Finite Group Databasebeta

← #723123

Subgroups — #723123

Order 512.

4378 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#554C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2580C211yesmaximal, characteristic, complemented
4C2×#2328C211yesmaximal, characteristic
5#16600C214yesmaximal, complemented
6#15951C218yesmaximal, complemented
7#15229C212yesmaximal, complemented
8#15228C212yesmaximal, complemented
9#14777C214yesmaximal, complemented
10#14282C211yesmaximal, characteristic, complemented
11#14025C212yesmaximal
12#13977C212yesmaximal, complemented
13#13941C211yesmaximal, characteristic, complemented
14#13850C211yesmaximal, characteristic, complemented
15C2×#554—28no—
16C2×#554—28no—
17C2×#554—28no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#127C2211yescharacteristic
20C22×#125C2212yes—
21C22×C22⋊Q8C2214yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2214yes—
24C22×C4×Q8C2211yescharacteristic
25C22×C4×D4C2211yescharacteristic
26C22×C4×D4C2212yes—
27C22×C4×D4C2211yescharacteristic
28C23×C4⋊C4C2212yes—
29#2580—28no—
30#2542—28no—
31C2×#399C2212yes—
32C2×#399C2214yes—
33C2×#393C2212yes—
34#2328—28no—
35C2×#390C2211yescharacteristic
36C2×#390C2211yescharacteristic
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#390C2212yes—
40C2×#389C2212yes—
41C2×#386C2214yes—
42C2×#386C2214yes—
43C2×#386C2218yes—
44C2×#385C2214yes—
45C2×#385C2218yes—
46C2×#384C2212yes—
47C2×#382C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×#376C2211yescharacteristic
51#1446C2212yes—
52#1446C2214yes—
53#1445C2212yes—
54#1445C2214yes—
55#1444C2218yes—
56#1441C2214yes—
57#1441C2218yes—
58#1439C2214yes—
59#1439C2218yes—
60#1439C2218yes—
61#1439C2218yes—
62#1438C2218yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C22⋊#95C2214yes—
67C4×#95C2212yes—
68C4×#95C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C23×D4C2311yescharacteristic
73C23×D4C2314yes—
74C24×C4C2314yes—
75C24×C4C2312yes—
76C24×C4C2312yes—
77#554—2128no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8D418yescomplemented
81C2×#127—24no—
82C2×#127—24no—
83C2×#127Q818yescomplemented
84C2×#125—28no—
85C2×#125—28no—
86C2×#125—28no—
87C2×C22⋊Q8—216no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—216no—
91C2×C22⋊Q8—216no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—216no—
94C2×C22⋊Q8—416no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8D418yescomplemented
98C2×C4×D4—28no—
99C2×C4×D4—28no—
100C2×C4×D4—28no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4D418yes—
107C22×C4⋊C4—28no—
108C22×C4⋊C4—28no—
109C22×C4⋊C4—28no—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2318yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2318yes—
125C22×C22⋊C4C2318yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#399—28no—
137#399—216no—
138#393—28no—
139#390—24no—
140#390—28no—
141#390—24no—
142#390—28no—
143#390—28no—
144#389—28no—
145#386—216no—
146#386—216no—
147#386—416no—
148#385—216no—
149#385—416no—
150#384—28no—
151#382—24no—
152#382—24no—
153C4×C4⋊C4—24no—
154#376—24no—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2318yes—
167C2×#95C2318yes—
168C2×#95C2318yes—
169C25C2412yes—
170C25C2412yes—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—44no—
176C22×Q8—48no—
177C22×Q8C2×D414yes—
178C22×Q8C2×D418yes—
179C22×D4—28no—
180C22×D4—28no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—48no—
192C22×D4—48no—
193C22×D4C2×D414yes—
194C22×D4C2×Q814yes—
195C22×D4C2×Q814yes—
196C22×D4C2×Q8116yes—
197C22×D4#4114yes—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—28no—
202C23×C4—28no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—48no—
207C23×C4—48no—
208C23×C4—48no—
209C23×C4—48no—
210C23×C4C2×D418yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C4⋊Q8—232no—
224#127—232no—
225#125—264no—
226C22⋊Q8—264no—
227C22⋊Q8—2128no—
228C22⋊Q8—464no—
229C4×Q8—232no—
230C4×D4—264no—
231C4×D4—232no—
232C4×D4—416no—
233C2×C4⋊C4—264no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—216no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—216no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D418yes—
265C2×C4⋊C4C2×D418yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D418yes—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—216no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4C2×D418yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42C2×D418yes—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×Q814yes—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—44no—
