Finite Group Databasebeta

← #4942014

Subgroups — #4942014

Order 512.

2494 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2379C211yesmaximal, characteristic
3#16694C212yesmaximal, complemented
4#16307C214yesmaximal, complemented
5#16110C212yesmaximal, complemented
6#15783C212yesmaximal, complemented
7#15557C214yesmaximal, complemented
8#15341C214yesmaximal, complemented
9#15085C212yesmaximal, complemented
10#14728C214yesmaximal, complemented
11#14473C212yesmaximal
12#14473C211yesmaximal, characteristic
13#14015C212yesmaximal
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2212yes—
19#2547—28no—
20C2×#400C2212yes—
21C2×#400C2212yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#397C2214yes—
25#2379—28no—
26C2×#391C2212yes—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#388C2214yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#385C2212yes—
35C2×#382C2212yes—
36C2×#380C2214yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1446C2214yes—
39#1446C2212yes—
40#1445C2212yes—
41#1445C2214yes—
42#1445C2212yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1436C2214yes—
59#1436C2212yes—
60#1436C2212yes—
61#1434C2212yes—
62#1434C2214yes—
63#1434C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C24×C4C2312yes—
74C24×C4C2312yes—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—42no—
77C2×C4⋊Q8—42no—
78C2×#124—28no—
79C2×#124—24no—
80C2×#124—28no—
81C2×#124D418yes—
82C2×C4×D4—24no—
83C2×C4×D4—42no—
84C2×C4×D4—42no—
85C22×C4⋊C4—48no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108#400—44no—
109#400—44no—
110#398—44no—
111#398—44no—
112#397—216no—
113#391—28no—
114#391—24no—
115#391—24no—
116#388—216no—
117#386—28no—
118#386—28no—
119#385—216no—
120#385—44no—
121#385—48no—
122#382—28no—
123#380—216no—
124C4×C4⋊C4—24no—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C25C2412yes—
152C22×Q8—22no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8C2×D414yes—
162C22×D4—41no—
163C22×D4—41no—
164C22×D4—42no—
165C22×D4C2×D414yes—
166C22×D4#4118yes—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—28no—
171C23×C4—24no—
172C23×C4—28no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—42no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—42no—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C4⋊Q8—416no—
195#124—232no—
196#124—416no—
197C4×D4—416no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
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—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—432no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4C2×D4116yes—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—44no—
258C2×C42C2×D414yes—
259#95—24no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—42no—
278#95—42no—
279#95—44no—
280#95—44no—
281#95—42no—
282#95—42no—
283#95—44no—
284#95—44no—
285C24—22no—
286C24—24no—
287C24—22no—
288C24—24no—
289C24—24no—
290C24—22no—
291C24—24no—
292C24—22no—
293C24C22×D414yes—
294C24C2×#4114yes—
295C24C2×#4114yes—
296C24C2511yescharacteristic
297C2×Q8—28no—
298C2×Q8—44no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8—84no—
306C2×Q8—84no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4#123116yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
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—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
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—416no—
373C22×C4—416no—
374C22×C4—48no—
375C22×C4—416no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—84no—
381C22×C4—88no—
382C22×C4—84no—
383C22×C4—88no—
384C22×C4—84no—
385C22×C4—88no—
386C22×C4C22×D414yes—
387C22×C4C22×D414yes—
388C22×C4C22×D412yes—
389C22×C4C2×#4114yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C4H(5,𝔽2)14yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C42−1+418yes—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—48no—
397C4⋊C4—44no—
398C4⋊C4—416no—
399C4⋊C4—416no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—88no—
404C22⋊C4—232no—
405C22⋊C4—416no—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—416no—
409C22⋊C4—84no—
410C22⋊C4—88no—
411C42—28no—
412C42—28no—
413C42—44no—
414C42—84no—
415C23—24no—
416C23—24no—
417C23—28no—
418C23—28no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—28no—
424C23—28no—
425C23—28no—
426C23—28no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23C2×#12318yes—
438C23C23×D411yescharacteristic
439C23C22×#4112yes—
440C23C2×H(5,𝔽2)11yescharacteristic
441C23C2×H(5,𝔽2)11yescharacteristic
442C23C2×H(5,𝔽2)12yes—
443C23C2×2−1+414yes—
444C23#58512yes—
445C23#58512yes—
446Q8—88no—
447Q8—88no—
448D4—416no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—432no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—816no—
516C2×C4—816no—
517C2×C4C2×#12314yes—
518C2×C4H(3,𝔽4)14yes—
519C22—28no—
520C22—216no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—48no—
525C22—44no—
526C22—44no—
527C22—48no—
528C22—48no—
529C22—44no—
530C22C22×#12311yescharacteristic
531C22#348411yescharacteristic
532C22#348511yescharacteristic
533C22#348812yes—
534C22#349912yes—
535C22#350512yes—
536C22#350512yes—
537C22#351514yes—
538C22#352014yes—
539C22C2×H(3,𝔽4)11yescharacteristic
540C22#353812yes—
541C22#354114yes—
542C22#354412yes—
543C22#354511yescharacteristic
544C22#355812yes—
545C22#356514yes—
546C4—24no—
547C4—24no—
548C4—44no—
549C4—42no—
550C4—48no—
551C4—44no—
552C4—82no—
553C4—84no—
554C4—84no—
555C4—82no—
556C4—84no—
557C4—82no—
558C2—48no—
559C2#3535111yescharacteristic
560C2#3599812yes—
561C2#4566712yes—
562C2#4597711yescharacteristic
563C2#4853414yes—
564C2#4903212yes—
565C2#5106912yes—
566C2#5154211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16694C2semidirect
4. #16307C2semidirect
5. #16110C2semidirect
6. #15783C2semidirect
7. #15557C2semidirect
8. #15341C2semidirect
9. #15085C2semidirect
10. #14728C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#23792
#166942
#163072
#161102
#157832
#155572
#153412
#150852
#147282
#144732
#140152