Finite Group Databasebeta

← #556178

Subgroups — #556178

Order 512.

6462 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#534C211yesmaximal, characteristic
2C2×#2308C211yesmaximal, characteristic, complemented
3C22×#390C211yesmaximal, characteristic, complemented
4#14706C212yesmaximal, complemented
5#14505C214yesmaximal, complemented
6#14363C218yesmaximal, complemented
7#14357C214yesmaximal, complemented
8#14256C212yesmaximal, complemented
9#14134C214yesmaximal, complemented
10#13869C212yesmaximal, complemented
11#13867C212yesmaximal, complemented
12C24×D4C2211yescharacteristic
13C24×D4C2211yescharacteristic
14C2×#534—28no—
15C2×#534—28no—
16C2×#534C4116yescomplemented
17C22×#123C2212yes—
18C22×#123C2212yes—
19C22×#121C2214yes—
20C22×#120C2212yes—
21C22×#120C2212yes—
22C23×C4⋊C4C2211yescharacteristic, complemented
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2212yes—
25C23×C22⋊C4C2212yescomplemented
26C23×C22⋊C4C2212yescomplemented
27C23×C42C2211yescharacteristic, complemented
28#2308—28no—
29C2×#390—28no—
30C2×#390C2214yes—
31C2×#390C2214yescomplemented
32C2×#386C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#386C2214yescomplemented
36C2×#386C2218yescomplemented
37C2×#381C2212yescomplemented
38C2×#381C2214yescomplemented
39C2×#379C2212yes—
40C2×#379C2214yescomplemented
41#1441C2218yescomplemented
42#1440C2218yescomplemented
43#1440C2218yescomplemented
44#1439C2218yescomplemented
45#1438C2218yescomplemented
46#1438C2218yescomplemented
47#1437C2218yescomplemented
48#1436C2218yescomplemented
49C22⋊#95C2214yescomplemented
50C22⋊#95C2218yescomplemented
51C22⋊#95C2214yescomplemented
52C22⋊#95C2214yescomplemented
53C22⋊#95C2214yescomplemented
54C26C2311yescharacteristic
55C26C2311yescharacteristic
56C26C2311yescharacteristic
57C23×D4—24no—
58C23×D4—24no—
59C23×D4—44no—
60C23×D4—44no—
61C23×D4—44no—
62C23×D4—44no—
63C23×D4C2×C418yes—
64C23×D4C2×C418yescomplemented
65C23×D4D418yes—
66C23×D4D418yes—
67C23×D4C2312yes—
68C23×D4C2312yes—
69C23×D4C2314yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C24×C4C2312yes—
73C24×C4C2311yescharacteristic
74C24×C4C2312yes—
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2314yes—
78#534—2128no—
79C2×#123—28no—
80C2×#123—28no—
81C2×#123—28no—
82C2×#123—28no—
83C2×#123C2×C4116yescomplemented
84C2×#123C2×C4116yescomplemented
85C2×#121—216no—
86C2×#121—216no—
87C2×#121C2×C4132yescomplemented
88C2×#120—28no—
89C2×#120—28no—
90C2×#120—28no—
91C2×#120—28no—
92C2×#120C2×C4116yes—
93C2×#120C2×C4116yescomplemented
94C22×C4⋊C4—24no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—48no—
104C22×C22⋊C4—48no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4C2×C418yescomplemented
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2318yes—
120C22×C22⋊C4C2318yes—
121C22×C42—24no—
122C22×C42—44no—
123C22×C42C2314yes—
124#390—216no—
125#390—216no—
126#386—216no—
127#386—216no—
128#386—416no—
129#386—48no—
130#386—48no—
131#381—28no—
132#381—48no—
133#379—28no—
134#379—48no—
135C2×#95C2318yes—
136C2×#95C2318yes—
137C2×#95C2318yes—
138C2×#95C2318yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C25—24no—
143C25—24no—
144C25—24no—
145C25—24no—
146C25—22no—
147C25—22no—
148C25—44no—
149C25—44no—
150C25—42no—
151C25—42no—
152C25—44no—
153C25—44no—
154C25—44no—
155C25—44no—
156C25C22×C414yes—
157C25C22×C414yes—
158C25C22×C414yes—
159C25C2×D414yes—
160C25C2×D414yes—
161C25C2×D414yes—
162C25C2×D414yes—
163C25C2412yes—
164C25C2411yescharacteristic
165C25C2411yescharacteristic
166C25C2412yes—
167C25C2411yescharacteristic
168C22×D4—232no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—28no—
175C22×D4—28no—
176C22×D4—28no—
177C22×D4—432no—
178C22×D4—416no—
179C22×D4—416no—
180C22×D4—432no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4—48no—
192C22×D4—48no—