316#95—48no—
317#95—44no—
318#95—48no—
319#95—48no—
320C24—22no—
321C24—22no—
322C24—28no—
323C24—22no—
324C24—22no—
325C24—28no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—28no—
331C24—22no—
332C24—22no—
333C24—28no—
334C24C22×D414yes—
335C24C22×Q814yes—
336C24C22×Q814yes—
337C24C2×#4114yes—
338C24C2511yescharacteristic
339C2×Q8—28no—
340C2×Q8—216no—
341C2×Q8—48no—
342C2×Q8—416no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—88no—
346C2×Q8—88no—
347C2×D4—232no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—28no—
354C2×D4—28no—
355C2×D4—28no—
356C2×D4—28no—
357C2×D4—432no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—88no—
363C2×D4—88no—
364C2×D4—816no—
365C2×D4C22⋊Q8116yescomplemented
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—232no—
394C22×C4—216no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—416no—
413C22×C4—48no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4—816no—
417C22×C4—88no—
418C22×C4—88no—
419C22×C4—816no—
420C22×C4C22×D414yes—
421C22×C4C22×D414yes—
422C22×C4C22×D414yes—
423C22×C4C22×D418yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×Q814yes—
434C22×C4C22×Q812yes—
435C22×C4C2×#4112yes—
436C22×C4H(5,𝔽2)18yes—
437C22×C42−1+412yes—
438C22×C42−1+412yes—
439C4⋊C4—232no—
440C4⋊C4—232no—
441C4⋊C4—216no—
442C4⋊C4—28no—
443C4⋊C4—216no—
444C4⋊C4—28no—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—432no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—88no—
451C22⋊C4—232no—
452C22⋊C4—216no—
453C22⋊C4—432no—
454C22⋊C4—432no—
455C22⋊C4—416no—
456C22⋊C4—48no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C42—216no—
461C42—28no—
462C42—44no—
463C42—48no—
464C42#120116yes—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—28no—
472C23—28no—
473C23—216no—
474C23—216no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—216no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—48no—
484C23—48no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—48no—
496C23—48no—
497C23—48no—
498C23C2×C22⋊Q818yes—
499C23C23×D411yescharacteristic
500C23C23×D411yescharacteristic
501C23C23×D411yescharacteristic
502C23C23×Q811yescharacteristic
503C23C22×#4111yescharacteristic
504C23C2×H(5,𝔽2)14yes—
505C23C2×2−1+411yescharacteristic
506C23C2×2−1+411yescharacteristic
507C23#58514yes—
508Q8—416no—
509Q8—816no—
510D4—232no—
511D4—416no—
512D4—832no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—28no—
527C2×C4—216no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—432no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—44no—
563C2×C4—48no—
564C2×C4—44no—
565C2×C4—816no—
566C2×C4—832no—
567C2×C4C2×#12014yes—
568C2×C4C2×#12014yes—
569C2×C4C2×#12014yes—
570C2×C4C2×C22⋊Q814yes—
571C2×C4#53614yes—
572C2×C4#54814yes—
573C2×C4#54814yes—
574C2×C4D4×Q814yes—
575C2×C4D4×Q814yes—
576C22—28no—
577C22—216no—
578C22—48no—
579C22—44no—
580C22—48no—
581C22—48no—
582C22—44no—
583C22—48no—
584C22—48no—
585C22—48no—
586C22—44no—
587C22—48no—
588C22—48no—
589C22—82no—
590C22—84no—
591C22—82no—
592C22—84no—
593C22—84no—
594C22C22×#12011yescharacteristic
595C22C22×C22⋊Q811yescharacteristic
596C22C2×#53611yescharacteristic
597C22#344912yes—
598C22C2×#54811yescharacteristic
599C22C2×#54811yescharacteristic
600C22C2×D4×Q811yescharacteristic
601C22C2×D4×Q811yescharacteristic
602C22#348412yes—
603C22#348512yes—
604C22#348712yes—
605C22#349812yes—
606C22#349812yes—
607C22#350114yes—
608C22#351314yes—
609C22#355414yes—
610C22#355914yes—
611C4—24no—
612C4—24no—
613C4—44no—
614C4—44no—
615C4—44no—
616C4—48no—
617C4—48no—
618C4—44no—
619C4—82no—
620C4—82no—
621C4—84no—
622C4#240918yes—
623C2—48no—
624C2—84no—
625C2C2×#240911yescharacteristic
626C2#3514911yescharacteristic
627C2#3552711yescharacteristic
628C2#3583612yes—
629C2#3916112yes—
630C2#3961112yes—
631C2#4265112yes—
632C2#4527112yes—
633C2#4535712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#554C2semidirect
2. C2×#2542C2semidirect
3. C2×#2580C2semidirect
5. #16600C2semidirect
6. #15951C2semidirect
7. #15229C2semidirect
8. #15228C2semidirect
9. #14777C2semidirect
10. #14282C2semidirect
12. #13977C2semidirect
13. #13941C2semidirect
14. #13850C2semidirect
80. C2×C4⋊Q8D4semidirect
83. C2×#127Q8semidirect
97. C2×C4×Q8D4semidirect
365. C2×D4C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5542
C2×#25422
C2×#25802
C2×#23282
#166002
#159512
#152292
#152282
#147772
#142822
#140252
#139772
#139412
#138502