Finite Group Databasebeta

← #4308956

Subgroups — #4308956

Order 512.

2502 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic
3#16598C214yesmaximal, complemented
4#16597C214yesmaximal, complemented
5#16554C214yesmaximal, complemented
6#16553C214yesmaximal, complemented
7#15015C211yesmaximal, characteristic
8#14920C212yesmaximal
9#14914C211yesmaximal, characteristic
10#14822C211yesmaximal, characteristic
11#14728C212yesmaximal
12#14720C211yesmaximal, characteristic
13#14705C212yesmaximal
14#14342C212yesmaximal
15#14339C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2627—28no—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#389—28no—
31C2×#389C2214yes—
32C2×#389C2214yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42#1446C2212yes—
43#1445C2214yes—
44#1445C2214yes—
45#1445C2214yes—
46#1445C2214yes—
47#1445C2214yes—
48#1445C2212yes—
49#1444C2212yes—
50#1444C2214yes—
51#1444C2214yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—44no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—44no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4D418yes—
95C22×C4⋊C4Q818yescomplemented
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C42—24no—
116C22×C42—44no—
117C22×C42C2314yes—
118#400—28no—
119#400—44no—
120#398—28no—
121#398—28no—
122#398—48no—
123#395—28no—
124#395—44no—
125#389—216no—
126#389—216no—
127#385—28no—
128#385—44no—
129#385—44no—
130#381—28no—
131#381—28no—
132#380—24no—
133#380—44no—
134C2×#95—24no—
135C2×#95—24no—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C25C2411yescharacteristic
168C22×Q8—22no—
169C22×Q8—42no—
170C22×Q8—42no—
171C22×Q8—42no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4C2×D414yes—
199C23×C4C2×Q814yes—
200C23×C4C2×Q814yes—
201C23×C4C2×Q814yes—
202C23×C4#4114yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C22⋊Q8—416no—
218C2×C4⋊C4—232no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—416no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C2×D418yes—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×Q8116yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C42—28no—
274C2×C42—44no—
275C2×C42—44no—
276C2×C42—416no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—42no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—42no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24C22×D412yes—
313C24C22×Q812yes—
314C24C2×#4112yes—
315C24H(5,𝔽2)14yes—
316C242−1+414yes—
317C24C2511yescharacteristic
318C2×Q8—44no—
319C2×Q8—82no—
320C2×Q8—82no—
321C2×Q8—88no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—216no—
325C22×C4—216no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
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—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—416no—
403C22×C4—416no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4C22⋊Q8116yes—
417C22×C4C22×D414yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D414yes—
421C22×C4C22×Q814yes—
422C22×C4C22×Q814yes—
423C22×C4C22×Q814yes—
424C22×C4C2×#4114yes—
425C22×C4H(5,𝔽2)14yes—
426C22×C42−1+414yes—
427C4⋊C4—232no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—416no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C4⋊C4—82no—
436C4⋊C4—84no—
437C4⋊C4—82no—
438C4⋊C4—84no—
439C4⋊C4—88no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C22⋊C4—48no—
443C22⋊C4—48no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C42—416no—
447C23—24no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23C2×C22⋊Q814yes—
460C23#53414yes—
461C23#53614yes—
462C23#53918yes—
463C23#54418yes—
464C23C23×D411yescharacteristic
465C23C23×Q811yescharacteristic
466C23C22×#4111yescharacteristic
467C23C2×H(5,𝔽2)12yes—
468C23C2×H(5,𝔽2)12yes—
469C23C2×2−1+412yes—
470C23C2×2−1+412yes—
471C23#58512yes—
472C23#58512yes—
473Q8—88no—
474C2×C4—232no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—416no—
500C2×C4—416no—
501C2×C4—416no—
502C2×C4—416no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
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—48no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—48no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—88no—
553C2×C4—88no—
554C2×C4C2×C22⋊Q818yes—
555C22—24no—
556C22—216no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—24no—
561C22—216no—
562C22—24no—
563C22#262718yes—
564C22C22×C22⋊Q811yescharacteristic
565C22C2×#53411yescharacteristic
566C22C2×#53611yescharacteristic
567C22C2×#53912yes—
568C22C2×#54412yes—
569C22#346012yes—
570C22#346112yes—
571C22#348411yescharacteristic
572C22#348511yescharacteristic
573C22#348611yescharacteristic
574C22#348711yescharacteristic
575C22#350112yes—
576C22#350212yes—
577C22#353414yes—
578C22#353514yes—
579C22#355214yes—
580C22#355314yes—
581C4—28no—
582C4—44no—
583C4—44no—
584C4—44no—
585C4—44no—
586C4—42no—
587C4—44no—
588C4—42no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—84no—
593C4—84no—
594C4—82no—
595C2—28no—
596C2C2×#262711yescharacteristic
597C2#3515411yescharacteristic
598C2#3517411yescharacteristic
599C2#3711711yescharacteristic
600C2#3712011yescharacteristic
601C2#3713514yes—
602C2#3718611yescharacteristic
603C2#3723311yescharacteristic
604C2#3752514yes—
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. #16598C2semidirect
4. #16597C2semidirect
5. #16554C2semidirect
6. #16553C2semidirect
95. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C22×#3892
#165982
#165972
#165542
#165532
#150152
#149202
#149142
#148222
#147282
#147202
#147052
#143422
#143392