Finite Group Databasebeta

← #4386957

Subgroups — #4386957

Order 512.

2784 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2816C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3C2×#2501C211yesmaximal, characteristic
4C2×#2313C211yesmaximal, characteristic, complemented
5#16257C214yesmaximal, complemented
6#15855C214yesmaximal, complemented
7#15765C214yesmaximal, complemented
8#15661C214yesmaximal, complemented
9#15629C214yesmaximal, complemented
10#14761C214yesmaximal, complemented
11#14292C211yesmaximal, characteristic, complemented
12#14290C211yesmaximal, characteristic
13#14165C211yesmaximal, characteristic, complemented
14C22×#123C2212yes—
15C22×#123C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2816—28no—
20#2502—28no—
21#2501—28no—
22C2×#399C2212yes—
23C2×#399C2214yes—
24C2×#398C2212yes—
25C2×#398C2214yes—
26C2×#398C2214yes—
27C2×#396C2214yes—
28C2×#394C2212yes—
29C2×#394C2214yes—
30C2×#393C2212yes—
31#2313—28no—
32C2×#391C2212yes—
33C2×#390C2212yes—
34C2×#389C2214yes—
35C2×#388C2214yes—
36C2×#387C2214yes—
37C2×#387C2214yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#384C2212yes—
41C2×#384C2212yes—
42C2×#381C2211yescharacteristic
43C2×#381C2211yescharacteristic
44C2×#380C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48#1446C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1444C2214yes—
52#1444C2214yes—
53#1442C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2212yes—
64#1438C2214yes—
65#1437C2214yes—
66#1437C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C23×Q8C2311yescharacteristic
72C23×D4C2312yes—
73C23×D4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C2×#123—28no—
78C2×#123—28no—
79C2×#123—48no—
80C2×#123—48no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×#121—24no—
85C2×#121—24no—
86C2×#121—24no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4D418yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2314yes—
117#399—216no—
118#399—28no—
119#398—216no—
120#398—28no—
121#398—48no—
122#396—48no—
123#394—216no—
124#394—28no—
125#393—28no—
126#391—28no—
127#390—28no—
128#389—48no—
129#388—48no—
130#387—216no—
131#387—216no—
132#385—28no—
133#385—28no—
134#384—28no—
135#384—28no—
136#381—24no—
137#381D418yes—
138#380—28no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—28no—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C25C2412yes—
163C25C2411yescharacteristic
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—22no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—22no—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—42no—
178C22×D4—42no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4C2×D414yes—
188C23×C4C2×D414yes—
189C23×C4C2×D414yes—
190C23×C4C2×D414yes—
191C23×C4#4114yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203#123—432no—
204#123—432no—
205C22⋊Q8—416no—
206#121—416no—
207C2×C4⋊C4—232no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×D418yes—
243C2×C4⋊C4C2×D418yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C2×D418yes—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—42no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—44no—
286C2×C42—44no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95C2×D418yes—
306#95C2×D418yes—
307C24—24no—
308C24—24no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24C22×D412yes—
321C24C22×D412yes—
322C24C2×#4112yes—
323C24C2511yescharacteristic
324C2×Q8—48no—
325C2×Q8—44no—
326C2×Q8—84no—
327C2×Q8—82no—
328C2×Q8—82no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—44no—
334C2×D4—84no—
335C2×D4—82no—
336C2×D4—82no—
337C2×D4—88no—
338C2×D4—88no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
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—28no—
370C22×C4—28no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
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—44no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—84no—
403C22×C4#121116yes—
404C22×C4C22×D418yes—
405C22×C4C22×D414yes—
406C22×C4C22×D414yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4H(5,𝔽2)18yes—
414C22×C42−1+418yes—
415C4⋊C4—216no—
416C4⋊C4—216no—
417C4⋊C4—416no—
418C4⋊C4—416no—
419C4⋊C4—416no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—88no—
424C4⋊C4—88no—
425C4⋊C4—88no—
426C4⋊C4—88no—
427C22⋊C4—216no—
428C22⋊C4—216no—
429C22⋊C4—48no—
430C22⋊C4—416no—
431C22⋊C4—416no—
432C22⋊C4—416no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—48no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C42—48no—
440C42—48no—
441C42—88no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—42no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23C2×#12114yes—
478C23C23×D411yescharacteristic
479C23C23×D411yescharacteristic
480C23C22×#4111yescharacteristic
481C23C2×H(5,𝔽2)14yes—
482C23C2×2−1+414yes—
483C23#58512yes—
484C23#58512yes—
485Q8—88no—
486D4—88no—
487D4—816no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—416no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
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—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
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×C4C2×#12114yes—
553C2×C4C2×#12114yes—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—44no—
558C22—44no—
559C22—48no—
560C22—48no—
561C22—84no—
562C22—82no—
563C22—82no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—82no—
568C22—84no—
569C22—84no—
570C22C22×#12111yescharacteristic
571C22#344912yes—
572C22#348412yes—
573C22#348512yes—
574C22#348612yes—
575C22#349111yescharacteristic
576C22#349111yescharacteristic
577C22#349812yes—
578C22#349912yes—
579C22#350812yes—
580C22#350912yes—
581C22#353514yes—
582C22#354114yes—
583C22#355414yes—
584C22#355914yes—
585C4—24no—
586C4—24no—
587C4—48no—
588C4—48no—
589C4—44no—
590C4—44no—
591C4—48no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—84no—
596C2—44no—
597C2—162no—
598C2#3508911yescharacteristic
599C2#3552811yescharacteristic
600C2#3566711yescharacteristic
601C2#4059914yes—
602C2#4685412yes—
603C2#4722412yes—
604C2#5064814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2502C2semidirect
4. C2×#2313C2semidirect
5. #16257C2semidirect
6. #15855C2semidirect
7. #15765C2semidirect
8. #15661C2semidirect
9. #15629C2semidirect
10. #14761C2semidirect
11. #14292C2semidirect
13. #14165C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28162
C2×#25022
C2×#25012
C2×#23132
#162572
#158552
#157652
#156612
#156292
#147612
#142922
#142902
#141652