Finite Group Databasebeta

← #682153

Subgroups — #682153

Order 512.

2558 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2C2×#2351C211yesmaximal, characteristic, complemented
3#15806C214yesmaximal, complemented
4#15701C214yesmaximal
5#15183C212yesmaximal, complemented
6#15132C212yesmaximal, complemented
7#15025C214yesmaximal, complemented
8#14689C211yesmaximal, characteristic, complemented
9#14249C214yesmaximal, complemented
10#13969C212yesmaximal, complemented
11#13969C212yesmaximal, complemented
12#13955C211yesmaximal, characteristic, complemented
13#13905C211yesmaximal, characteristic
14C4×#391C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic
16C22×C22⋊Q8C2212yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2536—28no—
21C2×#398C2212yes—
22C2×#397C2214yes—
23C2×#394C2212yes—
24#2351—28no—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#389C2214yes—
30C2×#388C2214yes—
31C2×#382C2211yescharacteristic
32C2×#382C2212yes—
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C43C2211yescharacteristic
43#1445C2214yes—
44#1445C2212yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2212yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2212yes—
61#1437C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73#1434C2214yes—
74#1434C2212yes—
75#1434C2212yes—
76C4×#95C2214yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—28no—
82C2×C22⋊Q8—28no—
83C2×C4×Q8—28no—
84C2×C4×Q8—24no—
85C2×C4×Q8—24no—
86C2×C4×Q8—44no—
87C2×C4×Q8—44no—
88C2×C4×Q8D418yes—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2314yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121#398—28no—
122#397—216no—
123#394—28no—
124#391—24no—
125#391—24no—
126#390—28no—
127#389—28no—
128#389—216no—
129#388—216no—
130#382—28no—
131#382—24no—
132#382—24no—
133#380—28no—
134#380—44no—
135#380—44no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—24no—
139C4×C22⋊C4—44no—
140C4×C22⋊C4—44no—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C43—24no—
169C25C2412yes—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8C2×D414yes—
175C22×Q8#4114yes—
176C22×Q8#4114yes—
177C22×D4—24no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—42no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C22⋊Q8—432no—
204C4×Q8—232no—
205C4×Q8—432no—
206C4×D4—416no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4C2×D418yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4#4118yes—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42C2×D418yes—
290C2×C42C2×D414yes—
291C2×C42#4114yes—
292C2×C42#4114yes—
293#95—22no—
294#95—24no—
295#95—22no—
296#95—28no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24C2×#4114yes—
335C24C2511yescharacteristic
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—28no—
341C2×Q8—28no—
342C2×Q8#124116yes—
343C2×D4—28no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—44no—
347C2×D4—44no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
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—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D414yes—
427C22×C4C22×D412yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+414yes—
443C4⋊C4—216no—
444C4⋊C4—28no—
445C4⋊C4—48no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—416no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—416no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—416no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C42—216no—
464C42—48no—
465C42—48no—
466C42—416no—
467C42—48no—
468C42—44no—
469C42—44no—
470C42—84no—
471C42#123116yes—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23C23×D411yescharacteristic
508C23C22×#4111yescharacteristic
509C23C22×#4111yescharacteristic
510C23C22×#4111yescharacteristic
511C23C22×#4111yescharacteristic
512C23C22×#4112yes—
513C23C2×H(5,𝔽2)11yescharacteristic
514C23C2×H(5,𝔽2)11yescharacteristic
515C23C2×2−1+412yes—
516C23#58512yes—
517C23#58512yes—
518Q8—232no—
519D4—416no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
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—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—816no—
599C2×C4C2×#12314yes—
600C2×C4C2×#12314yes—
601C2×C4C2×#12314yes—
602C2×C4C2×#12418yes—
603C2×C4C2×#12414yes—
604C2×C4#53514yes—
605C2×C4#55314yes—
606C2×C4#55314yes—
607C2×C4#55314yes—
608C2×C4#55314yes—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22C22×#12311yescharacteristic
625C22C22×#12411yescharacteristic
626C22#344411yescharacteristic
627C22C2×#53511yescharacteristic
628C22#345511yescharacteristic
629C22D4×#4112yes—
630C22C2×#55311yescharacteristic
631C22C2×#55311yescharacteristic
632C22C2×#55311yescharacteristic
633C22C2×#55311yescharacteristic
634C22#347512yes—
635C22#348012yes—
636C22#348212yes—
637C22#349512yes—
638C22#349712yes—
639C22#350512yes—
640C22#350911yescharacteristic
641C22#351011yescharacteristic
642C22#351612yes—
643C22#352012yes—
644C22#352412yes—
645C22#354712yes—
646C22#355112yes—
647C4—24no—
648C4—24no—
649C4—28no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—44no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4—84no—
658C4—82no—
659C4—82no—
660C4#250418yes—
661C2—84no—
662C2#3474211yescharacteristic
663C2C2×#250411yescharacteristic
664C2#3567311yescharacteristic
665C2#3646712yes—
666C2#3650911yescharacteristic
667C2#3653911yescharacteristic
668C2#3918611yescharacteristic
669C2#3918811yescharacteristic
670C2#3995012yes—
671C2#5921612yes—
672C2#5922112yes—
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
2. C2×#2351C2semidirect
3. #15806C2semidirect
5. #15183C2semidirect
6. #15132C2semidirect
7. #15025C2semidirect
8. #14689C2semidirect
9. #14249C2semidirect
10. #13969C2semidirect
11. #13969C2semidirect
12. #13955C2semidirect
14. C4×#391C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
C2×#23512
#158062
#157012
#151832
#151322
#150252
#146892
#142492
#139692
#139552
#139052
C4×#3912
C4×#3902