Finite Group Databasebeta

← #641194

Subgroups — #641194

Order 512.

3750 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C2×#2417C214yesmaximal, complemented
3C2×C4×#121C211yesmaximal, characteristic
4#17075C212yesmaximal, complemented
5#17071C212yesmaximal, complemented
6#16984C214yesmaximal, complemented
7#15193C212yesmaximal, complemented
8#15178C212yesmaximal, complemented
9#15177C212yesmaximal, complemented
10#15135C212yesmaximal, complemented
11#14836C214yesmaximal, complemented
12#14353C212yesmaximal, complemented
13#14271C212yesmaximal
14C2×C4⋊#95C211yesmaximal, characteristic, complemented
15C22×C4⋊Q8C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×C22⋊Q8C2214yes—
18C22×#121C2211yescharacteristic
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C42C2211yescharacteristic
23#2857—28no—
24#2417—232no—
25C2×#399C2214yes—
26C2×#398C2218yes—
27C2×#396C2218yes—
28C4×#121—28no—
29C2×#391C2212yes—
30C2×#391C2214yes—
31C2×#390C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#389C2211yescharacteristic
36C2×#388C2212yes—
37C2×#386C2218yes—
38C2×#386C2212yes—
39C2×#384C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2218yes—
44C2×#380C2214yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C22×#95C2212yes—
48C22×#95C2212yes—
49#1445C2214yes—
50#1445C2218yes—
51#1445C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1444C2214yes—
55#1444C2214yes—
56#1444C2214yes—
57#1439C2214yes—
58#1439C2218yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C23×Q8C2312yes—
67C23×D4C2312yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2314yes—
73C24×C4C2311yescharacteristic
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—44no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—216no—
78C2×C22⋊Q8—216no—
79C2×C22⋊Q8—48no—
80C2×C22⋊Q8—48no—
81C2×#121—24no—
82C2×#121—24no—
83C2×#121Q818yescomplemented
84C2×C4×D4—216no—
85C2×C4×D4—28no—
86C2×C4×D4—24no—
87C2×C4×D4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4D4116yescomplemented
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2318yes—
98C22×C4⋊C4C2318yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2318yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2312yes—
113C22×C42—44no—
114C22×C42D418yes—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#399—48no—
121#398—232no—
122#396—232no—
123#391—216no—
124#391—28no—
125#390—24no—
126#389—216no—
127#389—28no—
128#389—28no—
129#389—24no—
130#388—28no—
131#386—232no—
132#386—28no—
133#384—24no—
134#380—28no—
135#380—28no—
136#380—232no—
137#380—24no—
138#380—216no—
139C4×C4⋊C4—24no—
140C4×C22⋊C4—28no—
141C2×#95—28no—
142C2×#95—44no—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C2×#95C2318yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C25C2412yes—
157C25C2411yescharacteristic
158C22×Q8—28no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—28no—
162C22×Q8—24no—
163C22×D4—24no—
164C22×D4—28no—
165C22×D4—24no—
166C22×D4—28no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4C2×Q818yes—
172C22×D4C2×Q814yes—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4C2×D414yes—
194C23×C4C2×D418yes—
195C23×C4C2×D414yes—
196C23×C4C2×D4116yes—
197C23×C4C2×D414yes—
198C23×C4C2×Q814yes—
199C23×C4#4114yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2418yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C4⋊Q8—416no—
213C22⋊Q8—464no—
214C22⋊Q8—432no—
215#121—232no—
216C4×D4—416no—
217C4×D4—432no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—432no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4C2×D4116yes—
251C2×C4⋊C4C2×D4116yes—
252C2×C4⋊C4C2×Q814yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—216no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—216no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4C2×Q818yes—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—28no—
277C2×C42—416no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42C2×D414yes—
281C2×C42C2×D414yes—
282C2×C42C2×D414yes—
283C2×C42C2×D414yes—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—216no—
288#95—28no—
289#95—28no—
290#95—216no—
291#95—216no—
292#95—216no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—44no—
297C24—22no—
298C24—21no—
299C24—21no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—24no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24C22×D412yes—
314C24C22×D414yes—
315C24C22×Q814yes—
316C24C22×Q812yes—
317C24C2×#4112yes—
318C242−1+414yes—
319C24C2511yescharacteristic
320C2×Q8—48no—
321C2×Q8—416no—
322C2×Q8—48no—
323C2×Q8—416no—
324C2×Q8—48no—
325C2×D4—216no—
326C2×D4—232no—
327C2×D4—28no—
