Finite Group Databasebeta

← #162415

Subgroups — #162415

Order 512.

923 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#35008C211yesmaximal, characteristic
2#22696C211yesmaximal, characteristic, complemented
3#21996C211yesmaximal, characteristic, complemented
4#7732C211yesmaximal, characteristic, complemented
5#7728C211yesmaximal, characteristic, complemented
6#7723C211yesmaximal, characteristic, complemented
7#7714C211yesmaximal, characteristic, complemented
8#7672C211yesmaximal, characteristic, complemented
9#7668C211yesmaximal, characteristic, complemented
10#7663C211yesmaximal, characteristic, complemented
11#7654C211yesmaximal, characteristic, complemented
12#7586C211yesmaximal, characteristic, complemented
13#7573C211yesmaximal, characteristic, complemented
14#7565C211yesmaximal, characteristic, complemented
15C2×#1271C211yesmaximal, characteristic, complemented
16C2×#539—21no—
17C2×C8×D4—21no—
18#2860C2211yescharacteristic
19#2859C2211yescharacteristic
20#2857C2211yescharacteristic
21#2854C2211yescharacteristic
22#2664—21no—
23#2663—21no—
24#2556—21no—
25#2553—21no—
26#2506C411yescharacteristic
27#2501C411yescharacteristic
28#2496C411yescharacteristic
29#2488—21no—
30#2486C411yescharacteristic
31#2452C411yescharacteristic
32#2440C411yescharacteristic
33#2425—21no—
34#2422C411yescharacteristic
35#2416C411yescharacteristic
36#2322—21no—
37C4×#121C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C42×D4C2211yescharacteristic
40#1939—21no—
41#1923—21no—
42#1916—21no—
43#1869—21no—
44#1850—21no—
45#1850—21no—
46#1850—21no—
47#1839C2211yescharacteristic
48#1831C2211yescharacteristic
49#1831C2211yescharacteristic
50#1817C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53#1797C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C4×C22⋊C8C2211yescharacteristic
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1271C2211yescharacteristic
60#1271C2211yescharacteristic
61#1271C2211yescharacteristic
62#1271C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C23×C8—21no—
76C23×C8—21no—
77C23×C8—21no—
78#539—44no—
79#539—44no—
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×#126—21no—
82C2×#126—21no—
83C2×#125—21no—
84C2×#125—21no—
85C2×#123—21no—
86C2×#123—21no—
87C2×#123—21no—
88C2×#123—21no—
89C2×#123—21no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×#121—21no—
94C2×#121C2×C411yescharacteristic
95C2×#121C2×C411yescharacteristic
96C2×C4×D4—21no—
97C2×C4×D4C2×C411yescharacteristic
98C2×C4×D4C2×C411yescharacteristic
99C2×C4×D4D411yescharacteristic
100C2×C4×D4D411yescharacteristic
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2×C411yescharacteristic
103C22×C4⋊C4C2×C411yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C8×D4—44no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120#400—21no—
121#400—21no—
122#399—21no—
123#399—21no—
124#398—21no—
125#398—21no—
126#398—21no—
127#397—21no—
128#397—21no—
129#397—21no—
130#396—21no—
131#396—21no—
132#395—21no—
133#394—21no—
134#394—21no—
135#393—21no—
136#391—21no—
137#390C2×C411yescharacteristic
138#390C2×C411yescharacteristic
139#390C2×C411yescharacteristic
140#390C2311yescharacteristic
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#389C2×C411yescharacteristic
153#389C2311yescharacteristic
154#389C2311yescharacteristic
155#389C2311yescharacteristic
156#388—21no—
157#388—21no—
158#388—21no—
159#388—21no—
160#388—21no—
161#388—21no—
162#387—21no—
163#387—21no—
164#386—21no—
165#386—21no—
166#386—21no—
167#385—21no—
168#385—21no—
169#385—21no—
170#384C2×C411yescharacteristic
171#384C2×C411yescharacteristic
172#384C2×C411yescharacteristic
173#382—21no—
174#381—21no—
175#380—21no—
176#380C2×C411yescharacteristic
177#380C2×C411yescharacteristic
178#380C2×C411yescharacteristic
179#380C2×C411yescharacteristic
180C4×C4⋊C4C2×C411yescharacteristic
181C4×C4⋊C4C2×C411yescharacteristic
182C4×C4⋊C4D411yescharacteristic
183C4×C4⋊C4D411yescharacteristic
184C4×C22⋊C4—21no—
185C4×C22⋊C4C2×C411yescharacteristic
186C4×C22⋊C4C2×C411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C4×C22⋊C4D411yescharacteristic
191C2×#95—21no—
192C43C2311yescharacteristic
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336—21no—
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208#336C2311yescharacteristic
209C22×Q8—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C22⋊C411yescharacteristic
213C22×D4C22×C411yescharacteristic
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4C22×C411yescharacteristic
218C23×C4C22×C411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—42no—
232C22×C8—42no—
233C22×C8—42no—
234C22×C8—42no—
235C22×C8—42no—
236C22×C8—81no—
237C22×C8—81no—
238C22×C8—81no—
239C22×C8—81no—
240C4⋊Q8—81no—
241#126—44no—
242#126—42no—
243#126—42no—
244#125—44no—
245#125—42no—
246#125—42no—
247#123—44no—
248#123—44no—
249#123—42no—
250#123—42no—
251#123—44no—
252#123—44no—
253C22⋊Q8—42no—
254C22⋊Q8—42no—
255C22⋊Q8—44no—
256C22⋊Q8—44no—
257#121—24no—
258#121—44no—
259#121—81no—
260C4×D4—24no—
261C4×D4—42no—
