Finite Group Databasebeta

← #4607233

Subgroups — #4607233

Order 512.

2326 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2804C211yesmaximal, characteristic, complemented
2C2×#2504C211yesmaximal, characteristic
3#17142C214yesmaximal, complemented
4#16694C212yesmaximal, complemented
5#16689C212yesmaximal, complemented
6#16368C214yesmaximal, complemented
7#16307C214yesmaximal, complemented
8#16074C212yesmaximal, complemented
9#15269C214yesmaximal, complemented
10#15260C212yesmaximal
11#15050C212yesmaximal, complemented
12#14822C212yesmaximal, complemented
13#14667C211yesmaximal, characteristic, complemented
14C22×#123C2212yes—
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2804—28no—
18#2504—28no—
19C2×#400C2212yes—
20C2×#400C2212yes—
21C2×#397C2214yes—
22C2×#397C2212yes—
23C2×#394C2214yes—
24C2×#393C2214yes—
25C2×#393C2212yes—
26C2×#391C2212yes—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#386C2212yes—
30C2×#386C2214yes—
31C2×#385C2214yes—
32C2×#385C2212yes—
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#380C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1446C2214yes—
38#1446C2212yes—
39#1445C2214yes—
40#1445C2212yes—
41#1444C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2212yes—
53#1440C2214yes—
54#1440C2212yes—
55#1440C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2212yes—
62#1438C2212yes—
63#1434C2214yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2312yes—
75C2×#123—44no—
76C2×#123—48no—
77C2×#121—24no—
78C2×#121—42no—
79C2×#121—42no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4—42no—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105#400—28no—
106#400—44no—
107#397—216no—
108#397—44no—
109#394—48no—
110#393—216no—
111#393—44no—
112#391—28no—
113#390—24no—
114#390D418yes—
115#386—28no—
116#386—216no—
117#385—216no—
118#385—44no—
119#382—24no—
120#382—24no—
121#380—44no—
122C4×C22⋊C4—24no—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C25C2412yes—
149C25C2411yescharacteristic
150C22×Q8—24no—
151C22×Q8—44no—
152C22×Q8—44no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—28no—
158C22×D4—24no—
159C22×D4—41no—
160C22×D4—41no—
161C22×D4—42no—
162C22×D4—42no—
163C22×D4—44no—
164C22×D4C2×D414yes—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—42no—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191#123—432no—
192#121—416no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—42no—
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—416no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
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⋊C4C2×D414yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
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⋊C4C2×D4116yes—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—42no—
253C2×C42C2×D414yes—
254#95—24no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—24no—
261#95—28no—
262#95—28no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—42no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—44no—
277#95—42no—
278#95—44no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—24no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—22no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24C22×D414yes—
294C24H(5,𝔽2)14yes—
295C24H(5,𝔽2)12yes—
296C24C2511yescharacteristic
297C2×Q8—48no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8—88no—
301C2×Q8—88no—
302C2×D4—28no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—48no—
306C2×D4—416no—
307C2×D4—48no—
308C2×D4—84no—
309C2×D4—84no—
310C2×D4—84no—
311C2×D4—88no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—28no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4—48no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—88no—
379C22×C4—84no—
380C22×C4—84no—
381C22×C4—88no—
382C22×C4—88no—
383C22×C4—84no—
384C22×C4—88no—
385C22×C4C22×D414yes—
386C22×C4C22×D414yes—
387C22×C4C22×D412yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C42−1+414yes—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—416no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4—88no—
399C22⋊C4—416no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—416no—
405C22⋊C4—88no—
406C22⋊C4—84no—
407C22⋊C4—88no—
408C22⋊C4—88no—
409C22⋊C4—88no—
410C42—44no—
411C42—44no—
412C23—22no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—24no—
418C23—28no—
419C23—28no—
420C23—24no—
421C23—24no—
422C23—28no—
423C23—24no—
424C23—28no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23H(3,𝔽4)14yes—
449C23C23×D411yescharacteristic
450C23C2×H(5,𝔽2)11yescharacteristic
451C23C2×H(5,𝔽2)11yescharacteristic
452C23C2×H(5,𝔽2)12yes—
453C23C2×2−1+412yes—
454C23#58514yes—
455C23#58512yes—
456C23#58512yes—
457Q8—816no—
458D4—88no—
459D4—816no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—416no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—816no—
524C2×C4—816no—
525C2×C4#53614yes—
526C22—28no—
527C22—24no—
528C22—42no—
529C22—48no—
530C22—42no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—48no—
538C22—48no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—84no—
543C22C2×#53611yescharacteristic
544C22#348411yescharacteristic
545C22#348512yes—
546C22#348511yescharacteristic
547C22#348812yes—
548C22C2×H(3,𝔽4)11yescharacteristic
549C22#353312yes—
550C22#353512yes—
551C22#353512yes—
552C22#354012yes—
553C22#354511yescharacteristic
554C22#355812yes—
555C22#355914yes—
556C22#356014yes—
557C22#356514yes—
558C22#356614yes—
559C4—24no—
560C4—44no—
561C4—42no—
562C4—44no—
563C4—84no—
564C4—82no—
565C4—84no—
566C4—84no—
567C4—82no—
568C4—82no—
569C4—84no—
570C2—44no—
571C2—84no—
572C2#3672011yescharacteristic
573C2#4268212yes—
574C2#4597711yescharacteristic
575C2#5017012yes—
576C2#5205411yescharacteristic
577C2#5275312yes—
578C2#5385714yes—
579C2#5782412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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. #17142C2semidirect
4. #16694C2semidirect
5. #16689C2semidirect
6. #16368C2semidirect
7. #16307C2semidirect
8. #16074C2semidirect
9. #15269C2semidirect
11. #15050C2semidirect
12. #14822C2semidirect
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×#25042
#171422
#166942
#166892
#163682
#163072
#160742
#152692
#152602
#150502
#148222
#146672