Finite Group Databasebeta

← #4866009

Subgroups — #4866009

Order 512.

2056 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2611C211yesmaximal, characteristic
2C2×#2621C211yesmaximal, characteristic
3#17229C212yesmaximal
4#17160C212yesmaximal
5#17075C212yesmaximal, complemented
6#17031C212yesmaximal, complemented
7#17013C212yesmaximal, complemented
8#16608C212yesmaximal, complemented
9#16603C212yesmaximal, complemented
10#16454C214yesmaximal
11#16264C212yesmaximal
12#15676C214yesmaximal, complemented
13#15119C212yesmaximal, complemented
14#14763C212yesmaximal
15#13939C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×#117C2211yescharacteristic
20#2621—28no—
21#2611—28no—
22C2×#400C2211yescharacteristic
23C2×#399C2211yescharacteristic
24C2×#398C2211yescharacteristic
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#397C2212yes—
29C2×#397C2211yescharacteristic
30C2×#397C2212yes—
31C2×#395C2212yes—
32C2×#395C2212yes—
33C2×#391C2212yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#388C2212yes—
37C2×#387C2212yes—
38C2×#385C2212yes—
39C2×#382C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×#376C2211yescharacteristic
43#1446C2212yes—
44#1446C2212yes—
45#1445C2214yes—
46#1445C2214yes—
47#1444C2212yes—
48#1444C2214yes—
49#1444C2214yes—
50#1444C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2212yes—
64#1440C2214yes—
65#1440C2212yes—
66#1440C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74#1434C2214yes—
75#1434C2212yes—
76#1434C2212yes—
77C23×Q8C2312yes—
78C24×C4C2311yescharacteristic
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8—42no—
82C2×#125—24no—
83C2×#125—42no—
84C2×#125—42no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—48no—
87C2×#117—42no—
88C2×#117—42no—
89C2×#117D418yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117#400—24no—
118#399—24no—
119#398—24no—
120#398—28no—
121#398—28no—
122#398—48no—
123#397—24no—
124#397—28no—
125#397—28no—
126#395—44no—
127#395—44no—
128#391—28no—
129#389—28no—
130#389—48no—
131#388—28no—
132#387—28no—
133#385—28no—
134#382—24no—
135#380—42no—
136C4×C22⋊C4—42no—
137#376—24no—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25C2411yescharacteristic
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—44no—
173C22×Q8—44no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—42no—
178C23×C4C2×D414yes—
179C23×C4C2412yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2412yes—
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C4⋊Q8—416no—
194#125—416no—
195C22⋊Q8—432no—
196#117—416no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D414yes—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4C2×D414yes—
260C2×C22⋊C4C2×D414yes—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42C2×D414yes—
272C2×C42C2×D414yes—
273#95—24no—
274#95—22no—
275#95—22no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
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—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303C24—21no—
304C24—21no—
305C24—22no—
306C24—22no—
307C24—24no—
308C24—22no—
309C24—22no—
310C24C22×D412yes—
311C24C2511yescharacteristic
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—88no—
316C2×Q8—88no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
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—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—44no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C4H(5,𝔽2)12yes—
397C22×C4H(5,𝔽2)14yes—
398C22×C42−1+418yes—
399C22×C42−1+414yes—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—48no—
403C4⋊C4—416no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—416no—
413C4⋊C4—84no—
414C4⋊C4—88no—
415C4⋊C4—88no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—416no—
422C22⋊C4—84no—
423C42—28no—
424C42—28no—
425C42—48no—
426C42—84no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—44no—
436C23—42no—
437C23—44no—
438C23—41no—
439C23—42no—
440C23—41no—
441C23—44no—
442C23—44no—
443C23C23×D411yescharacteristic
444C23C2×H(5,𝔽2)11yescharacteristic
445C23C2×H(5,𝔽2)11yescharacteristic
446C23C2×H(5,𝔽2)12yes—
447C23C2×2−1+412yes—
448C23C2×2−1+414yes—
449C23#58512yes—
450C23#58512yes—
451Q8—816no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—416no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
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—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4#53414yes—
530C2×C4#53514yes—
531C22—42no—
532C22—42no—
533C22—44no—
534C22—42no—
535C22—44no—
536C22—84no—
537C22—82no—
538C22—82no—
539C22C2×#53411yescharacteristic
540C22C2×#53511yescharacteristic
541C22#344911yescharacteristic
542C22#348912yes—
543C22#349912yes—
544C22#353412yes—
545C22#353412yes—
546C22#353812yes—
547C22#353814yes—
548C22#354712yes—
549C22#354714yes—
550C22#355011yescharacteristic
551C22#355012yes—
552C22#355111yescharacteristic
553C22#355314yes—
554C22#355514yes—
555C4—24no—
556C4—24no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—48no—
562C4—44no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4—84no—
567C4—84no—
568C4—84no—
569C2—82no—
570C2#3737611yescharacteristic
571C2#3738611yescharacteristic
572C2#3744611yescharacteristic
573C2#4230512yes—
574C2#4299412yes—
575C2#5429614yes—
576C2#5646312yes—
577C2#5873112yes—
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
5. #17075C2semidirect
6. #17031C2semidirect
7. #17013C2semidirect
8. #16608C2semidirect
9. #16603C2semidirect
12. #15676C2semidirect
13. #15119C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26112
C2×#26212
#172292
#171602
#170752
#170312
#170132
#166082
#166032
#164542
#162642
#156762
#151192
#147632
#139392