Finite Group Databasebeta

← #681067

Subgroups — #681067

Order 512.

3334 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3#15391C218yesmaximal, complemented
4#15233C212yesmaximal, complemented
5#15219C212yesmaximal, complemented
6#15183C214yesmaximal, complemented
7#14687C211yesmaximal, characteristic, complemented
8#14137C212yesmaximal, complemented
9#13970C214yesmaximal, complemented
10#13924C212yesmaximal, complemented
11#13911C211yesmaximal, characteristic, complemented
12#13906C211yesmaximal, characteristic, complemented
13C4×#390C211yesmaximal, characteristic, complemented
14C4×#390C211yesmaximal, characteristic, complemented
15C22×#121C2212yes—
16C22×#121C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×C4×D4C2212yes—
20C23×C4⋊C4C2212yes—
21#2537—28no—
22C2×#394C2214yes—
23D4×C4⋊C4—28no—
24C2×#390C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#386C2218yes—
30C2×#385C2214yes—
31C2×#384C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#382C2214yes—
34C2×#382C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2214yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2214yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C43C2211yescharacteristic
45#1445C2218yes—
46#1443C2218yes—
47#1442C2214yes—
48#1440C2218yes—
49#1440C2218yes—
50#1439C2214yes—
51#1439C2218yes—
52#1438C2218yes—
53#1437C2214yes—
54#1437C2214yes—
55C4⋊#95C2212yes—
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2214yes—
59C4⋊#95C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64C22⋊#95C2218yes—
65C23×D4C2311yescharacteristic
66C23×D4C2314yes—
67C23×D4C2311yescharacteristic
68C24×C4C2312yes—
69C24×C4C2312yes—
70C24×C4C2314yes—
71C2×#121—28no—
72C2×#121—28no—
73C2×#121—216no—
74C2×#121—28no—
75C2×#121—44no—
76C2×C4×D4—28no—
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4—48no—
83C2×C4×D4D418yes—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—28no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2318yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116#394—216no—
117#390—24no—
118#390—28no—
119#390—24no—
120#390—44no—
121#389—28no—
122#386—416no—
123#385—216no—
124#384—24no—
125#384—24no—
126#382—48no—
127#382—44no—
128#380—28no—
129#380—28no—
130#380—28no—
131#380—48no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C22⋊C4—48no—
135C4×C22⋊C4—44no—
136C4×C22⋊C4—44no—
137C2×#95C2314yes—
138C2×#95C2318yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2318yes—
150C43—24no—
151C25C2412yes—
152C25C2412yes—
153C22×D4—28no—
154C22×D4—28no—
155C22×D4—216no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—24no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—44no—
166C22×D4—48no—
167C22×D4C2×D414yes—
168C22×D4#4114yes—
169C22×D4#4118yes—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—28no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—28no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—44no—
184C23×C4—48no—
185C23×C4—48no—
186C23×C4C2×D418yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199#121—432no—
200#121—432no—
201C4×D4—232no—
202C4×D4—432no—
203C4×D4—416no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—432no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4#4114yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—216no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—48no—
262C2×C22⋊C4—48no—
263C2×C22⋊C4—48no—
264C2×C22⋊C4C2×D418yes—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42C2×D414yes—
283C2×C42#4114yes—
284C2×C42#4114yes—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—216no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—48no—
295#95—48no—
296#95—48no—
297#95—48no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—24no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—24no—
317C24C22×D414yes—
318C24C2×#4114yes—
319C24C2×#4118yes—
320C24C2511yescharacteristic
321C2×D4—28no—
322C2×D4—28no—
323C2×D4—28no—
324C2×D4—28no—
325C2×D4—216no—
326C2×D4—216no—
327C2×D4—28no—
328C2×D4—416no—
329C2×D4—432no—
