Finite Group Databasebeta

← #366254

Subgroups — #366254

Order 512.

1092 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1927C211yesmaximal, characteristic
2#13921C211yesmaximal, characteristic
3#13817C211yesmaximal, characteristic
4#11378C211yesmaximal, characteristic
5#11256C211yesmaximal, characteristic
6#11190C211yesmaximal, characteristic
7#11124C211yesmaximal, characteristic
8#10926C211yesmaximal, characteristic
9#10908C211yesmaximal, characteristic
10#10907C211yesmaximal, characteristic
11#10806C211yesmaximal, characteristic
12#10797C211yesmaximal, characteristic
13#10789C211yesmaximal, characteristic
14#10789C211yesmaximal, characteristic
15C4×#340C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C22×#107C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26C2×#383—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1927C418yescomplemented
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46#1443—21no—
47#1443—21no—
48#1436—21no—
49#1436—21no—
50C4⋊#95C2211yescharacteristic
51C4⋊#95C2211yescharacteristic
52C4⋊#95C2211yescharacteristic
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56#1434—21no—
57#1434—21no—
58C4×#95—21no—
59C4×#95C2211yescharacteristic
60C23×Q8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—41no—
70C2×C4×Q8—41no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C42—21no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×#107—22no—
87C2×#107—41no—
88C2×#107—41no—
89C2×#106—41no—
90C2×#106—41no—
91C2×#106C2×C414yes—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—41no—
99C2×#103—41no—
100C2×#103—41no—
101C2×#103—41no—
102C2×#103C2×C414yes—
103C2×#103C2×C414yes—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—41no—
107#391—22no—
108#391—22no—
109#391—22no—
110#391C2×C414yes—
111#383—44no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—41no—
116C4×C4⋊C4—41no—
117C4×C4⋊C4C2×C414yes—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95D411yescharacteristic
138C2×#95D411yescharacteristic
139C2×#95D411yescharacteristic
140C2×#95D411yescharacteristic
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—41no—
149#340—41no—
150#340C2×C414yes—
151#336—22no—
152#336—41no—
153#336—41no—
154#336—41no—
155#336C2×C414yes—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8C22×C412yes—
168C22×Q8#4112yes—
169C22×Q8#4112yes—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
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×C8#4112yes—
212C22×C8#4112yes—
213C4×Q8—44no—
214C4×Q8—44no—
215C4×Q8—44no—
216C4×Q8—44no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
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—24no—
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—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4C22×C412yes—
256C2×C4⋊C4C22×C412yes—
257C2×C4⋊C4C22×C412yes—
258C2×C4⋊C4#4112yes—
259C2×C4⋊C4#4112yes—
260C2×C4⋊C4#4112yes—
261C2×C4⋊C4#4112yes—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—81no—
289C2×C42—81no—
290C2×C42C22×C412yes—
291C2×C42#4112yes—
292#107—44no—
293#106—44no—
294#103—28no—
295#103—44no—
296#103—44no—
297#103—44no—
298C4×C8—44no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—42no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—81no—
319#95—81no—
320#95—81no—
321#95—81no—
322C24C22×D411yescharacteristic
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8#11714yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
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—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4C4×D412yes—
444C22×C4C4×D412yes—
445C22×C4#13512yes—
446C22×C4#13512yes—
447C22×C4AGL(1,ℤ/8)12yes—
448C22×C4#13712yes—
449C22×C4C23×C411yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4C2×#4111yescharacteristic
453C22×C4C2×#4111yescharacteristic
454C22×C4C2×#4111yescharacteristic
455C22×C4C2×#4111yescharacteristic
456C22×C4C2×#4111yescharacteristic
457C22×C4C2×#4111yescharacteristic
458C22×C4C2×#4111yescharacteristic
459C22×C4H(5,𝔽2)11yescharacteristic
460C22×C4H(5,𝔽2)11yescharacteristic
461C22×C4H(5,𝔽2)11yescharacteristic
462C22×C42−1+411yescharacteristic
