Finite Group Databasebeta

← #825722

Subgroups — #825722

Order 512.

2430 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic
3#17080C212yesmaximal
4#15625C214yesmaximal
5#15510C214yesmaximal
6#15202C211yesmaximal, characteristic
7#15199C212yesmaximal
8#15196C212yesmaximal
9#15143C211yesmaximal, characteristic
10#15128C211yesmaximal, characteristic
11#14241C212yesmaximal
12#14053C212yesmaximal
13#14045C212yesmaximal
14#14005C212yesmaximal
15#13910C212yesmaximal
16C4×#395C211yesmaximal, characteristic
17#13818C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#2567—28no—
22C2×#395C2214yes—
23C2×#395C2212yes—
24C2×#395C2214yes—
25C2×#395C2211yescharacteristic
26#2369—28no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2214yes—
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#384C2212yes—
41C2×#384C2214yes—
42C2×#384C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50#1446C2212yes—
51#1446C2211yescharacteristic
52#1446C2212yes—
53#1446C2214yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1444C2212yes—
59#1444C2211yescharacteristic
60#1444C2212yes—
61#1444C2212yes—
62#1444C2214yes—
63#1444C2212yes—
64#1444C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2211yescharacteristic
69#1441C2212yes—
70#1441C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2214yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4×#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—44no—
92C2×C4⋊Q8Q818yescomplemented
93C2×C4×Q8—24no—
94C2×C4×Q8—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127#395—216no—
128#395—44no—
129#395—48no—
130#395Q818yescomplemented
131#391—28no—
132#391—28no—
133#391—28no—
134#391—48no—
135#389—28no—
136#389—28no—
137#389—24no—
138#389—24no—
139#389—28no—
140#389—48no—
141#389—44no—
142#389—44no—
143#389Q8116yescomplemented
144#384—28no—
145#384—44no—
146#384—48no—
147C4×C4⋊C4—24no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—44no—
153C4×C4⋊C4D418yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2314yes—
184C22×Q8—22no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—22no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8—44no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×Q8—42no—
195C22×Q8—44no—
196C22×Q8—42no—
197C22×Q8C2×Q818yes—
198C22×Q8C2×Q814yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C4⋊Q8—232no—
217C4⋊Q8—416no—
218C4×Q8—416no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×Q814yes—
274C2×C4⋊C4C2×Q818yes—
275C2×C4⋊C4C2×Q818yes—
276C2×C4⋊C4C2×Q814yes—
277C2×C4⋊C4C2×Q8116yes—
278C2×C4⋊C4C2×Q818yes—
279C2×C4⋊C4C2×Q814yes—
280C2×C4⋊C4C2×Q818yes—
281C2×C4⋊C4#4114yes—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—28no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—42no—
300C2×C42—44no—
301C2×C42C2×D414yes—
302C2×C42C2×D418yes—
303C2×C42C2×Q814yes—
304C2×C42C2×Q818yes—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—24no—
310#95—22no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—22no—
320#95—24no—
321#95—44no—
322#95—42no—
323#95—42no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—42no—
331#95C2×D418yes—
332#95C2×Q8116yes—
333#95C2×Q818yes—
334#95C2×Q814yes—
335C24C2511yescharacteristic
336C2×Q8—216no—
337C2×Q8—28no—
338C2×Q8—48no—
339C2×Q8—416no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×Q8—88no—
347C2×Q8—84no—
348C2×Q8—84no—
349C2×Q8—88no—
350C2×Q8—88no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4C22×D412yes—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×Q818yes—
430C22×C4C22×Q814yes—
431C22×C4C22×Q814yes—
432C22×C4C22×Q812yes—
433C22×C4C22×Q812yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q814yes—
436C22×C4C22×Q818yes—
437C22×C4C22×Q812yes—
438C22×C4C22×Q814yes—
439C22×C4C22×Q812yes—
440C22×C4C22×Q814yes—
441C22×C4C22×Q812yes—
442C22×C4C22×Q812yes—
443C22×C4C22×Q814yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)14yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C42−1+412yes—
450C4⋊C4—216no—
451C4⋊C4—28no—
452C4⋊C4—28no—
453C4⋊C4—216no—
454C4⋊C4—416no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—48no—
458C4⋊C4—44no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—88no—
464C4⋊C4—84no—
465C4⋊C4—88no—
466C4⋊C4—88no—
467C4⋊C4—84no—
468C4⋊C4—88no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C4⋊C4C22⋊Q8116yes—
472C4⋊C4C4⋊Q8116yes—
473C42—28no—
474C42—28no—
475C42—216no—
476C42—48no—
477C42—48no—
478C42—416no—
479C42—44no—
480C42—48no—
481C42—84no—
482C23C23×D411yescharacteristic
483C23C23×Q811yescharacteristic
484C23C23×Q812yes—
485C23C23×Q811yescharacteristic
486C23C22×#4111yescharacteristic
487C23C2×H(5,𝔽2)12yes—
488C23C2×H(5,𝔽2)11yescharacteristic
489C23C2×H(5,𝔽2)12yes—
490C23C2×2−1+411yescharacteristic
491C23#58511yescharacteristic
492C23#58512yes—
493Q8—416no—
494Q8—816no—
495Q8—816no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4C2×C22⋊Q818yes—
586C2×C4C2×C22⋊Q814yes—
587C2×C4C2×C4⋊Q818yes—
588C2×C4C2×C4⋊Q814yes—
589C2×C4#53514yes—
590C2×C4D4×Q814yes—
591C2×C4#55418yes—
592C2×C4Q8218yes—
593C2×C4Q8214yes—
594C2×C4Q8218yes—
595C22C22×C22⋊Q811yescharacteristic
596C22C22×C4⋊Q811yescharacteristic
597C22C2×#53511yescharacteristic
598C22C2×D4×Q811yescharacteristic
599C22C2×#55412yes—
600C22C2×Q8212yes—
601C22C2×Q8211yescharacteristic
602C22C2×Q8212yes—
603C22Q8×#4111yescharacteristic
604C22#348511yescharacteristic
605C22#348811yescharacteristic
606C22#349811yescharacteristic
607C22#350112yes—
608C22#350112yes—
609C22#350112yes—
610C22#350211yescharacteristic
611C22#350312yes—
612C22#350412yes—
613C22#351312yes—
614C22#352211yescharacteristic
615C22#354112yes—
616C22#354212yes—
617C22#354912yes—
618C4—28no—
619C4—28no—
620C4—24no—
621C4—24no—
622C4—28no—
623C4—44no—
624C4—48no—
625C4—42no—
626C4—44no—
627C4—42no—
628C4—44no—
629C4—42no—
630C4—82no—
631C4—81no—
632C4—82no—
633C4—84no—
634C4#256718yes—
635C2C2×#256711yescharacteristic
636C2#3583611yescharacteristic
637C2#3595011yescharacteristic
638C2Q8×C22⋊Q811yescharacteristic
639C2Q8×C4⋊Q811yescharacteristic
640C2#3959411yescharacteristic
641C2#4296311yescharacteristic
642C2#4494612yes—
643C2#4495012yes—
644C2#5991812yes—
645C2#5992412yes—
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
92. C2×C4⋊Q8Q8semidirect
130. #395Q8semidirect
143. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#23692
#170802
#156252
#155102
#152022
#151992
#151962
#151432
#151282
#142412
#140532
#140452
#140052
#139102
C4×#3952
#138182