Finite Group Databasebeta

← #4375189

Subgroups — #4375189

Order 512.

2252 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C2×#2326C211yesmaximal, characteristic
3#16815C212yesmaximal, complemented
4#16591C214yesmaximal, complemented
5#16394C212yesmaximal, complemented
6#16074C212yesmaximal
7#15884C212yesmaximal, complemented
8#15798C212yesmaximal, complemented
9#15784C212yesmaximal
10#15121C212yesmaximal, complemented
11#14884C212yesmaximal, complemented
12#14843C212yesmaximal, complemented
13#14841C212yesmaximal, complemented
14#14542C214yesmaximal, complemented
15#13940C211yesmaximal, characteristic
16C22×#124C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2487—28no—
21C2×#400C2212yes—
22C2×#400C2212yes—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#397C2214yes—
26#2326—28no—
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#388C2212yes—
31C2×#388C2212yes—
32C2×#386C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#384C2211yescharacteristic
37C2×#381C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×#376C2211yescharacteristic
42#1446C2214yes—
43#1445C2212yes—
44#1445C2212yes—
45#1444C2212yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2212yes—
56#1441C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1440C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1438C2212yes—
65#1437C2212yes—
66#1436C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73#1434C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C2×#124—24no—
83C2×#124—28no—
84C2×C4×Q8—24no—
85C2×C4×Q8—42no—
86C2×C4×Q8—42no—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C2×C4×D4D418yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—42no—
92C22×C4⋊C4—42no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2312yes—
118C22×C42C2312yes—
119#400—28no—
120#400—44no—
121#400—44no—
122#399—28no—
123#397—216no—
124#391—24no—
125#390—24no—
126#389—28no—
127#388—28no—
128#388—28no—
129#386—28no—
130#385—28no—
131#385—28no—
132#385—28no—
133#384—24no—
134#381—42no—
135#380—28no—
136#380—24no—
137C4×C22⋊C4—42no—
138#376—24no—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
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—22no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4C2×D414yes—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—44no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4C2×D414yes—
191C23×C4C2×D414yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205#124—232no—
206C4×Q8—416no—
207C4×D4—416no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
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—28no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—416no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C2×D414yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4C2×D414yes—
262C2×C22⋊C4C2×D414yes—
263C2×C22⋊C4#4118yes—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—44no—
275C2×C42C2×D414yes—
276#95—24no—
277#95—24no—
278#95—28no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—24no—
287#95—24no—
288#95—28no—
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—21no—
305C24—21no—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—22no—
310C24—21no—
311C24—21no—
312C24—22no—
313C24—24no—
314C24—24no—
315C24C22×D412yes—
316C24C22×D412yes—
317C24C2×#4114yes—
318C24C2511yescharacteristic
319C2×Q8—28no—
320C2×Q8—48no—
321C2×Q8—48no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×D4—28no—
325C2×D4—28no—
326C2×D4—44no—
327C2×D4—48no—
328C2×D4—48no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
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—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4—88no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—88no—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C2×#4114yes—
409C22×C4C2×#4114yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C42−1+414yes—
413C4⋊C4—44no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—44no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4—88no—
422C4⋊C4—88no—
423C22⋊C4—216no—
424C22⋊C4—216no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—84no—
434C42—28no—
435C42—28no—
436C42—48no—
437C42—44no—
438C42—84no—
439C23—22no—
440C23—22no—
441C23—24no—
442C23—24no—
443C23—28no—
444C23—24no—
445C23—24no—
446C23—28no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23#52514yes—
469C23C23×D411yescharacteristic
470C23C22×#4112yes—
471C23C2×H(5,𝔽2)11yescharacteristic
472C23C2×H(5,𝔽2)11yescharacteristic
473C23C2×2−1+412yes—
474C23#58514yes—
475C23#58514yes—
476Q8—416no—
477D4—416no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—28no—
496C2×C4—416no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
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—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—88no—
549C2×C4—816no—
550C2×C4#53414yes—
551C2×C4#53614yes—
552C22—24no—
553C22—28no—
554C22—48no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—48no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—44no—
563C22—44no—
564C22—84no—
565C22—84no—
566C22C2×#52511yescharacteristic
567C22C2×#53411yescharacteristic
568C22C2×#53611yescharacteristic
569C22#348512yes—
570C22#350012yes—
571C22#351214yes—
572C22#352012yes—
573C22#352012yes—
574C22#352114yes—
575C22#353312yes—
576C22#353312yes—
577C22#354012yes—
578C22#355112yes—
579C22#355714yes—
580C22#356514yes—
581C4—24no—
582C4—24no—
583C4—44no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—44no—
588C4—84no—
589C4—84no—
590C4—82no—
591C4—82no—
592C4—84no—
593C4—84no—
594C2—44no—
595C2—82no—
596C2#3508411yescharacteristic
597C2#3603212yes—
598C2#3695511yescharacteristic
599C2#3741911yescharacteristic
600C2#4264712yes—
601C2#4713014yes—
602C2#5143514yes—
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. #16815C2semidirect
4. #16591C2semidirect
5. #16394C2semidirect
7. #15884C2semidirect
8. #15798C2semidirect
10. #15121C2semidirect
11. #14884C2semidirect
12. #14843C2semidirect
13. #14841C2semidirect
14. #14542C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×#23262
#168152
#165912
#163942
#160742
#158842
#157982
#157842
#151212
#148842
#148432
#148412
#145422
#139402