Finite Group Databasebeta

← #4866448

Subgroups — #4866448

Order 512.

1974 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2599C211yesmaximal, characteristic
2C2×#2628C211yesmaximal, characteristic
3#17227C212yesmaximal, complemented
4#17144C212yesmaximal, complemented
5#17048C212yesmaximal, complemented
6#16962C212yesmaximal, complemented
7#16831C212yesmaximal
8#16823C212yesmaximal
9#16617C214yesmaximal, complemented
10#16560C212yesmaximal, complemented
11#16394C212yesmaximal
12#15828C214yesmaximal
13#14884C212yesmaximal, complemented
14#14826C212yesmaximal
15#13958C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#2628—28no—
20#2599—28no—
21C2×#400C2212yes—
22C2×#400C2212yes—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#398C2211yescharacteristic
26C2×#397C2211yescharacteristic
27C2×#395C2212yes—
28C2×#395C2211yescharacteristic
29C2×#393C2212yes—
30C2×#393C2211yescharacteristic
31C2×#389C2212yes—
32C2×#388C2212yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#385C2212yes—
36C2×#383C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×#381C2211yescharacteristic
39C2×#380C2212yes—
40C22×#95C2211yescharacteristic
41#1446C2214yes—
42#1446C2214yes—
43#1446C2214yes—
44#1445C2212yes—
45#1445C2212yes—
46#1445C2212yes—
47#1445C2214yes—
48#1445C2212yes—
49#1444C2212yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1440C2214yes—
68#1439C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1434C2212yes—
72#1434C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×Q8C2311yescharacteristic
78C23×Q8C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8—42no—
84C2×C4×Q8—24no—
85C2×C4×Q8—42no—
86C2×C4×Q8—42no—
87C2×#117—42no—
88C2×#117—42no—
89C2×#117D418yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2312yes—
110#400—28no—
111#400—44no—
112#400—44no—
113#399—28no—
114#398—24no—
115#397—24no—
116#395—24no—
117#395—28no—
118#393—24no—
119#393—28no—
120#389—28no—
121#388—28no—
122#387—28no—
123#387—28no—
124#385—28no—
125#383—24no—
126#382—24no—
127#381—42no—
128#380—28no—
129C2×#95—42no—
130C2×#95C2312yes—
131C2×#95C2311yescharacteristic
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2311yescharacteristic
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C25C2411yescharacteristic
166C22×Q8—22no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—41no—
170C22×Q8—41no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8—44no—
175C22×Q8—44no—
176C23×C4—24no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—24no—
180C23×C4—42no—
181C23×C4—44no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4C2×D414yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C4⋊Q8—416no—
201C4×Q8—416no—
202#117—416no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4C2×D414yes—
250C2×C22⋊C4C2×D414yes—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C2×D414yes—
259C2×C42C2×D414yes—
260#95—28no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—22no—
266#95—22no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—28no—
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#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—42no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—24no—
299C24—22no—
300C24—22no—
301C24C22×D412yes—
302C242−1+414yes—
303C24C2511yescharacteristic
304C2×Q8—48no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—48no—
308C2×Q8—48no—
309C2×Q8—44no—
310C2×Q8—88no—
311C2×Q8—84no—
312C2×Q8—84no—
313C2×Q8—88no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—28no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
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—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—44no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—84no—
390C22×C4—88no—
391C22×C4—84no—
392C22×C4—88no—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C42−1+414yes—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—88no—
412C4⋊C4—88no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—88no—
418C22⋊C4—84no—
419C42—28no—
420C42—28no—
421C42—48no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—28no—
429C23—28no—
430C23—41no—
431C23—42no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—44no—
436C23—44no—
437C23#53614yes—
438C23C23×D411yescharacteristic
439C23C2×H(5,𝔽2)11yescharacteristic
440C23C2×H(5,𝔽2)11yescharacteristic
441C23C2×2−1+412yes—
442C23C2×2−1+412yes—
443C23#58512yes—
444C23#58512yes—
445C23#58514yes—
446Q8—416no—
447Q8—816no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—416no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
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—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—816no—
527C2×C4#53414yes—
528C22—24no—
529C22—28no—
530C22—44no—
531C22—42no—
532C22—48no—
533C22—44no—
534C22—44no—
535C22—42no—
536C22C2×#53411yescharacteristic
537C22C2×#53611yescharacteristic
538C22#344911yescharacteristic
539C22#348912yes—
540C22#350012yes—
541C22#353512yes—
542C22#353512yes—
543C22#353811yescharacteristic
544C22#354012yes—
545C22#354011yescharacteristic
546C22#354112yes—
547C22#355112yes—
548C22#355714yes—
549C22#356314yes—
550C22#356414yes—
551C22#356814yes—
552C4—24no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—44no—
559C4—84no—
560C4—82no—
561C4—84no—
562C4—84no—
563C4—82no—
564C4—82no—
565C4—84no—
566C2—44no—
567C2#3737411yescharacteristic
568C2#3741911yescharacteristic
569C2#3744711yescharacteristic
570C2#4215012yes—
571C2#4215612yes—
572C2#5845814yes—
573C2#5875512yes—
574C2#5891512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17227C2semidirect
4. #17144C2semidirect
5. #17048C2semidirect
6. #16962C2semidirect
9. #16617C2semidirect
10. #16560C2semidirect
13. #14884C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25992
C2×#26282
#172272
#171442
#170482
#169622
#168312
#168232
#166172
#165602
#163942
#158282
#148842
#148262
#139582