Finite Group Databasebeta

← #877576

Subgroups — #877576

Order 512.

2254 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2806C212yesmaximal
2#17078C212yesmaximal
3#17073C212yesmaximal
4#15746C214yesmaximal
5#15623C214yesmaximal
6#15145C212yesmaximal
7#15143C212yesmaximal
8#15133C212yesmaximal
9#14714C211yesmaximal, characteristic
10#14712C211yesmaximal, characteristic
11#14467C212yesmaximal
12#14029C212yesmaximal
13#14005C212yesmaximal
14#13926C212yesmaximal
15C4×#391C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18#2806—216no—
19C2×#398C2214yes—
20C2×#398C2214yes—
21C2×#398C2214yes—
22C2×#395C2214yes—
23C2×#395C2214yes—
24C2×#391C2214yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#389C2214yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#387C2214yes—
36C2×#384C2212yes—
37C2×#384C2212yes—
38C2×#384C2211yescharacteristic
39C2×#384C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C43C2211yescharacteristic
44#1445C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1436C2212yes—
58#1436C2214yes—
59#1436C2212yes—
60#1436C2214yes—
61C4⋊#95C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2211yescharacteristic
74C4×#95C2212yes—
75C4×#95C2212yes—
76C23×Q8C2312yes—
77C23×Q8C2311yescharacteristic
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—42no—
80C2×C4⋊Q8—42no—
81C2×C4×Q8—42no—
82C2×C4×Q8—42no—
83C2×C4×Q8Q818yescomplemented
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#398—216no—
117#398—216no—
118#398—48no—
119#395—216no—
120#395—48no—
121#391—216no—
122#391—28no—
123#391—44no—
124#391D418yes—
125#391Q818yescomplemented
126#389—28no—
127#389—28no—
128#389—216no—
129#389—44no—
130#389—44no—
131#389—44no—
132#387—216no—
133#384—44no—
134#384—42no—
135#384—42no—
136#384—44no—
137#384Q818yescomplemented
138C4×C4⋊C4—42no—
139C4×C4⋊C4—44no—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C43—42no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8C2×D414yes—
178C22×Q8C2×Q814yes—
179C22×Q8C2×Q814yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C4⋊Q8—416no—
196C4×Q8—416no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×Q818yes—
258C2×C4⋊C4C2×Q818yes—
259C2×C4⋊C4C2×Q818yes—
260C2×C4⋊C4C2×Q814yes—
261C2×C4⋊C4C2×Q818yes—
262C2×C4⋊C4C2×Q814yes—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—28no—
279C2×C42—41no—
280C2×C42—41no—
281C2×C42—41no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42—41no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42C2×D414yes—
288C2×C42C2×Q818yes—
289C2×C42C2×Q814yes—
290C2×C42C2×Q814yes—
291C2×C42C2×Q814yes—
292C2×C42#4114yes—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95C2×D418yes—
315#95C2×D418yes—
316#95C2×Q818yes—
317#95C2×Q818yes—
318C24C2511yescharacteristic
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—84no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×Q8C4⋊Q8116yescomplemented
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—21no—
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—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4C22×D414yes—
409C22×C4C22×D412yes—
410C22×C4C22×D414yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×Q814yes—
414C22×C4C22×Q814yes—
415C22×C4C22×Q812yes—
416C22×C4C22×Q818yes—
417C22×C4C22×Q814yes—
418C22×C4C22×Q814yes—
419C22×C4C22×Q812yes—
420C22×C4C22×Q812yes—
421C22×C4C22×Q812yes—
422C22×C4C22×Q812yes—
423C22×C4C22×Q812yes—
424C22×C4C22×Q812yes—
425C22×C4C22×Q812yes—
