Finite Group Databasebeta

← #4607307

Subgroups — #4607307

Order 512.

2540 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2548C211yesmaximal, characteristic
3#17098C214yesmaximal, complemented
4#16599C212yesmaximal, complemented
5#16559C212yesmaximal, complemented
6#16374C214yesmaximal, complemented
7#16273C214yesmaximal, complemented
8#15784C212yesmaximal, complemented
9#15271C214yesmaximal, complemented
10#15261C212yesmaximal
11#15097C212yesmaximal, complemented
12#14884C212yesmaximal, complemented
13#14667C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×#123C2212yes—
18C22×C4×Q8C2211yescharacteristic
19#2804—28no—
20#2548—28no—
21C2×#400C2212yes—
22C2×#400C2212yes—
23C2×#398C2212yes—
24C2×#397C2214yes—
25C2×#395C2212yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#393C2214yes—
29C2×#393C2214yes—
30C2×#391C2212yes—
31C2×#390C2214yes—
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#389C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#385C2214yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39#1446C2214yes—
40#1446C2212yes—
41#1445C2212yes—
42#1442C2214yes—
43#1442C2214yes—
44#1441C2212yes—
45#1441C2212yes—
46#1441C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2212yes—
62#1438C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1434C2212yes—
67#1434C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C23×D4C2312yes—
74C23×D4C2311yescharacteristic
75C24×C4C2312yes—
76C2×#127—24no—
77C2×#127—42no—
78C2×#127—42no—
79C2×#124—24no—
80C2×#124—24no—
81C2×#124—44no—
82C2×#124—44no—
83C2×#123—44no—
84C2×#123—48no—
85C2×C4×Q8—24no—
86C2×C4×Q8—42no—
87C2×C4×Q8—42no—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113#400—28no—
114#400—44no—
115#398—44no—
116#397—48no—
117#395—44no—
118#394—216no—
119#394—48no—
120#393—216no—
121#393—48no—
122#391—28no—
123#390—24no—
124#390—216no—
125#390D418yes—
126#389—44no—
127#386—28no—
128#386—44no—
129#385—216no—
130C4×C4⋊C4—24no—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C25C2412yes—
153C25C2412yes—
154C22×Q8—22no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—44no—
161C22×Q8—42no—
162C22×Q8—44no—
163C22×Q8—42no—
164C22×D4—24no—
165C22×D4—24no—
166C22×D4—22no—
167C22×D4—24no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4—42no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4C2×D414yes—
184C23×C4—24no—
185C23×C4—42no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—42no—
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200#127—416no—
201#124—416no—
202#124—416no—
203#123—432no—
204C4×Q8—416no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C2×D414yes—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
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—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4C2×D4116yes—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C2×D414yes—
274#95—24no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—24no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—42no—
290#95—44no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95—42no—
295C24—24no—
296C24—24no—
297C24—22no—
298C24—22no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—22no—
304C24—24no—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24C22×D414yes—
313C24C2511yescharacteristic
314C2×Q8—28no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—44no—
318C2×Q8—48no—
319C2×Q8—44no—
320C2×Q8—84no—
321C2×Q8—84no—
322C2×Q8—88no—
323C2×Q8—88no—
324C2×D4—28no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—416no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—48no—
333C2×D4—44no—
334C2×D4—48no—
335C2×D4—84no—
336C2×D4—84no—
337C2×D4—84no—
338C2×D4—84no—
339C2×D4—84no—
340C2×D4—88no—
341C2×D4—84no—
342C2×D4—88no—
343C2×D4—88no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
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—22no—
389C22×C4—24no—
390C22×C4—24no—
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—24no—
400C22×C4—48no—
401C22×C4—84no—
402C22×C4—88no—
403C22×C4—84no—
404C22×C4—88no—
405C22×C4C22×D414yes—
406C22×C4C22×D414yes—
407C22×C4C22×D412yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C4H(5,𝔽2)14yes—
410C22×C4H(5,𝔽2)14yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C42−1+414yes—
413C4⋊C4—44no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—416no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—88no—
421C22⋊C4—416no—
422C22⋊C4—416no—
423C22⋊C4—416no—
424C22⋊C4—48no—
425C22⋊C4—416no—
426C22⋊C4—88no—
427C22⋊C4—84no—
428C22⋊C4—88no—
429C22⋊C4—84no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C42—28no—
435C42—44no—
436C42—48no—
437C42—44no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—28no—
442C23—28no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23C23×D411yescharacteristic
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×H(5,𝔽2)11yescharacteristic
483C23C2×H(5,𝔽2)12yes—
484C23C2×H(5,𝔽2)12yes—
485C23C2×2−1+412yes—
486C23#58514yes—
487C23#58512yes—
488Q8—416no—
489Q8—816no—
490D4—88no—
491D4—816no—
492D4—88no—
493D4—816no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
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—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—816no—
557C2×C4#53614yes—
558C2×C4H(3,𝔽4)14yes—
559C22—44no—
560C22—44no—
561C22—48no—
562C22—44no—
563C22—48no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22—82no—
569C22—84no—
570C22—82no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22—82no—
575C22—84no—
576C22—84no—
577C22C2×#53611yescharacteristic
578C22#348411yescharacteristic
579C22#348411yescharacteristic
580C22#348512yes—
581C22#348812yes—
582C22C2×H(3,𝔽4)11yescharacteristic
583C22#353312yes—
584C22#353612yes—
585C22#353612yes—
586C22#354012yes—
587C22#354511yescharacteristic
588C22#354614yes—
589C22#355414yes—
590C22#355714yes—
591C22#355812yes—
592C22#356214yes—
593C4—24no—
594C4—24no—
595C4—44no—
596C4—42no—
597C4—44no—
598C4—44no—
599C4—84no—
600C4—84no—
601C4—82no—
602C4—84no—
603C4—84no—
604C2—84no—
605C2—162no—
606C2#3671811yescharacteristic
607C2#4267812yes—
608C2#4597711yescharacteristic
609C2#4955112yes—
610C2#5205411yescharacteristic
611C2#5308012yes—
612C2#5387114yes—
613C2#5778312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2804C2semidirect
3. #17098C2semidirect
4. #16599C2semidirect
5. #16559C2semidirect
6. #16374C2semidirect
7. #16273C2semidirect
8. #15784C2semidirect
9. #15271C2semidirect
11. #15097C2semidirect
12. #14884C2semidirect
13. #14667C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28042
C2×#25482
#170982
#165992
#165592
#163742
#162732
#157842
#152712
#152612
#150972
#148842
#146672