Finite Group Databasebeta

← #4378652

Subgroups — #4378652

Order 512.

2260 conjugacy classes of subgroups.

The subgroup lattice has 607 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2322C211yesmaximal, characteristic
3#16560C212yesmaximal, complemented
4#16447C214yesmaximal, complemented
5#16394C212yesmaximal, complemented
6#16074C212yesmaximal
7#15983C212yesmaximal
8#15930C212yesmaximal, complemented
9#15568C212yesmaximal, complemented
10#15080C212yesmaximal, complemented
11#15063C212yesmaximal, complemented
12#14841C212yesmaximal, complemented
13#14822C212yesmaximal, complemented
14#14394C214yesmaximal, complemented
15#13940C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2563—28no—
21C2×#400C2212yes—
22C2×#400C2212yes—
23C2×#393C2212yes—
24C2×#393C2212yes—
25#2322—28no—
26C2×#390C2212yes—
27C2×#389C2212yes—
28C2×#388C2212yes—
29C2×#387C2212yes—
30C2×#387C2214yes—
31C2×#386C2211yescharacteristic
32C2×#385C2212yes—
33C2×#385C2211yescharacteristic
34C2×#382C2212yes—
35C2×#381C2211yescharacteristic
36C2×#381C2211yescharacteristic
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42#1446C2214yes—
43#1446C2214yes—
44#1445C2212yes—
45#1445C2212yes—
46#1444C2212yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2212yes—
54#1441C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1441C2212yes—
58#1440C2214yes—
59#1440C2214yes—
60#1440C2212yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1438C2212yes—
66#1437C2212yes—
67#1436C2212yes—
68#1436C2214yes—
69#1436C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73#1434C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—42no—
85C2×C4⋊Q8—42no—
86C2×#123—24no—
87C2×#123—28no—
88C2×C4×D4—42no—
89C2×C4×D4—42no—
90C2×C4×D4D418yes—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4—42no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C42C2312yes—
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2312yes—
118#400—44no—
119#400—44no—
120#393—28no—
121#393—28no—
122#390—28no—
123#389—28no—
124#388—28no—
125#387—216no—
126#387—28no—
127#386—24no—
128#385—28no—
129#385—24no—
130#382—28no—
131#381—24no—
132#381—42no—
133#380—28no—
134#380—28no—
135C4×C4⋊C4—24no—
136C4×C22⋊C4—24no—
137C4×C22⋊C4—42no—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C25C2411yescharacteristic
168C25C2411yescharacteristic
169C22×Q8—24no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—44no—
173C22×Q8—44no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4C2×D414yes—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4—41no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4C2×D414yes—
195C23×C4C2×D414yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C4⋊Q8—416no—
210#123—232no—
211C4×D4—416no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—416no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4#4118yes—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4C2×D414yes—
265C2×C22⋊C4C2×D414yes—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—28no—
270C2×C42—24no—
271C2×C42—42no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42C2×D414yes—
275#95—24no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—24no—
288#95—24no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—42no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304C24—22no—
305C24—21no—
306C24—21no—
307C24—24no—
308C24—24no—
309C24—22no—
310C24—21no—
311C24—21no—
312C24—22no—
313C24—24no—
314C24—24no—
315C24C22×D412yes—
316C24C22×D412yes—
317C242−1+414yes—
318C24C2511yescharacteristic
319C2×Q8—48no—
320C2×Q8—88no—
321C2×Q8—88no—
322C2×Q8—84no—
323C2×Q8—84no—
324C2×D4—28no—
325C2×D4—28no—
326C2×D4—44no—
327C2×D4—48no—
328C2×D4—48no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—28no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
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—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—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4—84no—
404C22×C4—84no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4114yes—
414C22×C4C2×#4114yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)12yes—
417C4⋊C4—216no—
418C4⋊C4—44no—
419C4⋊C4—48no—
420C4⋊C4—416no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—84no—
426C22⋊C4—28no—
427C22⋊C4—216no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—48no—
431C22⋊C4—44no—
432C22⋊C4—84no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—84no—
437C42—28no—
438C42—48no—
439C42—84no—
440C42—84no—
441C23—22no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—28no—
448C23—28no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—24no—
453C23—24no—
454C23—44no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—44no—
469C23—44no—
470C23#53614yes—
471C23C23×D411yescharacteristic
472C23C22×#4112yes—
473C23C2×H(5,𝔽2)11yescharacteristic
474C23C2×H(5,𝔽2)11yescharacteristic
475C23C2×2−1+412yes—
476C23#58514yes—
477C23#58514yes—
478Q8—816no—
479D4—416no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—416no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
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—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—88no—
551C2×C4—88no—
552C2×C4—88no—
553C2×C4#52514yes—
554C2×C4#53414yes—
555C22—24no—
556C22—28no—
557C22—48no—
558C22—42no—
559C22—48no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—44no—
565C22—44no—
566C22—42no—
567C22—84no—
568C22—84no—
569C22C2×#52511yescharacteristic
570C22C2×#53411yescharacteristic
571C22C2×#53611yescharacteristic
572C22#348512yes—
573C22#349314yes—
574C22#350012yes—
575C22#352012yes—
576C22#352012yes—
577C22#352414yes—
578C22#354012yes—
579C22#354512yes—
580C22#354512yes—
581C22#355112yes—
582C22#355314yes—
583C22#356414yes—
584C4—24no—
585C4—24no—
586C4—44no—
587C4—42no—
588C4—44no—
589C4—42no—
590C4—42no—
591C4—84no—
592C4—84no—
593C4—84no—
594C4—82no—
595C4—84no—
596C4—82no—
597C2—44no—
598C2—82no—
599C2#3508411yescharacteristic
600C2#3601112yes—
601C2#3711911yescharacteristic
602C2#3741911yescharacteristic
603C2#4137412yes—
604C2#4698414yes—
605C2#4941514yes—
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. #16560C2semidirect
4. #16447C2semidirect
5. #16394C2semidirect
8. #15930C2semidirect
9. #15568C2semidirect
10. #15080C2semidirect
11. #15063C2semidirect
12. #14841C2semidirect
13. #14822C2semidirect
14. #14394C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#23222
#165602
#164472
#163942
#160742
#159832
#159302
#155682
#150802
#150632
#148412
#148222
#143942
#139402