Finite Group Databasebeta

← #613114

Subgroups — #613114

Order 512.

2714 conjugacy classes of subgroups.

The subgroup lattice has 655 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2C2×C42×Q8C211yesmaximal, characteristic, complemented
3#15200C212yesmaximal, complemented
4#15147C212yesmaximal
5#14882C214yesmaximal, complemented
6#14785C214yesmaximal, complemented
7#14707C211yesmaximal, characteristic, complemented
8#14700C212yesmaximal
9#14699C212yesmaximal, complemented
10#14177C214yesmaximal, complemented
11#14032C214yesmaximal, complemented
12#13955C211yesmaximal, characteristic, complemented
13#13941C211yesmaximal, characteristic
14#13925C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic
16C22×C4×Q8C2212yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#2580—28no—
20C2×#399C2212yes—
21C2×#397C2214yes—
22C2×#393C2212yes—
23C2×#390C2211yescharacteristic
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#387C2214yes—
32C2×#385C2214yes—
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C42×Q8—28no—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C22⋊C4C2214yes—
43C2×C43C2212yes—
44C2×C43C2211yescharacteristic
45#1446C2212yes—
46#1445C2214yes—
47#1445C2212yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72#1434C2214yes—
73#1434C2212yes—
74#1434C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4×Q8—28no—
81C2×C4×Q8—28no—
82C2×C4×Q8—28no—
83C2×C4×Q8—24no—
84C2×C4×Q8—24no—
85C2×C4×Q8D418yes—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#399—28no—
121#397—216no—
122#393—28no—
123#390—24no—
124#390—28no—
125#390—28no—
126#390—24no—
127#389—28no—
128#389—28no—
129#388—216no—
130#388—48no—
131#387—48no—
132#385—216no—
133#382—24no—
134#382—24no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—28no—
137C4×C4⋊C4—28no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4—24no—
141C4×C22⋊C4—48no—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C43—28no—
168C43—24no—
169C25C2412yes—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8C2×D414yes—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4#4114yes—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C4×Q8—264no—
202C4×Q8—232no—
203C4×D4—416no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4C2×D418yes—
238C2×C4⋊C4C2×D414yes—
239C2×C4⋊C4#4118yes—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—24no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—44no—
290C2×C42C2×D418yes—
291C2×C42C2×D414yes—
292C2×C42#4118yes—
293C2×C42#4114yes—
294C2×C42#4114yes—
295C2×C42#4114yes—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321C24—22no—
322C24—24no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24C2×#4114yes—
332C24C2511yescharacteristic
333C2×Q8—28no—
334C2×Q8—28no—
335C2×Q8—216no—
336C2×Q8—28no—
337C2×Q8—216no—
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—28no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
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—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
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—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—416no—
411C22×C4—48no—
412C22×C4—88no—
413C22×C4—84no—
414C22×C4—84no—
415C22×C4C22×D414yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D414yes—
419C22×C4C22×D412yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4114yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4114yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4H(5,𝔽2)14yes—
436C22×C42−1+412yes—
437C22×C42−1+412yes—
438C4⋊C4—216no—
439C4⋊C4—216no—
440C4⋊C4—216no—
441C4⋊C4—28no—
442C4⋊C4—216no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—416no—
449C4⋊C4—88no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—48no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—88no—
457C42—216no—
458C42—216no—
459C42—216no—
460C42—216no—
461C42—28no—
462C42—48no—
463C42—48no—
464C42—48no—
465C42—48no—
466C42—44no—
467C42—84no—
468C42#123116yes—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—42no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—42no—
480C23—44no—
481C23—42no—
482C23—42no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23C23×D411yescharacteristic
493C23C22×#4111yescharacteristic
494C23C22×#4111yescharacteristic
495C23C22×#4112yes—
496C23C22×#4112yes—
497C23C22×#4111yescharacteristic
498C23C22×#4111yescharacteristic
499C23C2×H(5,𝔽2)12yes—
500C23C2×2−1+411yescharacteristic
501C23C2×2−1+411yescharacteristic
502C23#58512yes—
503Q8—232no—
504D4—416no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—816no—
583C2×C4C2×#12314yes—
584C2×C4C2×#12314yes—
585C2×C4C2×#12314yes—
586C2×C4#52518yes—
587C2×C4#52914yes—
588C2×C4#54114yes—
589C2×C4#54814yes—
590C2×C4#54814yes—
591C2×C4#55214yes—
592C2×C4#55614yes—
593C2×C4#55918yes—
594C22—44no—
595C22—44no—
596C22—48no—
597C22—48no—
598C22—44no—
599C22—82no—
600C22—82no—
601C22—84no—
602C22—82no—
603C22—84no—
604C22—82no—
605C22C22×#12311yescharacteristic
606C22C2×#52512yes—
607C22C2×#52911yescharacteristic
608C22#344411yescharacteristic
609C22#344411yescharacteristic
610C22#344412yes—
611C22C2×#54111yescharacteristic
612C22#346012yes—
613C22C2×#54811yescharacteristic
614C22C2×#54811yescharacteristic
615C22C2×#55211yescharacteristic
616C22C2×#55611yescharacteristic
617C22#347512yes—
618C22C2×#55912yes—
619C22#348012yes—
620C22#348112yes—
621C22#348812yes—
622C22#350712yes—
623C22#350811yescharacteristic
624C22#351011yescharacteristic
625C22#351112yes—
626C22#351512yes—
627C22#352212yes—
628C4—24no—
629C4—28no—
630C4—24no—
631C4—28no—
632C4—44no—
633C4—44no—
634C4—44no—
635C4—44no—
636C4—82no—
637C4—82no—
638C4—84no—
639C4—82no—
640C4—84no—
641C4#242118yes—
642C2—84no—
643C2C2×#242111yescharacteristic
644C2#3491812yes—
645C2#3511612yes—
646C2#3525411yescharacteristic
647C2#3637011yescharacteristic
648C2#3639411yescharacteristic
649C2#3649212yes—
650C2#3658012yes—
651C2#3660111yescharacteristic
652C2#3660911yescharacteristic
653C2#3662611yescharacteristic
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
2. C2×C42×Q8C2semidirect
3. #15200C2semidirect
5. #14882C2semidirect
6. #14785C2semidirect
7. #14707C2semidirect
9. #14699C2semidirect
10. #14177C2semidirect
11. #14032C2semidirect
12. #13955C2semidirect
14. #13925C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×C42×Q82
#152002
#151472
#148822
#147852
#147072
#147002
#146992
#141772
#140322
#139552
#139412
#139252
C4×#3902