Finite Group Databasebeta

← #406833

Subgroups — #406833

Order 512.

1056 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2088C211yesmaximal, characteristic
2C2×#2058C211yesmaximal, characteristic
3#14053C211yesmaximal, characteristic
4#13933C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11378C211yesmaximal, characteristic
7#11292C211yesmaximal, characteristic
8#11292C211yesmaximal, characteristic
9#11231C211yesmaximal, characteristic
10#11220C211yesmaximal, characteristic
11#10908C211yesmaximal, characteristic
12#10876C211yesmaximal, characteristic
13#10861C211yesmaximal, characteristic
14#10807C211yesmaximal, characteristic
15C4×#340C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#391—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#2088—24no—
29#2058—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C2×#336C2211yescharacteristic
47#1446—21no—
48#1442—21no—
49#1442—21no—
50#1441—21no—
51#1436—21no—
52C4⋊#95—21no—
53C4⋊#95C2211yescharacteristic
54C4⋊#95C2211yescharacteristic
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59#1434—21no—
60C4×#95C2211yescharacteristic
61C23×Q8—21no—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—41no—
71C2×C4⋊Q8—41no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×#106—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#106—22no—
97C2×#106—41no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8—41no—
104#391—44no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—41no—
109#389—41no—
110#389—41no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—41no—
114C4×C4⋊C4Q814yescomplemented
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
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×#95D411yescharacteristic
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135C2×#95D411yescharacteristic
136#340—22no—
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—41no—
145#340—41no—
146#340—41no—
147#340D414yes—
148#340Q814yescomplemented
149#336—22no—
150#336—22no—
151#336—22no—
152#336—41no—
153C22×Q8—24no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—81no—
159C22×Q8—81no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2411yescharacteristic
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
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×C8C2×D412yes—
199C22×C8C2×Q812yes—
200C22×C8#4112yes—
201C22×C8#4112yes—
202C22×C8#4112yes—
203C4⋊Q8—44no—
204C4⋊Q8—44no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
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—24no—
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—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—81no—
256C2×C4⋊C4—81no—
257C2×C4⋊C4—81no—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×Q812yes—
261C2×C4⋊C4C2×Q812yes—
262C2×C4⋊C4C2×Q812yes—
263C2×C4⋊C4C2×Q812yes—
264C2×C4⋊C4#4112yes—
265C2×C4⋊C4#4112yes—
266C2×C4⋊C4#4112yes—
267C2×C42—21no—
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—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—24no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—81no—
287C2×C42—81no—
288C2×C42—81no—
289C2×C42C2×Q812yes—
290C2×C42#4112yes—
291#106—44no—
292#106—44no—
293#106—44no—
294C4⋊C8—44no—
295C4×C8—44no—
296#95—24no—
297#95—22no—
298#95—22no—
299#95—41no—
300#95—42no—
301#95—42no—
302#95—42no—
303#95—42no—
304#95—42no—
305#95—42no—
306#95—42no—
307#95—42no—
308#95—42no—
309#95—81no—
310#95—81no—
311#95—81no—
312#95—81no—
313#95—81no—
314#95—81no—
315#95—81no—
316#95SD1614yes—
317C24C22×D411yescharacteristic
318C2×Q8—48no—
319C2×Q8—48no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8—84no—
323C2×Q8—84no—
324C2×Q8—84no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
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—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×C4C22⋊Q812yes—
417C22×C4C22⋊Q812yes—
418C22×C4C2×SD1612yes—
419C22×C4C2×SD1612yes—
420C22×C4C2×SD1612yes—
421C22×C4#13512yes—
422C22×C4#13712yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×Q811yescharacteristic
425C22×C4C22×Q811yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C22×C4C2×#4111yescharacteristic
432C22×C4C2×#4111yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C42−1+411yescharacteristic
436C22×C42−1+411yescharacteristic
