Finite Group Databasebeta

← #791709

Subgroups — #791709

Order 512.

2446 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#16990C212yesmaximal
4#15779C214yesmaximal
5#15625C214yesmaximal
6#15211C212yesmaximal
7#15194C211yesmaximal, characteristic
8#15143C211yesmaximal, characteristic
9#15128C211yesmaximal, characteristic
10#15098C212yesmaximal
11#14226C212yesmaximal
12#14017C212yesmaximal
13#14005C212yesmaximal
14#13976C212yesmaximal
15#13910C212yesmaximal
16C4×#398C211yesmaximal, characteristic
17#13817C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22#2567—28no—
23C2×#398C2211yescharacteristic
24C2×#395C2214yes—
25C2×#395C2212yes—
26C2×#395C2214yes—
27Q8×C4⋊C4—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2214yes—
38C2×#389C2214yes—
39C2×#389C2212yes—
40C2×#389C2214yes—
41C2×#384C2212yes—
42C2×#384C2211yescharacteristic
43C2×#384C2212yes—
44C2×#384C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50#1446C2212yes—
51#1445C2211yescharacteristic
52#1445C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2214yes—
56#1444C2214yes—
57#1444C2212yes—
58#1444C2211yescharacteristic
59#1444C2212yes—
60#1444C2212yes—
61#1444C2214yes—
62#1444C2212yes—
63#1444C2214yes—
64#1442C2212yes—
65#1442C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2212yes—
69#1441C2211yescharacteristic
70#1441C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82C4×#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—44no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—44no—
94C2×C4×Q8—44no—
95C2×C4×Q8D418yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2312yes—
130#398Q818yescomplemented
131#395—216no—
132#395—44no—
133#395—48no—
134#391—28no—
135#391Q8116yescomplemented
136#389—24no—
137#389—28no—
138#389—28no—
139#389—28no—
140#389—28no—
141#389—24no—
142#389—28no—
143#389—48no—
144#389—48no—
145#389—44no—
146#389—48no—
147#384—28no—
148#384—24no—
149#384—28no—
150#384—44no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—44no—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—44no—
190C22×Q8—42no—
191C22×Q8—44no—
192C22×Q8—42no—
193C22×Q8C2×D414yes—
194C22×Q8C2×Q818yes—
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C4⋊Q8—416no—
213C4×Q8—232no—
214C4×Q8—416no—
215C4×Q8—416no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—41no—
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⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×Q818yes—
278C2×C4⋊C4C2×Q814yes—
279C2×C4⋊C4C2×Q814yes—
280C2×C4⋊C4C2×Q818yes—
281C2×C4⋊C4#4114yes—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C2×D414yes—
302C2×C42C2×D418yes—
303C2×C42C2×Q818yes—
304C2×C42#4114yes—
305#95—22no—
306#95—24no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—22no—
321#95—24no—
322#95—42no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—42no—
329#95—44no—
330#95C2×Q8116yes—
331#95C2×Q8116yes—
332#95C2×Q818yes—
333#95C2×Q814yes—
334C24C2511yescharacteristic
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—216no—
338C2×Q8—216no—
339C2×Q8—48no—
340C2×Q8—84no—
341C2×Q8—84no—
342C2×Q8—88no—
343C2×Q8—88no—
344C2×Q8C4⋊Q8116yes—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4C22×D412yes—
420C22×C4C22×D414yes—
421C22×C4C22×D414yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×Q814yes—
425C22×C4C22×Q812yes—
426C22×C4C22×Q818yes—
427C22×C4C22×Q814yes—
428C22×C4C22×Q818yes—
429C22×C4C22×Q812yes—
430C22×C4C22×Q812yes—
431C22×C4C22×Q814yes—
432C22×C4C22×Q814yes—
433C22×C4C22×Q814yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4H(5,𝔽2)12yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C4H(5,𝔽2)14yes—
442C22×C42−1+414yes—
443C22×C42−1+412yes—
444C4⋊C4—28no—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—28no—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—28no—
451C4⋊C4—216no—
452C4⋊C4—216no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—416no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C4⋊C4—88no—
466C4⋊C4—84no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C42—216no—
470C42—28no—
471C42—28no—
472C42—216no—
473C42—48no—
474C42—48no—
475C42—48no—
476C42—44no—
477C42—88no—
478C23C23×D411yescharacteristic
479C23C23×Q812yes—
480C23C23×Q811yescharacteristic
481C23C22×#4111yescharacteristic
482C23C22×#4111yescharacteristic
483C23C2×H(5,𝔽2)12yes—
484C23C2×H(5,𝔽2)11yescharacteristic
485C23C2×H(5,𝔽2)12yes—
486C23C2×2−1+411yescharacteristic
487C23C2×2−1+412yes—
488C23#58511yescharacteristic
489Q8—232no—
490Q8—816no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4#52514yes—
582C2×C4C2×C4⋊Q818yes—
583C2×C4C2×C4⋊Q814yes—
584C2×C4#53414yes—
585C2×C4D4×Q814yes—
586C2×C4#55414yes—
587C2×C4#55418yes—
588C2×C4#55418yes—
589C2×C4Q8218yes—
590C22C2×#52511yescharacteristic
591C22C22×C4⋊Q811yescharacteristic
592C22C2×#53411yescharacteristic
593C22C2×D4×Q811yescharacteristic
594C22C2×#55412yes—
595C22C2×#55412yes—
596C22C2×#55411yescharacteristic
597C22C2×Q8212yes—
598C22Q8×#4111yescharacteristic
599C22#348611yescharacteristic
600C22#348811yescharacteristic
601C22#349212yes—
602C22#349811yescharacteristic
603C22#350212yes—
604C22#350312yes—
605C22#350312yes—
606C22#350312yes—
607C22#350711yescharacteristic
608C22#352012yes—
609C22#352211yescharacteristic
610C22#354312yes—
611C22#354612yes—
612C22#355212yes—
613C4—28no—
614C4—24no—
615C4—28no—
616C4—24no—
617C4—48no—
618C4—42no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—42no—
623C4—44no—
624C4—44no—
625C4—42no—
626C4—81no—
627C4—82no—
628C4—84no—
629C4#258318yes—
630C2C2×#258311yescharacteristic
631C2#3570111yescharacteristic
632C2#3593711yescharacteristic
633C2#3654411yescharacteristic
634C2Q8×C4⋊Q811yescharacteristic
635C2#3959311yescharacteristic
636C2#4296911yescharacteristic
637C2#4492312yes—
638C2#4508612yes—
639C2#5932212yes—
640C2#5945912yes—
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
130. #398Q8semidirect
135. #391Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×Q8×C4⋊C42
#169902
#157792
#156252
#152112
#151942
#151432
#151282
#150982
#142262
#140172
#140052
#139762
#139102
C4×#3982
#138172