Finite Group Databasebeta

← #657591

Subgroups — #657591

Order 512.

4132 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2501C211yesmaximal, characteristic
2C2×#2501C211yesmaximal, characteristic
3C2×#2413C214yesmaximal, complemented
4C2×#2344C211yesmaximal, characteristic, complemented
5C22×#384C211yesmaximal, characteristic, complemented
6#15765C218yesmaximal, complemented
7#15145C212yesmaximal, complemented
8#15134C212yesmaximal, complemented
9#15069C214yesmaximal, complemented
10#14698C211yesmaximal, characteristic
11#14001C212yesmaximal, complemented
12#13910C212yesmaximal, complemented
13#13904C211yesmaximal, characteristic, complemented
14C4×#385C211yesmaximal, characteristic, complemented
15C22×#123C2212yes—
16C22×#123C2214yes—
17C22×#123C2212yes—
18C22×#121C2211yescharacteristic
19C22×#117C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2214yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2501—28no—
25#2501—28no—
26#2413—232no—
27C2×#399C2214yes—
28C2×#399C2214yes—
29C2×#398C2218yes—
30C2×#396C2218yes—
31#2344—28no—
32C2×#390C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2211yescharacteristic
36C2×#389C2212yes—
37C2×#389C2218yes—
38C2×#388C2218yes—
39C2×#387C2214yes—
40C2×#386C2214yes—
41C2×#385C2212yes—
42C2×#384—28no—
43C2×#384C2212yes—
44C2×#384C2211yescharacteristic
45C2×#384C2212yes—
46C2×#384C2211yescharacteristic
47C2×#384C2212yes—
48C2×#384C2214yes—
49C2×#381C2212yes—
50C2×#381C2214yes—
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56#1445C2214yes—
57#1444C2218yes—
58#1444C2218yes—
59#1439C2218yes—
60#1439C2218yes—
61#1437C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C22⋊#95C2212yes—
67C4×#95C2212yes—
68C23×D4C2312yes—
69C23×D4C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2311yescharacteristic
74C2×#123—216no—
75C2×#123—216no—
76C2×#123—216no—
77C2×#123—28no—
78C2×#123—28no—
79C2×#123—48no—
80C2×#123—48no—
81C2×#121—24no—
82C2×#121—24no—
83C2×#121—24no—
84C2×#117—28no—
85C2×#117—216no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—216no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—216no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—48no—
94C22×C4⋊C4D418yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2318yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2318yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C42—24no—
121C22×C42—28no—
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2312yes—
128#399—216no—
129#399—216no—
130#398—232no—
131#396—232no—
132#390—24no—
133#390—24no—
134#390—28no—
135#390—28no—
136#389—232no—
137#389—28no—
138#388—232no—
139#387—216no—
140#386—216no—
141#385D4116yescomplemented
142#384—24no—
143#384—24no—
144#384—28no—
145#384—28no—
146#384—216no—
147#384—28no—
148#381—28no—
149#381—216no—
150#380—24no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—28no—
153C4×C22⋊C4—24no—
154C4×C22⋊C4D418yes—
155C2×#95C2318yes—
156C2×#95C2318yes—
157C2×#95C2318yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C25C2412yes—
167C25C2411yescharacteristic
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—28no—
171C22×D4—24no—
172C22×D4—28no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—24no—
177C22×D4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—28no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4C2×D418yes—
194C23×C4C2×D418yes—
195C23×C4C2×D414yes—
196C23×C4C2×D414yes—
197C23×C4#4114yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210#123—464no—
211#123—432no—
212#123—432no—
213#121—416no—
214#117—432no—
215C2×C4⋊C4—232no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—216no—
230C2×C4⋊C4—216no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—216no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—48no—
254C2×C4⋊C4—464no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4C2×D4116yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4C2×D414yes—
282C2×C22⋊C4C2×D418yes—
283C2×C22⋊C4C2×D414yes—
284C2×C22⋊C4C2×D418yes—
285C2×C42—232no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—216no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42C2×D418yes—
299C2×C42#4118yes—
300#95—24no—
301#95—28no—
302#95—24no—
303#95—28no—
304#95—216no—
305#95—216no—
306#95—216no—
307#95—28no—
308#95—28no—
309#95C2×D418yes—
310#95C2×D4116yes—
311C24—22no—
312C24—22no—
