Finite Group Databasebeta

← #949912

Subgroups — #949912

Order 512.

2038 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17273C212yesmaximal
2#17272C212yesmaximal
3#16951C214yesmaximal, complemented
4#16915C214yesmaximal, complemented
5#16245C212yesmaximal, complemented
6#16214C212yesmaximal, complemented
7#15232C212yesmaximal, complemented
8#15216C212yesmaximal, complemented
9#15216C212yesmaximal, complemented
10#14264C212yesmaximal, complemented
11#14129C212yesmaximal, complemented
12#14054C212yesmaximal, complemented
13#13905C211yesmaximal, characteristic
14#13871C211yesmaximal, characteristic
15#13839C211yesmaximal, characteristic
16C22×#125C2212yes—
17C22×C4×D4C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2212yes—
29C2×C4×C22⋊C4C2211yescharacteristic
30#1445C2212yes—
31#1445C2214yes—
32#1445C2214yes—
33#1444C2214yes—
34#1443C2214yes—
35#1443C2214yes—
36#1442C2214yes—
37#1442C2214yes—
38#1442C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1441C2212yes—
43#1441C2214yes—
44#1440C2211yescharacteristic
45#1440C2211yescharacteristic
46#1440C2212yes—
47#1439C2212yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2211yescharacteristic
57#1438C2211yescharacteristic
58#1438C2212yes—
59#1438C2212yes—
60#1437C2212yes—
61#1437C2212yes—
62#1437C2212yes—
63#1436C2212yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4×#95C2212yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C2×#125—28no—
79C2×#125—44no—
80C2×#125—44no—
81C2×C4×D4—42no—
82C2×C4×D4—42no—
83C2×C4×D4Q818yes—
84C2×#117—44no—
85C2×#117—42no—
86C2×#117—44no—
87C2×#117—42no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2311yescharacteristic
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2312yes—
116#389—44no—
117#389—44no—
118#382—42no—
119#380—28no—
120#380—28no—
121#380—44no—
122#380—42no—
123C4×C4⋊C4—42no—
124C4×C22⋊C4—44no—
125C4×C22⋊C4—42no—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C25C2411yescharacteristic
155C25C2411yescharacteristic
156C22×D4—41no—
157C22×D4—41no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4C2×Q814yes—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4C2×Q814yes—
170C23×C4C2×Q814yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2414yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C23×C4C2414yes—
183C23×C4C2411yescharacteristic
184#125—432no—
185C4×D4—416no—
186#117—416no—
187#117—416no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
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⋊C4C2×Q814yes—
217C2×C4⋊C4#4118yes—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
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⋊C4—42no—
249C2×C22⋊C4C2×Q814yes—
250C2×C22⋊C4C2×Q814yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—44no—
260C2×C42—42no—
261C2×C42—42no—
262C2×C42—44no—
263C2×C42C2×Q814yes—
264#95—28no—
265#95—28no—
266#95—42no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—42no—
271#95—44no—
272#95—42no—
273#95—42no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—42no—
284#95—42no—
285#95—42no—
286#95—42no—
287#95—44no—
288#95—42no—
289#95—42no—
290#95—44no—
291#95—44no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—21no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24C22×Q812yes—
309C24C22×Q812yes—
310C24C2511yescharacteristic
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—48no—
315C2×D4—48no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4C22×Q812yes—
398C22×C4C22×Q812yes—
399C22×C4C22×Q812yes—
400C22×C4C22×Q812yes—
401C22×C4C22×Q812yes—
402C22×C4C2×#4118yes—
403C22×C4C2×#4114yes—
404C22×C4H(5,𝔽2)12yes—
405C22×C4H(5,𝔽2)14yes—
406C22×C4H(5,𝔽2)12yes—
407C4⋊C4—216no—
408C4⋊C4—44no—
409C4⋊C4—416no—
410C4⋊C4—416no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—48no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—88no—
427C22⋊C4—88no—
428C22⋊C4—84no—
429C22⋊C4—88no—
430C42—28no—
431C42—48no—
432C42—48no—
433C42—48no—
434C42—84no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23C23×Q811yescharacteristic
478C23C22×#4112yes—
479C23C2×H(5,𝔽2)11yescharacteristic
480C23C2×H(5,𝔽2)12yes—
481C23C2×H(5,𝔽2)11yescharacteristic
482C23#58512yes—
483C23#58512yes—
484C23#58512yes—
485C23#58512yes—
486D4—416no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—416no—
515C2×C4—416no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4#53818yes—
566C2×C4#54314yes—
567C2×C4#55418yes—
568C2×C4#55914yes—
569C2×C4H(3,𝔽4)14yes—
570C22—42no—
571C22—42no—
572C22—44no—
573C22—44no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22C2×#53812yes—
589C22C2×#54311yescharacteristic
590C22#345811yescharacteristic
591C22#345811yescharacteristic
592C22C2×#55412yes—
593C22C2×#55911yescharacteristic
594C22#350412yes—
595C22#352112yes—
596C22#352112yes—
597C22#352412yes—
598C22#352412yes—
599C22C2×H(3,𝔽4)11yescharacteristic
600C22#353912yes—
601C22#353912yes—
602C22#353912yes—
603C22#353912yes—
604C22#354011yescharacteristic
605C22#354011yescharacteristic
606C22#355112yes—
607C22#355112yes—
608C22#355112yes—
609C4—48no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—84no—
618C4—84no—
619C4—82no—
620C4—84no—
621C4#280418yes—
622C2—82no—
623C2—82no—
624C2#3742611yescharacteristic
625C2#3748111yescharacteristic
626C2#3810712yes—
627C2#3896112yes—
628C2#4276012yes—
629C2#4290612yes—
630C2#4520512yes—
631C2C2×#280411yescharacteristic
632C2#5306512yes—
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
3. #16951C2semidirect
4. #16915C2semidirect
5. #16245C2semidirect
6. #16214C2semidirect
7. #15232C2semidirect
8. #15216C2semidirect
9. #15216C2semidirect
10. #14264C2semidirect
11. #14129C2semidirect
12. #14054C2semidirect

Maximal subgroups

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

subgroupindex
#172732
#172722
#169512
#169152
#162452
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