Finite Group Databasebeta

← #522207

Subgroups — #522207

Order 512.

2458 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#14459C214yesmaximal, complemented
3#14449C212yesmaximal
4#14448C212yesmaximal
5#14297C212yesmaximal
6#14132C212yesmaximal, complemented
7#14058C214yesmaximal, complemented
8#14053C212yesmaximal, complemented
9#14029C212yesmaximal, complemented
10#13993C212yesmaximal
11#13973C212yesmaximal
12#13818C212yesmaximal, complemented
13#13780C212yesmaximal
14C4×#380C211yesmaximal, characteristic
15C4×#381C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C23×C42C2211yescharacteristic
19#2857C4116yescomplemented
20C2×#391C2214yes—
21C2×#391C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#387C2214yes—
27C2×#384C2211yescharacteristic
28C2×#381C2211yescharacteristic
29C2×#381C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40#1446C2214yes—
41#1445C2214yes—
42#1445C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1441C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1437C2214yes—
51#1437C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64#1434C2214yes—
65#1434C2212yes—
66#1434C2212yes—
67#1434C2214yes—
68#1434C2212yes—
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2212yes—
71C22⋊#95C2211yescharacteristic
72C4×#95C2212yes—
73C4×#95C2212yes—
74C4×#95C2212yes—
75C4×#95C2214yes—
76C23×Q8C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8C2×C418yes—
83C2×C22⋊Q8—48no—
84C2×C22⋊Q8C2×C4116yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2311yescharacteristic
101C22×C42—42no—
102C22×C42—42no—
103C22×C42C2×C418yes—
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2314yes—
111C22×C42C2314yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114#391—48no—
115#391C2×C4116yes—
116#389—28no—
117#389—44no—
118#389C2×C4116yes—
119#389C2×C4116yes—
120#387—48no—
121#384C2×C418yes—
122#381—24no—
123#381Q818yescomplemented
124#380—28no—
125#380C2×C4116yes—
126#380C2×C4116yes—
127C4×C4⋊C4—28no—
128C4×C4⋊C4—42no—
129C4×C4⋊C4—42no—
130C4×C4⋊C4—44no—
131C4×C22⋊C4—28no—
132C4×C22⋊C4—42no—
133C4×C22⋊C4—42no—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C25C2411yescharacteristic
164C22×Q8—28no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—44no—
168C22×Q8C22×C418yes—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—41no—
175C23×C4—41no—
176C23×C4—42no—
177C23×C4—41no—
178C23×C4—41no—
179C23×C4—41no—
180C23×C4—41no—
181C23×C4C22×C414yes—
182C23×C4C22×C414yes—
183C23×C4C22×C414yes—
184C23×C4C2×Q814yes—
185C23×C4#4114yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C4⋊Q8—416no—
201C22⋊Q8—432no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
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—28no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C22×C418yes—
225C2×C4⋊C4C22×C418yes—
226C2×C4⋊C4C22×C418yes—
227C2×C4⋊C4C22×C4116yes—
228C2×C4⋊C4C22×C418yes—
229C2×C4⋊C4C22×C418yes—
230C2×C4⋊C4#4118yes—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4C22×C4116yes—
247C2×C22⋊C4C2×Q814yes—
248C2×C22⋊C4C2×Q814yes—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—28no—
262C2×C42—416no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42—41no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42C22×C414yes—
277C2×C42C22×C414yes—
278C2×C42C22×C414yes—
279C2×C42C22×C414yes—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—24no—
284#95—24no—
285#95—28no—
286#95—24no—
287#95—28no—
288#95—24no—
289#95—42no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—42no—
294#95—44no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95C22×C4116yes—
306#95C22×C4116yes—
307#95C2×Q818yes—
