Finite Group Databasebeta

← #533696

Subgroups — #533696

Order 512.

2766 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2#14614C212yesmaximal, complemented
3#14515C212yesmaximal, complemented
4#14252C212yesmaximal, complemented
5#14240C214yesmaximal, complemented
6#14068C214yesmaximal
7#13994C212yesmaximal, complemented
8#13993C212yesmaximal, complemented
9#13982C212yesmaximal, complemented
10#13905C211yesmaximal, characteristic
11#13823C212yesmaximal, complemented
12#13821C212yesmaximal, complemented
13C22⋊C4×C4⋊C4C212yesmaximal, complemented
14C4×#380C212yesmaximal, complemented
15C4×#386C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2487C4116yescomplemented
20C2×#400C2212yes—
21C2×#399C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#388C2214yes—
25C2×#386C2212yes—
26C2×#386C2212yes—
27C2×#385C2212yes—
28C2×#382C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2214yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2214yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2214yes—
42#1445C2212yes—
43#1445C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1438C2212yes—
50#1437C2212yes—
51#1437C2212yes—
52#1437C2214yes—
53#1437C2211yescharacteristic
54#1437C2211yescharacteristic
55#1436C2211yescharacteristic
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2211yescharacteristic
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C4×#95C2212yes—
75C4×#95C2214yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2312yes—
81C2×C4×D4—42no—
82C2×C4×D4—42no—
83C2×C4×D4C2×C418yes—
84C22×C4⋊C4—42no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4C2×C418yescomplemented
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C42—28no—
105C22×C42—42no—
106C22×C42C2314yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2314yes—
113C22×C42C2314yes—
114C22×C42C2312yes—
115#400C2×C4116yescomplemented
116#399C2×C4116yes—
117#389—28no—
118#389C2×C4116yescomplemented
119#388—216no—
120#386—28no—
121#386C2×C4116yes—
122#385C2×C4116yescomplemented
123#382—42no—
124#380—28no—
125#380—28no—
126#380C2×C4116yes—
127#380C2×C418yescomplemented
128C4×C4⋊C4—28no—
129C4×C4⋊C4—48no—
130C4×C4⋊C4—42no—
131C4×C4⋊C4—44no—
132C4×C4⋊C4—44no—
133C4×C22⋊C4—28no—
134C4×C22⋊C4—48no—
135C4×C22⋊C4—42no—
136C4×C22⋊C4—48no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C25C2411yescharacteristic
166C25C2411yescharacteristic
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4C22×C414yes—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4C22×C414yes—
192C23×C4C22×C414yes—
193C23×C4C22×C418yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C4×D4—416no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—416no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4C22×C418yes—
228C2×C4⋊C4C22×C418yes—
229C2×C4⋊C4C22×C418yes—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C22×C4116yes—
232C2×C4⋊C4#4118yes—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4C22×C418yes—
255C2×C22⋊C4C22×C414yes—
256C2×C22⋊C4C22×C414yes—
257C2×C22⋊C4C22×C4116yes—
258C2×C22⋊C4#4118yes—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—28no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—28no—
272C2×C42—416no—
273C2×C42—41no—
274C2×C42—41no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42—42no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42C22×C414yes—
283C2×C42#4118yes—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—24no—
296#95—44no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—41no—
303#95—41no—
304#95—44no—
305#95—42no—
306#95—44no—
307#95—42no—
308#95—42no—
309#95C22×C4116yes—
310#95C22×C418yes—
311#95C22×C4116yes—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24C23×C412yes—
327C24C23×C412yes—
328C24C2×#4114yes—
329C242−1+414yes—
330C24C2511yescharacteristic
331C2×D4—28no—
332C2×D4—28no—
333C2×D4—44no—
334C2×D4—48no—
335C2×D4—48no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—216no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—28no—
390C22×C4—28no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—416no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—88no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4C23×C412yes—
423C22×C4C23×C412yes—
424C22×C4C23×C418yes—
425C22×C4C23×C412yes—
426C22×C4C23×C418yes—
427C22×C4C23×C412yes—
428C22×C4C23×C412yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4118yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4114yes—
435C22×C4C2×#4114yes—
436C22×C4H(5,𝔽2)14yes—
437C22×C4H(5,𝔽2)12yes—
438C22×C4H(5,𝔽2)12yes—
439C4⋊C4—216no—
440C4⋊C4—216no—
441C4⋊C4—216no—
442C4⋊C4—48no—
443C4⋊C4—416no—
444C4⋊C4—44no—
445C4⋊C4—416no—
446C4⋊C4—48no—
447C22⋊C4—216no—
448C22⋊C4—216no—
449C22⋊C4—416no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—48no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—48no—
456C42—28no—
457C42—216no—
458C42—44no—
459C42—44no—
460C42—416no—
461C42—48no—
462C42—88no—
463C42—84no—
464C42—88no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—24no—
484C23—24no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23C4×#4114yes—
506C23#51914yes—
507C23C24×C411yescharacteristic
508C23C22×#4112yes—
509C23C22×#4112yes—
510C23C22×#4112yes—
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)12yes—
514C23C2×2−1+412yes—
515C23#58512yes—
516D4—416no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—28no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—432no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—416no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—816no—
596C2×C4C4×#4118yes—
597C2×C4#51814yes—
598C2×C4#55114yes—
599C2×C4#55318yes—
600C2×C4#55318yes—
601C2×C4#55914yes—
602C22—28no—
603C22—24no—
604C22—24no—
605C22—28no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—42no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22C2×C4×#4112yes—
621C22C2×C4×#4111yescharacteristic
622C22C2×#51811yescharacteristic
623C22C2×#51911yescharacteristic
624C22C4×H(5,𝔽2)12yes—
625C22C2×#55111yescharacteristic
626C22C2×#55312yes—
627C22C2×#55312yes—
628C22#347512yes—
629C22C2×#55911yescharacteristic
630C22#348012yes—
631C22#348112yes—
632C22#349212yes—
633C22#349512yes—
634C22#351612yes—
635C22#352012yes—
636C22#352012yes—
637C22#352012yes—
638C22#354212yes—
639C22#354911yescharacteristic
640C22#355111yescharacteristic
641C4—28no—
642C4—44no—
643C4—44no—
644C4—48no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—48no—
649C4—48no—
650C4—42no—
651C4—82no—
652C4—84no—
653C4#237418yes—
654C2—44no—
655C2—44no—
656C2C2×#237411yescharacteristic
657C2C4×#55312yes—
658C2#3424511yescharacteristic
659C2#3429912yes—
660C2#3441511yescharacteristic
661C2#5920612yes—
662C2#5922012yes—
663C2#5922112yes—
664C2#5963412yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14614C2semidirect
3. #14515C2semidirect
4. #14252C2semidirect
5. #14240C2semidirect
7. #13994C2semidirect
8. #13993C2semidirect
9. #13982C2semidirect
11. #13823C2semidirect
12. #13821C2semidirect
13. C22⋊C4×C4⋊C4C2semidirect
14. C4×#380C2semidirect
19. #2487C4semidirect
86. C22×C4⋊C4C2×C4semidirect
115. #400C2×C4semidirect
118. #389C2×C4semidirect
122. #385C2×C4semidirect
127. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
#146142
#145152
#142522
#142402
#140682
#139942
#139932
#139822
#139052
#138232
#138212
C22⋊C4×C4⋊C42
C4×#3802
C4×#3862