Finite Group Databasebeta

← #521077

Subgroups — #521077

Order 512.

3722 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2397C211yesmaximal, characteristic
2C2×#2290C212yesmaximal, complemented
3#14584C214yesmaximal, complemented
4#14578C214yesmaximal, complemented
5#14520C214yesmaximal, complemented
6#14519C214yesmaximal, complemented
7#13854C211yesmaximal, characteristic, complemented
8#13839C212yesmaximal, complemented
9#13839C212yesmaximal, complemented
10C4×#390C212yesmaximal, complemented
11C4×#386C212yesmaximal, complemented
12C4×#386C212yesmaximal, complemented
13#13757C211yesmaximal, characteristic, complemented
14C22×C4×D4C2212yes—
15C22×C4×D4C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×#117C2212yes—
19C23×C42C2212yes—
20#2397C4116yescomplemented
21C2×#399C2212yes—
22C2×#399C2212yes—
23C2×#392C2212yes—
24C2×#392C2212yes—
25C2×#390C2214yes—
26#2290—216no—
27C2×#388C2214yes—
28C2×#387C2214yes—
29C2×#386C2214yes—
30C2×#386C2214yes—
31C2×#386C2214yes—
32C2×#385C2214yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C22⋊C4C2212yes—
35C2×C4×C22⋊C4C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×#376C2211yescharacteristic
41C2×#376C2211yescharacteristic
42C2×C43C2211yescharacteristic
43#1442C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1438C2214yes—
52#1438C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59C4⋊#95C2214yes—
60#1434C2214yes—
61#1434C2214yes—
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C4×#95C2211yescharacteristic
65C4×#95C2212yes—
66C4×#95C2212yes—
67C4×#95C2212yes—
68C4×#95C2211yescharacteristic
69C23×D4C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C24×C4C2314yes—
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×D4C2×C418yes—
83C2×C4×D4C2×C4116yescomplemented
84C2×C4×D4C2×C418yescomplemented
85C2×C4×D4C2×C4116yescomplemented
86C2×#117—28no—
87C2×#117—44no—
88C2×#117—44no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C42—28no—
109C22×C42—44no—
110C22×C42—44no—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2314yes—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2314yes—
119C22×C42C2311yescharacteristic
120#399C2×C4116yescomplemented
121#399C2×C4116yescomplemented
122#392C2×C4116yes—
123#392C2×C4116yescomplemented
124#390—216no—
125#388—216no—
126#387—216no—
127#386—216no—
128#386—216no—
129#386—216no—
130#385—216no—
131C4×C4⋊C4—28no—
132C4×C22⋊C4—28no—
133C4×C22⋊C4—28no—
134C4×C22⋊C4—44no—
135C4×C22⋊C4—44no—
136C4×C22⋊C4—44no—
137C4×C22⋊C4—44no—
138#376C2×C418yescomplemented
139#376D418yescomplemented
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C43—24no—
159C25C2412yes—
160C25C2412yes—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C22×D4C22×C414yes—
174C22×D4C22×C418yes—
175C22×D4C22×C418yes—
176C22×D4C22×C414yes—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4C22×C418yes—
197C23×C4C22×C418yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C4×D4—264no—
210C4×D4—416no—
211C4×D4—416no—
212C4×D4—432no—
213#117—432no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4C22×C418yes—
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C22×C418yes—
229C2×C4⋊C4C22×C414yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
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—28no—
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—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4C22×C418yes—
263C2×C22⋊C4C22×C418yes—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C22×C418yes—
267C2×C22⋊C4C22×C418yes—
268C2×C22⋊C4#41116yes—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—28no—
276C2×C42—28no—
277C2×C42—24no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—432no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42C22×C414yes—
290C2×C42C22×C414yes—
291C2×C42C22×C414yes—
292C2×C42C2×D418yes—
293C2×C42C2×D414yes—
