Finite Group Databasebeta

← #538276

Subgroups — #538276

Order 512.

2566 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2#14619C214yesmaximal, complemented
3#14610C212yesmaximal, complemented
4#14492C212yesmaximal
5#14253C212yesmaximal, complemented
6#14169C212yesmaximal
7#14090C214yesmaximal, complemented
8#14053C212yesmaximal, complemented
9#13993C212yesmaximal, complemented
10#13973C212yesmaximal, complemented
11#13871C211yesmaximal, characteristic
12#13839C211yesmaximal, characteristic
13#13810C212yesmaximal, complemented
14#13801C212yesmaximal, complemented
15#13777C212yesmaximal, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#2547C4116yescomplemented
22C2×#397C2212yes—
23C2×#397C2212yes—
24C2×#391C2214yes—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37C2×#376C2214yes—
38#1446C2214yes—
39#1443C2214yes—
40#1442C2214yes—
41#1441C2212yes—
42#1441C2214yes—
43#1440C2211yescharacteristic
44#1440C2211yescharacteristic
45#1440C2212yes—
46#1439C2212yes—
47#1439C2214yes—
48#1439C2214yes—
49#1438C2211yescharacteristic
50#1438C2211yescharacteristic
51#1438C2212yes—
52#1437C2212yes—
53#1437C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1436C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73#1434C2214yes—
74C4×#95C2214yes—
75C4×#95C2212yes—
76C23×Q8C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8C2×C418yes—
83C2×#124—48no—
84C2×#124C2×C4116yes—
85C2×C4×D4—42no—
86C2×C4×D4—42no—
87C2×C4×D4C2×C418yes—
88C2×#117—28no—
89C2×#117—28no—
90C2×#117—42no—
91C2×#117—42no—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2314yes—
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2314yes—
120C22×C42C2314yes—
121#397C2×C4116yes—
122#397C2×C4116yes—
123#391—48no—
124#391C2×C4116yes—
125#389—44no—
126#380—28no—
127#380—42no—
128#380C2×C4116yes—
129#380C2×C4116yes—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—28no—
132C4×C4⋊C4—44no—
133C4×C4⋊C4C2×C418yes—
134C4×C22⋊C4—28no—
135C4×C22⋊C4—42no—
136#376—48no—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C25C2411yescharacteristic
164C25C2411yescharacteristic
165C22×Q8—28no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—44no—
169C22×Q8C22×C418yes—
170C22×D4—24no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—42no—
174C22×D4C22×C414yes—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4C22×C414yes—
184C23×C4C22×C414yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2411yescharacteristic
189C23×C4C2414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C4⋊Q8—416no—
199#124—432no—
200C4×D4—416no—
201#117—416no—
202#117—416no—
203C2×C4⋊C4—24no—
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—24no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4C22×C418yes—
223C2×C4⋊C4C22×C418yes—
224C2×C4⋊C4C22×C414yes—
225C2×C4⋊C4C22×C418yes—
226C2×C4⋊C4#4118yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4C22×C414yes—
257C2×C22⋊C4C22×C414yes—
258C2×C22⋊C4C22×C4116yes—
259C2×C22⋊C4C22×C4116yes—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—28no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—28no—
268C2×C42—28no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—28no—
272C2×C42—42no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42C22×C414yes—
280C2×C42C22×C418yes—
281C2×C42#4118yes—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—24no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—44no—
292#95—42no—
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—42no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95C22×C418yes—
306#95C22×C4116yes—
307#95C22×C4116yes—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—21no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24C23×C412yes—
325C24C23×C412yes—
326C24C2511yescharacteristic
327C2×Q8—216no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—88no—
331C2×Q8—88no—
332C2×D4—28no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—48no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—216no—
389C22×C4—216no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4C23×C412yes—
413C22×C4C23×C412yes—
414C22×C4C23×C418yes—
415C22×C4C23×C412yes—
416C22×C4C23×C418yes—
417C22×C4C23×C412yes—
418C22×C4C23×C412yes—
419C22×C4C2×#4114yes—
420C22×C4C2×#4118yes—
421C22×C4C2×#4114yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4114yes—
424C22×C4H(5,𝔽2)14yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C42−1+414yes—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—216no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—88no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—48no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—44no—
447C22⋊C4—88no—
448C22⋊C4—84no—
449C42—28no—
450C42—216no—
451C42—216no—
452C42—416no—
453C42—48no—
454C42—416no—
455C42—48no—
456C42—88no—
457C23—24no—
458C23—24no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23C24×C411yescharacteristic
500C23C22×#4112yes—
501C23C22×#4112yes—
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×H(5,𝔽2)11yescharacteristic
504C23C2×H(5,𝔽2)12yes—
505C23C2×2−1+412yes—
506C23#58512yes—
507C23#58512yes—
508Q8—816no—
509D4—416no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—416no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4C4×#4118yes—
588C2×C4#51814yes—
589C2×C4#51814yes—
590C2×C4#55114yes—
591C2×C4#55114yes—
592C2×C4#55518yes—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22C2×C4×#4112yes—
612C22C2×#51811yescharacteristic
613C22C2×#51811yescharacteristic
614C22#342811yescharacteristic
615C22C4×2−1+412yes—
616C22C2×#55111yescharacteristic
617C22C2×#55111yescharacteristic
618C22C2×#55512yes—
619C22#349312yes—
620C22#349512yes—
621C22#350612yes—
622C22#351112yes—
623C22#351712yes—
624C22#351812yes—
625C22#352012yes—
626C22#352112yes—
627C22#353811yescharacteristic
628C22#354011yescharacteristic
629C22#354212yes—
630C22#354212yes—
631C22#354712yes—
632C4—24no—
633C4—48no—
634C4—44no—
635C4—48no—
636C4—48no—
637C4—44no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—82no—
643C4—84no—
644C4#237218yes—
645C2—82no—
646C2—82no—
647C2C2×#237211yescharacteristic
648C2#3430712yes—
649C2#3431612yes—
650C2#3436311yescharacteristic
651C2#3441211yescharacteristic
652C2#4489812yes—
653C2#4490912yes—
654C2#4507812yes—
655C2#4508712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14619C2semidirect
3. #14610C2semidirect
5. #14253C2semidirect
7. #14090C2semidirect
8. #14053C2semidirect
9. #13993C2semidirect
10. #13973C2semidirect
13. #13810C2semidirect
14. #13801C2semidirect
15. #13777C2semidirect
21. #2547C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
#146192
#146102
#144922
#142532
#141692
#140902
#140532
#139932
#139732
#138712
#138392
#138102
#138012
#137772