Finite Group Databasebeta

← #763439

Subgroups — #763439

Order 512.

2684 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
1C2×#2451C212yesmaximal, complemented
2C2×#2323C212yesmaximal
3#15981C212yesmaximal, complemented
4#15965C212yesmaximal, complemented
5#15779C212yesmaximal, complemented
6#15778C212yesmaximal, complemented
7#15457C214yesmaximal
8#15427C214yesmaximal
9#15179C212yesmaximal
10#15178C211yesmaximal, characteristic
11#15125C211yesmaximal, characteristic
12#14133C212yesmaximal, complemented
13#14024C212yesmaximal, complemented
14#14021C212yesmaximal, complemented
15#13844C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2212yes—
20C22×#117C2211yescharacteristic
21#2451—216no—
22C2×#398C2214yes—
23C2×#395C2214yes—
24C2×#393C2212yes—
25#2323—216no—
26C2×#391C2212yes—
27C2×#391C2214yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#389C2214yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#387C2214yes—
37C2×#387C2214yes—
38C2×#387C2214yes—
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44#1446C2212yes—
45#1444C2211yescharacteristic
46#1444C2211yescharacteristic
47#1444C2212yes—
48#1444C2212yes—
49#1443C2214yes—
50#1443C2214yes—
51#1442C2211yescharacteristic
52#1442C2211yescharacteristic
53#1442C2212yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1439C2212yes—
62#1439C2212yes—
63#1439C2212yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2211yescharacteristic
74C4⋊#95C2212yes—
75#1434C2214yes—
76C4×#95C2212yes—
77C23×Q8C2312yes—
78C23×Q8C2312yes—
79C24×C4C2311yescharacteristic
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8—42no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—28no—
87C2×C4×Q8—28no—
88C2×C4×Q8—28no—
89C2×C4×Q8—44no—
90C2×#117—28no—
91C2×#117—42no—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2314yes—
118#398—216no—
119#395—216no—
120#393—44no—
121#391—28no—
122#391—28no—
123#391—28no—
124#391—28no—
125#391—216no—
126#389—216no—
127#389—28no—
128#389D4116yes—
129#388—216no—
130#388—216no—
131#387—216no—
132#387—216no—
133#387—48no—
134#380—28no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—28no—
137C4×C22⋊C4—24no—
138C4×C22⋊C4—42no—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C25C2411yescharacteristic
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—28no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—42no—
179C22×Q8—44no—
180C22×Q8—42no—
181C22×Q8#4118yes—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4#4114yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C4⋊Q8—416no—
202C4⋊Q8—416no—
203C22⋊Q8—264no—
204C4×Q8—432no—
205#117—416no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C2×D418yes—
241C2×C4⋊C4C2×D418yes—
242C2×C4⋊C4C2×D4116yes—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—28no—
264C2×C42—28no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—28no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42C2×D418yes—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—22no—
289#95—22no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—24no—
300#95—42no—
301#95—44no—
302#95—41no—
303#95—41no—
304#95—44no—
305#95—42no—
306#95C2×D4116yes—
307C24—22no—
308C24—22no—
309C24—21no—
310C24—21no—
311C24—22no—
312C24—21no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—22no—
317C24C2×#4112yes—
318C24C2511yescharacteristic
319C2×Q8—216no—
320C2×Q8—216no—
321C2×Q8—216no—
322C2×Q8—48no—
323C2×Q8—416no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—88no—
329C2×Q8—84no—
330C2×Q8—84no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—24no—
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—22no—
387C22×C4—216no—
388C22×C4—28no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—44no—
404C22×C4—48no—
405C22×C4—44no—
406C22×C4C22×D418yes—
407C22×C4C22×D414yes—
408C22×C4C22×D418yes—
409C22×C4C22×D414yes—
410C22×C4C22×D414yes—
411C22×C4C2×#4118yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4114yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)12yes—
417C22×C42−1+412yes—
418C22×C42−1+414yes—
419C22×C42−1+412yes—
420C22×C42−1+412yes—
421C22×C42−1+412yes—
422C4⋊C4—216no—
423C4⋊C4—232no—
424C4⋊C4—216no—
425C4⋊C4—216no—
426C4⋊C4—416no—
427C4⋊C4—48no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—416no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—88no—
436C22⋊C4—232no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—88no—
442C42—216no—
443C42—416no—
444C42—48no—
445C42—44no—
446C42—44no—
447C42—88no—
448C23—24no—
449C23—22no—
450C23—24no—
451C23—22no—
452C23—22no—
453C23—41no—
454C23—42no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—41no—
471C23C23×D412yes—
472C23C22×#4111yescharacteristic
473C23C22×#4112yes—
474C23C2×H(5,𝔽2)11yescharacteristic
475C23C2×H(5,𝔽2)11yescharacteristic
476C23C2×2−1+411yescharacteristic
477C23C2×2−1+411yescharacteristic
478C23C2×2−1+412yes—
479C23C2×2−1+411yescharacteristic
480C23C2×2−1+411yescharacteristic
481C23#58512yes—
482Q8—432no—
483Q8—816no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—416no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4#52518yes—
567C2×C4#53914yes—
568C2×C4D4214yes—
569C2×C4#54818yes—
570C2×C4#55114yes—
571C22—42no—
572C22—44no—
573C22—44no—
574C22—42no—
575C22—42no—
576C22—81no—
577C22—82no—
578C22—81no—
579C22—81no—
580C22—82no—
581C22—81no—
582C22C2×#52512yes—
583C22C2×#53911yescharacteristic
584C22C2×D4211yescharacteristic
585C22C2×#54812yes—
586C22C2×#55111yescharacteristic
587C22#348612yes—
588C22#348912yes—
589C22#349512yes—
590C22#349612yes—
591C22#349912yes—
592C22#350911yescharacteristic
593C22#351111yescharacteristic
594C22#351612yes—
595C22#351812yes—
596C22#352212yes—
597C22#352312yes—
598C22#353611yescharacteristic
599C22#353811yescharacteristic
600C22#355011yescharacteristic
601C22#355011yescharacteristic
602C22#355111yescharacteristic
603C22#355512yes—
604C22#355611yescharacteristic
605C4—44no—
606C4—42no—
607C4—48no—
608C4—42no—
609C4—44no—
610C4—48no—
611C4—42no—
612C4—44no—
613C4—42no—
614C4—48no—
615C4—42no—
616C4—82no—
617C4—84no—
618C4#246318yes—
619C2—82no—
620C2C2×#246311yescharacteristic
621C2#3560612yes—
622C2#3566512yes—
623C2#3732311yescharacteristic
624C2#3738311yescharacteristic
625C2#4489612yes—
626C2#4510311yescharacteristic
627C2#4530811yescharacteristic
628C2#4532111yescharacteristic
629C2#4534312yes—
630C2#4535111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2451C2semidirect
3. #15981C2semidirect
4. #15965C2semidirect
5. #15779C2semidirect
6. #15778C2semidirect
12. #14133C2semidirect
13. #14024C2semidirect
14. #14021C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24512
C2×#23232
#159812
#159652
#157792
#157782
#154572
#154272
#151792
#151782
#151252
#141332
#140242
#140212
#138442