Finite Group Databasebeta

← #520886

Subgroups — #520886

Order 512.

3026 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#121C211yesmaximal, characteristic
2#14503C212yesmaximal, complemented
3#14497C214yesmaximal, complemented
4#14487C212yesmaximal, complemented
5#14453C212yesmaximal, complemented
6#14418C212yesmaximal
7#14376C214yesmaximal, complemented
8#13842C212yesmaximal, complemented
9#13839C212yesmaximal, complemented
10#13839C212yesmaximal, complemented
11C4×#381C211yesmaximal, characteristic, complemented
12C4×#390C212yesmaximal, complemented
13C4×#387C212yesmaximal, complemented
14#13761C212yesmaximal, complemented
15#13753C211yesmaximal, characteristic, complemented
16C22×#121C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2212yes—
19C22×#117C2212yes—
20C23×C42C2211yescharacteristic
21C4×#121C4116yescomplemented
22C2×#390C2214yes—
23C2×#390C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388C2212yes—
26C2×#387C2214yes—
27C2×#386C2212yes—
28C2×#381C2212yes—
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2212yes—
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C2×#376C2211yescharacteristic
40C2×#376C2211yescharacteristic
41C2×C43C2212yes—
42#1443C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2214yes—
53#1438C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2214yes—
63#1434C2214yes—
64#1434C2212yes—
65#1434C2212yes—
66#1434C2212yes—
67#1434C2212yes—
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C4×#95C2212yes—
72C4×#95C2212yes—
73C4×#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2212yes—
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C2×#121—28no—
83C2×#121C2×C418yes—
84C2×C4×D4—44no—
85C2×C4×D4—44no—
86C2×C4×D4—42no—
87C2×C4×D4—42no—
88C2×C4×D4C2×C4116yes—
89C2×C4×D4C2×C418yes—
90C2×#117—28no—
91C2×#117—44no—
92C2×#117—44no—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C42—42no—
112C22×C42—42no—
113C22×C42C2×C418yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125#390—216no—
126#390C2×C418yes—
127#389C2×C418yes—
128#388C2×C4116yes—
129#387—216no—
130#386C2×C4116yes—
131#381—28no—
132#380C2×C418yes—
133C4×C4⋊C4—44no—
134C4×C4⋊C4C2×C418yes—
135C4×C22⋊C4—28no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—42no—
138C4×C22⋊C4—44no—
139C4×C22⋊C4—44no—
140C4×C22⋊C4—42no—
141C4×C22⋊C4C2×C4116yes—
142#376—42no—
143#376—42no—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C43—44no—
169C25C2412yes—
170C25C2411yescharacteristic
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—42no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4C22×C418yes—
180C22×D4C22×C414yes—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—41no—
196C23×C4—41no—
197C23×C4—41no—
198C23×C4—41no—
199C23×C4C22×C418yes—
200C23×C4C22×C414yes—
201C23×C4C22×C414yes—
202C23×C4C22×C414yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216#121—232no—
217C4×D4—416no—
218C4×D4—432no—
219#117—432no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C22×C414yes—
232C2×C4⋊C4C22×C418yes—
233C2×C4⋊C4C22×C414yes—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4#4118yes—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
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—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C22×C418yes—
267C2×C22⋊C4C22×C418yes—
268C2×C22⋊C4C22×C418yes—
269C2×C22⋊C4#4118yes—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—28no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—416no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—41no—
293C2×C42—41no—
294C2×C42—41no—
295C2×C42—41no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42C22×C418yes—
299C2×C42C22×C414yes—
300C2×C42C22×C414yes—
301C2×C42C22×C414yes—
302C2×C42C22×C414yes—
303C2×C42#4118yes—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—42no—
317#95—42no—
318#95—42no—
319#95—42no—
320#95—42no—
321#95—44no—
322#95—42no—
323#95—44no—
324#95—44no—
325#95C22×C418yes—
326#95C22×C418yes—
327#95C22×C418yes—
328C24—22no—
329C24—22no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24C23×C414yes—
346C24C23×C412yes—
347C24C2×#4114yes—
348C24C2511yescharacteristic
349C2×D4—216no—
350C2×D4—216no—
351C2×D4—28no—
352C2×D4—44no—
353C2×D4—48no—
354C2×D4—44no—
355C2×D4—416no—
356C2×D4—416no—
357C2×D4—48no—
358C2×D4—48no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—216no—
410C22×C4—216no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4C23×C412yes—
440C22×C4C23×C412yes—
441C22×C4C23×C412yes—
442C22×C4C23×C414yes—
443C22×C4C23×C414yes—
444C22×C4C23×C412yes—
445C22×C4C23×C414yes—
446C22×C4C23×C412yes—
447C22×C4C23×C412yes—
448C22×C4C2×#4118yes—
449C22×C4C2×#4118yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4118yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C4H(5,𝔽2)12yes—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—416no—
460C4⋊C4—416no—
461C4⋊C4#117116yescomplemented
462C22⋊C4—216no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C42—28no—
475C42—28no—
476C42—216no—
477C42—216no—
478C42—44no—
479C42—44no—
480C42—416no—
481C42—416no—
482C42—48no—
483C42—48no—
484C42—48no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—28no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23C4×#4114yes—
525C23C24×C411yescharacteristic
526C23C22×#4112yes—
527C23C22×#4112yes—
528C23C22×#4112yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)12yes—
531C23C2×H(5,𝔽2)11yescharacteristic
532C23#58512yes—
533C23#58512yes—
534D4—432no—
535D4—416no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—416no—
569C2×C4—416no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—416no—
593C2×C4—432no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4C2×#11718yes—
609C2×C4C2×#11714yes—
610C2×C4C4×#4114yes—
611C2×C4#51814yes—
612C2×C4#51814yes—
613C2×C4#53818yes—
614C2×C4#54018yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—48no—
622C22—48no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22C22×#11711yescharacteristic
636C22C2×C4×#4111yescharacteristic
637C22C2×C4×#4111yescharacteristic
638C22C2×#51811yescharacteristic
639C22C2×#51811yescharacteristic
640C22#342811yescharacteristic
641C22#342811yescharacteristic
642C22C2×#53812yes—
643C22C2×#54012yes—
644C22#345912yes—
645C22#346012yes—
646C22#346012yes—
647C22#346012yes—
648C22#347514yes—
649C22#348014yes—
650C22#348114yes—
651C22#352914yes—
652C4—24no—
653C4—28no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—48no—
661C4—48no—
662C4—84no—
663C4—84no—
664C4#233318yes—
665C2—44no—
666C2—84no—
667C2#3405811yescharacteristic
668C2C2×#233311yescharacteristic
669C2#3408211yescharacteristic
670C2#3412511yescharacteristic
671C2#3412611yescharacteristic
672C2#3414911yescharacteristic
673C2#3434611yescharacteristic
674C2#5227714yes—
675C2#5228214yes—
subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14503C2semidirect
3. #14497C2semidirect
4. #14487C2semidirect
5. #14453C2semidirect
7. #14376C2semidirect
8. #13842C2semidirect
9. #13839C2semidirect
10. #13839C2semidirect
11. C4×#381C2semidirect
12. C4×#390C2semidirect
13. C4×#387C2semidirect
14. #13761C2semidirect
15. #13753C2semidirect
21. C4×#121C4semidirect
461. C4⋊C4#117semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1212
#145032
#144972
#144872
#144532
#144182
#143762
#138422
#138392
C4×#3812
C4×#3902
C4×#3872
#137612
#137532