193C22×D4—432no—
194C22×D4—432no—
195C22×D4—416no—
196C22×D4—416no—
197C22×D4C22⋊C4164yescomplemented
198C22×D4C22×C418yes—
199C22×D4C22×C4116yes—
200C22×D4C22×C418yes—
201C22×D4C22×C418yes—
202C22×D4C22×C418yes—
203C22×D4C2×D418yes—
204C22×D4C2×D418yes—
205C22×D4C2×D418yes—
206C22×D4C2×D418yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—44no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—48no—
229C23×C4—48no—
230C23×C4C22×C414yes—
231C23×C4C22×C418yes—
232C23×C4C22×C414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2414yes—
236C23×C4C2418yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241#123—264no—
242#123—264no—
243#121—2128no—
244#120—264no—
245#120—264no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—416no—
252C2×C4⋊C4C22×C4116yes—
253C2×C22⋊C4—232no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—216no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—432no—
281C2×C22⋊C4—432no—
282C2×C22⋊C4—432no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4C22×C418yes—
293C2×C22⋊C4C22×C418yes—
294C2×C22⋊C4C22×C4116yes—
295C2×C22⋊C4C22×C4116yes—
296C2×C42—28no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—416no—
300#95—216no—
301#95—216no—
302#95—48no—
303#95—48no—
304#95—48no—
305#95—48no—
306#95—48no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—28no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—28no—
323C24—216no—
324C24—216no—
325C24—216no—
326C24—216no—
327C24—28no—
328C24—28no—
329C24—28no—
330C24—28no—
331C24—22no—
332C24—22no—
333C24—28no—
334C24—24no—
335C24—24no—
336C24—28no—
337C24—28no—
338C24—48no—
339C24—48no—
340C24—48no—
341C24—416no—
342C24—416no—
343C24—48no—
344C24—48no—
345C24—416no—
346C24—416no—
347C24—416no—
348C24—48no—
349C24—44no—
350C24—44no—
351C24—816no—
352C24—816no—
353C24—88no—
354C24—88no—
355C24—88no—
356C24—88no—
357C24—84no—
358C24—84no—
359C24—88no—
360C24—816no—
361C24—88no—
362C24—88no—
363C24C2×C22⋊C4116yes—
364C24C2×C22⋊C4116yes—
365C24C23×C414yes—
366C24C23×C412yes—
367C24C23×C414yes—
368C24C23×C412yes—
369C24C23×C412yes—
370C24C22×D412yes—
371C24C22×D412yes—
372C24C22×D414yes—
373C24C22×D414yes—
374C24C22×D412yes—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C22×D412yes—
378C24H(5,𝔽2)14yes—
379C24H(5,𝔽2)14yes—
380C24C2511yescharacteristic
381C2×D4—216no—
382C2×D4—216no—
383C2×D4—232no—
384C2×D4—232no—
385C2×D4—4128no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—416no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—48no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—432no—
400C2×D4—4128no—
401C2×D4—416no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—416no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—88no—
408C2×D4—88no—
409C2×D4—816no—
410C2×D4C2×C22⋊C4132yes—
411C2×D4C2×C22⋊C4132yes—
412C22×C4—216no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—216no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—216no—
421C22×C4—28no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—24no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—216no—
435C22×C4—28no—
436C22×C4—24no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—48no—
448C22×C4—416no—
449C22×C4—432no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—88no—
462C22×C4—816no—
463C22×C4—816no—
464C22×C4—816no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4C2×C22⋊C4116yes—
468C22×C4C23×C418yes—
469C22×C4C23×C414yes—
470C22×C4C23×C414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4H(5,𝔽2)18yes—
474C4⋊C4—232no—
475C4⋊C4—416no—
476C22⋊C4—216no—
477C22⋊C4—216no—
478C22⋊C4—232no—
479C22⋊C4—232no—
480C22⋊C4—416no—
481C22⋊C4—432no—
482C22⋊C4—416no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—816no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—816no—
