Finite Group Databasebeta

← #359309

Subgroups — #359309

Order 512.

1412 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2092C211yesmaximal, characteristic
2C2×#1776C211yesmaximal, characteristic, complemented
3C2×C4×#107C211yesmaximal, characteristic, complemented
4C2×#1762C211yesmaximal, characteristic
5C2×C4×#102C211yesmaximal, characteristic
6#13785C211yesmaximal, characteristic
7#13779C211yesmaximal, characteristic, complemented
8#11378C211yesmaximal, characteristic, complemented
9#11377C211yesmaximal, characteristic, complemented
10#11372C211yesmaximal, characteristic, complemented
11#11324C211yesmaximal, characteristic, complemented
12#11123C211yesmaximal, characteristic
13#11123C211yesmaximal, characteristic
14#11049C211yesmaximal, characteristic
15#10694C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#107C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#102C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×#376C2211yescharacteristic
35#2092C418yescomplemented
36#1776—24no—
37C4×#107—24no—
38#1762—24no—
39C4×#102—24no—
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52#1437—22no—
53#1436—22no—
54#1436—22no—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C22⋊#95—22no—
60C23×D4C2311yescharacteristic
61C24×C4—21no—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4Q814yes—
72C22×C4⋊C4—22no—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—22no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#107—22no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—22no—
94C2×#107—22no—
95C2×#107—22no—
96C2×#107—22no—
97C2×#107C2×C414yescomplemented
98C2×#106—22no—
99C2×#106—22no—
100C2×#106C2×C414yescomplemented
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—22no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—22no—
108C2×#102—22no—
109C2×#102—22no—
110C2×#102—22no—
111C2×#102C2×C414yes—
112C2×#102C2×C414yes—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97D414yes—
116C2×C4×C8—22no—
117C2×C4×C8—22no—
118C2×C4×C8D414yes—
119#390—22no—
120#390—22no—
121#390—22no—
122#390C2×C414yes—
123#384—22no—
124#384—22no—
125#382—22no—
126#382C2×C414yescomplemented
127C4×C4⋊C4—22no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133#376—22no—
134#376—22no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95D412yes—
144C2×#95D412yes—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#336—22no—
154#336—22no—
155#336—22no—
156#336C2×C414yescomplemented
157C25—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4C22×C412yes—
170C22×D4C2×Q812yes—
171C22×D4#4112yes—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—24no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—82no—
183C23×C4—82no—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2411yescharacteristic
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8C22×C412yes—
210C22×C8C22×C412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214C22×C8C2×D412yes—
215C22×C8#4112yes—
216C22×C8#4112yes—
217C4×D4—28no—
218C4×D4—28no—
219C4×D4—44no—
220C4×D4—44no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—82no—
258C2×C4⋊C4—82no—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C22×C412yes—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C2×Q812yes—
263C2×C4⋊C4#4112yes—
264C2×C4⋊C4#4112yes—
265C2×C4⋊C4#4112yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—82no—
272C2×C22⋊C4—82no—
273C2×C22⋊C4—82no—
274C2×C22⋊C4—82no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—82no—
289C2×C42—82no—
290C2×C42—82no—
291C2×C42C22×C412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×Q812yes—
295C2×C42#4112yes—
296C2×C42#4112yes—
297C2×C42#4112yes—
298C2×C42#4112yes—
299#107—28no—
300#107—28no—
301#107—28no—
302#106—28no—
303#102—28no—
304#102—28no—
305#102—28no—
306#102—28no—
307#97—28no—
308C4×C8—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95—82no—
316#95—82no—
317#95—82no—
318#95—82no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—41no—
323C24—41no—
324C24—41no—
325C24—41no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24—42no—
330C24C22×D411yescharacteristic
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4C4×Q814yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—48no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4C4×D414yes—
417C22×C4C22⋊Q812yes—