328C2×D4—48no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—416no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—28no—
345C22×C4—216no—
346C22×C4—28no—
347C22×C4—216no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—22no—
353C22×C4—28no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—28no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—232no—
364C22×C4—28no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—28no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—416no—
391C22×C4—416no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—416no—
397C22×C4—48no—
398C22×C4—416no—
399C22×C4—88no—
400C22×C4—88no—
401C22×C4—88no—
402C22×C4—88no—
403C22×C4C22⋊Q8116yescomplemented
404C22×C4#127116yes—
405C22×C4C22×D412yes—
406C22×C4C22×D4116yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D414yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D414yes—
414C22×C4C22×Q814yes—
415C22×C4C22×Q812yes—
416C22×C4C22×Q812yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C4H(5,𝔽2)14yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C42−1+412yes—
423C22×C42−1+412yes—
424C4⋊C4—28no—
425C4⋊C4—232no—
426C4⋊C4—416no—
427C4⋊C4—48no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—432no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—416no—
436C4⋊C4—48no—
437C4⋊C4—816no—
438C22⋊C4—216no—
439C22⋊C4—48no—
440C22⋊C4—432no—
441C22⋊C4—48no—
442C22⋊C4—416no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—816no—
446C42—28no—
447C42—28no—
448C42—44no—
449C42—44no—
450C42—48no—
451C23—24no—
452C23—22no—
453C23—24no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—28no—
472C23—24no—
473C23—24no—
474C23—42no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23C2×C22⋊Q814yes—
493C23C2×#12714yes—
494C23#53614yes—
495C23#54718yes—
496C23D4×Q818yes—
497C23C23×D411yescharacteristic
498C23C23×D412yes—
499C23C23×Q811yescharacteristic
500C23C22×#4111yescharacteristic
501C23C2×H(5,𝔽2)12yes—
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×H(5,𝔽2)11yescharacteristic
504C23C2×2−1+412yes—
505C23C2×2−1+411yescharacteristic
506C23C2×2−1+411yescharacteristic
507C23#58512yes—
508Q8—816no—
509D4—416no—
510D4—432no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—232no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—216no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—216no—
534C2×C4—28no—
535C2×C4—216no—
536C2×C4—416no—
537C2×C4—416no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—48no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—48no—
567C2×C4—832no—
568C2×C4—816no—
569C2×C4C2×C22⋊Q814yes—
570C2×C4C2×C22⋊Q814yes—
571C2×C4C2×#12714yes—
572C2×C4C2×#12714yes—
573C2×C4#53414yes—
574C2×C4#53418yes—
575C2×C4#53714yes—
576C2×C4#53718yes—
577C2×C4#53714yes—
578C22—24no—
579C22—28no—
580C22—28no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—28no—
585C22—28no—
586C22—24no—
587C22—24no—
588C22—44no—
589C22—48no—
590C22—48no—
591C22—44no—
592C22—44no—
593C22—82no—
594C22—84no—
595C22—84no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22#254218yes—
600C22C22×C22⋊Q811yescharacteristic
601C22C22×#12711yescharacteristic
602C22C2×#53412yes—
603C22C2×#53411yescharacteristic
604C22C2×#53611yescharacteristic
605C22C2×#53712yes—
606C22C2×#53711yescharacteristic
607C22C2×#53711yescharacteristic
608C22#344911yescharacteristic
609C22C2×#54712yes—
610C22C2×D4×Q812yes—
611C22#349612yes—
612C22#349712yes—
613C22#350112yes—
614C22#350311yescharacteristic
615C22#350311yescharacteristic
616C22#350512yes—
617C22#350811yescharacteristic
618C22#350911yescharacteristic
619C22#353612yes—
620C22#353812yes—
621C22#356112yes—
622C22#356312yes—
623C4—24no—
624C4—24no—
625C4—24no—
626C4—24no—
627C4—416no—
628C4—42no—
629C4—44no—
630C4—82no—
631C4—84no—
632C4—82no—
633C4—84no—
634C4#265618yes—
635C2—28no—
636C2—84no—
637C2C2×#265611yescharacteristic
638C2C2×#254211yescharacteristic
639C2#3501011yescharacteristic
640C2#3516611yescharacteristic
641C2#3528011yescharacteristic
642C2#3744211yescharacteristic
643C2#3755511yescharacteristic
644C2#3948612yes—
645C2#3948712yes—
646C2#3961412yes—
647C2#3961512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2857C2semidirect
2. C2×#2417C2semidirect
4. #17075C2semidirect
5. #17071C2semidirect
6. #16984C2semidirect
7. #15193C2semidirect
8. #15178C2semidirect
9. #15177C2semidirect
10. #15135C2semidirect
11. #14836C2semidirect
12. #14353C2semidirect
14. C2×C4⋊#95C2semidirect
83. C2×#121Q8semidirect
90. C22×C4⋊C4D4semidirect
403. C22×C4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#24172
C2×C4×#1212
#170752
#170712
#169842
#151932
#151782
#151772
#151352
#148362
#143532
#142712
C2×C4⋊#952