Finite Group Databasebeta

← #582449

Subgroups — #582449

Order 512.

2706 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2641C211yesmaximal, characteristic
2#14641C214yesmaximal, complemented
3#14622C214yesmaximal, complemented
4#14540C214yesmaximal, complemented
5#14535C214yesmaximal, complemented
6#14348C211yesmaximal, characteristic, complemented
7#14054C212yesmaximal, complemented
8#14025C212yesmaximal
9#14022C212yesmaximal, complemented
10#14018C212yesmaximal, complemented
11#14002C212yesmaximal, complemented
12#13977C212yesmaximal, complemented
13#13854C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×C4×D4C2212yes—
17#2641C4116yescomplemented
18C2×#394C2214yes—
19C2×#394C2214yes—
20C2×#390C2212yes—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#388C2214yes—
27C2×#386C2214yes—
28C2×#386C2214yes—
29C2×#385C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×#376C2211yescharacteristic
38#1445C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2212yes—
42#1442C2212yes—
43#1442C2212yes—
44#1442C2212yes—
45#1441C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2214yes—
55#1438C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1436C2214yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2214yes—
65C22⋊#95C2214yes—
66C4×#95C2212yes—
67C4×#95C2212yes—
68C23×D4C2312yes—
69C23×D4C2314yes—
70C24×C4C2314yes—
71C2×#127—42no—
72C2×#127—42no—
73C2×#127C2×C418yes—
74C2×#125—42no—
75C2×#125—42no—
76C2×#125C2×C418yescomplemented
77C2×C4×D4—44no—
78C2×C4×D4—44no—
79C2×C4×D4—44no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
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×C42C2312yes—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2314yes—
107#394C2×C4132yescomplemented
108#394C2×C4132yescomplemented
109#390—44no—
110#390—44no—
111#390—44no—
112#390C2×C4116yescomplemented
113#390C2×C4116yescomplemented
114#389—44no—
115#388—48no—
116#386—48no—
117#386—48no—
118#385—48no—
119#385—48no—
120#385—48no—
121#384—42no—
122#382—42no—
123C4×C4⋊C4—42no—
124C4×C4⋊C4—42no—
125C4×C4⋊C4—44no—
126#376C2×C418yescomplemented
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C25C2414yes—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4—44no—
162C22×D4—44no—
163C22×D4C22×C418yes—
164C22×D4C22×C4116yes—
165C23×C4—28no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4C2412yes—
173C23×C4C2414yes—
174C23×C4C2412yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184#127—416no—
185#125—416no—
186C4×D4—432no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C22×C418yes—
212C2×C4⋊C4C22×C4116yes—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4C22×C4116yes—
243C2×C22⋊C4C22×C4116yes—
244C2×C22⋊C4C22×C4116yes—
245C2×C22⋊C4C22×C4116yes—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42C22×C414yes—
262C2×C42C22×C414yes—
263C2×C42C22×C414yes—
264#95—24no—
265#95—24no—
266#95—28no—
267#95—24no—
268#95—24no—
269#95—42no—
270#95—42no—
271#95—44no—
272#95—42no—
273#95—42no—
274#95—42no—
275#95—42no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95C22×C418yes—
286#95C22×C418yes—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24C23×C418yes—
302C24C2511yescharacteristic
303C2×D4—216no—
304C2×D4—216no—
305C2×D4—216no—
306C2×D4—416no—
307C2×D4—416no—
308C2×D4—416no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—88no—
314C2×D4—88no—
315C2×D4—88no—
316C2×D4—88no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—416no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4—88no—
389C22×C4—88no—
390C22×C4C23×C412yes—
391C22×C4C23×C412yes—
392C22×C4C23×C414yes—
393C22×C4C23×C414yes—
394C22×C4C23×C418yes—
395C22×C4C23×C412yes—
396C22×C4H(5,𝔽2)14yes—
397C22×C4H(5,𝔽2)14yes—
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)12yes—
402C22×C42−1+412yes—
403C22×C42−1+412yes—
404C4⋊C4—216no—
405C4⋊C4—416no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—88no—
413C4⋊C4—88no—
414C22⋊C4—416no—
415C22⋊C4—416no—
416C22⋊C4—416no—
417C22⋊C4—416no—
418C22⋊C4—416no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C22⋊C4—88no—
424C22⋊C4—88no—
425C42—28no—
426C42—28no—
427C42—28no—
428C42—48no—
429C42—48no—
430C42—88no—
431C23—28no—
432C23—28no—
433C23—28no—
434C23—28no—
435C23—28no—
436C23—28no—
437C23—28no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23C24×C411yescharacteristic
467C23C2×H(5,𝔽2)11yescharacteristic
468C23C2×H(5,𝔽2)11yescharacteristic
469C23C2×H(5,𝔽2)11yescharacteristic
470C23C2×H(5,𝔽2)11yescharacteristic
471C23C2×H(5,𝔽2)12yes—
472C23C2×H(5,𝔽2)12yes—
473C23C2×2−1+411yescharacteristic
474C23C2×2−1+411yescharacteristic
475C23#58512yes—
476C23#58512yes—
477D4—432no—
478D4—832no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—28no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—832no—
553C2×C4#51814yes—
554C2×C4#52014yes—
555C2×C4#52014yes—
556C2×C4H(3,𝔽4)14yes—
557C2×C4#56214yes—
558C2×C4#56214yes—
559C2×C4#56314yes—
560C22—48no—
561C22—48no—
562C22—48no—
563C22—48no—
564C22—48no—
565C22—48no—
566C22—48no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22C2×#51811yescharacteristic
576C22C2×#52011yescharacteristic
577C22C2×#52011yescharacteristic
578C22C4×H(5,𝔽2)12yes—
579C22C4×H(5,𝔽2)12yes—
580C22C2×H(3,𝔽4)11yescharacteristic
581C22C2×#56211yescharacteristic
582C22C2×#56211yescharacteristic
583C22C2×#56311yescharacteristic
584C22#353711yescharacteristic
585C22#353711yescharacteristic
586C22#353911yescharacteristic
587C22#353911yescharacteristic
588C22#354312yes—
589C22#354312yes—
590C22#354612yes—
591C22#354712yes—
592C22#355412yes—
593C22#355412yes—
594C22#355812yes—
595C22#355912yes—
596C22#356212yes—
597C22#356212yes—
598C4—42no—
599C4—42no—
600C4—44no—
601C4—48no—
602C4—44no—
603C4—44no—
604C4—44no—
605C4—84no—
606C4—82no—
607C4—82no—
608C4—84no—
609C4#237918yes—
610C2—88no—
611C2C2×#237911yescharacteristic
612C2#3441811yescharacteristic
613C2#3441811yescharacteristic
614C2#3443112yes—
615C2#3443812yes—
616C2#5298511yescharacteristic
617C2#5298611yescharacteristic
618C2#5299512yes—
619C2#5299712yes—
620C2#5307411yescharacteristic
621C2#5307611yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14641C2semidirect
3. #14622C2semidirect
4. #14540C2semidirect
5. #14535C2semidirect
6. #14348C2semidirect
7. #14054C2semidirect
9. #14022C2semidirect
10. #14018C2semidirect
11. #14002C2semidirect
12. #13977C2semidirect
13. #13854C2semidirect
17. #2641C4semidirect
76. C2×#125C2×C4semidirect
107. #394C2×C4semidirect
108. #394C2×C4semidirect
112. #390C2×C4semidirect
113. #390C2×C4semidirect
126. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26412
#146412
#146222
#145402
#145352
#143482
#140542
#140252
#140222
#140182
#140022
#139772
#138542