418C22×C4C22⋊Q812yes—
419C22×C4#12312yes—
420C22×C4#12312yes—
421C22×C4#12712yes—
422C22×C4#12712yes—
423C22×C4C23×C411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×Q811yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4C2×#4111yescharacteristic
434C22×C4C2×#4111yescharacteristic
435C22×C4C2×#4111yescharacteristic
436C22×C4C2×#4111yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
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×C8C4×D414yes—
464C2×C8C4×D414yes—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—88no—
494C22⋊C4—48no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—84no—
506C42—84no—
507C42—84no—
508C42C22⋊Q814yes—
509C42C22⋊Q814yes—
510C42#12314yes—
511C42#12314yes—
512C23—24no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—82no—
526C23—82no—
527C23—82no—
528C23—82no—
529C23—81no—
530C23—81no—
531C23—81no—
532C23—81no—
533C23—82no—
534C23—82no—
535C23C2×C4×D411yescharacteristic
536C23C2×C22⋊Q811yescharacteristic
537C23C2×#12311yescharacteristic
538C23#52511yescharacteristic
539C23#52511yescharacteristic
540C23C2×#12711yescharacteristic
541C23#53211yescharacteristic
542C23#57612yes—
543C23#57612yes—
544C23#57712yes—
545C23#57812yes—
546D4—24no—
547D4—24no—
548D4—42no—
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—816no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—81no—
598C2×C4—81no—
599C2×C4—81no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4#39914yes—
613C2×C4#39914yes—
614C2×C4C2×C4×D411yescharacteristic
615C2×C4C2×C4×D411yescharacteristic
616C2×C4C2×C4×Q811yescharacteristic
617C2×C4C4×#4111yescharacteristic
618C2×C4C4×#4111yescharacteristic
619C2×C4C4×#4111yescharacteristic
620C2×C4C2×C22⋊Q811yescharacteristic
621C2×C4C2×C22⋊Q811yescharacteristic
622C2×C4C2×#12311yescharacteristic
623C2×C4C2×#12311yescharacteristic
624C2×C4#52511yescharacteristic
625C2×C4#52511yescharacteristic
626C2×C4#52511yescharacteristic
627C2×C4#52511yescharacteristic
628C2×C4#52911yescharacteristic
629C2×C4#52911yescharacteristic
630C2×C4#52911yescharacteristic
631C2×C4#52911yescharacteristic
632C2×C4#52911yescharacteristic
633C2×C4#52911yescharacteristic
634C2×C4#52911yescharacteristic
635C2×C4#52911yescharacteristic
636C2×C4#52911yescharacteristic
637C2×C4#53211yescharacteristic
638C2×C4#53211yescharacteristic
639C2×C4#53311yescharacteristic
640C2×C4#53311yescharacteristic
641C2×C4#53311yescharacteristic
642C8—24no—
643C8—24no—
644C8—42no—
645C8—42no—
646C22—44no—
647C22—48no—
648C22—82no—
649C22—82no—
650C22—84no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—84no—
656C22—84no—
657C22C4×#12311yescharacteristic
658C22C4×C22⋊Q811yescharacteristic
659C22C4×#12711yescharacteristic
660C22C2×#39911yescharacteristic
661C22#239511yescharacteristic
662C22#239611yescharacteristic
663C22#239711yescharacteristic
664C22#294612yes—
665C22#295112yes—
666C22#308112yes—
667C22#308312yes—
668C22#309612yes—
669C22#309812yes—
670C22#312012yes—
671C22#312112yes—
672C22#312212yes—
673C22#312212yes—
674C22#315411yescharacteristic
675C22#315511yescharacteristic
676C22#315511yescharacteristic
677C22#315611yescharacteristic
678C22#317512yes—
679C22#317912yes—
680C4—24no—
681C4—24no—
682C4—44no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C4C4×#12311yescharacteristic
692C4C4×#12311yescharacteristic
693C4C4×C22⋊Q811yescharacteristic
694C4C4×C22⋊Q811yescharacteristic
695C4#239511yescharacteristic
696C4#239511yescharacteristic
697C4#239611yescharacteristic
698C4#239611yescharacteristic
699C2—84no—
700C2C4×#39911yescharacteristic
701C2#2159012yes—
702C2#2162012yes—
703C2#2169611yescharacteristic
704C2#2170411yescharacteristic
705C2#2373511yescharacteristic
706C2#2373711yescharacteristic
707C2#2376712yes—
708C2#2378712yes—
709C2#2380911yescharacteristic
710C2#2381011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1776C2semidirect
3. C2×C4×#107C2semidirect
7. #13779C2semidirect
8. #11378C2semidirect
9. #11377C2semidirect
10. #11372C2semidirect
11. #11324C2semidirect
15. #10694C2semidirect
35. #2092C4semidirect
97. C2×#107C2×C4semidirect
100. C2×#106C2×C4semidirect
126. #382C2×C4semidirect
156. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20922
C2×#17762
C2×C4×#1072
C2×#17622
C2×C4×#1022
#137852
#137792
#113782
#113772
#113722
#113242
#111232
#110492
#106942