Finite Group Databasebeta

← #2745800

Subgroups — #2745800

Order 512.

2014 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17306C212yesmaximal, complemented
2#17269C214yesmaximal, complemented
3#17132C212yesmaximal, complemented
4#16520C214yesmaximal
5#16305C212yesmaximal
6#15196C212yesmaximal, complemented
7#15187C212yesmaximal, complemented
8#15133C212yesmaximal, complemented
9#15131C212yesmaximal
10#15128C212yesmaximal, complemented
11#15125C211yesmaximal, characteristic
12#14636C212yesmaximal
13#13924C211yesmaximal, characteristic
14#13907C212yesmaximal
15C4×#399C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#399C2211yescharacteristic
19C2×#398C2214yes—
20C2×#396C2214yes—
21C2×#395C2214yes—
22C2×#393C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#384C2212yes—
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30C2×#380C2214yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35#1446C2212yes—
36#1446C2214yes—
37#1445C2212yes—
38#1445C2211yescharacteristic
39#1445C2214yes—
40#1445C2212yes—
41#1445C2212yes—
42#1444C2214yes—
43#1444C2211yescharacteristic
44#1444C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1439C2212yes—
61#1439C2214yes—
62#1439C2212yes—
63#1439C2211yescharacteristic
64C4⋊#95C2212yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C4×#95C2211yescharacteristic
75C23×Q8C2312yes—
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C2×C4⋊Q8—42no—
79C2×C4⋊Q8—44no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4Q818yescomplemented
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109#399Q818yescomplemented
110#398—48no—
111#396—48no—
112#395—48no—
113#393—44no—
114#389—44no—
115#389—44no—
116#389—44no—
117#389—44no—
118#384—44no—
119#380—48no—
120#380—44no—
121#380—42no—
122C4×C4⋊C4—44no—
123C4×C4⋊C4—42no—
124C4×C22⋊C4—42no—
125C4×C22⋊C4—44no—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2311yescharacteristic
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C25C2411yescharacteristic
157C22×Q8—24no—
158C22×Q8—42no—
159C22×Q8—44no—
160C22×Q8—42no—
161C22×Q8—44no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4C2×Q814yes—
171C23×C4C2×Q818yes—
172C23×C4C2414yes—
173C23×C4C2414yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181C23×C4C2414yes—
182C23×C4C2411yescharacteristic
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C4⋊Q8—416no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—416no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C2×Q818yes—
231C2×C4⋊C4C2×Q818yes—
232C2×C4⋊C4C2×Q818yes—
233C2×C4⋊C4C2×Q814yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4C2×Q818yes—
251C2×C22⋊C4C2×Q814yes—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262#95—28no—
263#95—42no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—42no—
268#95—41no—
269#95—42no—
270#95—42no—
271#95—41no—
272#95—41no—
273#95—42no—
274#95—42no—
275#95—42no—
276#95—44no—
277#95—44no—
278#95—42no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—42no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95C2×Q814yes—
292C24—22no—
293C24—22no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—21no—
299C24—22no—
300C24C22×Q812yes—
301C24C22×Q812yes—
302C24H(5,𝔽2)12yes—
303C24C2511yescharacteristic
304C2×Q8—48no—
305C2×Q8—84no—
306C2×Q8—88no—
307C2×Q8—88no—
308C2×Q8—84no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—216no—
351C22×C4—24no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
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—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—48no—
385C22×C4—416no—
386C22×C4—44no—
387C22×C4—48no—
388C22×C4—44no—
389C22×C4—84no—
390C22×C4—88no—
391C22×C4—84no—
392C22×C4C22×Q814yes—
393C22×C4C22×Q812yes—
394C22×C4C22×Q814yes—
395C22×C4C22×Q812yes—
396C22×C4C22×Q818yes—
397C22×C4C22×Q812yes—
398C22×C4C22×Q812yes—
399C22×C4H(5,𝔽2)14yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)12yes—
402C22×C4H(5,𝔽2)14yes—
403C22×C42−1+412yes—
404C22×C42−1+414yes—
405C4⋊C4—216no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—44no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C4⋊C4—88no—
418C4⋊C4—88no—
419C4⋊C4—88no—
420C4⋊C4—88no—
421C22⋊C4—48no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—48no—
425C22⋊C4—88no—
426C22⋊C4—84no—
427C42—48no—
428C42—44no—
429C42—48no—
430C42—44no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—24no—
436C23—22no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—24no—
443C23—41no—
444C23—42no—
445C23—41no—
446C23—42no—
447C23—41no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23Q8214yes—
454C23C23×Q811yescharacteristic
455C23C23×Q811yescharacteristic
456C23C2×H(5,𝔽2)11yescharacteristic
457C23C2×H(5,𝔽2)11yescharacteristic
458C23C2×H(5,𝔽2)12yes—
459C23C2×H(5,𝔽2)11yescharacteristic
460C23C2×H(5,𝔽2)12yes—
461C23C2×2−1+412yes—
462C23C2×2−1+411yescharacteristic
463C23#58512yes—
464C23#58511yescharacteristic
465Q8—816no—
466C2×C4—28no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—416no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—41no—
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—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—816no—
549C2×C4#54314yes—
550C2×C4#54418yes—
551C2×C4#56014yes—
552C2×C4#56218yes—
553C2×C4#56314yes—
554C22—28no—
555C22—24no—
556C22—42no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—42no—
565C22C2×#54311yescharacteristic
566C22C2×#54412yes—
567C22C2×Q8211yescharacteristic
568C22#350112yes—
569C22#350111yescharacteristic
570C22#350112yes—
571C22#350211yescharacteristic
572C22#350311yescharacteristic
573C22#350312yes—
574C22C2×#56011yescharacteristic
575C22C2×#56212yes—
576C22C2×#56311yescharacteristic
577C22#353612yes—
578C22#353611yescharacteristic
579C22#353711yescharacteristic
580C22#353712yes—
581C22#354612yes—
582C22#354712yes—
583C22#355111yescharacteristic
584C22#355112yes—
585C22#355512yes—
586C22#355811yescharacteristic
587C22#355912yes—
588C4—42no—
589C4—44no—
590C4—42no—
591C4—48no—
592C4—42no—
593C4—44no—
594C4—44no—
595C4—42no—
596C4—81no—
597C4—82no—
598C4—84no—
599C4—84no—
600C4—84no—
601C4#284118yes—
602C2—44no—
603C2C2×#284111yescharacteristic
604C2#3962312yes—
605C2#3962411yescharacteristic
606C2#4249011yescharacteristic
607C2#4249212yes—
608C2#4550112yes—
609C2#4550211yescharacteristic
610C2#4633011yescharacteristic
611C2#5296111yescharacteristic
612C2#5297012yes—
613C2#5298811yescharacteristic
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
1. #17306C2semidirect
2. #17269C2semidirect
3. #17132C2semidirect
6. #15196C2semidirect
7. #15187C2semidirect
8. #15133C2semidirect
10. #15128C2semidirect
82. C22×C4⋊C4Q8semidirect
109. #399Q8semidirect

Maximal subgroups

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

subgroupindex
#173062
#172692
#171322
#165202
#163052
#151962
#151872
#151332
#151312
#151282
#151252
#146362
#139242
#139072
C4×#3992