Finite Group Databasebeta

← #4452742

Subgroups — #4452742

Order 512.

2550 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C22×#398C211yesmaximal, characteristic
3#17073C214yesmaximal, complemented
4#17036C214yesmaximal, complemented
5#16907C214yesmaximal, complemented
6#16827C214yesmaximal, complemented
7#15092C212yesmaximal
8#15014C211yesmaximal, characteristic
9#14914C211yesmaximal, characteristic
10#14821C211yesmaximal, characteristic
11#14786C212yesmaximal
12#14720C211yesmaximal, characteristic
13#13962C212yesmaximal
14#13962C212yesmaximal
15#13795C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2627—28no—
23C2×#400C2212yes—
24C2×#398—28no—
25C2×#398C2214yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#398C2212yes—
29C2×#398C2212yes—
30C2×#398C2214yes—
31C2×#395C2212yes—
32C2×#395C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#381C2212yes—
39C2×#380C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43#1446C2212yes—
44#1445C2214yes—
45#1445C2212yes—
46#1444C2212yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2212yes—
51#1444C2214yes—
52#1442C2214yes—
53#1441C2212yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70C22⋊#95C2212yes—
71C22⋊#95C2211yescharacteristic
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
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—
87C2×#117—44no—
88C2×#117—44no—
89C2×#117Q8116yescomplemented
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4D418yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2314yes—
120#400—28no—
121#398—28no—
122#398—28no—
123#398—216no—
124#398—216no—
125#398—48no—
126#398—44no—
127#395—28no—
128#395—44no—
129#385—28no—
130#385—44no—
131#385—44no—
132#385—44no—
133#385—44no—
134#381—28no—
135#380—24no—
136C2×#95—24no—
137C2×#95—24no—
138C2×#95—24no—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25C2411yescharacteristic
169C22×Q8—22no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—42no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×D414yes—
203C23×C4C2×Q818yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C22⋊Q8—416no—
219#117—432no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
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—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—416no—
251C2×C4⋊C4—416no—
252C2×C4⋊C4—416no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×Q8116yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4C2×Q818yes—
281C2×C22⋊C4C2×Q818yes—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42C2×Q8116yes—
285#95—28no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—42no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—44no—
311#95—42no—
312#95—44no—
313#95—44no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24C22×D412yes—
319C24C22×Q814yes—
320C24H(5,𝔽2)14yes—
321C24H(5,𝔽2)12yes—
322C242−1+412yes—
323C24C2511yescharacteristic
324C2×Q8—44no—
325C2×Q8—82no—
326C2×Q8—82no—
327C2×Q8—88no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—216no—
331C22×C4—216no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—28no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—48no—
407C22×C4—416no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—416no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—88no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—88no—
424C22×C4C4⋊Q8116yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C22×Q818yes—
430C22×C4C22×Q818yes—
431C22×C4C22×Q814yes—
432C22×C4C22×Q814yes—
433C22×C4H(5,𝔽2)14yes—
434C22×C42−1+414yes—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—416no—
438C4⋊C4—416no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—416no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—82no—
445C4⋊C4—84no—
446C4⋊C4—82no—
447C4⋊C4—84no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C4⋊C4—88no—
452C22⋊C4—216no—
453C22⋊C4—216no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C42—416no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—28no—
462C23—28no—
463C23—24no—
464C23—24no—
465C23—28no—
466C23—28no—
467C23—24no—
468C23—24no—
469C23C2×C4⋊Q814yes—
470C23#53414yes—
471C23#53714yes—
472C23#54318yes—
473C23#54418yes—
474C23C23×D411yescharacteristic
475C23C23×Q812yes—
476C23C2×H(5,𝔽2)12yes—
477C23C2×H(5,𝔽2)11yescharacteristic
478C23C2×H(5,𝔽2)12yes—
479C23C2×2−1+411yescharacteristic
480C23C2×2−1+412yes—
481C23#58512yes—
482C23#58512yes—
483Q8—88no—
484C2×C4—232no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—416no—
512C2×C4—416no—
513C2×C4—416no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—88no—
564C2×C4—88no—
565C2×C4—88no—
566C2×C4C2×C4⋊Q818yes—
567C22—216no—
568C22—216no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22#265918yes—
576C22C22×C4⋊Q811yescharacteristic
577C22C2×#53411yescharacteristic
578C22C2×#53711yescharacteristic
579C22C2×#54312yes—
580C22C2×#54412yes—
581C22#348411yescharacteristic
582C22#348511yescharacteristic
583C22#348611yescharacteristic
584C22#348711yescharacteristic
585C22#350314yes—
586C22#350414yes—
587C22#353412yes—
588C22#353514yes—
589C22#353512yes—
590C22#354214yes—
591C22#355912yes—
592C22#356312yes—
593C4—28no—
594C4—48no—
595C4—48no—
596C4—44no—
597C4—44no—
598C4—42no—
599C4—42no—
600C4—44no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—82no—
607C2—28no—
608C2C2×#265911yescharacteristic
609C2#3557611yescharacteristic
610C2#3564011yescharacteristic
611C2#3671511yescharacteristic
612C2#3672211yescharacteristic
613C2#3696811yescharacteristic
614C2#3718611yescharacteristic
615C2#4199814yes—
616C2#4200014yes—
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. #17073C2semidirect
4. #17036C2semidirect
5. #16907C2semidirect
6. #16827C2semidirect
89. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C22×#3982
#170732
#170362
#169072
#168272
#150922
#150142
#149142
#148212
#147862
#147202
#139622
#137952