Finite Group Databasebeta

← #530953

Subgroups — #530953

Order 512.

2682 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2#14688C211yesmaximal, characteristic, complemented
3#14509C214yesmaximal, complemented
4#14477C212yesmaximal, complemented
5#14473C212yesmaximal
6#14284C212yesmaximal, complemented
7#13990C214yesmaximal, complemented
8#13925C212yesmaximal, complemented
9#13881C212yesmaximal, complemented
10#13875C212yesmaximal, complemented
11#13875C212yesmaximal, complemented
12#13872C212yesmaximal
13C22⋊C4×C4⋊C4C212yesmaximal, complemented
14#13791C212yesmaximal
15C4×#382C211yesmaximal, characteristic, complemented
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C4×#124C4116yescomplemented
21C2×#391C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#388C2212yes—
24C2×#388C2212yes—
25C2×#386C2214yes—
26C2×#383C2212yes—
27C2×#382C2212yes—
28C2×#382C2212yes—
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2214yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C43C2211yescharacteristic
40#1446C2214yes—
41#1445C2214yes—
42#1443C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1437C2214yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2211yescharacteristic
62C4⋊#95C2212yes—
63#1434C2214yes—
64#1434C2214yes—
65#1434C2212yes—
66#1434C2212yes—
67#1434C2212yes—
68#1434C2212yes—
69#1434C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C4×#95C2212yes—
75C4×#95C2212yes—
76C23×Q8C2311yescharacteristic
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2314yes—
80C2×#124—28no—
81C2×#124C2×C418yes—
82C2×C4×Q8—44no—
83C2×C4×Q8C2×C418yescomplemented
84C2×C4×D4—44no—
85C2×C4×D4C2×C418yes—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#391C2×C418yescomplemented
117#390C2×C418yes—
118#388C2×C4116yescomplemented
119#388C2×C4116yescomplemented
120#386—216no—
121#383—44no—
122#382—28no—
123#382—44no—
124#382C2×C418yescomplemented
125#380—28no—
126#380—44no—
127#380—44no—
128C4×C4⋊C4—44no—
129C4×C4⋊C4—42no—
130C4×C4⋊C4—42no—
131C4×C22⋊C4—48no—
132C4×C22⋊C4C2×C4116yescomplemented
133C4×C22⋊C4C2×C4116yescomplemented
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C43C2×C418yescomplemented
163C25C2412yes—
164C22×Q8—24no—
165C22×Q8—42no—
166C22×Q8—41no—
167C22×Q8—41no—
168C22×Q8C22×C414yes—
169C22×D4—42no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4C22×C414yes—
173C22×D4#4118yes—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—28no—
181C23×C4—28no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C22×C418yes—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202#124—232no—
203C4×Q8—416no—
204C4×D4—416no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
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—24no—
217C2×C4⋊C4—432no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4C22×C414yes—
225C2×C4⋊C4C22×C414yes—
226C2×C4⋊C4C22×C418yes—
227C2×C4⋊C4#4118yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4C22×C418yes—
248C2×C22⋊C4C22×C418yes—
249C2×C22⋊C4C22×C418yes—
250C2×C22⋊C4C22×C418yes—
251C2×C22⋊C4C22×C418yes—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42—42no—
266C2×C42—44no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—44no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—44no—
273C2×C42C22×C414yes—
274C2×C42C22×C418yes—
275C2×C42C22×C414yes—
276C2×C42C22×C414yes—
277C2×C42C22×C414yes—
278C2×C42C22×C414yes—
279C2×C42C22×C414yes—
280C2×C42C22×C414yes—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—44no—
288#95—44no—
289#95—42no—
290#95—42no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—42no—
296#95—42no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—42no—
303#95—44no—
304#95—42no—
305#95C22×C418yes—
306#95C22×C418yes—
307#95C22×C418yes—
308#95C22×C418yes—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24C23×C414yes—
