Finite Group Databasebeta

← #950068

Subgroups — #950068

Order 512.

1998 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17273C212yesmaximal
2#17271C212yesmaximal
3#16954C214yesmaximal, complemented
4#16725C214yesmaximal, complemented
5#16176C212yesmaximal, complemented
6#16171C212yesmaximal, complemented
7#15223C212yesmaximal, complemented
8#15221C212yesmaximal, complemented
9#15220C212yesmaximal, complemented
10#14266C212yesmaximal, complemented
11#14129C212yesmaximal, complemented
12#13951C212yesmaximal, complemented
13#13871C211yesmaximal, characteristic
14#13871C211yesmaximal, characteristic
15#13847C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#382C2212yes—
24C2×#380C2212yes—
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C2×#376C2211yescharacteristic
32#1445C2211yescharacteristic
33#1445C2211yescharacteristic
34#1445C2214yes—
35#1445C2214yes—
36#1445C2214yes—
37#1445C2214yes—
38#1444C2214yes—
39#1443C2214yes—
40#1443C2214yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2214yes—
45#1441C2214yes—
46#1440C2212yes—
47#1440C2212yes—
48#1439C2212yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2211yescharacteristic
59#1438C2211yescharacteristic
60#1438C2212yes—
61#1438C2212yes—
62#1437C2212yes—
63#1437C2212yes—
64#1436C2212yes—
65#1436C2212yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C2×C4×D4—42no—
79C2×C4×D4—42no—
80C2×C4×D4Q818yes—
81C2×#117—28no—
82C2×#117—42no—
83C2×#117—44no—
84C2×#117—42no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112#389—28no—
113#389—44no—
114#389—44no—
115#389—44no—
116#382—44no—
117#380—28no—
118#380—44no—
119#380—42no—
120#380—42no—
121C4×C4⋊C4—42no—
122C4×C22⋊C4—44no—
123C4×C22⋊C4—44no—
124#376—42no—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
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—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4C2×Q814yes—
170C23×C4C2×Q814yes—
171C23×C4C2414yes—
172C23×C4C2414yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C23×C4C2414yes—
183C23×C4C2411yescharacteristic
184C4×D4—416no—
185#117—416no—
186#117—416no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
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—28no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
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⋊C4C2×Q814yes—
216C2×C4⋊C4#4118yes—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—42no—
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—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4C2×Q814yes—
248C2×C22⋊C4C2×Q814yes—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—28no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—42no—
259C2×C42—44no—
260C2×C42—42no—
261C2×C42C2×Q814yes—
262#95—28no—
263#95—28no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—42no—
275#95—41no—
276#95—41no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—42no—
281#95—41no—
282#95—41no—
283#95—44no—
284#95—42no—
285#95—42no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—21no—
296C24—21no—
297C24—21no—
298C24—21no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24C22×Q812yes—
308C24C22×Q812yes—
309C24C2511yescharacteristic
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—48no—
314C2×D4—48no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—216no—
340C22×C4—216no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
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—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4C22×Q812yes—
397C22×C4C22×Q812yes—
398C22×C4C22×Q812yes—
399C22×C4C22×Q812yes—
400C22×C4C22×Q812yes—
401C22×C4C2×#4118yes—
402C22×C4C2×#4114yes—
403C22×C4H(5,𝔽2)12yes—
404C22×C4H(5,𝔽2)14yes—
405C22×C4H(5,𝔽2)12yes—
406C4⋊C4—216no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—88no—
413C4⋊C4—88no—
414C4⋊C4—88no—
415C4⋊C4—88no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—88no—
425C22⋊C4—88no—
426C22⋊C4—84no—
427C22⋊C4—84no—
428C42—28no—
429C42—48no—
430C42—48no—
431C42—48no—
432C42—48no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23C23×Q811yescharacteristic
476C23C22×#4112yes—
477C23C2×H(5,𝔽2)11yescharacteristic
478C23C2×H(5,𝔽2)12yes—
479C23C2×H(5,𝔽2)11yescharacteristic
480C23#58512yes—
481C23#58512yes—
482C23#58512yes—
483C23#58512yes—
484D4—416no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—416no—
513C2×C4—416no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
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—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
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—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4#54018yes—
564C2×C4#54314yes—
565C2×C4#55114yes—
566C2×C4#55418yes—
567C2×C4#56014yes—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22C2×#54012yes—
587C22C2×#54311yescharacteristic
588C22#345811yescharacteristic
589C22#345811yescharacteristic
590C22C2×#55111yescharacteristic
591C22C2×#55412yes—
592C22#350412yes—
593C22#352112yes—
594C22#352112yes—
595C22#352412yes—
596C22#352412yes—
597C22C2×#56011yescharacteristic
598C22#353912yes—
599C22#353912yes—
600C22#353912yes—
601C22#354012yes—
602C22#354012yes—
603C22#354012yes—
604C22#354012yes—
605C22#355111yescharacteristic
606C22#355111yescharacteristic
607C4—42no—
608C4—48no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—82no—
613C4—84no—
614C4—82no—
615C4—84no—
616C4—84no—
617C4—82no—
618C4—82no—
619C4#274818yes—
620C2—82no—
621C2—82no—
622C2#3742511yescharacteristic
623C2#3748011yescharacteristic
624C2#3810612yes—
625C2#3810712yes—
626C2#4276112yes—
627C2#4277812yes—
628C2#4540112yes—
629C2C2×#274811yescharacteristic
630C2#5307812yes—
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. #16954C2semidirect
4. #16725C2semidirect
5. #16176C2semidirect
6. #16171C2semidirect
7. #15223C2semidirect
8. #15221C2semidirect
9. #15220C2semidirect
10. #14266C2semidirect
11. #14129C2semidirect
12. #13951C2semidirect

Maximal subgroups

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

subgroupindex
#172732
#172712
#169542
#167252
#161762
#161712
#152232
#152212
#152202
#142662
#141292
#139512
#138712
#138472