Finite Group Databasebeta

← #217170

Subgroups — #217170

Order 512.

828 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34916C211yesmaximal, characteristic
2#21349C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#9868C211yesmaximal, characteristic, complemented
5#9864C211yesmaximal, characteristic, complemented
6#9677C211yesmaximal, characteristic, complemented
7#9673C211yesmaximal, characteristic, complemented
8#9075C211yesmaximal, characteristic
9#9075C211yesmaximal, characteristic
10#9071C211yesmaximal, characteristic
11#9071C211yesmaximal, characteristic
12#8568C211yesmaximal, characteristic, complemented
13#8568C211yesmaximal, characteristic, complemented
14#8564C211yesmaximal, characteristic, complemented
15#8564C211yesmaximal, characteristic, complemented
16C2×#433—21no—
17C2×#431C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2590—21no—
20#2589—21no—
21#2588—21no—
22#2587—21no—
23#2511C411yescharacteristic
24#2507C411yescharacteristic
25#2497C411yescharacteristic
26#2492C411yescharacteristic
27#2451C411yescharacteristic
28#2446C411yescharacteristic
29#2429C411yescharacteristic
30#2423C411yescharacteristic
31#2402—21no—
32#2396—21no—
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35C4×C22⋊Q8C2211yescharacteristic
36C4×C22⋊Q8C2211yescharacteristic
37#2298—21no—
38#2295—21no—
39#2295C2211yescharacteristic
40C42×Q8C2211yescharacteristic
41C4×#117C2211yescharacteristic
42#1939—21no—
43#1939—21no—
44#1850—21no—
45#1850—21no—
46#1842C2211yescharacteristic
47#1842C2211yescharacteristic
48#1832—21no—
49#1798—21no—
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1768—21no—
53#1767C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C4×C4⋊C8C2211yescharacteristic
56C4×C4⋊C8C2211yescharacteristic
57C4×C4⋊C8C2211yescharacteristic
58C4×C4⋊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
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71#1279C2211yescharacteristic
72#1279C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C23×C8—21no—
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×C22⋊Q8C2×C411yescharacteristic
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×C4×Q8—21no—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×#117D411yescharacteristic
86C2×#117D411yescharacteristic
87C22×C42C2311yescharacteristic
88#433—44no—
89#431—41no—
90#431—41no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×#97—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—41no—
111C2×C4×C8—41no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#400—21no—
117#400—21no—
118#400—21no—
119#400—21no—
120#400—21no—
121#399—21no—
122#398—21no—
123#398—21no—
124#398—21no—
125#398—21no—
126#398—21no—
127#398—21no—
128#396—21no—
129#396—21no—
130#395—21no—
131#393—21no—
132#391—21no—
133#391—21no—
134#391—21no—
135#391C2×C411yescharacteristic
136#391C2×C411yescharacteristic
137#391C2×C411yescharacteristic
138#391C2×C411yescharacteristic
139#391C2×C411yescharacteristic
140#391C2×C411yescharacteristic
141#389—21no—
142#389—21no—
143#389—21no—
144#389—21no—
145#389—21no—
146#389C2×C411yescharacteristic
147#389C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#389C2×C411yescharacteristic
151#389C2×C411yescharacteristic
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#387—21no—
166#387—21no—
167#387—21no—
168#385—21no—
169#385—21no—
170#384C2×C411yescharacteristic
171#384C2×C411yescharacteristic
172#380C2×C411yescharacteristic
173#380C2×C411yescharacteristic
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4—21no—
182C4×C4⋊C4—21no—
183C4×C4⋊C4C2×C411yescharacteristic
184C4×C4⋊C4C2×C411yescharacteristic
185C4×C4⋊C4C2×C411yescharacteristic
186C4×C4⋊C4C2×C411yescharacteristic
187C4×C4⋊C4D411yescharacteristic
188C4×C4⋊C4D411yescharacteristic
189C4×C4⋊C4C2311yescharacteristic
190C4×C4⋊C4C2311yescharacteristic
191C4×C22⋊C4—21no—
192C4×C22⋊C4—21no—
193C4×C22⋊C4C2×C411yescharacteristic
194C4×C22⋊C4C2×C411yescharacteristic
195C4×C22⋊C4D411yescharacteristic
196C4×C22⋊C4D411yescharacteristic
197#376—21no—
198#376D411yescharacteristic
199#376D411yescharacteristic
200C43—21no—
201C43C2311yescharacteristic
202C43C2311yescharacteristic
203#336—21no—
204#336—21no—
205#336—21no—
206#336—21no—
207#336—21no—
208#336—21no—
209#336—21no—
210#336—21no—
211Q8⋊C8—41no—
212Q8⋊C8—41no—
213Q8⋊C8—41no—
214Q8⋊C8—41no—
215Q8⋊C8—41no—
216Q8⋊C8—41no—
217Q8⋊C8—41no—
218Q8⋊C8—41no—
219C22×Q8—21no—
220C22×Q8C22×C411yescharacteristic
221C22×Q8C22×C411yescharacteristic
222C23×C4C22×C411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—42no—
234C22×C8—42no—
235C22×C8—42no—
236C4⋊Q8—22no—
237C4⋊Q8—22no—
238C4⋊Q8—41no—
239C4⋊Q8—41no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C22⋊Q8—41no—
243C22⋊Q8—41no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—41no—
255C4×Q8—41no—
256C4×Q8—42no—
257C4×Q8—42no—
258#117—41no—
259#117—41no—
260#117SD1614yes—
261#117Q1614yes—
