Finite Group Databasebeta

← #627159

Subgroups — #627159

Order 512.

3066 conjugacy classes of subgroups.

The subgroup lattice has 669 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic
3#15964C214yesmaximal, complemented
4#15963C214yesmaximal, complemented
5#15337C214yesmaximal, complemented
6#15333C214yesmaximal, complemented
7#14920C212yesmaximal
8#14728C212yesmaximal
9#14702C211yesmaximal, characteristic
10#14697C211yesmaximal, characteristic
11#14353C211yesmaximal, characteristic
12#14281C211yesmaximal, characteristic
13#14275C211yesmaximal, characteristic
14#14149C212yesmaximal
15#13922C212yesmaximal
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20C2×#398C2214yes—
21#2329—28no—
22C2×#391—28no—
23C2×#391C2212yes—
24C2×#391C2214yes—
25C2×#391C2214yes—
26C2×#391C2214yes—
27C2×#391C2212yes—
28C2×#391C2214yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#387C2212yes—
32C2×#387C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1446C2214yes—
47#1446C2212yes—
48#1446C2214yes—
49#1445C2214yes—
50#1445C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2212yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—24no—
79C23×Q8—44no—
80C23×Q8—44no—
81C23×Q8Q818yescomplemented
82C23×Q8C2314yes—
83C23×Q8C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8Q818yescomplemented
94C22×C4⋊C4—24no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C42—24no—
114C22×C42—44no—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119#398—48no—
120#391—28no—
121#391—216no—
122#391—216no—
123#391—44no—
124#391—48no—
125#391—48no—
126#389—28no—
127#389—44no—
128#387—28no—
129#387—28no—
130#385—28no—
131#385—44no—
132#385—44no—
133#380—24no—
134#380—24no—
135#380—24no—
136#380—28no—
137#380—44no—
138C4×C22⋊C4—28no—
139C2×#95—24no—
140C2×#95—24no—
141C2×#95—24no—
142C2×#95D418yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C25C2411yescharacteristic
173C22×Q8—232no—
174C22×Q8—28no—
175C22×Q8—22no—
176C22×Q8—28no—
177C22×Q8—22no—
178C22×Q8—28no—
179C22×Q8—28no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—432no—
185C22×Q8—416no—
186C22×Q8—416no—
187C22×Q8—432no—
188C22×Q8C2×Q8116yes—
189C22×Q8C2×Q814yescomplemented
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2×Q814yes—
220C23×C4C2×Q814yes—
221C23×C4C2×Q814yes—
222C23×C4C2×Q814yes—
223C23×C4#4114yes—
224C23×C4#4114yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2412yes—
239C22⋊Q8—232no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
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—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×Q814yescomplemented
275C2×C4⋊C4C2×Q818yescomplemented
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4C2×Q818yes—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—416no—
297C2×C42—44no—
298C2×C42—44no—
299#95—28no—
300#95—28no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95C2×D418yes—
327#95C2×D418yes—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24C22×D412yes—
333C24C22×Q812yes—
334C24C22×Q812yes—
335C24C2×#4112yes—
336C24C2×#4112yes—
337C24H(5,𝔽2)12yes—
338C242−1+412yes—
339C24C2511yescharacteristic
340C2×Q8—232no—
341C2×Q8—28no—
342C2×Q8—232no—
343C2×Q8—416no—
344C2×Q8—48no—
345C2×Q8—416no—
346C2×Q8—416no—
347C2×Q8—416no—
348C2×Q8—416no—
349C2×Q8—4128no—
350C2×Q8—82no—
351C2×Q8—82no—
352C2×Q8—84no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—216no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—216no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
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—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4C22⋊Q8116yes—
448C22×C4C22⋊Q8116yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×Q814yes—
454C22×C4C22×Q814yes—
455C22×C4C22×Q814yes—
456C22×C4C22×Q812yes—
457C22×C4C22×Q812yes—
458C22×C4C22×Q814yes—
459C22×C4C22×Q814yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4114yes—
463C22×C4H(5,𝔽2)14yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C42−1+414yes—
466C4⋊C4—28no—
467C4⋊C4—216no—
468C4⋊C4—416no—
469C4⋊C4—48no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4—88no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—84no—
476C4⋊C4—88no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C22⋊C4—216no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C42—48no—
484C42—48no—
485C42—84no—
486C42—84no—
487C23—24no—
488C23—24no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23C2×C22⋊Q814yes—
504C23C2×C22⋊Q814yes—
505C23#53714yes—
506C23#55714yes—
507C23#55714yes—
508C23Q8214yes—
509C23#55914yes—
510C23C23×D411yescharacteristic
511C23C23×Q811yescharacteristic
512C23C23×Q811yescharacteristic
513C23C22×#4111yescharacteristic
514C23C22×#4111yescharacteristic
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)12yes—
517C23C2×H(5,𝔽2)12yes—
518C23C2×2−1+411yescharacteristic
519C23C2×2−1+412yes—
520C23#58512yes—
521Q8—464no—
522Q8—88no—
523C2×C4—232no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—416no—
555C2×C4—416no—
556C2×C4—416no—
557C2×C4—416no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—88no—
603C2×C4—88no—
604C2×C4C2×C22⋊Q814yes—
605C2×C4C2×C22⋊Q814yes—
606C2×C4C2×C22⋊Q818yes—
607C2×C4#54318yes—
608C22—24no—
609C22—28no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—28no—
614C22—24no—
615C22—28no—
616C22—24no—
617C22—28no—
618C22#258218yes—
619C22C22×C22⋊Q811yescharacteristic
620C22C22×C22⋊Q811yescharacteristic
621C22C2×#53711yescharacteristic
622C22C2×#54312yes—
623C22#346112yes—
624C22C2×#55711yescharacteristic
625C22C2×#55711yescharacteristic
626C22C2×Q8211yescharacteristic
627C22C2×#55911yescharacteristic
628C22#348411yescharacteristic
629C22#348411yescharacteristic
630C22#348511yescharacteristic
631C22#348611yescharacteristic
632C22#349012yes—
633C22#350112yes—
634C22#350212yes—
635C22#350312yes—
636C22#350512yes—
637C22#351112yes—
638C22#353612yes—
639C22#353812yes—
640C22#355812yes—
641C22#356112yes—
642C4—24no—
643C4—24no—
644C4—28no—
645C4—28no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—84no—
655C4—84no—
656C2—28no—
657C2C2×#258211yescharacteristic
658C2#3498211yescharacteristic
659C2#3515411yescharacteristic
660C2#3534911yescharacteristic
661C2#3557411yescharacteristic
662C2#3671911yescharacteristic
663C2#3695711yescharacteristic
664C2#3962412yes—
665C2#3967612yes—
666C2#3975212yes—
667C2#3980412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #15964C2semidirect
4. #15963C2semidirect
5. #15337C2semidirect
6. #15333C2semidirect
81. C23×Q8Q8semidirect
93. C2×C22⋊Q8Q8semidirect
189. C22×Q8C2×Q8semidirect
274. C2×C4⋊C4C2×Q8semidirect
275. C2×C4⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
C22×#3912
#159642
#159632
#153372
#153332
#149202
#147282
#147022
#146972
#143532
#142812
#142752
#141492
#139222