489C42—416no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—216no—
495C23—28no—
496C23—28no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—216no—
503C23—28no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—24no—
520C23—28no—
521C23—28no—
522C23—216no—
523C23—232no—
524C23—232no—
525C23—24no—
526C23—28no—
527C23—216no—
528C23—432no—
529C23—432no—
530C23—416no—
531C23—48no—
532C23—48no—
533C23—48no—
534C23—48no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—48no—
549C23—48no—
550C23—48no—
551C23—48no—
552C23—48no—
553C23—832no—
554C23—832no—
555C23—832no—
556C23—832no—
557C23—88no—
558C23—88no—
559C23—816no—
560C23—816no—
561C23—88no—
562C23—88no—
563C23—816no—
564C23—832no—
565C23—832no—
566C23—88no—
567C23—816no—
568C23—88no—
569C23C22×C22⋊C414yes—
570C23C22×C22⋊C418yes—
571C23C22×C22⋊C414yes—
572C23C22×C22⋊C414yes—
573C23#51818yes—
574C23#53418yes—
575C23#53414yes—
576C23#53414yes—
577C23C24×C411yescharacteristic
578C23C23×D411yescharacteristic
579C23C23×D411yescharacteristic
580C23C2×H(5,𝔽2)12yes—
581C23C2×H(5,𝔽2)12yes—
582C23C2×H(5,𝔽2)14yes—
583C23#58514yes—
584D4—432no—
585D4—432no—
586D4—432no—
587D4—432no—
588D4—832no—
589C2×C4—232no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—216no—
600C2×C4—28no—
601C2×C4—216no—
602C2×C4—432no—
603C2×C4—416no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—48no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—816no—
626C2×C4—816no—
627C2×C4—832no—
628C2×C4C22×C22⋊C418yes—
629C22—216no—
630C22—216no—
631C22—24no—
632C22—28no—
633C22—28no—
634C22—24no—
635C22—28no—
636C22—216no—
637C22—28no—
638C22—24no—
639C22—24no—
640C22—28no—
641C22—48no—
642C22—48no—
643C22—48no—
644C22—48no—
645C22—48no—
646C22—48no—
647C22—48no—
648C22—48no—
649C22—42no—
650C22—44no—
651C22—42no—
652C22—44no—
653C22—48no—
654C22—44no—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—416no—
659C22—48no—
660C22—44no—
661C22—832no—
662C22—832no—
663C22—832no—
664C22—88no—
665C22—82no—
666C22—84no—
667C22—82no—
668C22#230818yes—
669C22C23×C22⋊C411yescharacteristic
670C22C2×#51812yes—
671C22C4×H(5,𝔽2)14yes—
672C22C2×#53412yes—
673C22C2×#53411yescharacteristic
674C22C2×#53411yescharacteristic
675C22#348412yes—
676C22#348512yes—
677C22#349614yes—
678C22#353418yes—
679C22#354314yes—
680C22#354514yes—
681C4—28no—
682C4—48no—
683C4—44no—
684C4—48no—
685C4—88no—
686C4—84no—
687C2—28no—
688C2—44no—
689C2—48no—
690C2—44no—
691C2—84no—
692C2C2×#230811yescharacteristic
693C2#3425612yes—
694C2#3440814yes—
695C2#3711712yes—
696C2#3711911yescharacteristic
697C2#3712914yes—
698C2#3723211yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#534C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2308C2semidirect
3. C22×#390C2semidirect
4. #14706C2semidirect
5. #14505C2semidirect
6. #14363C2semidirect
7. #14357C2semidirect
8. #14256C2semidirect
9. #14134C2semidirect
10. #13869C2semidirect
11. #13867C2semidirect
16. C2×#534C4semidirect
22. C23×C4⋊C4C22semidirect
25. C23×C22⋊C4C22semidirect
26. C23×C22⋊C4C22semidirect
27. C23×C42C22semidirect
31. C2×#390C22semidirect
35. C2×#386C22semidirect
36. C2×#386C22semidirect
37. C2×#381C22semidirect
38. C2×#381C22semidirect
40. C2×#379C22semidirect
41. #1441C22semidirect
42. #1440C22semidirect
43. #1440C22semidirect
44. #1439C22semidirect
45. #1438C22semidirect
46. #1438C22semidirect
47. #1437C22semidirect
48. #1436C22semidirect
49. C22⋊#95C22semidirect
50. C22⋊#95C22semidirect
51. C22⋊#95C22semidirect
52. C22⋊#95C22semidirect
53. C22⋊#95C22semidirect
64. C23×D4C2×C4semidirect
83. C2×#123C2×C4semidirect
84. C2×#123C2×C4semidirect
87. C2×#121C2×C4semidirect
93. C2×#120C2×C4semidirect
107. C22×C22⋊C4C2×C4semidirect
197. C22×D4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5342
C2×#23082
C22×#3902
#147062
#145052
#143632
#143572
#142562
#141342
#138692
#138672