Finite Group Databasebeta

← #1494753

Subgroups — #1494753

Order 512.

3032 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
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2432C211yesmaximal, characteristic
3#15884C212yesmaximal, complemented
4#15670C212yesmaximal, complemented
5#15554C214yesmaximal, complemented
6#15371C214yesmaximal, complemented
7#15191C212yesmaximal, complemented
8#14817C214yesmaximal, complemented
9#14473C212yesmaximal, complemented
10#14243C212yesmaximal, complemented
11#13956C212yesmaximal
12#13896C214yesmaximal, complemented
13#13824C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18#2804—28no—
19#2432—28no—
20C2×#400C2212yes—
21C2×#397C2212yes—
22C2×#394C2214yes—
23C2×#393C2214yes—
24C2×#393C2212yes—
25C2×#391C2211yescharacteristic
26C2×#391C2212yes—
27C2×#391C2214yes—
28C2×#390C2211yescharacteristic
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#385C2214yes—
34C2×#385C2214yes—
35C2×#382C2212yes—
36C2×#381C2212yes—
37C2×C4×C22⋊C4C2214yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C22×#95C2212yes—
40#1446C2212yes—
41#1446C2214yes—
42#1446C2214yes—
43#1445C2212yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2212yes—
50#1442C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2212yes—
55#1438C2214yes—
56#1437C2214yes—
57#1437C2212yes—
58#1437C2214yes—
59#1436C2212yes—
60C4⋊#95C2214yes—
61#1434C2214yes—
62#1434C2214yes—
63#1434C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C4×#95C2212yes—
69C22×#41—24no—
70C22×#41—28no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41—44no—
74C23×Q8C2311yescharacteristic
75C23×Q8C2312yes—
76C23×Q8C2311yescharacteristic
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—44no—
85C2×#124—24no—
86C2×#124—28no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—42no—
89C2×C22⋊Q8—42no—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2314yes—
108C22×C42C2311yescharacteristic
109#400—28no—
110#397—44no—
111#394—216no—
112#393—216no—
113#393—28no—
114#391—24no—
115#391—28no—
116#391—48no—
117#390D418yes—
118#386—28no—
119#386—28no—
120#386—44no—
121#386—44no—
122#385—216no—
123#385—216no—
124#382—44no—
125#381—44no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—216no—
128C2×#95—44no—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C25C2411yescharacteristic
155C25C2412yes—
156C2×#41—416no—
157C2×#41—432no—
158C2×#41—432no—
159C2×#41—416no—
160C2×#41—416no—
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8—42no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—42no—
182C22×D4—41no—
183C22×D4—41no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4C2×D414yes—
187C22×D4#4118yes—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C4⋊Q8—416no—
222#124—416no—
223C22⋊Q8—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4C2×D414yes—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4C2×D4116yes—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—28no—
271C2×C42—28no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42C2×D414yes—
277#95—24no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—24no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—42no—
300#95—44no—
301#95#4118yescomplemented
302C24—24no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—21no—
311C24—22no—
312C24—22no—
313C24—21no—
314C24—22no—
315C24—22no—
316C24—24no—
317C24C22×D414yes—
318C24C2×#4114yes—
319C24C2×#4114yes—
320C24H(5,𝔽2)12yes—
321C24C2511yescharacteristic
322#41—4128no—
323C2×Q8—28no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—44no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—416no—
330C2×Q8—44no—
331C2×Q8—44no—
332C2×Q8—48no—
333C2×Q8—84no—
334C2×Q8—84no—
335C2×Q8—84no—
336C2×Q8—84no—
337C2×Q8—88no—
338C2×D4—216no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4#124116yes—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—216no—
357C22×C4—21no—
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—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—44no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—88no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4—88no—
425C22×C4—84no—
426C22×C4—84no—
427C22×C4—88no—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D414yes—
431C22×C4C2×#4118yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4H(5,𝔽2)12yes—
435C22×C4H(5,𝔽2)14yes—
436C22×C42−1+414yes—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—416no—
440C4⋊C4—84no—
441C4⋊C4—84no—
442C22⋊C4—232no—
443C22⋊C4—416no—
444C22⋊C4—48no—
445C22⋊C4—416no—
446C22⋊C4—48no—
447C22⋊C4—88no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C42—44no—
453C42—48no—
454C42—44no—
455C42—88no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—42no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—42no—
482C23—41no—
483C23—42no—
484C23—44no—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—44no—
493C23—41no—
494C23C2×#12418yes—
495C23C23×D411yescharacteristic
496C23C22×#4112yes—
497C23C22×#4112yes—
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)12yes—
500C23C2×H(5,𝔽2)11yescharacteristic
501C23C2×2−1+412yes—
502C23#58514yes—
503Q8—416no—
504Q8—88no—
505Q8—816no—
506D4—416no—
507D4—432no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—416no—
527C2×C4—432no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
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—816no—
574C2×C4—816no—
575C2×C4C2×#12414yes—
576C2×C4#54718yes—
577C22—28no—
578C22—216no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—48no—
585C22—48no—
586C22—48no—
587C22—44no—
588C22—42no—
589C22—42no—
590C22—48no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22C22×#12411yescharacteristic
597C22C2×#54712yes—
598C22#347514yes—
599C22#348111yescharacteristic
600C22#348411yescharacteristic
601C22#348411yescharacteristic
602C22#349012yes—
603C22#349012yes—
604C22#350012yes—
605C22#351414yes—
606C22#351514yes—
607C22#352212yes—
608C22#354012yes—
609C22#354114yes—
610C22#354512yes—
611C22#355011yescharacteristic
612C4—24no—
613C4—28no—
614C4—42no—
615C4—44no—
616C4—48no—
617C4—44no—
618C4—84no—
619C4—82no—
620C4—82no—
621C4—84no—
622C4—84no—
623C2—48no—
624C2—82no—
625C2#3536711yescharacteristic
626C2#3616412yes—
627C2#4366014yes—
628C2#4389212yes—
629C2#5023712yes—
630C2#5174911yescharacteristic
631C2#5907011yescharacteristic
632C2#5998812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2804C2semidirect
3. #15884C2semidirect
4. #15670C2semidirect
5. #15554C2semidirect
6. #15371C2semidirect
7. #15191C2semidirect
8. #14817C2semidirect
9. #14473C2semidirect
10. #14243C2semidirect
12. #13896C2semidirect
13. #13824C2semidirect
301. #95#41semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#24322
#158842
#156702
#155542
#153712
#151912
#148172
#144732
#142432
#139562
#138962
#138242