Finite Group Databasebeta

← #732199

Subgroups — #732199

Order 512.

2566 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#16568C214yesmaximal, complemented
4#16057C214yesmaximal, complemented
5#15296C214yesmaximal, complemented
6#15225C212yesmaximal, complemented
7#15139C212yesmaximal, complemented
8#14822C214yesmaximal, complemented
9#14276C211yesmaximal, characteristic, complemented
10#14150C211yesmaximal, characteristic, complemented
11#13979C212yesmaximal
12#13941C211yesmaximal, characteristic
13#13920C212yesmaximal
14#13854C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19#2571—28no—
20C2×#400C2214yes—
21C2×#400C2214yes—
22D4×C4⋊C4—28no—
23C2×#390C2211yescharacteristic
24C2×#390C2212yes—
25C2×#390C2211yescharacteristic
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#385C2214yes—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#382C2212yes—
35C2×#382C2212yes—
36C2×#380C2212yes—
37C2×#380C2214yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×#376C2211yescharacteristic
42#1445C2214yes—
43#1444C2212yes—
44#1443C2214yes—
45#1442C2212yes—
46#1442C2212yes—
47#1442C2212yes—
48#1442C2212yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2212yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2214yes—
63#1437C2214yes—
64#1436C2214yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C23×D4C2311yescharacteristic
78C24×C4C2314yes—
79C24×C4C2312yes—
80C2×C4×D4—28no—
81C2×C4×D4—28no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4—44no—
85C2×C4×D4Q818yes—
86C22×C4⋊C4—28no—
87C22×C4⋊C4—48no—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115#400—216no—
116#400—48no—
117#390—24no—
118#390—24no—
119#390—28no—
120#389—28no—
121#389—28no—
122#389—28no—
123#389—44no—
124#389—44no—
125#385—216no—
126#384D418yes—
127#382—28no—
128#382—28no—
129#382—24no—
130#380—28no—
131#380—48no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—24no—
135#376—24no—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C25C2412yes—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4C2×Q814yes—
171C22×D4#4114yes—
172C22×D4#4114yes—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—28no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4C2×Q818yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C4×D4—232no—
200C4×D4—432no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—432no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C2×D418yes—
238C2×C4⋊C4C2×D418yes—
239C2×C4⋊C4C2×D418yes—
240C2×C4⋊C4C2×Q814yes—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4C2×Q818yes—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—44no—
273C2×C42C2×D414yes—
274C2×C42C2×Q814yes—
275C2×C42#4114yes—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—24no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—42no—
294#95—42no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—44no—
299#95—42no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305C24—22no—
306C24—22no—
307C24—24no—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24C22×Q814yes—
314C24C2×#4114yes—
315C24C2×#4114yes—
316C24C2511yescharacteristic
317C2×D4—28no—
318C2×D4—28no—
319C2×D4—28no—
320C2×D4—28no—
321C2×D4—28no—
322C2×D4—28no—
323C2×D4#125116yes—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—216no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
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—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—416no—
398C22×C4—416no—
399C22×C4—48no—
400C22×C4—416no—
401C22×C4—48no—
402C22×C4—416no—
403C22×C4—416no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4C22×D418yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×Q814yes—
413C22×C4C22×Q812yes—
414C22×C4C22×Q812yes—
415C22×C4C22×Q812yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4H(5,𝔽2)14yes—
422C22×C4H(5,𝔽2)12yes—
423C22×C42−1+412yes—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—216no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—416no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—88no—
437C4⋊C4—88no—
438C4⋊C4—88no—
439C22⋊C4—216no—
440C22⋊C4—416no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C42—48no—
446C42—48no—
447C42—48no—
448C42—44no—
449C42—44no—
450C42C22⋊Q8116yes—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23C2×#12518yes—
474C23C23×D411yescharacteristic
475C23C23×Q811yescharacteristic
476C23C22×#4111yescharacteristic
477C23C22×#4111yescharacteristic
478C23C22×#4111yescharacteristic
479C23C2×H(5,𝔽2)11yescharacteristic
480C23C2×H(5,𝔽2)12yes—
481C23C2×2−1+411yescharacteristic
482C23#58512yes—
483C23#58512yes—
484C23#58512yes—
485D4—232no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—432no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—832no—
567C2×C4C2×C22⋊Q814yes—
568C2×C4C2×C22⋊Q814yes—
569C2×C4C2×C22⋊Q814yes—
570C2×C4C2×#12514yes—
571C2×C4#54414yes—
572C2×C4#54614yes—
573C2×C4#54814yes—
574C2×C4#55214yes—
575C2×C4#55314yes—
576C22—216no—
577C22—28no—
578C22—48no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—48no—
584C22—48no—
585C22—44no—
586C22—44no—
587C22C22×C22⋊Q811yescharacteristic
588C22C22×#12511yescharacteristic
589C22C2×#54411yescharacteristic
590C22#346012yes—
591C22C2×#54611yescharacteristic
592C22C2×#54811yescharacteristic
593C22C2×#55211yescharacteristic
594C22C2×#55311yescharacteristic
595C22#348411yescharacteristic
596C22#348511yescharacteristic
597C22#348711yescharacteristic
598C22#348711yescharacteristic
599C22#350112yes—
600C22#350412yes—
601C22#351212yes—
602C22#351312yes—
603C22#351412yes—
604C22#352112yes—
605C22#352112yes—
606C22#354712yes—
607C22#355812yes—
608C22#356212yes—
609C22#356512yes—
610C4—24no—
611C4—24no—
612C4—48no—
613C4—44no—
614C4—44no—
615C4—82no—
616C4—82no—
617C4—84no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4#244218yes—
624C2—48no—
625C2C2×#244211yescharacteristic
626C2#3517411yescharacteristic
627C2#3563911yescharacteristic
628C2#3596112yes—
629C2#3827812yes—
630C2#3928412yes—
631C2#4250012yes—
632C2#4493811yescharacteristic
633C2#4514011yescharacteristic
634C2#4514211yescharacteristic
635C2#4526911yescharacteristic
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×#2571C2semidirect
3. #16568C2semidirect
4. #16057C2semidirect
5. #15296C2semidirect
6. #15225C2semidirect
7. #15139C2semidirect
8. #14822C2semidirect
9. #14276C2semidirect
10. #14150C2semidirect
14. #13854C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×D4×C4⋊C42
#165682
#160572
#152962
#152252
#151392
#148222
#142762
#141502
#139792
#139412
#139202
#138542
#137852