Finite Group Databasebeta

← #217121

Subgroups — #217121

Order 512.

832 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34913C211yesmaximal, characteristic
2#21430C211yesmaximal, characteristic, complemented
3#20755C211yesmaximal, characteristic, complemented
4#9875C211yesmaximal, characteristic, complemented
5#9871C211yesmaximal, characteristic, complemented
6#9645C211yesmaximal, characteristic, complemented
7#9639C211yesmaximal, characteristic, complemented
8#8944C211yesmaximal, characteristic
9#8942C211yesmaximal, characteristic, complemented
10#8939C211yesmaximal, characteristic
11#8937C211yesmaximal, characteristic, complemented
12#8570C211yesmaximal, characteristic, complemented
13#8570C211yesmaximal, characteristic, complemented
14#8565C211yesmaximal, characteristic, complemented
15#8565C211yesmaximal, characteristic, complemented
16C2×#423C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18#2578—21no—
19#2577—21no—
20#2576—21no—
21#2575—21no—
22#2514C411yescharacteristic
23#2508C411yescharacteristic
24#2495C411yescharacteristic
25#2489C411yescharacteristic
26#2454C411yescharacteristic
27#2447C411yescharacteristic
28#2427C411yescharacteristic
29#2420C411yescharacteristic
30#2401—21no—
31#2395—21no—
32C4×#125C2211yescharacteristic
33C4×#124C2211yescharacteristic
34C4×C22⋊Q8C2211yescharacteristic
35C4×#121C2211yescharacteristic
36#2295C2211yescharacteristic
37#2290—21no—
38#2283—21no—
39C42×D4C2211yescharacteristic
40C4×#117C2211yescharacteristic
41#1923—21no—
42#1923—21no—
43#1916—21no—
44#1916—21no—
45#1832—21no—
46#1832—21no—
47#1831C2211yescharacteristic
48#1831C2211yescharacteristic
49#1831C2211yescharacteristic
50#1831C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55#1744C2211yescharacteristic
56#1729—21no—
57#1729—21no—
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
75C22×M16—21no—
76C23×C8—21no—
77C2×#125C2×C411yescharacteristic
78C2×#124C2×C411yescharacteristic
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×#121C2×C411yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4C2×C411yescharacteristic
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4C2311yescharacteristic
86C2×#117D411yescharacteristic
87C2×#117D411yescharacteristic
88C22×C42—21no—
89C22×C42C2311yescharacteristic
90#423—41no—
91#423—41no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—41no—
97C2×C4⋊C8—41no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#401—21no—
111#400—21no—
112#400—21no—
113#400—21no—
114#400—21no—
115#399—21no—
116#398—21no—
117#397—21no—
118#397—21no—
119#397—21no—
120#397—21no—
121#397—21no—
122#397—21no—
123#396—21no—
124#394—21no—
125#394—21no—
126#391C2×C411yescharacteristic
127#391C2×C411yescharacteristic
128#391C2×C411yescharacteristic
129#391C2×C411yescharacteristic
130#390C2×C411yescharacteristic
131#390C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#389C2×C411yescharacteristic
138#388—21no—
139#388—21no—
140#388—21no—
141#388—21no—
142#388—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#386—21no—
153#385—21no—
154#385—21no—
155#385—21no—
156#383—21no—
157#383—21no—
158#382—21no—
159#382—21no—
160#382C2×C411yescharacteristic
161#382C2×C411yescharacteristic
162#381—21no—
163#381—21no—
164#380—21no—
165#380—21no—
166#380C2×C411yescharacteristic
167#380C2×C411yescharacteristic
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4D411yescharacteristic
177C4×C4⋊C4D411yescharacteristic
178C4×C4⋊C4C2311yescharacteristic
179C4×C4⋊C4C2311yescharacteristic
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4—21no—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4C2×C411yescharacteristic
188C4×C22⋊C4C2×C411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191#376—21no—
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—
210Q8⋊C8—41no—
211Q8⋊C8—41no—
212Q8⋊C8—41no—
213Q8⋊C8—41no—
214D4⋊C8—41no—
215D4⋊C8—41no—
216D4⋊C8—41no—
217D4⋊C8—41no—
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
227C2×M16—42no—
228C2×M16—42no—
229C2×M16—42no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—42no—
239C22×C8—42no—
240C22×C8—42no—
241#125—22no—
242#125—41no—
243#124—22no—
244#124—41no—
245C22⋊Q8—22no—
246C22⋊Q8—41no—
247#121—22no—
248#121—41no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—41no—
260C4×D4—41no—
261C4×D4—42no—
262C4×D4—42no—
263#117—22no—
264#117—22no—
265#117—41no—
266#117—41no—
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⋊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×C22⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
301C2×C22⋊C4C22⋊C411yescharacteristic