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⋊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⋊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×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313C2×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
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—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—42no—
336C4⋊C8—42no—
337C4⋊C8—42no—
338C4⋊C8—42no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341C4⋊C8—42no—
342C4⋊C8—42no—
343C4⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—42no—
346C22⋊C8—42no—
347#97—42no—
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—21no—
351C4×C8—21no—
352C4×C8—21no—
353C4×C8—21no—
354C4×C8—21no—
355C4×C8—21no—
356C4×C8—42no—
357C4×C8—42no—
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#95—21no—
381#95—21no—
382#95C22⋊C411yescharacteristic
383#95C22⋊C411yescharacteristic
384#95C22⋊C411yescharacteristic
385#95C22⋊C411yescharacteristic
386#95C22⋊C411yescharacteristic
387#95C22⋊C411yescharacteristic
388#95C22⋊C411yescharacteristic
389#95C22⋊C411yescharacteristic
390C24C2×C22⋊C411yescharacteristic
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—21no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—81no—
405C2×Q8#13112yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
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—41no—
427C22×C4—41no—
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
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×SD1612yes—
440C22×C4C2×Q1612yes—
441C22×C4C23×C411yescharacteristic
442C22×C4C22×D411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—48no—
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—
468C2×C8—41no—
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—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4#10314yes—
504C4⋊C4#10314yes—
505C4⋊C4#13112yes—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—81no—
511C22⋊C4—81no—
512C22⋊C4#10314yes—
513C22⋊C4#10314yes—
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—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—42no—
537C42—42no—
538C42—41no—
539C42—41no—
540C42—41no—
541C42—41no—
542C42—41no—
543C42—41no—
544C42—41no—
545C42—41no—
546C42—81no—
547C42—81no—
548C42C2×SD1612yes—
549C42C2×SD1612yes—
550C42C2×Q1612yes—
551C42C2×Q1612yes—
552C42H(5,𝔽2)11yescharacteristic
553C42H(5,𝔽2)11yescharacteristic
554C422−1+411yescharacteristic
555C422−1+411yescharacteristic
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23C2×#10312yes—
563C23C22×C22⋊C411yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—41no—
567Q8—41no—
568Q8—41no—
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—84no—
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—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
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—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—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4C2×#10312yes—
638C2×C4C2×#10312yes—
639C2×C4C2×#10312yes—
640C2×C4C2×#10312yes—
641C2×C4C2×#10312yes—
642C2×C4#52011yescharacteristic
643C2×C4#52011yescharacteristic
644C2×C4#53511yescharacteristic
645C2×C4#53611yescharacteristic
646C2×C4#53711yescharacteristic
647C2×C4#53711yescharacteristic
648C2×C4C2×#13111yescharacteristic
649C2×C4C2×#13111yescharacteristic
650C2×C4#56811yescharacteristic
651C2×C4#56811yescharacteristic
652C2×C4C22×SD1611yescharacteristic
653C2×C4C22×Q1611yescharacteristic
654C2×C4#57511yescharacteristic
655C2×C4#57511yescharacteristic
656C8—42no—
657C8—42no—
658C8—81no—
659C8—81no—
660C8—81no—
661C8—81no—
662C8—81no—
663C8—81no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22#231111yescharacteristic
672C22#288011yescharacteristic
673C22#288011yescharacteristic
674C22C22×#10311yescharacteristic
675C22#289611yescharacteristic
676C22#290111yescharacteristic
677C22#290111yescharacteristic
678C4—22no—
679C4—22no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—21no—
684C4—21no—
685C4—41no—
686C4—42no—
687C4—41no—
688C4—41no—
689C4—42no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4—81no—
699C4#299211yescharacteristic
700C4#299311yescharacteristic
701C4#299411yescharacteristic
702C4#299511yescharacteristic
703C4#324611yescharacteristic
704C4#324611yescharacteristic
705C4#324911yescharacteristic
706C4#324911yescharacteristic
707C2—42no—
708C2#1750911yescharacteristic
709C2#1750911yescharacteristic
710C2#1751311yescharacteristic
711C2#1751611yescharacteristic
712C2#2185511yescharacteristic
713C2#2188511yescharacteristic
714C2#2190111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊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
4. #9868C2semidirect
5. #9864C2semidirect
6. #9677C2semidirect
7. #9673C2semidirect
12. #8568C2semidirect
13. #8568C2semidirect
14. #8564C2semidirect
15. #8564C2semidirect

Maximal subgroups

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

subgroupindex
#349162
#213492
#208122
#98682
#98642
#96772
#96732
#90752
#90712
#85682
#85642