308#95C2×Q818yes—
309C24—22no—
310C24—22no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—21no—
316C24—21no—
317C24C23×C412yes—
318C24C22×Q812yes—
319C24C2×#4112yes—
320C24C2511yescharacteristic
321C2×Q8—216no—
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—88no—
325C2×Q8—88no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4—48no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4C4×Q8116yes—
413C22×C4C23×C412yes—
414C22×C4C23×C418yes—
415C22×C4C23×C418yes—
416C22×C4C23×C412yes—
417C22×C4C23×C412yes—
418C22×C4C23×C412yes—
419C22×C4C23×C412yes—
420C22×C4C23×C412yes—
421C22×C4C22×Q814yes—
422C22×C4C22×Q814yes—
423C22×C4C22×Q812yes—
424C22×C4C22×Q812yes—
425C22×C4C2×#4118yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4112yes—
428C22×C4C2×#4114yes—
429C22×C4H(5,𝔽2)14yes—
430C22×C4H(5,𝔽2)12yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+414yes—
433C4⋊C4—216no—
434C4⋊C4—216no—
435C4⋊C4—216no—
436C4⋊C4—216no—
437C4⋊C4—48no—
438C4⋊C4—416no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—88no—
442C4⋊C4—88no—
443C22⋊C4—416no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—416no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C42—28no—
450C42—28no—
451C42—44no—
452C42—44no—
453C42—48no—
454C42—48no—
455C42—84no—
456C42—84no—
457C42—88no—
458C42—88no—
459C23—22no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23C2×C4×Q814yes—
482C23C24×C411yescharacteristic
483C23C23×Q811yescharacteristic
484C23C22×#4111yescharacteristic
485C23C22×#4112yes—
486C23C2×H(5,𝔽2)11yescharacteristic
487C23C2×H(5,𝔽2)11yescharacteristic
488C23C2×H(5,𝔽2)12yes—
489C23C2×2−1+412yes—
490C23#58512yes—
491C23#58511yescharacteristic
492C23#58511yescharacteristic
493Q8—816no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—416no—
532C2×C4—416no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—416no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4C2×C4×Q814yes—
580C2×C4C2×C4×Q814yes—
581C2×C4C4×#4114yes—
582C2×C4#51814yes—
583C2×C4#51814yes—
584C2×C4#51914yes—
585C2×C4#53918yes—
586C2×C4#54218yes—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—44no—
594C22—44no—
595C22—42no—
596C22—44no—
597C22—44no—
598C22C22×C4×Q811yescharacteristic
599C22C2×C4×#4111yescharacteristic
600C22C2×#51811yescharacteristic
601C22C2×#51811yescharacteristic
602C22C2×#51911yescharacteristic
603C22#342811yescharacteristic
604C22#342811yescharacteristic
605C22C2×#53912yes—
606C22C2×#54212yes—
607C22#347512yes—
608C22Q8×#4112yes—
609C22#350111yescharacteristic
610C22#350111yescharacteristic
611C22#350312yes—
612C22#350511yescharacteristic
613C22#350511yescharacteristic
614C22#351112yes—
615C22#351412yes—
616C22#351712yes—
617C22#354212yes—
618C22#354212yes—
619C22#356512yes—
620C22#356512yes—
621C4—24no—
622C4—24no—
623C4—24no—
624C4—42no—
625C4—42no—
626C4—48no—
627C4—48no—
628C4—48no—
629C4—44no—
630C4—44no—
631C4—82no—
632C4—82no—
633C4—84no—
634C4#233218yes—
635C2—44no—
636C2C2×#233211yescharacteristic
637C2#3414511yescharacteristic
638C2#3421211yescharacteristic
639C2#3421611yescharacteristic
640C2#3423211yescharacteristic
641C2#3436311yescharacteristic
642C2#3440311yescharacteristic
643C2#3952912yes—
644C2#3953012yes—
645C2#3965712yes—
646C2#3965812yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14459C2semidirect
6. #14132C2semidirect
7. #14058C2semidirect
8. #14053C2semidirect
9. #14029C2semidirect
12. #13818C2semidirect
19. #2857C4semidirect
123. #381Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#144592
#144492
#144482
#142972
#141322
#140582
#140532
#140292
#139932
#139732
#138182
#137802
C4×#3802
C4×#3812