Finite Group Databasebeta

← #612303

Subgroups — #612303

Order 512.

3054 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2398C212yesmaximal, complemented
2#17077C212yesmaximal, complemented
3#17076C211yesmaximal, characteristic, complemented
4#17068C211yesmaximal, characteristic
5#16932C212yesmaximal, complemented
6#16867C214yesmaximal, complemented
7#16652C212yesmaximal, complemented
8#16530C214yesmaximal, complemented
9C4×#394C212yesmaximal, complemented
10C4×#392C214yesmaximal, complemented
11#13842C214yesmaximal, complemented
12C4×#391C211yesmaximal, characteristic, complemented
13C4×#390C212yesmaximal, complemented
14C22×C4×Q8C2211yescharacteristic
15C22×C4×D4C2212yes—
16C22×C4×D4C2212yes—
17C22×C4×D4C2212yes—
18#2398—216no—
19C2×#400C2212yes—
20C2×#400C2212yes—
21C2×#397C2214yes—
22C2×#394C2212yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25C2×#393C2214yes—
26C2×#392C2214yes—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2214yes—
34C2×C4×C22⋊C4C2214yes—
35C2×C4×C22⋊C4C2214yes—
36C2×C4×C22⋊C4C2214yes—
37C2×#376C2212yes—
38C2×C43C2211yescharacteristic
39#1444C2214yes—
40#1443C2214yes—
41#1443C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2212yes—
45#1441C2212yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2214yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2212yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4×#95C2212yes—
67C4×#95C2214yes—
68C23×Q8C2311yescharacteristic
69C23×D4C2312yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C24×C4C2314yes—
73C2×C4×Q8—42no—
74C2×C4×Q8—42no—
75C2×C4×Q8D418yescomplemented
76C2×C4×D4—28no—
77C2×C4×D4—28no—
78C2×C4×D4—44no—
79C2×C4×D4—44no—
80C2×C4×D4—44no—
81C2×C4×D4—44no—
82C2×C4×D4—48no—
83C2×C4×D4—44no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C42C2312yes—
105C22×C42C2314yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#400—28no—
113#400—28no—
114#397—216no—
115#394—28no—
116#394—28no—
117#394—28no—
118#393—216no—
119#392—216no—
120#391—24no—
121#391—24no—
122#390—28no—
123#390—28no—
124C4×C4⋊C4—44no—
125C4×C4⋊C4—42no—
126C4×C22⋊C4—216no—
127C4×C22⋊C4—216no—
128C4×C22⋊C4—216no—
129C4×C22⋊C4—48no—
130#376—28no—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C43—42no—
152C25C2414yes—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8C2×D414yes—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—44no—
169C22×D4—42no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—44no—
175C22×D4—42no—
176C22×D4#4118yes—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—28no—
180C23×C4—28no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2411yescharacteristic
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C4×Q8—416no—
197C4×D4—432no—
198C4×D4—432no—
199C4×D4—432no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D418yes—
225C2×C4⋊C4#4114yes—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—28no—
266C2×C42—28no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—28no—
272C2×C42—41no—
273C2×C42—42no—
274C2×C42—41no—
275C2×C42—41no—
276C2×C42—42no—
277C2×C42—41no—
278C2×C42C2×D418yes—
279C2×C42C2×D414yes—
280C2×C42#4118yes—
281C2×C42#4114yes—
282#95—28no—
283#95—24no—
284#95—24no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—44no—
296#95—42no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—42no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24C2×#4118yes—
317C24C2511yescharacteristic
318C2×Q8—28no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×D4—216no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—416no—
329C2×D4—416no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—416no—
339C2×D4—416no—
340C2×D4—48no—
341C2×D4—48no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
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—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4—416no—
403C22×C4—416no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4C22×D412yes—
408C22×C4C22×D414yes—
409C22×C4C22×D414yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4114yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4114yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)14yes—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C22⋊C4—416no—
433C22⋊C4—416no—
434C22⋊C4—416no—
435C22⋊C4—416no—
436C22⋊C4—416no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C42—216no—
444C42—28no—
445C42—216no—
446C42—44no—
447C42—48no—
448C42—44no—
449C42—416no—
450C42—48no—
451C23—28no—
452C23—28no—
453C23—28no—
454C23—28no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23C23×D411yescharacteristic
487C23C22×#4111yescharacteristic
488C23C22×#4112yes—
489C23C22×#4112yes—
490C23C22×#4111yescharacteristic
491C23C2×H(5,𝔽2)11yescharacteristic
492C23C2×H(5,𝔽2)11yescharacteristic
493C23C2×H(5,𝔽2)12yes—
494C23#58511yescharacteristic
495C23#58511yescharacteristic
496C23#58512yes—
497Q8—416no—
498D4—432no—
499D4—432no—
500D4—432no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
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—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—832no—
569C2×C4#52918yes—
570C2×C4#53214yes—
571C2×C4#53414yes—
572C2×C4#53818yes—
573C2×C4#55018yes—
574C2×C4#55514yes—
575C2×C4H(3,𝔽4)14yes—
576C22—48no—
577C22—48no—
578C22—48no—
579C22—48no—
580C22—48no—
581C22—48no—
582C22—48no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22C2×#52912yes—
592C22C2×#53211yescharacteristic
593C22C2×#53411yescharacteristic
594C22#344911yescharacteristic
595C22C2×#53812yes—
596C22#345512yes—
597C22#345911yescharacteristic
598C22#345912yes—
599C22#345912yes—
600C22#345911yescharacteristic
601C22#346012yes—
602C22#346011yescharacteristic
603C22#346011yescharacteristic
604C22C2×#55012yes—
605C22D4×#4112yes—
606C22C2×#55511yescharacteristic
607C22#347512yes—
608C22#348012yes—
609C22#348112yes—
610C22#348111yescharacteristic
611C22C2×H(3,𝔽4)11yescharacteristic
612C22#352911yescharacteristic
613C22#352912yes—
614C4—28no—
615C4—24no—
616C4—44no—
617C4—44no—
618C4—44no—
619C4—42no—
620C4—44no—
621C4—82no—
622C4—84no—
623C4—84no—
624C4—84no—
625C4#273318yes—
626C2—88no—
627C2#3457012yes—
628C2#3457511yescharacteristic
629C2#3493412yes—
630C2#3494311yescharacteristic
631C2#3669411yescharacteristic
632C2C2×#273311yescharacteristic
633C2#5216012yes—
634C2#5216311yescharacteristic
635C2#5227011yescharacteristic
636C2#5227812yes—
637C2#5230111yescharacteristic
subgroupG/H
centerC23×C4C24
commutator, Frattini, socle, cosocleC24C25
prime radical#17068C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2398C2semidirect
2. #17077C2semidirect
3. #17076C2semidirect
5. #16932C2semidirect
6. #16867C2semidirect
7. #16652C2semidirect
8. #16530C2semidirect
9. C4×#394C2semidirect
10. C4×#392C2semidirect
11. #13842C2semidirect
12. C4×#391C2semidirect
13. C4×#390C2semidirect
75. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23982
#170772
#170762
#170682
#169322
#168672
#166522
#165302
C4×#3942
C4×#3922
#138422
C4×#3912
C4×#3902