302C2×C22⋊C4C22⋊C411yescharacteristic
303C2×C22⋊C4C22⋊C411yescharacteristic
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2411yescharacteristic
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8—21no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341C4⋊C8—42no—
342C4⋊C8—42no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C4×C8—21no—
346C4×C8—21no—
347C4×C8—21no—
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—21no—
351C4×C8—21no—
352C4×C8—21no—
353C4×C8—42no—
354C4×C8—42no—
355#95—21no—
356#95—21no—
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#95C22⋊C411yescharacteristic
380#95C22⋊C411yescharacteristic
381#95C22⋊C411yescharacteristic
382#95C22⋊C411yescharacteristic
383#95C22⋊C411yescharacteristic
384#95C22⋊C411yescharacteristic
385#95C22⋊C411yescharacteristic
386#95C22⋊C411yescharacteristic
387C24—21no—
388C24C2×C22⋊C411yescharacteristic
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—42no—
401C2×D4—42no—
402C2×D4—81no—
403C2×D4#13112yes—
404C22×C4—21no—
405C22×C4—21no—
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—41no—
424C22×C4—41no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
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×C4#13512yes—
449C22×C4#13512yes—
450C22×C4C23×C411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453M16—84no—
454C2×C8—22no—
455C2×C8—22no—
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—
477C2×C8—41no—
478C2×C8—84no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—42no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—81no—
505C4⋊C4#13112yes—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—42no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C22⋊C4—81no—
523C22⋊C4—81no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—42no—
551C42—42no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42—41no—
558C42H(5,𝔽2)11yescharacteristic
559C42H(5,𝔽2)11yescharacteristic
560C422−1+411yescharacteristic
561C422−1+411yescharacteristic
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23#41612yes—
576C23C22×C22⋊C411yescharacteristic
577Q8—41no—
578Q8—41no—
579Q8—41no—
580Q8—41no—
581D4—22no—
582D4—22no—
583D4—41no—
584D4—41no—
585D4—84no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
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—22no—
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—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×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—81no—
650C2×C4—81no—
651C2×C4—84no—
652C2×C4—84no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4—81no—
657C2×C4#41612yes—
658C2×C4#52011yescharacteristic
659C2×C4#52011yescharacteristic
660C2×C4#53511yescharacteristic
661C2×C4#53611yescharacteristic
662C2×C4#53711yescharacteristic
663C2×C4#53711yescharacteristic
664C2×C4C2×#13111yescharacteristic
665C2×C4C2×#13111yescharacteristic
666C2×C4#56811yescharacteristic
667C2×C4#56811yescharacteristic
668C2×C4C2×#13511yescharacteristic
669C2×C4C2×#13511yescharacteristic
670C2×C4#57511yescharacteristic
671C2×C4#57511yescharacteristic
672C8—42no—
673C8—42no—
674C8—81no—
675C8—81no—
676C8—81no—
677C8—81no—
678C8—81no—
679C8—81no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22#231111yescharacteristic
695C22#288011yescharacteristic
696C22#288011yescharacteristic
697C22C2×#41611yescharacteristic
698C22#289611yescharacteristic
699C22#290111yescharacteristic
700C22#290111yescharacteristic
701C4—21no—
702C4—21no—
703C4—22no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—42no—
715C4—81no—
716C4—81no—
717C4—81no—
718C4—81no—
719C4—81no—
720C4—81no—
721C4#299211yescharacteristic
722C4#299311yescharacteristic
723C4#299411yescharacteristic
724C4#299511yescharacteristic
725C4#321111yescharacteristic
726C4#321111yescharacteristic
727C4#323211yescharacteristic
728C4#323211yescharacteristic
729C2—42no—
730C2—82no—
731C2#1750811yescharacteristic
732C2#1751011yescharacteristic
733C2#1751311yescharacteristic
734C2#1751311yescharacteristic
735C2#2185511yescharacteristic
736C2#2185811yescharacteristic
737C2#2186111yescharacteristic
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. #21430C2semidirect
3. #20755C2semidirect
4. #9875C2semidirect
5. #9871C2semidirect
6. #9645C2semidirect
7. #9639C2semidirect
9. #8942C2semidirect
11. #8937C2semidirect
12. #8570C2semidirect
13. #8570C2semidirect
14. #8565C2semidirect
15. #8565C2semidirect

Maximal subgroups

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

subgroupindex
#349132
#214302
#207552
#98752
#98712
#96452
#96392
#89442
#89422
#89392
#89372
#85702
#85652