Finite Group Databasebeta

← #522506

Subgroups — #522506

Order 512.

2914 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2362C212yesmaximal
3#14423C212yesmaximal
4#14422C212yesmaximal
5#14369C214yesmaximal
6#14280C212yesmaximal, complemented
7#14138C212yesmaximal
8#14083C214yesmaximal, complemented
9#14029C212yesmaximal, complemented
10#14024C212yesmaximal, complemented
11#14021C212yesmaximal, complemented
12#14017C212yesmaximal, complemented
13#13790C212yesmaximal, complemented
14C4×#380C211yesmaximal, characteristic
15#13756C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2212yes—
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2857C4116yescomplemented
23#2362—216no—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#384C2211yescharacteristic
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2214yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2212yes—
44#1445C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67#1434C2214yes—
68#1434C2211yescharacteristic
69#1434C2211yescharacteristic
70C22⋊#95C2212yes—
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C4×#95C2212yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C23×Q8C2312yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8C2×C418yes—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8C2×C4116yes—
87C2×C4×Q8—28no—
88C2×C4×Q8—44no—
89C2×C4×Q8—44no—
90C2×#117—24no—
91C2×#117—42no—
92C2×#117—42no—
93C2×#117—42no—
94C2×#117—42no—
95C2×#117D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C42—42no—
114C22×C42—42no—
115C22×C42C2×C418yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#391—28no—
127#391—44no—
128#391—44no—
129#391C2×C4116yes—
130#389—28no—
131#389C2×C4116yes—
132#389C2×C4116yes—
133#388—216no—
134#388—216no—
135#387—216no—
136#387—216no—
137#384C2×C418yes—
138#380—28no—
139#380C2×C4116yes—
140#380C2×C4116yes—
141C4×C4⋊C4—28no—
142C4×C4⋊C4—216no—
143C4×C4⋊C4—42no—
144C4×C4⋊C4—42no—
145C4×C22⋊C4—28no—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C25C2411yescharacteristic
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8C22×C418yes—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—42no—
190C23×C4—41no—
191C23×C4—41no—
192C23×C4—41no—
193C23×C4—41no—
194C23×C4C22×C414yes—
195C23×C4C22×C414yes—
196C23×C4C22×C414yes—
197C23×C4C2×D414yes—
198C23×C4#4114yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C4⋊Q8—416no—
214C22⋊Q8—264no—
215C4×Q8—432no—
216#117—416no—
217#117—416no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C4116yes—
247C2×C4⋊C4C22×C418yes—
248C2×C4⋊C4C22×C418yes—
249C2×C4⋊C4C22×C418yes—
250C2×C4⋊C4C22×C418yes—
251C2×C4⋊C4C2×D418yes—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4C22×C4116yes—
266C2×C22⋊C4C2×D418yes—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—28no—
273C2×C42—24no—
274C2×C42—28no—
275C2×C42—24no—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—416no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—41no—
289C2×C42—41no—
290C2×C42—41no—
291C2×C42—41no—
292C2×C42—44no—
293C2×C42C22×C414yes—
294C2×C42C22×C414yes—
295C2×C42C22×C414yes—
296C2×C42C22×C414yes—
297C2×C42C2×D418yes—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—41no—
316#95—41no—
317#95—41no—
318#95—41no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95C22×C4116yes—
324#95C22×C4116yes—
325#95#4118yes—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24—21no—
333C24—22no—
334C24C23×C412yes—
335C24C22×D412yes—
336C24C2×#4112yes—
337C24C2511yescharacteristic
338C2×Q8—216no—
339C2×Q8—216no—
340C2×Q8—216no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—48no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4C4×D4116yes—
430C22×C4C23×C412yes—
431C22×C4C23×C418yes—
432C22×C4C23×C418yes—
433C22×C4C23×C412yes—
434C22×C4C23×C412yes—
435C22×C4C23×C412yes—
436C22×C4C23×C412yes—
437C22×C4C23×C412yes—
438C22×C4C22×D418yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4112yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C4H(5,𝔽2)12yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C42−1+414yes—
450C22×C42−1+414yes—
451C22×C42−1+412yes—
452C4⋊C4—216no—
453C4⋊C4—232no—
454C4⋊C4—216no—
455C4⋊C4—216no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C4⋊C4—416no—
463C4⋊C4—48no—
464C22⋊C4—232no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C42—28no—
470C42—28no—
471C42—44no—
472C42—44no—
473C42—48no—
474C42—416no—
475C42—48no—
476C42—416no—
477C42—48no—
478C42—48no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23C2×C4×D414yes—
502C23C24×C411yescharacteristic
503C23C23×D411yescharacteristic
504C23C22×#4111yescharacteristic
505C23C22×#4112yes—
506C23C2×H(5,𝔽2)11yescharacteristic
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×2−1+412yes—
510C23C2×2−1+412yes—
511C23C2×2−1+411yescharacteristic
512C23#58512yes—
513Q8—432no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—232no—
533C2×C4—28no—
534C2×C4—28no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—416no—
552C2×C4—416no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4C2×C4×D414yes—
597C2×C4C2×C4×D414yes—
598C2×C4C4×#4114yes—
599C2×C4#51814yes—
600C2×C4#51914yes—
601C2×C4#51914yes—
602C2×C4#52014yes—
603C2×C4#53414yes—
604C2×C4#53714yes—
605C2×C4#53814yes—
606C2×C4#53914yes—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—44no—
611C22—44no—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22C22×C4×D411yescharacteristic
619C22C2×C4×#4111yescharacteristic
620C22C2×#51811yescharacteristic
621C22C2×#51911yescharacteristic
622C22C2×#51911yescharacteristic
623C22C2×#52011yescharacteristic
624C22#342811yescharacteristic
625C22C2×#53411yescharacteristic
626C22C2×#53711yescharacteristic
627C22C2×#53811yescharacteristic
628C22C2×#53911yescharacteristic
629C22D4×#4112yes—
630C22#348012yes—
631C22#349912yes—
632C22#350912yes—
633C22#350912yes—
634C22#351112yes—
635C22#352012yes—
636C22#352012yes—
637C22#354112yes—
638C22#354112yes—
639C22#355412yes—
640C22#355512yes—
641C4—24no—
642C4—24no—
643C4—24no—
644C4—24no—
645C4—48no—
646C4—48no—
647C4—44no—
648C4—44no—
649C4—42no—
650C4—48no—
651C4—82no—
652C4—82no—
653C4—84no—
654C4#232518yes—
655C2—44no—
656C2C2×#232511yescharacteristic
657C2#3414111yescharacteristic
658C2#3420911yescharacteristic
659C2#3422511yescharacteristic
660C2#3424511yescharacteristic
661C2#3437011yescharacteristic
662C2#3437311yescharacteristic
663C2#3995612yes—
664C2#3995912yes—
665C2#4002812yes—
666C2#4003112yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #14280C2semidirect
8. #14083C2semidirect
9. #14029C2semidirect
10. #14024C2semidirect
11. #14021C2semidirect
12. #14017C2semidirect
13. #13790C2semidirect
22. #2857C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#23622
#144232
#144222
#143692
#142802
#141382
#140832
#140292
#140242
#140212
#140172
#137902
C4×#3802
#137562