8⋊C8—22no—
211Q8⋊C8—41no—
212Q8⋊C8—41no—
213Q8⋊C8—41no—
214Q8⋊C8—41no—
215D4⋊C8—22no—
216D4⋊C8—22no—
217D4⋊C8—41no—
218D4⋊C8—41no—
219C22×Q8C22⋊C411yescharacteristic
220C22×Q8C22×C411yescharacteristic
221C22×Q8C22×C411yescharacteristic
222C22×D4C22×C411yescharacteristic
223C23×C4—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C4⋊Q8—22no—
233C4⋊Q8—41no—
234#126—44no—
235#124—22no—
236#124—22no—
237#124—22no—
238#124—41no—
239C4×Q8—24no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—41no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260#117—42no—
261#117—42no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
275C2×C4⋊C4C22⋊C411yescharacteristic
276C2×C4⋊C4C22⋊C411yescharacteristic
277C2×C4⋊C4C22⋊C411yescharacteristic
278C2×C4⋊C4C22⋊C411yescharacteristic
279C2×C4⋊C4C22⋊C411yescharacteristic
280C2×C4⋊C4C22⋊C411yescharacteristic
281C2×C4⋊C4C22⋊C411yescharacteristic
282C2×C4⋊C4C22⋊C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C4⋊C4C22×C411yescharacteristic
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
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×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313C2×C42C22×C411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2411yescharacteristic
327C4⋊C8—24no—
328C4⋊C8—24no—
329C4⋊C8—24no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—42no—
338C4⋊C8—81no—
339C4⋊C8#4112yes—
340#97—42no—
341#97—81no—
342C4×C8—24no—
343C4×C8—24no—
344C4×C8—24no—
345C4×C8—21no—
346C4×C8—21no—
347C4×C8—21no—
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—21no—
351C4×C8—21no—
352C4×C8—42no—
353C4×C8—42no—
354C4×C8—42no—
355C4×C8—81no—
356C4×C8#4112yes—
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—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95C22⋊C411yescharacteristic
381#95C22⋊C411yescharacteristic
382#95C22⋊C411yescharacteristic
383#95C22⋊C411yescharacteristic
384#95C22⋊C411yescharacteristic
385#95C22⋊C411yescharacteristic
386C24—21no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8#13112yes—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C22⋊C411yescharacteristic
432C22×C4C2×C22⋊C411yescharacteristic
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
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—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4#10414yes—
493C22⋊C4—42no—
494C22⋊C4—42no—
495C22⋊C4—41no—
496C22⋊C4—41no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—81no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—22no—
528C42—22no—
529C42—22no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—42no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42—42no—
541C42—42no—
542C42—81no—
543C42#13112yes—
544C42#13112yes—
545C42#13512yes—
546C42#13512yes—
547C42C2×#4111yescharacteristic
548C42C2×#4111yescharacteristic
549C42H(5,𝔽2)11yescharacteristic
550C422−1+411yescharacteristic
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C22×C22⋊C411yescharacteristic
559Q8—28no—
560Q8—22no—
561Q8—22no—
562Q8—41no—
563Q8—41no—
564Q8—41no—
565Q8—41no—
566Q8—41no—
567Q8—41no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
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—22no—
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—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
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—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4#40812yes—
629C2×C4#40812yes—
630C2×C4#41612yes—
631C2×C4#41612yes—
632C2×C4C2×#10412yes—
633C2×C4C2×#10412yes—
634C2×C4C2×#10412yes—
635C2×C4C4×#4111yescharacteristic
636C2×C4#52011yescharacteristic
637C2×C4#54811yescharacteristic
638C2×C4#54811yescharacteristic
639C2×C4#55011yescharacteristic
640C2×C4#55011yescharacteristic
641C2×C4C2×#13111yescharacteristic
642C2×C4C2×#13111yescharacteristic
643C2×C4C2×#13111yescharacteristic
644C2×C4#56811yescharacteristic
645C2×C4C2×#13511yescharacteristic
646C2×C4C2×#13511yescharacteristic
647C2×C4#57711yescharacteristic
648C2×C4#57811yescharacteristic
649C8—24no—
650C8—42no—
651C8—42no—
652C8—42no—
653C8—42no—
654C8—81no—
655C8—81no—
656C8—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22#234111yescharacteristic
665C22C2×#40811yescharacteristic
666C22#288011yescharacteristic
667C22C2×#41611yescharacteristic
668C22#289811yescharacteristic
669C22C22×#10411yescharacteristic
670C22#290111yescharacteristic
671C4—21no—
672C4—21no—
673C4—22no—
674C4—22no—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—22no—
679C4—22no—
680C4—42no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—81no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4—81no—
692C4#296611yescharacteristic
693C4#296911yescharacteristic
694C4#299311yescharacteristic
695C4#299411yescharacteristic
696C4#336011yescharacteristic
697C4#336411yescharacteristic
698C4#339311yescharacteristic
699C4#339411yescharacteristic
700C2—161no—
701C2#1747311yescharacteristic
702C2#1748411yescharacteristic
703C2#1751011yescharacteristic
704C2#1751911yescharacteristic
705C2#2253111yescharacteristic
706C2#2254111yescharacteristic
707C2#2255011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22823C2semidirect
3. C4×C8×Q8C2semidirect
4. #9677C2semidirect
9. #9637C2semidirect
10. #8570C2semidirect
12. #8566C2semidirect
14. #8176C2semidirect
15. C4×Q8⋊C8C2semidirect

Maximal subgroups

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

subgroupindex
#349022
#228232
C4×C8×Q82
#96772
#96762
#96752
#96392
#96372
#85702
#85672
#85662
#81772
#81762
C4×Q8⋊C82