Finite Group Databasebeta

← #797468

Subgroups — #797468

Order 512.

2730 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C212yesmaximal
2C2×#2546C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic
4#16820C212yesmaximal
5#15676C214yesmaximal
6#15518C214yesmaximal
7#15192C211yesmaximal, characteristic
8#15190C211yesmaximal, characteristic
9#15180C211yesmaximal, characteristic
10#15166C212yesmaximal
11#15012C212yesmaximal
12#14226C212yesmaximal
13#14024C212yesmaximal
14#14024C212yesmaximal
15#14021C212yesmaximal
16C4×#398C211yesmaximal, characteristic
17Q8×#95C211yesmaximal, characteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4×Q8C2212yes—
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25#2546—216no—
26#2546—28no—
27C2×#398C2214yes—
28C2×#398C2214yes—
29C2×#398C2214yes—
30C2×#398C2211yescharacteristic
31#2365—28no—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#389C2212yes—
40C2×#389C2214yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2214yes—
44C2×#387C2212yes—
45C2×#387C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51#1445C2211yescharacteristic
52#1445C2212yes—
53#1444C2211yescharacteristic
54#1444C2214yes—
55#1444C2212yes—
56#1444C2212yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2212yes—
60#1442C2212yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2214yes—
64#1441C2212yes—
65#1441C2211yescharacteristic
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2212yes—
71#1441C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C4×#95C2212yes—
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—28no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—24no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—44no—
98C2×C4⋊Q8—44no—
99C2×C4⋊Q8D418yes—
100C2×C4×Q8—28no—
101C2×C4×Q8—24no—
102C2×C4×Q8—28no—
103C2×C4×Q8—44no—
104C2×C4×Q8—44no—
105C2×C4×Q8—44no—
106C2×C4×Q8D418yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136#398—216no—
137#398—216no—
138#398—216no—
139#398Q818yescomplemented
140#391—24no—
141#391—28no—
142#391—28no—
143#391—24no—
144#391—28no—
145#391—28no—
146#391—44no—
147#389—28no—
148#389—216no—
149#389—216no—
150#389—28no—
151#389—28no—
152#387—28no—
153#387—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—24no—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8C2×D414yes—
206C22×Q8C2×D414yes—
207C22×Q8C2×D414yes—
208C22×Q8#4114yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C4⋊Q8—232no—
227C4⋊Q8—432no—
228C4⋊Q8—416no—
229C4⋊Q8—432no—
230C4×Q8—416no—
231C4×Q8—432no—
232C4×Q8—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D418yes—
281C2×C4⋊C4C2×D418yes—
282C2×C4⋊C4C2×D418yes—
283C2×C4⋊C4C2×Q814yes—
284C2×C4⋊C4C2×Q814yes—
285C2×C4⋊C4C2×Q818yes—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—42no—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D418yes—
308C2×C42#4118yes—
309#95—22no—
310#95—24no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—24no—
328#95—22no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—28no—
333#95—24no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—42no—
338#95—44no—
339#95—44no—
340#95C2×Q818yes—
341#95C2×Q814yes—
342C24C2511yescharacteristic
343C2×Q8—28no—
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—216no—
347C2×Q8—216no—
348C2×Q8—216no—
349C2×Q8—48no—
350C2×Q8—44no—
351C2×Q8—48no—
352C2×Q8—44no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8#121116yes—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4C22×D412yes—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q814yes—
446C22×C4C22×Q812yes—
447C22×C4C22×Q812yes—
448C22×C4C22×Q814yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4114yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C42−1+414yes—
456C22×C42−1+412yes—
457C22×C42−1+412yes—
458C22×C42−1+414yes—
459C4⋊C4—28no—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—216no—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—216no—
466C4⋊C4—28no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—416no—
470C4⋊C4—416no—
471C4⋊C4—416no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—416no—
476C4⋊C4—416no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C42—28no—
480C42—28no—
481C42—216no—
482C42—48no—
483C42—48no—
484C42—416no—
485C42—48no—
486C42—48no—
487C42—44no—
488C23C23×D411yescharacteristic
489C23C23×D411yescharacteristic
490C23C23×Q811yescharacteristic
491C23C22×#4111yescharacteristic
492C23C22×#4112yes—
493C23C2×H(5,𝔽2)12yes—
494C23C2×2−1+411yescharacteristic
495C23C2×2−1+411yescharacteristic
496C23C2×2−1+412yes—
497C23C2×2−1+412yes—
498C23#58511yescharacteristic
499Q8—232no—
500Q8—416no—
501Q8—816no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4C2×#12118yes—
593C2×C4C2×#12114yes—
594C2×C4#53214yes—
595C2×C4#53414yes—
596C2×C4#54618yes—
597C2×C4D4×Q814yes—
598C2×C4D4×Q814yes—
599C2×C4#55318yes—
600C2×C4#55718yes—
601C22C22×#12111yescharacteristic
602C22C2×#53211yescharacteristic
603C22C2×#53411yescharacteristic
604C22C2×#54612yes—
605C22C2×D4×Q811yescharacteristic
606C22C2×D4×Q811yescharacteristic
607C22C2×#55312yes—
608C22C2×#55712yes—
609C22Q8×#4111yescharacteristic
610C22#348612yes—
611C22#348611yescharacteristic
612C22#348911yescharacteristic
613C22#349711yescharacteristic
614C22#349911yescharacteristic
615C22#350312yes—
616C22#350712yes—
617C22#350912yes—
618C22#350912yes—
619C22#352311yescharacteristic
620C22#352312yes—
621C22#355212yes—
622C22#355212yes—
623C22#355412yes—
624C4—28no—
625C4—28no—
626C4—24no—
627C4—24no—
628C4—44no—
629C4—44no—
630C4—42no—
631C4—44no—
632C4—42no—
633C4—44no—
634C4—42no—
635C4—44no—
636C4—84no—
637C4—84no—
638C4—82no—
639C4—81no—
640C4#254318yes—
641C2C2×#254311yescharacteristic
642C2#3570011yescharacteristic
643C2#3593811yescharacteristic
644C2Q8×#12111yescharacteristic
645C2#3656011yescharacteristic
646C2#3959311yescharacteristic
647C2#4291711yescharacteristic
648C2#4532312yes—
649C2#4534412yes—
650C2#5914412yes—
651C2#5968112yes—
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
139. #398Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#23652
#168202
#156762
#155182
#151922
#151902
#151802
#151662
#150122
#142262
#140242
#140212
C4×#3982
Q8×#952