294C2×C42#4114yes—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95C22×C418yes—
311#95C22×C418yes—
312#95C2×D4116yes—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24C23×C414yes—
333C24C23×C414yes—
334C24C2×#4118yes—
335C24C2511yescharacteristic
336C2×D4—216no—
337C2×D4—216no—
338C2×D4—216no—
339C2×D4—216no—
340C2×D4—44no—
341C2×D4—48no—
342C2×D4—44no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—416no—
347C2×D4—416no—
348C2×D4—416no—
349C2×D4—416no—
350C2×D4—416no—
351C2×D4—416no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
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—216no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—216no—
396C22×C4—216no—
397C22×C4—216no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—416no—
409C22×C4—416no—
410C22×C4—416no—
411C22×C4—416no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4C23×C412yes—
421C22×C4C23×C414yes—
422C22×C4C23×C412yes—
423C22×C4C23×C414yes—
424C22×C4C23×C414yes—
425C22×C4C23×C414yes—
426C22×C4C23×C412yes—
427C22×C4C22×D418yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C2×#4118yes—
431C22×C4C2×#4118yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4112yes—
434C22×C4H(5,𝔽2)14yes—
435C22×C4H(5,𝔽2)14yes—
436C4⋊C4—216no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—416no—
441C4⋊C4—416no—
442C22⋊C4—216no—
443C22⋊C4—216no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—416no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C42—28no—
456C42—28no—
457C42—216no—
458C42—416no—
459C42—416no—
460C42—48no—
461C42—48no—
462C42—48no—
463C42C4×D4116yes—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23C4×#4118yes—
510C23C24×C411yescharacteristic
511C23C23×D411yescharacteristic
512C23C22×#4111yescharacteristic
513C23C22×#4114yes—
514C23C2×H(5,𝔽2)12yes—
515C23C2×H(5,𝔽2)12yes—
516C23#58514yes—
517D4—432no—
518D4—416no—
519D4—432no—
520D4—416no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—464no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
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—432no—
573C2×C4—432no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4C2×C4×D418yes—
582C2×C4C2×C4×D414yes—
583C2×C4#51814yes—
584C2×C4#51814yes—
585C2×C4#53518yes—
586C2×C4#54018yes—
587C22—28no—
588C22—28no—
589C22—28no—
590C22—44no—
591C22—48no—
592C22—48no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—48no—
597C22—48no—
598C22—44no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22C22×C4×D411yescharacteristic
610C22C2×C4×#4112yes—
611C22C2×#51811yescharacteristic
612C22C2×#51811yescharacteristic
613C22#342812yes—
614C22C2×#53512yes—
615C22C2×#54012yes—
616C22#345914yes—
617C22#346014yes—
618C22D4×#4114yes—
619C22#347514yes—
620C22#348114yes—
621C22#352914yes—
622C4—28no—
623C4—44no—
624C4—44no—
625C4—48no—
626C4—44no—
627C4—42no—
628C4—42no—
629C4—48no—
630C4—48no—
631C4—84no—
632C4#232818yes—
633C2—48no—
634C2—84no—
635C2C2×#232811yescharacteristic
636C2#3408212yes—
637C2#3413612yes—
638C2#3415211yescharacteristic
639C2#3434211yescharacteristic
640C2#5216414yes—
641C2#5224914yes—
subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC2×#2397C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2290C2semidirect
3. #14584C2semidirect
4. #14578C2semidirect
5. #14520C2semidirect
6. #14519C2semidirect
7. #13854C2semidirect
8. #13839C2semidirect
9. #13839C2semidirect
10. C4×#390C2semidirect
11. C4×#386C2semidirect
12. C4×#386C2semidirect
13. #13757C2semidirect
20. #2397C4semidirect
83. C2×C4×D4C2×C4semidirect
84. C2×C4×D4C2×C4semidirect
85. C2×C4×D4C2×C4semidirect
120. #399C2×C4semidirect
121. #399C2×C4semidirect
123. #392C2×C4semidirect
138. #376C2×C4semidirect
139. #376D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23972
C2×#22902
#145842
#145782
#145202
#145192
#138542
#138392
C4×#3902
C4×#3862
#137572