Finite Group Databasebeta

← #640635

Subgroups — #640635

Order 512.

3146 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2856C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#16693C214yesmaximal, complemented
4#16361C214yesmaximal, complemented
5#15244C212yesmaximal, complemented
6#15224C212yesmaximal, complemented
7#15219C212yesmaximal, complemented
8#15130C212yesmaximal, complemented
9#15126C214yesmaximal, complemented
10#14260C214yesmaximal, complemented
11#14154C211yesmaximal, characteristic, complemented
12#14154C211yesmaximal, characteristic, complemented
13#13911C211yesmaximal, characteristic, complemented
14#13910C211yesmaximal, characteristic, complemented
15C4×#380C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#121C2212yes—
18C22×#121C2212yes—
19C22×#121C2212yes—
20C23×C42C2211yescharacteristic
21#2856—28no—
22C2×#394C2218yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2214yes—
27C2×#390C2214yes—
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#389C2214yes—
33C2×#384C2211yescharacteristic
34C2×#384C2212yes—
35#2283—28no—
36C2×#383C2212yes—
37C2×#382C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#382C2212yes—
40C2×#381C2212yes—
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×#376C2212yes—
51#1444C2214yes—
52#1442C2214yes—
53#1440C2218yes—
54#1440C2218yes—
55#1440C2218yes—
56#1439C2218yes—
57#1439C2218yes—
58#1439C2218yes—
59#1438C2218yes—
60#1438C2214yes—
61#1438C2214yes—
62C4⋊#95C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2212yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C23×D4C2312yes—
70C23×D4C2312yes—
71C23×D4C2314yes—
72C23×D4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×#127—24no—
77C2×#127—24no—
78C2×#127—24no—
79C2×#121—28no—
80C2×#121—28no—
81C2×#121—28no—
82C2×#121—44no—
83C2×#121—44no—
84C2×#121—48no—
85C2×#121—48no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2318yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2318yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2318yes—
104C22×C22⋊C4C2318yes—
105C22×C22⋊C4C2318yes—
106C22×C42—24no—
107C22×C42—24no—
108C22×C42D418yes—
109C22×C42C2312yes—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#394—416no—
118#390—216no—
119#390—28no—
120#390—28no—
121#390—28no—
122#390—44no—
123#390—44no—
124#390—44no—
125#390—44no—
126#390—48no—
127#389—48no—
128#384—24no—
129#384—44no—
130#383—28no—
131#382—24no—
132#382—24no—
133#382—44no—
134#381—28no—
135#380—24no—
136#380—24no—
137#380—28no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C4⋊C4—24no—
142C4×C22⋊C4—28no—
143C4×C22⋊C4—44no—
144#376—28no—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2318yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C25C2414yes—
162C25C2411yescharacteristic
163C22×D4—28no—
164C22×D4—216no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—28no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—28no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—48no—
183C22×D4—44no—
184C22×D4—44no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—42no—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215#127—416no—
216#121—432no—
217#121—432no—
218#121—432no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4#4118yes—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—216no—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4—48no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4—48no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—48no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—232no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42C2×D414yes—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293C2×C42C2×D414yes—
294C2×C42#4114yes—
295C2×C42#4114yes—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—44no—
308#95—44no—
309#95—48no—
310#95—48no—
311#95—48no—
312C24—22no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—21no—
320C24—22no—
321C24—24no—
322C24—28no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—28no—
329C24—28no—
330C24—24no—
331C24—28no—
332C24C22×D412yes—
333C24H(5,𝔽2)14yes—
334C24C2511yescharacteristic
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—416no—
338C2×D4—416no—
339C2×D4—416no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—416no—
352C2×D4—416no—
353C2×D4—88no—
354C2×D4—88no—
355C2×D4—816no—
356C2×D4—816no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C2×#4118yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4114yes—
442C22×C4H(5,𝔽2)12yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C4H(5,𝔽2)14yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C42−1+412yes—
447C4⋊C4—216no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—816no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—416no—
460C22⋊C4—416no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—816no—
464C22⋊C4—816no—
465C22⋊C4—816no—
466C22⋊C4—816no—
467C42—28no—
468C42—28no—
469C42—28no—
470C42—48no—
471C42—48no—
472C42—48no—
473C42—48no—
474C42#123116yes—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—48no—
512C23—48no—
513C23—48no—
514C23—48no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23—48no—
519C23—48no—
520C23#53514yes—
521C23C23×D411yescharacteristic
522C23C22×#4111yescharacteristic
523C23C22×#4111yescharacteristic
524C23C22×#4112yes—
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×H(5,𝔽2)11yescharacteristic
527C23C2×H(5,𝔽2)12yes—
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×2−1+411yescharacteristic
531C23#58512yes—
532D4—816no—
533D4—832no—
534D4—816no—
535D4—816no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—28no—
550C2×C4—28no—
551C2×C4—28no—
552C2×C4—416no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—416no—
566C2×C4—416no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
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—48no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4C2×#12314yes—
600C2×C4C2×#12314yes—
601C2×C4C2×#12314yes—
602C2×C4#53214yes—
603C2×C4#53514yes—
604C2×C4#53514yes—
605C2×C4#53714yes—
606C2×C4#54014yes—
607C2×C4#54014yes—
608C2×C4#54214yes—
609C2×C4#54214yes—
610C2×C4#55318yes—
611C2×C4#55518yes—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—88no—
624C22—88no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—88no—
631C22—88no—
632C22—84no—
633C22—84no—
634C22C22×#12311yescharacteristic
635C22C2×#53211yescharacteristic
636C22C2×#53511yescharacteristic
637C22C2×#53511yescharacteristic
638C22C2×#53511yescharacteristic
639C22C2×#53711yescharacteristic
640C22#344911yescharacteristic
641C22C2×#54011yescharacteristic
642C22C2×#54011yescharacteristic
643C22C2×#54211yescharacteristic
644C22C2×#54211yescharacteristic
645C22C2×#55312yes—
646C22C2×#55512yes—
647C22#349412yes—
648C22#349412yes—
649C22#351412yes—
650C22#351712yes—
651C22#351912yes—
652C22#352212yes—
653C22#354312yes—
654C22#354312yes—
655C22#354612yes—
656C22#355412yes—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—48no—
662C4—84no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4—82no—
668C4#250018yes—
669C4#262418yes—
670C2—44no—
671C2—164no—
672C2C2×#262411yescharacteristic
673C2C2×#250011yescharacteristic
674C2#3519111yescharacteristic
675C2#3523811yescharacteristic
676C2#3526211yescharacteristic
677C2#3761411yescharacteristic
678C2#3761611yescharacteristic
679C2#3894112yes—
680C2#3894212yes—
681C2#3914712yes—
682C2#3914812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2856C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2283C2semidirect
3. #16693C2semidirect
4. #16361C2semidirect
5. #15244C2semidirect
6. #15224C2semidirect
7. #15219C2semidirect
8. #15130C2semidirect
9. #15126C2semidirect
10. #14260C2semidirect
11. #14154C2semidirect
12. #14154C2semidirect
13. #13911C2semidirect
14. #13910C2semidirect
15. C4×#380C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28562
C2×#22832
#166932
#163612
#152442
#152242
#152192
#151302
#151262
#142602
#141542
#139112
#139102
C4×#3802