313C24—24no—
314C24—24no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—22no—
319C24—24no—
320C24—28no—
321C24—24no—
322C24C22×D412yes—
323C24C22×D412yes—
324C24C22×D414yes—
325C24C2×#4112yes—
326C24H(5,𝔽2)14yes—
327C24C2511yescharacteristic
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—416no—
332C2×D4—416no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—48no—
337C2×D4—48no—
338C22×C4—216no—
339C22×C4—216no—
340C22×C4—232no—
341C22×C4—216no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—232no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—232no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—24no—
380C22×C4—28no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—416no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4#121116yes—
402C22×C4C22×D414yes—
403C22×C4C22×D414yes—
404C22×C4C22×D414yes—
405C22×C4C22×D414yes—
406C22×C4C22×D414yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D414yes—
410C22×C4C22×D418yes—
411C22×C4C22×D418yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C2×#4112yes—
415C22×C4C2×#4118yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4114yes—
418C22×C42−1+418yes—
419C22×C42−1+414yes—
420C4⋊C4—216no—
421C4⋊C4—28no—
422C4⋊C4—28no—
423C4⋊C4—232no—
424C4⋊C4—432no—
425C4⋊C4—432no—
426C4⋊C4—416no—
427C4⋊C4—416no—
428C4⋊C4—416no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—44no—
432C4⋊C4—416no—
433C4⋊C4—416no—
434C4⋊C4—48no—
435C22⋊C4—28no—
436C22⋊C4—216no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—416no—
443C22⋊C4—432no—
444C22⋊C4#127116yes—
445C42—216no—
446C42—216no—
447C42—216no—
448C42—44no—
449C42—416no—
450C42—44no—
451C23—22no—
452C23—22no—
453C23—28no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—28no—
458C23—24no—
459C23—28no—
460C23—24no—
461C23—28no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—48no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—48no—
474C23—44no—
475C23—48no—
476C23—44no—
477C23—48no—
478C23C2×#12114yes—
479C23C2×#12714yes—
480C23#53514yes—
481C23D4218yes—
482C23#55018yes—
483C23C23×D411yescharacteristic
484C23C23×D411yescharacteristic
485C23C23×D412yes—
486C23C22×#4111yescharacteristic
487C23C22×#4112yes—
488C23C2×H(5,𝔽2)12yes—
489C23C2×2−1+412yes—
490C23C2×2−1+414yes—
491D4—816no—
492D4—88no—
493C2×C4—264no—
494C2×C4—22no—
495C2×C4—28no—
496C2×C4—22no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—232no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—432no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—48no—
543C2×C4—44no—
544C2×C4—48no—
545C2×C4—44no—
546C2×C4—48no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—816no—
551C2×C4C2×#12114yes—
552C2×C4C2×#12114yes—
553C2×C4#52514yes—
554C2×C4C2×#12718yes—
555C2×C4#53614yes—
556C2×C4#547116yes—
557C2×C4#54718yes—
558C2×C4#55218yes—
559C22—28no—
560C22—24no—
561C22—28no—
562C22—28no—
563C22—24no—
564C22—216no—
565C22—28no—
566C22—84no—
567C22—82no—
568C22—84no—
569C22—82no—
570C22—84no—
571C22—82no—
572C22—88no—
573C22—84no—
574C22#254118yes—
575C22C22×#12111yescharacteristic
576C22C2×#52511yescharacteristic
577C22C22×#12711yescharacteristic
578C22C2×#53511yescharacteristic
579C22C2×#53611yescharacteristic
580C22C2×D4212yes—
581C22C2×#54714yes—
582C22C2×#54712yes—
583C22C2×#55012yes—
584C22C2×#55212yes—
585C22#348612yes—
586C22#349214yes—
587C22#349912yes—
588C22#349914yes—
589C22#350912yes—
590C22#354414yes—
591C4—216no—
592C4—28no—
593C4—24no—
594C4—44no—
595C4—44no—
596C4—44no—
597C4—44no—
598C4—44no—
599C4—84no—
600C4—84no—
601C4—82no—
602C4#241318yescomplemented
603C2—28no—
604C2—162no—
605C2C2×#241311yescharacteristic
606C2C2×#254111yescharacteristic
607C2#3566711yescharacteristic
608C2#3610312yes—
609C2#3617912yes—
610C2#3786312yes—
611C2#3933712yes—
612C2#5931214yes—
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
3. C2×#2413C2semidirect
4. C2×#2344C2semidirect
5. C22×#384C2semidirect
6. #15765C2semidirect
7. #15145C2semidirect
8. #15134C2semidirect
9. #15069C2semidirect
11. #14001C2semidirect
12. #13910C2semidirect
13. #13904C2semidirect
14. C4×#385C2semidirect
141. #385D4semidirect
602. C4#2413semidirect

Maximal subgroups

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

subgroupindex
C2×#25012
C2×#24132
C2×#23442
C22×#3842
#157652
#151452
#151342
#150692
#146982
#140012
#139102
#139042
C4×#3852