322C24C2×#4114yes—
323C24C2×#4114yes—
324C24C2511yescharacteristic
325C2×Q8—28no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4#117116yes—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—216no—
383C22×C4—216no—
384C22×C4—216no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4—88no—
417C22×C4C23×C414yes—
418C22×C4C23×C412yes—
419C22×C4C23×C412yes—
420C22×C4C23×C412yes—
421C22×C4C23×C414yes—
422C22×C4C23×C414yes—
423C22×C4C23×C412yes—
424C22×C4C23×C412yes—
425C22×C4C23×C412yes—
426C22×C4C23×C412yes—
427C22×C4C2×#4118yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4114yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)14yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C42−1+412yes—
434C22×C42−1+412yes—
435C4⋊C4—216no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—416no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C22⋊C4—216no—
444C22⋊C4—216no—
445C22⋊C4—216no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—416no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C42—28no—
452C42—28no—
453C42—28no—
454C42—216no—
455C42—28no—
456C42—28no—
457C42—44no—
458C42—44no—
459C42—84no—
460C42—84no—
461C42—88no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
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—
496C23C2×#11718yes—
497C23C24×C411yescharacteristic
498C23C22×#4112yes—
499C23C22×#4112yes—
500C23C2×H(5,𝔽2)12yes—
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×2−1+411yescharacteristic
504C23C2×2−1+411yescharacteristic
505C23#58511yescharacteristic
506C23#58511yescharacteristic
507C23#58512yes—
508Q8—416no—
509D4—416no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—432no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
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—832no—
591C2×C4C2×#11714yes—
592C2×C4C4×#4114yes—
593C2×C4#51814yes—
594C2×C4#52014yes—
595C2×C4#52014yes—
596C2×C4#53818yes—
597C2×C4#54218yes—
598C22—28no—
599C22—28no—
600C22—28no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22C22×#11711yescharacteristic
614C22C2×C4×#4111yescharacteristic
615C22C2×#51811yescharacteristic
616C22C2×#52011yescharacteristic
617C22C2×#52011yescharacteristic
618C22#342811yescharacteristic
619C22#342811yescharacteristic
620C22C2×#53812yes—
621C22C2×#54212yes—
622C22#348112yes—
623C22#348212yes—
624C22#350512yes—
625C22#350512yes—
626C22#351412yes—
627C22#351512yes—
628C22#352112yes—
629C22#352412yes—
630C22#354612yes—
631C22#354712yes—
632C22#355811yescharacteristic
633C22#355911yescharacteristic
634C22#356111yescharacteristic
635C22#356311yescharacteristic
636C4—24no—
637C4—24no—
638C4—44no—
639C4—42no—
640C4—44no—
641C4—42no—
642C4—48no—
643C4—44no—
644C4—42no—
645C4—42no—
646C4—84no—
647C4—84no—
648C4—84no—
649C4#233718yes—
650C2—48no—
651C2C2×#233711yescharacteristic
652C2#3412711yescharacteristic
653C2#3418511yescharacteristic
654C2#3419111yescharacteristic
655C2#3424411yescharacteristic
656C2#3436711yescharacteristic
657C2#3438211yescharacteristic
658C2#4005612yes—
659C2#4005812yes—
660C2#4010412yes—
661C2#4010612yes—
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. #14688C2semidirect
3. #14509C2semidirect
4. #14477C2semidirect
6. #14284C2semidirect
7. #13990C2semidirect
8. #13925C2semidirect
9. #13881C2semidirect
10. #13875C2semidirect
11. #13875C2semidirect
13. C22⋊C4×C4⋊C4C2semidirect
15. C4×#382C2semidirect
20. C4×#124C4semidirect
83. C2×C4×Q8C2×C4semidirect
116. #391C2×C4semidirect
118. #388C2×C4semidirect
119. #388C2×C4semidirect
124. #382C2×C4semidirect
132. C4×C22⋊C4C2×C4semidirect
133. C4×C22⋊C4C2×C4semidirect
162. C43C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
#146882
#145092
#144772
#144732
#142842
#139902
#139252
#138812
#138752
#138722
C22⋊C4×C4⋊C42
#137912
C4×#3822