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—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#12414yes—
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—48no—
478C4⋊C4—41no—
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—41no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4#12414yes—
500C4⋊C4C4⋊Q814yes—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—84no—
514C42—84no—
515C23C2×C22⋊Q811yescharacteristic
516C23#52511yescharacteristic
517C23#53211yescharacteristic
518C23D4211yescharacteristic
519C23#54611yescharacteristic
520C23#54711yescharacteristic
521C23D4×Q811yescharacteristic
522C23C22×SD1611yescharacteristic
523C23C2×#13511yescharacteristic
524C23C2×#13711yescharacteristic
525C23#57511yescharacteristic
526C23#57611yescharacteristic
527C23#57711yescharacteristic
528C23#57711yescharacteristic
529C23#57711yescharacteristic
530Q8—816no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
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—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4#47514yes—
629C2×C4C2×#12411yescharacteristic
630C2×C4C2×#12411yescharacteristic
631C2×C4#52911yescharacteristic
632C2×C4#52911yescharacteristic
633C2×C4C2×C4⋊Q811yescharacteristic
634C2×C4#53311yescharacteristic
635C2×C4#53311yescharacteristic
636C2×C4#53311yescharacteristic
637C2×C4#53311yescharacteristic
638C2×C4#53311yescharacteristic
639C2×C4#53811yescharacteristic
640C2×C4#53911yescharacteristic
641C2×C4#54111yescharacteristic
642C2×C4#54211yescharacteristic
643C2×C4#54411yescharacteristic
644C2×C4#54411yescharacteristic
645C2×C4#54611yescharacteristic
646C2×C4#54711yescharacteristic
647C2×C4#54811yescharacteristic
648C2×C4#55111yescharacteristic
649C2×C4#55111yescharacteristic
650C2×C4#55211yescharacteristic
651C2×C4#55211yescharacteristic
652C2×C4#55211yescharacteristic
653C2×C4#55411yescharacteristic
654C2×C4#55411yescharacteristic
655C2×C4#55511yescharacteristic
656C2×C4#55711yescharacteristic
657C8—24no—
658C8—42no—
659C8—42no—
660C8—81no—
661C22#239611yescharacteristic
662C22#243811yescharacteristic
663C22#244011yescharacteristic
664C22#246211yescharacteristic
665C22#247211yescharacteristic
666C22#249111yescharacteristic
667C22#255611yescharacteristic
668C22C2×#47511yescharacteristic
669C22#307611yescharacteristic
670C22#308111yescharacteristic
671C22#308411yescharacteristic
672C22#310111yescharacteristic
673C22#310811yescharacteristic
674C22#311811yescharacteristic
675C22#312611yescharacteristic
676C22#316711yescharacteristic
677C22#317211yescharacteristic
678C22#317311yescharacteristic
679C22#317911yescharacteristic
680C22#327311yescharacteristic
681C22#327811yescharacteristic
682C22D4×SD1611yescharacteristic
683C22#328211yescharacteristic
684C22#329511yescharacteristic
685C22#330111yescharacteristic
686C22#331111yescharacteristic
687C22#332011yescharacteristic
688C22#333511yescharacteristic
689C22#334011yescharacteristic
690C22#334611yescharacteristic
691C22#335011yescharacteristic
692C22Q8×SD1611yescharacteristic
693C22#338011yescharacteristic
694C22#338711yescharacteristic
695C22#338811yescharacteristic
696C4—24no—
697C4—42no—
698C4—44no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—44no—
703C4—82no—
704C4—82no—
705C4—81no—
706C4—82no—
707C4—161no—
708C4—161no—
709C4—161no—
710C4#248811yescharacteristic
711C4#248911yescharacteristic
712C4#252011yescharacteristic
713C4#255011yescharacteristic
714C4#257411yescharacteristic
715C4#261211yescharacteristic
716C4#266611yescharacteristic
717C4#286211yescharacteristic
718C2#1493711yescharacteristic
719C2#2377611yescharacteristic
720C2#2378611yescharacteristic
721C2#2599111yescharacteristic
722C2#2599211yescharacteristic
723C2#2604211yescharacteristic
724C2#2605111yescharacteristic
725C2#2685811yescharacteristic
726C2#2686311yescharacteristic
727C2#2722411yescharacteristic
728C2#2722511yescharacteristic
729C2#2778711yescharacteristic
730C2#2779511yescharacteristic
731C2#2894911yescharacteristic
732C2#2896311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
114. C4×C4⋊C4Q8semidirect
148. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#20882
C2×#20582
#140532
#139332
#113842
#113782
#112922
#112312
#112202
#109082
#108762
#108612
#108072
C4×#3402