330C2×D4—416no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—88no—
336C2×D4—816no—
337C2×D4#124116yes—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—216no—
363C22×C4—216no—
364C22×C4—216no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—416no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—416no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—416no—
402C22×C4—416no—
403C22×C4—816no—
404C22×C4—88no—
405C22×C4—816no—
406C22×C4C22×D414yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C2×#4114yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4114yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4H(5,𝔽2)14yes—
421C22×C4H(5,𝔽2)14yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C42−1+414yes—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—416no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C22⋊C4—216no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C22⋊C4—416no—
445C22⋊C4—48no—
446C22⋊C4—816no—
447C42—216no—
448C42—48no—
449C42—48no—
450C42—44no—
451C42—48no—
452C42—44no—
453C42#123116yes—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—24no—
463C23—24no—
464C23—28no—
465C23—28no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—28no—
470C23—28no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23C2×#12418yes—
500C23C23×D411yescharacteristic
501C23C22×#4111yescharacteristic
502C23C22×#4111yescharacteristic
503C23C22×#4112yes—
504C23C22×#4111yescharacteristic
505C23C22×#4111yescharacteristic
506C23C2×H(5,𝔽2)12yes—
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×H(5,𝔽2)11yescharacteristic
509C23C2×H(5,𝔽2)12yes—
510C23C2×2−1+412yes—
511D4—232no—
512D4—416no—
513D4—832no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—432no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—48no—
565C2×C4—48no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—832no—
573C2×C4—816no—
574C2×C4C2×#12314yes—
575C2×C4C2×#12314yes—
576C2×C4C2×#12314yes—
577C2×C4C2×#12414yes—
578C2×C4#52914yes—
579C2×C4#53514yes—
580C2×C4#54014yes—
581C2×C4#54614yes—
582C2×C4#54614yes—
583C2×C4#55314yes—
584C2×C4#55318yes—
585C2×C4#55318yes—
586C2×C4#55514yes—
587C22—28no—
588C22—28no—
589C22—28no—
590C22—48no—
591C22—44no—
592C22—44no—
593C22—48no—
594C22—48no—
595C22—48no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—48no—
600C22—48no—
601C22—44no—
602C22—44no—
603C22—84no—
604C22—82no—
605C22—84no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22C22×#12311yescharacteristic
610C22C22×#12411yescharacteristic
611C22C2×#52911yescharacteristic
612C22C2×#53511yescharacteristic
613C22C2×#54011yescharacteristic
614C22C2×#54611yescharacteristic
615C22C2×#54611yescharacteristic
616C22C2×#55312yes—
617C22C2×#55312yes—
618C22C2×#55311yescharacteristic
619C22C2×#55511yescharacteristic
620C22#348112yes—
621C22#348212yes—
622C22#349212yes—
623C22#349811yescharacteristic
624C22#349911yescharacteristic
625C22#351112yes—
626C22#351912yes—
627C22#351912yes—
628C22#352012yes—
629C22#352012yes—
630C22#354812yes—
631C22#354912yes—
632C4—24no—
633C4—44no—
634C4—44no—
635C4—44no—
636C4—44no—
637C4—84no—
638C4—82no—
639C4—84no—
640C4—84no—
641C4—82no—
642C4#242018yes—
643C4#250418yes—
644C2—48no—
645C2—84no—
646C2C2×#242011yescharacteristic
647C2C2×#250411yescharacteristic
648C2#3553011yescharacteristic
649C2#3628911yescharacteristic
650C2#3630311yescharacteristic
651C2#3645312yes—
652C2#3895111yescharacteristic
653C2#3895311yescharacteristic
654C2#3919112yes—
655C2#5921712yes—
656C2#5922012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×D4×C4⋊C4C2semidirect
3. #15391C2semidirect
4. #15233C2semidirect
5. #15219C2semidirect
6. #15183C2semidirect
7. #14687C2semidirect
8. #14137C2semidirect
9. #13970C2semidirect
10. #13924C2semidirect
11. #13911C2semidirect
12. #13906C2semidirect
13. C4×#390C2semidirect
14. C4×#390C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×D4×C4⋊C42
#153912
#152332
#152192
#151832
#146872
#141372
#139702
#139242
#139112
#139062
C4×#3902