Finite Group Databasebeta

← #1010230

Subgroups — #1010230

Order 512.

2514 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×#2429C212yesmaximal, complemented
2#17196C212yesmaximal
3#17036C212yesmaximal, complemented
4#16419C214yesmaximal, complemented
5#16002C212yesmaximal, complemented
6#15741C214yesmaximal, complemented
7#15196C211yesmaximal, characteristic, complemented
8#15182C212yesmaximal, complemented
9#15177C212yesmaximal
10#14589C212yesmaximal
11#14155C211yesmaximal, characteristic, complemented
12#13973C212yesmaximal, complemented
13#13969C212yesmaximal, complemented
14#13931C212yesmaximal, complemented
15C4×#394C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21#2429—216no—
22C2×#398C2214yes—
23C2×#396C2214yes—
24C2×#394C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2214yes—
29C2×#389C2212yes—
30C2×#388C2214yes—
31C2×#384C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38#1445C2212yes—
39#1445C2214yes—
40#1445C2212yes—
41#1445C2214yes—
42#1444C2212yes—
43#1444C2212yes—
44#1444C2214yes—
45#1442C2211yescharacteristic
46#1442C2214yes—
47#1441C2211yescharacteristic
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2212yes—
52#1441C2212yes—
53#1440C2212yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2211yescharacteristic
62#1438C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72#1434C2214yes—
73#1434C2214yes—
74#1434C2214yes—
75C4×#95C2211yescharacteristic
76C23×Q8C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2312yes—
79C2×C4⋊Q8—42no—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8D418yescomplemented
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—44no—
86C2×C22⋊Q8—44no—
87C2×C22⋊Q8—44no—
88C2×C4×Q8—28no—
89C2×C4×Q8—28no—
90C2×C4×Q8—44no—
91C2×C4×D4—42no—
92C2×C4×D4—44no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#398—216no—
123#396—216no—
124#394Q818yescomplemented
125#391—28no—
126#391—44no—
127#391—44no—
128#389—216no—
129#389—28no—
130#388—216no—
131#384—42no—
132#380—28no—
133#380—44no—
134C4×C4⋊C4—42no—
135C4×C4⋊C4—42no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—44no—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C25C2412yes—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8C2×D418yes—
173C22×D4—42no—
174C22×D4—41no—
175C22×D4—42no—
176C22×D4—41no—
177C22×D4C2×Q814yes—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2411yescharacteristic
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C4⋊Q8—416no—
200C22⋊Q8—432no—
201C22⋊Q8—432no—
202C4×Q8—432no—
203C4×D4—416no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4C2×D418yes—
240C2×C4⋊C4C2×D418yes—
241C2×C4⋊C4C2×Q814yes—
242C2×C4⋊C4#4118yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4C2×Q818yes—
269C2×C22⋊C4C2×Q818yes—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—28no—
274C2×C42—28no—
275C2×C42—24no—
276C2×C42—22no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—44no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42C2×D414yes—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—42no—
294#95—44no—
295#95—42no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—41no—
301#95—44no—
302#95—42no—
303#95—42no—
304#95—41no—
305#95—41no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95C2×Q814yes—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24C22×Q814yes—
328C24C2511yescharacteristic
329C2×Q8—216no—
330C2×Q8—216no—
331C2×Q8—216no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×D4—44no—
337C2×D4—48no—
338C2×D4—44no—
339C2×D4—48no—
340C2×D4—44no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4C22×D418yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×Q814yes—
424C22×C4C22×Q812yes—
425C22×C4C22×Q812yes—
426C22×C4C22×Q812yes—
427C22×C4C2×#4118yes—
428C22×C4C2×#4114yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)14yes—
432C22×C42−1+412yes—
433C22×C42−1+412yes—
434C4⋊C4—216no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—416no—
440C4⋊C4—416no—
441C4⋊C4—416no—
442C4⋊C4—416no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C22⋊C4—216no—
449C22⋊C4—48no—
450C22⋊C4—416no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—416no—
454C22⋊C4—48no—
455C22⋊C4—88no—
456C22⋊C4—88no—
457C42—28no—
458C42—416no—
459C42—48no—
460C42—44no—
461C42—44no—
462C42—84no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
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—
498C23C23×D411yescharacteristic
499C23C23×Q811yescharacteristic
500C23C22×#4112yes—
501C23C2×H(5,𝔽2)11yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×H(5,𝔽2)12yes—
504C23C2×2−1+411yescharacteristic
505C23C2×2−1+411yescharacteristic
506C23#58512yes—
507C23#58512yes—
508C23#58511yescharacteristic
509Q8—432no—
510D4—416no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—816no—
592C2×C4#53414yes—
593C2×C4#54018yes—
594C2×C4#54618yes—
595C2×C4D4×Q814yes—
596C2×C4#55514yes—
597C2×C4#56014yes—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22C2×#53411yescharacteristic
614C22C2×#54012yes—
615C22C2×#54612yes—
616C22C2×D4×Q811yescharacteristic
617C22C2×#55511yescharacteristic
618C22Q8×#4112yes—
619C22#348712yes—
620C22#349711yescharacteristic
621C22#350111yescharacteristic
622C22#350112yes—
623C22#350311yescharacteristic
624C22#351512yes—
625C22#351812yes—
626C22#352112yes—
627C22C2×#56011yescharacteristic
628C22#353811yescharacteristic
629C22#353912yes—
630C22#354011yescharacteristic
631C22#354112yes—
632C22#354712yes—
633C22#355111yescharacteristic
634C22#356111yescharacteristic
635C22#356212yes—
636C4—24no—
637C4—42no—
638C4—42no—
639C4—48no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—48no—
644C4—81no—
645C4—84no—
646C4—84no—
647C4—82no—
648C4—82no—
649C4#269918yes—
650C2—84no—
651C2#3961511yescharacteristic
652C2#3965612yes—
653C2#4278412yes—
654C2#4288311yescharacteristic
655C2#4524312yes—
656C2#4541011yescharacteristic
657C2#4549311yescharacteristic
658C2C2×#269911yescharacteristic
659C2#5300611yescharacteristic
660C2#5903612yes—
661C2#5993511yescharacteristic
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
1. C2×#2429C2semidirect
3. #17036C2semidirect
4. #16419C2semidirect
5. #16002C2semidirect
6. #15741C2semidirect
7. #15196C2semidirect
8. #15182C2semidirect
11. #14155C2semidirect
12. #13973C2semidirect
13. #13969C2semidirect
14. #13931C2semidirect
81. C2×C4⋊Q8D4semidirect
124. #394Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#24292
#171962
#170362
#164192
#160022
#157412
#151962
#151822
#151772
#145892
#141552
#139732
#139692
#139312
C4×#3942