Finite Group Databasebeta

← #2737540

Subgroups — #2737540

Order 512.

2234 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2454C211yesmaximal, characteristic
3#17122C214yesmaximal, complemented
4#17106C214yesmaximal, complemented
5#17059C212yesmaximal, complemented
6#16815C212yesmaximal, complemented
7#16394C212yesmaximal, complemented
8#15232C212yesmaximal, complemented
9#15228C212yesmaximal, complemented
10#15200C211yesmaximal, characteristic, complemented
11#15143C211yesmaximal, characteristic, complemented
12#15138C212yesmaximal
13#15136C212yesmaximal
14#15128C211yesmaximal, characteristic, complemented
15#14674C212yesmaximal, complemented
16#13916C211yesmaximal, characteristic
17C4×#400C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2563—28no—
22#2454—28no—
23C2×#400C2211yescharacteristic
24C2×#399C2212yes—
25C2×#397C2214yes—
26C2×#397C2214yes—
27C2×#397C2214yes—
28C2×#396C2212yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#393C2214yes—
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#389C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#385C2212yes—
40C2×#384C2212yes—
41C2×#382C2212yes—
42C2×#381C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48#1446C2212yes—
49#1445C2212yes—
50#1445C2211yescharacteristic
51#1445C2214yes—
52#1444C2212yes—
53#1444C2212yes—
54#1444C2212yes—
55#1444C2214yes—
56#1444C2214yes—
57#1443C2214yes—
58#1443C2212yes—
59#1443C2214yes—
60#1442C2211yescharacteristic
61#1442C2212yes—
62#1442C2212yes—
63#1442C2214yes—
64#1442C2214yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2211yescharacteristic
68#1441C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C22⋊#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—44no—
90C2×#125—24no—
91C2×#125—44no—
92C2×C4×D4—44no—
93C2×C4×D4D418yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#400—24no—
126#399—28no—
127#397—48no—
128#397—48no—
129#397—48no—
130#396—28no—
131#395—48no—
132#395—44no—
133#393—216no—
134#390—28no—
135#390—24no—
136#390—24no—
137#390—44no—
138#389—24no—
139#389—24no—
140#389—44no—
141#385—28no—
142#384—44no—
143#382—28no—
144#381—28no—
145#380—28no—
146#380—28no—
147#380—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—24no—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C25C2412yes—
182C22×Q8—24no—
183C22×Q8—44no—
184C22×Q8—42no—
185C22×Q8—44no—
186C22×Q8—42no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4C2×D414yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C2×D418yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C4⋊Q8—416no—
214#125—416no—
215C4×D4—416no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
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—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4C2×D414yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4C2×D418yes—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42C2×D414yes—
295#95—22no—
296#95—24no—
297#95—22no—
298#95—28no—
299#95—28no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—22no—
313#95—44no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95#4114yes—
326C24—22no—
327C24—22no—
328C24—24no—
329C24—24no—
330C24—22no—
331C24—24no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24C22×D414yes—
337C24C2511yescharacteristic
338C2×Q8—48no—
339C2×Q8—84no—
340C2×Q8—84no—
341C2×Q8—88no—
342C2×Q8—88no—
343C2×D4—28no—
344C2×D4—28no—
345C2×D4—28no—
346C2×D4—48no—
347C2×D4—48no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—216no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4112yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)14yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C42−1+412yes—
444C22×C42−1+414yes—
445C22×C42−1+412yes—
446C22×C42−1+412yes—
447C4⋊C4—28no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—88no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—88no—
467C22⋊C4—88no—
468C22⋊C4—84no—
469C22⋊C4—88no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C22⋊C4—88no—
473C42—28no—
474C42—44no—
475C42—44no—
476C42—48no—
477C42—48no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23C23×D411yescharacteristic
502C23C22×#4111yescharacteristic
503C23C2×H(5,𝔽2)12yes—
504C23C2×H(5,𝔽2)11yescharacteristic
505C23C2×H(5,𝔽2)12yes—
506C23C2×2−1+412yes—
507C23C2×2−1+411yescharacteristic
508C23C2×2−1+411yescharacteristic
509C23C2×2−1+411yescharacteristic
510C23#58512yes—
511C23#58511yescharacteristic
512Q8—816no—
513D4—416no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—41no—
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—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—816no—
603C2×C4#53414yes—
604C2×C4#53614yes—
605C2×C4#53614yes—
606C2×C4#54714yes—
607C2×C4#56318yes—
608C2×C4#56318yes—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22C2×#53411yescharacteristic
621C22C2×#53611yescharacteristic
622C22C2×#53611yescharacteristic
623C22C2×#54711yescharacteristic
624C22#348511yescharacteristic
625C22#348811yescharacteristic
626C22#349811yescharacteristic
627C22#351112yes—
628C22#351412yes—
629C22#351511yescharacteristic
630C22#352211yescharacteristic
631C22C2×#56312yes—
632C22C2×#56312yes—
633C22#354312yes—
634C22#354411yescharacteristic
635C22#354412yes—
636C22#354612yes—
637C22#354712yes—
638C22#355412yes—
639C22#355511yescharacteristic
640C22#355912yes—
641C22#356112yes—
642C22#356512yes—
643C4—24no—
644C4—44no—
645C4—42no—
646C4—44no—
647C4—42no—
648C4—44no—
649C4—81no—
650C4—82no—
651C4—81no—
652C4—82no—
653C4—82no—
654C4—81no—
655C4—82no—
656C4—84no—
657C4—84no—
658C4#281718yes—
659C2—84no—
660C2C2×#281711yescharacteristic
661C2#3891811yescharacteristic
662C2#3892111yescharacteristic
663C2#3948911yescharacteristic
664C2#4265111yescharacteristic
665C2#4268111yescharacteristic
666C2#4296911yescharacteristic
667C2#4605612yes—
668C2#4606812yes—
669C2#5303512yes—
670C2#5303912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17122C2semidirect
4. #17106C2semidirect
5. #17059C2semidirect
6. #16815C2semidirect
7. #16394C2semidirect
8. #15232C2semidirect
9. #15228C2semidirect
10. #15200C2semidirect
11. #15143C2semidirect
14. #15128C2semidirect
15. #14674C2semidirect
17. C4×#400C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#24542
#171222
#171062
#170592
#168152
#163942
#152322
#152282
#152002
#151432
#151382
#151362
#151282
#146742
#139162
C4×#4002