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—48no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4#12614yes—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—48no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—48no—
534C42—84no—
535C42—84no—
536C23C2×C4×D411yescharacteristic
537C23#52511yescharacteristic
538C23#52511yescharacteristic
539C23#54611yescharacteristic
540C23#54611yescharacteristic
541C23#54711yescharacteristic
542C23#55011yescharacteristic
543C23C2×#13511yescharacteristic
544C23C2×#13511yescharacteristic
545C23C2×AGL(1,ℤ/8)11yescharacteristic
546C23C2×#13711yescharacteristic
547C23#57711yescharacteristic
548C23#57711yescharacteristic
549C23#57811yescharacteristic
550C23#57811yescharacteristic
551Q8—24no—
552Q8—24no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—42no—
557Q8—42no—
558Q8—42no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—81no—
643C2×C4—81no—
644C2×C4—81no—
645C2×C4—81no—
646C2×C4—81no—
647C2×C4—81no—
648C2×C4—81no—
649C2×C4—81no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—81no—
653C2×C4—81no—
654C2×C4—81no—
655C2×C4—81no—
656C2×C4#44014yes—
657C2×C4C2×#11711yescharacteristic
658C2×C4C4×#4111yescharacteristic
659C2×C4C4×#4111yescharacteristic
660C2×C4C4×#4111yescharacteristic
661C2×C4#51811yescharacteristic
662C2×C4#52011yescharacteristic
663C2×C4C2×#12611yescharacteristic
664C2×C4#52911yescharacteristic
665C2×C4#52911yescharacteristic
666C2×C4#52911yescharacteristic
667C2×C4#52911yescharacteristic
668C2×C4#52911yescharacteristic
669C2×C4#53811yescharacteristic
670C2×C4#53811yescharacteristic
671C2×C4#54211yescharacteristic
672C2×C4#54211yescharacteristic
673C2×C4#55111yescharacteristic
674C2×C4#55111yescharacteristic
675C2×C4#55111yescharacteristic
676C2×C4#55111yescharacteristic
677C2×C4#55111yescharacteristic
678C2×C4#55211yescharacteristic
679C2×C4#55211yescharacteristic
680C2×C4#55211yescharacteristic
681C2×C4#55311yescharacteristic
682C2×C4#55311yescharacteristic
683C2×C4#55311yescharacteristic
684C2×C4#55511yescharacteristic
685C8—42no—
686C8—42no—
687C8—42no—
688C8—42no—
689C22#229011yescharacteristic
690C22#235911yescharacteristic
691C22#236311yescharacteristic
692C22#246711yescharacteristic
693C22#246811yescharacteristic
694C22#247911yescharacteristic
695C22#248011yescharacteristic
696C22C4×#13511yescharacteristic
697C22C2×#44011yescharacteristic
698C22#294711yescharacteristic
699C22#294711yescharacteristic
700C22#310611yescharacteristic
701C22#310611yescharacteristic
702C22#310811yescharacteristic
703C22#310811yescharacteristic
704C22#312011yescharacteristic
705C22#312111yescharacteristic
706C22#312511yescharacteristic
707C22#312711yescharacteristic
708C22#329611yescharacteristic
709C22#330011yescharacteristic
710C22#330111yescharacteristic
711C22#330211yescharacteristic
712C22#330611yescharacteristic
713C22#331211yescharacteristic
714C22#332911yescharacteristic
715C22#333011yescharacteristic
716C22#333711yescharacteristic
717C22#334211yescharacteristic
718C22#334311yescharacteristic
719C22#334411yescharacteristic
720C22#339311yescharacteristic
721C22#339411yescharacteristic
722C22#339711yescharacteristic
723C22#339811yescharacteristic
724C4—24no—
725C4—42no—
726C4—42no—
727C4—44no—
728C4—44no—
729C4—42no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—81no—
734C4—82no—
735C4—81no—
736C4—161no—
737C4—161no—
738C4C4×#12611yescharacteristic
739C4#233711yescharacteristic
740C4#234311yescharacteristic
741C4#235311yescharacteristic
742C4#249511yescharacteristic
743C4#251511yescharacteristic
744C4#257211yescharacteristic
745C4#257811yescharacteristic
746C2#1399011yescharacteristic
747C2#2112011yescharacteristic
748C2#2112711yescharacteristic
749C2#2291911yescharacteristic
750C2#2293411yescharacteristic
751C2#2309711yescharacteristic
752C2#2310011yescharacteristic
753C2#2706711yescharacteristic
754C2#2707111yescharacteristic
755C2#2709911yescharacteristic
756C2#2710311yescharacteristic
757C2#2748711yescharacteristic
758C2#2749911yescharacteristic
759C2#2751911yescharacteristic
760C2#2753111yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
30. #1927C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19272
#139212
#138172
#113782
#112562
#111902
#111242
#109262
#109082
#109072
#108062
#107972
#107892
C4×#3402