Finite Group Databasebeta

← #949110

Subgroups — #949110

Order 512.

2366 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×#2441C212yesmaximal, complemented
2#17136C214yesmaximal
3#16612C214yesmaximal, complemented
4#16550C214yesmaximal, complemented
5#16103C214yesmaximal, complemented
6#15135C212yesmaximal, complemented
7#15133C212yesmaximal, complemented
8#15131C212yesmaximal, complemented
9#14283C211yesmaximal, characteristic, complemented
10#14276C211yesmaximal, characteristic, complemented
11#13879C212yesmaximal, complemented
12#13871C212yesmaximal
13#13847C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×#117C2212yes—
17#2441—216no—
18C2×#399C2214yes—
19C2×#399C2214yes—
20C2×#398C2214yes—
21C2×#396C2214yes—
22C2×#396C2214yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2214yes—
26C2×#389C2212yes—
27C2×#387C2214yes—
28C2×#384C2212yes—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×#376C2211yescharacteristic
39#1445C2212yes—
40#1445C2214yes—
41#1444C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1439C2212yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C2×#125—42no—
71C2×#125—42no—
72C2×#125D418yescomplemented
73C2×C4×D4—42no—
74C2×C4×D4—42no—
75C2×C4×D4Q818yes—
76C2×#117—28no—
77C2×#117—44no—
78C2×#117—44no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2314yes—
108#399—216no—
109#399—48no—
110#398—48no—
111#396—216no—
112#396—48no—
113#389—216no—
114#389—28no—
115#389—28no—
116#389—44no—
117#387—216no—
118#384—28no—
119#384—24no—
120#384—42no—
121#382—42no—
122#380—28no—
123#380—44no—
124#380—44no—
125C4×C4⋊C4—42no—
126C4×C22⋊C4—28no—
127C4×C22⋊C4—28no—
128#376—42no—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C25C2412yes—
152C22×D4—41no—
153C22×D4—41no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4C2×Q814yes—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4C2×Q818yes—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2414yes—
169C23×C4C2411yescharacteristic
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176#125—416no—
177C4×D4—416no—
178#117—432no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4C2×D418yes—
220C2×C4⋊C4C2×D4116yes—
221C2×C4⋊C4C2×Q814yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4C2×Q818yes—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—28no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42C2×D414yes—
262C2×C42C2×Q814yes—
263C2×C42#4114yes—
264#95—24no—
265#95—24no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—24no—
270#95—24no—
271#95—28no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24C22×Q814yes—
301C24C2511yescharacteristic
302C2×D4—44no—
303C2×D4—48no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—48no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—216no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4C22×D414yes—
380C22×C4C22×D418yes—
381C22×C4C22×D412yes—
382C22×C4C22×Q812yes—
383C22×C4C22×Q812yes—
384C22×C4C22×Q814yes—
385C22×C4C22×Q812yes—
386C22×C4C2×#4114yes—
387C22×C4C2×#4112yes—
388C22×C4H(5,𝔽2)14yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C4H(5,𝔽2)12yes—
392C22×C42−1+414yes—
393C22×C42−1+412yes—
394C4⋊C4—216no—
395C4⋊C4—416no—
396C4⋊C4—44no—
397C4⋊C4—416no—
398C4⋊C4—416no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—88no—
405C4⋊C4—88no—
406C4⋊C4—88no—
407C4⋊C4—88no—
408C4⋊C4—88no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—48no—
414C22⋊C4—416no—
415C22⋊C4—88no—
416C22⋊C4—88no—
417C22⋊C4—88no—
418C42—28no—
419C42—28no—
420C42—28no—
421C42—416no—
422C42—48no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23C23×D411yescharacteristic
459C23C23×Q811yescharacteristic
460C23C22×#4111yescharacteristic
461C23C2×H(5,𝔽2)11yescharacteristic
462C23C2×H(5,𝔽2)11yescharacteristic
463C23C2×H(5,𝔽2)11yescharacteristic
464C23C2×H(5,𝔽2)12yes—
465C23C2×2−1+411yescharacteristic
466C23C2×2−1+412yes—
467C23#58512yes—
468C23#58512yes—
469D4—416no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—432no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4#53414yes—
545C2×C4#53914yes—
546C2×C4#54314yes—
547C2×C4D4×Q814yes—
548C2×C4#55014yes—
549C2×C4#55414yes—
550C2×C4#56214yes—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22—82no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22C2×#53411yescharacteristic
567C22C2×#53911yescharacteristic
568C22C2×#54311yescharacteristic
569C22#345812yes—
570C22C2×D4×Q811yescharacteristic
571C22C2×#55011yescharacteristic
572C22C2×#55411yescharacteristic
573C22#348812yes—
574C22#348912yes—
575C22#349412yes—
576C22#349512yes—
577C22#350312yes—
578C22C2×#56211yescharacteristic
579C22#353711yescharacteristic
580C22#353811yescharacteristic
581C22#353911yescharacteristic
582C22#353911yescharacteristic
583C22#353912yes—
584C22#354612yes—
585C22#354712yes—
586C22#354812yes—
587C22#355412yes—
588C22#355512yes—
589C4—42no—
590C4—44no—
591C4—44no—
592C4—44no—
593C4—48no—
594C4—44no—
595C4—42no—
596C4—44no—
597C4—84no—
598C4—84no—
599C4—84no—
600C4#273218yes—
601C2—84no—
602C2#3730711yescharacteristic
603C2#3748111yescharacteristic
604C2#3769012yes—
605C2#3787812yes—
606C2#4292712yes—
607C2#4293512yes—
608C2#4542311yescharacteristic
609C2#4543211yescharacteristic
610C2C2×#273211yescharacteristic
611C2#5307611yescharacteristic
612C2#5307711yescharacteristic
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×#2441C2semidirect
3. #16612C2semidirect
4. #16550C2semidirect
5. #16103C2semidirect
6. #15135C2semidirect
7. #15133C2semidirect
8. #15131C2semidirect
9. #14283C2semidirect
10. #14276C2semidirect
11. #13879C2semidirect
72. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24412
#171362
#166122
#165502
#161032
#151352
#151332
#151312
#142832
#142762
#138792
#138712
#138472