262C4×D4—42no—
263C4×D4—42no—
264C4×D4—81no—
265C4×D4D814yes—
266C4×D4SD1614yes—
267C2×C4⋊C4—24no—
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⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—81no—
290C2×C4⋊C4C22⋊C411yescharacteristic
291C2×C4⋊C4C22⋊C411yescharacteristic
292C2×C4⋊C4C22⋊C411yescharacteristic
293C2×C4⋊C4C22×C411yescharacteristic
294C2×C4⋊C4C22×C411yescharacteristic
295C2×C4⋊C4C22×C411yescharacteristic
296C2×C4⋊C4C22×C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4C22⋊C411yescharacteristic
316C2×C22⋊C4C22⋊C411yescharacteristic
317C2×C22⋊C4C22⋊C411yescharacteristic
318C2×C22⋊C4C22⋊C411yescharacteristic
319C2×C22⋊C4C22⋊C411yescharacteristic
320C2×C22⋊C4C22⋊C411yescharacteristic
321C2×C42—21no—
322C2×C42—81no—
323C2×C42—81no—
324C2×C42C22×C411yescharacteristic
325C2×C42C22×C411yescharacteristic
326C2×C42C22×C411yescharacteristic
327C2×C42C22×C411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2411yescharacteristic
337C4⋊C8—42no—
338C4⋊C8—42no—
339C4⋊C8—81no—
340C4⋊C8—81no—
341C4⋊C8—81no—
342C4⋊C8—81no—
343C22⋊C8—42no—
344C22⋊C8—42no—
345C22⋊C8—42no—
346C22⋊C8—81no—
347C22⋊C8—81no—
348C4×C8—42no—
349C4×C8—81no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
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#95C22⋊C411yescharacteristic
366#95C22⋊C411yescharacteristic
367#95C22⋊C411yescharacteristic
368#95C22⋊C411yescharacteristic
369#95C22⋊C411yescharacteristic
370#95C22⋊C411yescharacteristic
371C24—21no—
372C24C2×C22⋊C411yescharacteristic
373C24C2×C22⋊C411yescharacteristic
374C2×Q8—42no—
375C2×Q8—81no—
376C2×Q8—81no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—81no—
387C2×D4—81no—
388C2×D4#10214yes—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—81no—
421C22×C4—81no—
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×C4#13114yes—
433C22×C4C2×D812yes—
434C22×C4C2×D812yes—
435C22×C4C2×SD1612yes—
436C22×C4C2×SD1612yes—
437C22×C4C23×C411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C2×C8—22no—
441C2×C8—41no—
442C2×C8—41no—
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—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—84no—
465C2×C8—84no—
466C2×C8—84no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
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—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—81no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4#10214yes—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—41no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—42no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—42no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4#10214yes—
521C22⋊C4#10214yes—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—42no—
536C42C2×D812yes—
537C42C2×SD1612yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23#40812yes—
556C23C2×#10212yes—
557C23C2×#10212yes—
558C23#42112yes—
559C23C22×C22⋊C411yescharacteristic
560Q8—162no—
561D4—28no—
562D4—82no—
563D4—84no—
564D4—162no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
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—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—41no—
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—82no—
627C2×C4—84no—
628C2×C4—162no—
629C2×C4—162no—
630C2×C4#40812yes—
631C2×C4C2×#10212yes—
632C2×C4C2×#10212yes—
633C2×C4C2×#10212yes—
634C2×C4C2×#10212yes—
635C2×C4C2×#13112yes—
636C2×C4C22×D811yescharacteristic
637C2×C4C22×SD1611yescharacteristic
638C2×C4#57711yescharacteristic
639C2×C4#57811yescharacteristic
640C8—42no—
641C8—81no—
642C8—81no—
643C8—81no—
644C8—81no—
645C8—81no—
646C8—81no—
647C8—81no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22#148014yes—
669C22C2×#40811yescharacteristic
670C22#288311yescharacteristic
671C22#288911yescharacteristic
672C22C22×#10211yescharacteristic
673C22#289811yescharacteristic
674C22C2×#42111yescharacteristic
675C22#290111yescharacteristic
676C4—22no—
677C4—22no—
678C4—21no—
679C4—21no—
680C4—21no—
681C4—21no—
682C4—22no—
683C4—22no—
684C4—22no—
685C4—22no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—81no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4—81no—
697C4—81no—
698C2—24no—
699C2—42no—
700C2—161no—
701C2#1738711yescharacteristic
702C2C2×#148011yescharacteristic
703C2#1748411yescharacteristic
704C2#1759811yescharacteristic
705C2#1761311yescharacteristic
706C2#1768011yescharacteristic
707C2#1772111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2416C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22696C2semidirect
3. #21996C2semidirect
4. #7732C2semidirect
5. #7728C2semidirect
6. #7723C2semidirect
7. #7714C2semidirect
8. #7672C2semidirect
9. #7668C2semidirect
10. #7663C2semidirect
11. #7654C2semidirect
12. #7586C2semidirect
13. #7573C2semidirect
14. #7565C2semidirect
15. C2×#1271C2semidirect

Maximal subgroups

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

subgroupindex
#350082
#226962
#219962
#77322
#77282
#77232
#77142
#76722
#76682
#76632
#76542
#75862
#75732
#75652
C2×#12712