Finite Group Databasebeta

← #217169

Subgroups — #217169

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 738 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34913C211yesmaximal, characteristic
2#21349C211yesmaximal, characteristic, complemented
3#20786C211yesmaximal, characteristic, complemented
4#9867C211yesmaximal, characteristic, complemented
5#9866C211yesmaximal, characteristic, complemented
6#9676C211yesmaximal, characteristic, complemented
7#9675C211yesmaximal, characteristic, complemented
8#9076C211yesmaximal, characteristic
9#9075C211yesmaximal, characteristic, complemented
10#9072C211yesmaximal, characteristic
11#9071C211yesmaximal, characteristic, complemented
12#8570C211yesmaximal, characteristic, complemented
13#8570C211yesmaximal, characteristic, complemented
14#8565C211yesmaximal, characteristic, complemented
15#8565C211yesmaximal, characteristic, complemented
16C2×#431C2211yescharacteristic
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#1939—21no—
42#1939—21no—
43#1850—21no—
44#1850—21no—
45#1840—21no—
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1818—21no—
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C4×C4⋊C8C2211yescharacteristic
56C4×C4⋊C8C2211yescharacteristic
57#1750—21no—
58#1746—21no—
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
75C23×C8—21no—
76C2×#125C2×C411yescharacteristic
77C2×#124C2×C411yescharacteristic
78C2×C22⋊Q8C2×C411yescharacteristic
79C2×#121C2×C411yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4C2×C411yescharacteristic
84C2×C4×D4C2311yescharacteristic
85C2×#117D411yescharacteristic
86C2×#117D411yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
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⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—41no—
107C2×C4×C8—41no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#401—21no—
113#400—21no—
114#400—21no—
115#400—21no—
116#400—21no—
117#399—21no—
118#398—21no—
119#397—21no—
120#397—21no—
121#397—21no—
122#397—21no—
123#397—21no—
124#397—21no—
125#396—21no—
126#394—21no—
127#394—21no—
128#391C2×C411yescharacteristic
129#391C2×C411yescharacteristic
130#391C2×C411yescharacteristic
131#391C2×C411yescharacteristic
132#390C2×C411yescharacteristic
133#390C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#389C2×C411yescharacteristic
138#389C2×C411yescharacteristic
139#389C2×C411yescharacteristic
140#388—21no—
141#388—21no—
142#388—21no—
143#388—21no—
144#388—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#387—21no—
151#387—21no—
152#386—21no—
153#386—21no—
154#386—21no—
155#385—21no—
156#385—21no—
157#385—21no—
158#383—21no—
159#383—21no—
160#382—21no—
161#382—21no—
162#382C2×C411yescharacteristic
163#382C2×C411yescharacteristic
164#381—21no—
165#381—21no—
166#380—21no—
167#380—21no—
168#380C2×C411yescharacteristic
169#380C2×C411yescharacteristic
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⋊C4C2311yescharacteristic
181C4×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
189C4×C22⋊C4C2×C411yescharacteristic
190C4×C22⋊C4C2×C411yescharacteristic
191C4×C22⋊C4D411yescharacteristic
192C4×C22⋊C4D411yescharacteristic
193#376—21no—
194#376D411yescharacteristic
195#376D411yescharacteristic
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—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
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—42no—
236C22×C8—42no—
237C22×C8—42no—
238#125—22no—
239#125—41no—
240#124—22no—
241#124—41no—
242C22⋊Q8—22no—
243C22⋊Q8—41no—
244#121—22no—
245#121—41no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4—21no—
256C4×D4—41no—
257C4×D4—41no—
258C4×D4—42no—
259C4×D4—42no—
260#117—22no—
261#117—22no—
262#117—41no—
263#117—41no—
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×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4C22⋊C411yescharacteristic
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—42no—
311C2×C42—42no—
312C2×C42C22×C411yescharacteristic
313C2×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
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×C42C2×D411yescharacteristic
327C2×C42C2411yescharacteristic
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—42no—
337C4⋊C8—42no—
338C4⋊C8—42no—
339C4⋊C8—42no—
340C4⋊C8—42no—
341C22⋊C8—42no—
342C22⋊C8—42no—
343C22⋊C8—42no—
344#97—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
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—48no—
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—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—21no—
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—41no—
492C4⋊C4—41no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4#13112yes—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—42no—
519C22⋊C4—81no—
520C22⋊C4—81no—
521C22⋊C4—81no—
522C22⋊C4—81no—
523C42—21no—
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—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—42no—
550C42—42no—
551C42—41no—
552C42—41no—
553C42—41no—
554C42—41no—
555C42—41no—
556C42—41no—
557C42H(5,𝔽2)11yescharacteristic
558C42H(5,𝔽2)11yescharacteristic
559C422−1+411yescharacteristic
560C422−1+411yescharacteristic
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23#41612yes—
575C23C22×C22⋊C411yescharacteristic
576Q8—41no—
577Q8—41no—
578Q8—41no—
579Q8—41no—
580D4—22no—
581D4—22no—
582D4—41no—
583D4—41no—
584D4—84no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
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—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—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—42no—
625C2×C4—42no—
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—41no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4#41612yes—
657C2×C4#52011yescharacteristic
658C2×C4#52011yescharacteristic
659C2×C4#53511yescharacteristic
660C2×C4#53611yescharacteristic
661C2×C4#53711yescharacteristic
662C2×C4#53711yescharacteristic
663C2×C4C2×#13111yescharacteristic
664C2×C4C2×#13111yescharacteristic
665C2×C4#56811yescharacteristic
666C2×C4#56811yescharacteristic
667C2×C4C2×#13511yescharacteristic
668C2×C4C2×#13511yescharacteristic
669C2×C4#57511yescharacteristic
670C2×C4#57511yescharacteristic
671C8—42no—
672C8—42no—
673C8—81no—
674C8—81no—
675C8—81no—
676C8—81no—
677C8—81no—
678C8—81no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22#231111yescharacteristic
694C22#288011yescharacteristic
695C22#288011yescharacteristic
696C22C2×#41611yescharacteristic
697C22#289611yescharacteristic
698C22#290111yescharacteristic
699C22#290111yescharacteristic
700C4—22no—
701C4—21no—
702C4—21no—
703C4—41no—
704C4—41no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—42no—
713C4—41no—
714C4—81no—
715C4—81no—
716C4—81no—
717C4—81no—
718C4—81no—
719C4—81no—
720C4#299211yescharacteristic
721C4#299311yescharacteristic
722C4#299411yescharacteristic
723C4#299511yescharacteristic
724C4#325011yescharacteristic
725C4#325011yescharacteristic
726C4#325011yescharacteristic
727C4#325011yescharacteristic
728C2—42no—
729C2—82no—
730C2#1750811yescharacteristic
731C2#1751011yescharacteristic
732C2#1751311yescharacteristic
733C2#1751311yescharacteristic
734C2#2185511yescharacteristic
735C2#2188611yescharacteristic
736C2#2190111yescharacteristic
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. #21349C2semidirect
3. #20786C2semidirect
4. #9867C2semidirect
5. #9866C2semidirect
6. #9676C2semidirect
7. #9675C2semidirect
9. #9075C2semidirect
11. #9071C2semidirect
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
#213492
#207862
#98672
#98662
#96762
#96752
#90762
#90752
#90722
#90712
#85702
#85652