426C22×C4C22×Q812yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C42−1+412yes—
436C22×C42−1+412yes—
437C22×C42−1+412yes—
438C4⋊C4—216no—
439C4⋊C4—216no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—416no—
449C4⋊C4—48no—
450C4⋊C4—84no—
451C4⋊C4—88no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4—88no—
456C4⋊C4—88no—
457C4⋊C4—88no—
458C4⋊C4—88no—
459C4⋊C4—88no—
460C42—28no—
461C42—28no—
462C42—216no—
463C42—28no—
464C42—44no—
465C42—44no—
466C42—44no—
467C42—44no—
468C42—48no—
469C42—88no—
470C23C23×D411yescharacteristic
471C23C23×Q811yescharacteristic
472C23C23×Q811yescharacteristic
473C23C23×Q811yescharacteristic
474C23C22×#4111yescharacteristic
475C23C2×H(5,𝔽2)11yescharacteristic
476C23C2×H(5,𝔽2)11yescharacteristic
477C23C2×H(5,𝔽2)11yescharacteristic
478C23C2×H(5,𝔽2)11yescharacteristic
479C23C2×H(5,𝔽2)11yescharacteristic
480C23C2×2−1+411yescharacteristic
481C23C2×2−1+411yescharacteristic
482C23C2×2−1+411yescharacteristic
483C23#58511yescharacteristic
484C23#58511yescharacteristic
485Q8—416no—
486Q8—816no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4C2×C4⋊Q814yes—
580C2×C4C2×C4⋊Q818yes—
581C2×C4#53814yes—
582C2×C4#54314yes—
583C2×C4#54414yes—
584C2×C4D4×Q814yes—
585C2×C4#55014yes—
586C2×C4#55414yes—
587C2×C4#55714yes—
588C2×C4#55714yes—
589C2×C4Q8214yes—
590C2×C4Q8214yes—
591C2×C4#56214yes—
592C22C22×C4⋊Q811yescharacteristic
593C22C2×#53811yescharacteristic
594C22C2×#54311yescharacteristic
595C22C2×#54411yescharacteristic
596C22#345811yescharacteristic
597C22C2×D4×Q811yescharacteristic
598C22C2×#55011yescharacteristic
599C22C2×#55411yescharacteristic
600C22C2×#55711yescharacteristic
601C22C2×#55711yescharacteristic
602C22C2×Q8211yescharacteristic
603C22C2×Q8211yescharacteristic
604C22#349611yescharacteristic
605C22#349611yescharacteristic
606C22#349811yescharacteristic
607C22#349911yescharacteristic
608C22#350111yescharacteristic
609C22#350111yescharacteristic
610C22#350111yescharacteristic
611C22#350211yescharacteristic
612C22#350311yescharacteristic
613C22#350311yescharacteristic
614C22#350311yescharacteristic
615C22#350311yescharacteristic
616C22#350611yescharacteristic
617C22#350711yescharacteristic
618C22C2×#56211yescharacteristic
619C22#353711yescharacteristic
620C22#353811yescharacteristic
621C22#354811yescharacteristic
622C22#354811yescharacteristic
623C22#355811yescharacteristic
624C22#355911yescharacteristic
625C22#356111yescharacteristic
626C22#356311yescharacteristic
627C4—24no—
628C4—28no—
629C4—24no—
630C4—24no—
631C4—24no—
632C4—24no—
633C4—44no—
634C4—48no—
635C4—44no—
636C4—42no—
637C4—44no—
638C4—84no—
639C4—84no—
640C4—84no—
641C4#280618yescomplemented
642C2#3500211yescharacteristic
643C2Q8×C4⋊Q811yescharacteristic
644C2#3659311yescharacteristic
645C2#3730711yescharacteristic
646C2#3936811yescharacteristic
647C2#3938311yescharacteristic
648C2#3962111yescharacteristic
649C2#3962311yescharacteristic
650C2#3964011yescharacteristic
651C2#3974911yescharacteristic
652C2#4599511yescharacteristic
653C2#4600511yescharacteristic
654C2C2×#280611yescharacteristic
655C2#5982311yescharacteristic
656C2#5987111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
83. C2×C4×Q8Q8semidirect
125. #391Q8semidirect
137. #384Q8semidirect
332. C2×Q8C4⋊Q8semidirect
641. C4#2806semidirect

Maximal subgroups

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

subgroupindex
C2×#28062
#170782
#170732
#157462
#156232
#151452
#151432
#151332
#147142
#147122
#144672
#140292
#140052
#139262
C4×#3912