Finite Group Databasebeta

← #2811414

Subgroups — #2811414

Order 512.

2978 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2537C211yesmaximal, characteristic
3#16520C214yesmaximal, complemented
4#16514C212yesmaximal, complemented
5#15833C212yesmaximal, complemented
6#15761C212yesmaximal, complemented
7#15673C214yesmaximal, complemented
8#15316C214yesmaximal, complemented
9#14883C212yesmaximal, complemented
10#14865C212yesmaximal, complemented
11#14454C211yesmaximal, characteristic, complemented
12#14392C214yesmaximal, complemented
13#14176C212yesmaximal, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#124C2212yes—
16C22×#121C2214yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2212yes—
19#2567—28no—
20#2537—28no—
21C2×#396C2214yes—
22C2×#395C2214yes—
23C2×#394C2212yes—
24C2×#394C2212yes—
25C2×#394C2212yes—
26C2×#394C2212yes—
27C2×#394C2214yes—
28C2×#394C2212yes—
29C2×#393C2214yes—
30C2×#393C2214yes—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#390C2212yes—
34C2×#389C2214yes—
35C2×#388C2214yes—
36C2×#385C2214yes—
37C2×#384C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C2×#380C2214yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41#1446C2212yes—
42#1445C2214yes—
43#1444C2212yes—
44#1442C2214yes—
45#1441C2212yes—
46#1441C2212yes—
47#1441C2212yes—
48#1441C2214yes—
49#1441C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2214yes—
56#1438C2212yes—
57#1438C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2212yes—
64#1434C2212yes—
65#1434C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C23×D4C2312yes—
74C24×C4C2314yes—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—42no—
77C2×C4⋊Q8—42no—
78C2×#124—48no—
79C2×#124—44no—
80C2×#121—216no—
81C2×#121—48no—
82C2×#121—48no—
83C2×C4×Q8—24no—
84C2×C4×Q8—42no—
85C2×C4×Q8—42no—
86C2×C4×D4—28no—
87C2×C4×D4—48no—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115#396—48no—
116#395—216no—
117#394—28no—
118#394—28no—
119#394—28no—
120#394—44no—
121#394—44no—
122#394—48no—
123#393—48no—
124#393—48no—
125#391—24no—
126#391—24no—
127#390—44no—
128#389—216no—
129#388—48no—
130#385—216no—
131#384—24no—
132#384D418yes—
133#380—216no—
134C4×C4⋊C4—24no—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25C2414yes—
156C22×Q8—22no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8—44no—
164C22×Q8—42no—
165C22×Q8—44no—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—28no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—42no—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C4⋊Q8—416no—
201#124—432no—
202#121—464no—
203C4×Q8—416no—
204C4×D4—432no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D4116yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4#4118yes—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
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—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—24no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42—42no—
280C2×C42C2×D414yes—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—28no—
290#95—24no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—44no—
299#95—42no—
300#95—44no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24H(5,𝔽2)18yes—
316C24C2511yescharacteristic
317C2×Q8—28no—
318C2×Q8—48no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—44no—
322C2×Q8—48no—
323C2×Q8—44no—
324C2×Q8—88no—
325C2×Q8—84no—
326C2×Q8—88no—
327C2×D4—216no—
328C2×D4—416no—
329C2×D4—416no—
330C2×D4—48no—
331C2×D4—416no—
332C2×D4—416no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—84no—
337C2×D4—84no—
338C2×D4—88no—
339C2×D4—88no—
340C2×D4—84no—
341C2×D4—84no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
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—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—416no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—88no—
401C22×C4—88no—
402C22×C4—88no—
403C22×C4—88no—
404C22×C4C22×D414yes—
405C22×C4C22×D414yes—
406C22×C4C22×D412yes—
407C22×C4C22×D414yes—
408C22×C4C2×#4114yes—
409C22×C4C2×#4114yes—
410C22×C4C2×#4114yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C4H(5,𝔽2)14yes—
413C22×C4H(5,𝔽2)12yes—
414C22×C42−1+414yes—
415C22×C42−1+414yes—
416C4⋊C4—216no—
417C4⋊C4—44no—
418C4⋊C4—416no—
419C4⋊C4—44no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C4⋊C4—88no—
425C4⋊C4—88no—
426C4⋊C4—84no—
427C22⋊C4—416no—
428C22⋊C4—416no—
429C22⋊C4—416no—
430C22⋊C4—416no—
431C22⋊C4—416no—
432C22⋊C4—416no—
433C22⋊C4—416no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C22⋊C4—88no—
438C42—28no—
439C42—28no—
440C42—48no—
441C42—48no—
442C42—44no—
443C23—28no—
444C23—28no—
445C23—28no—
446C23—28no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23C23×D411yescharacteristic
479C23C22×#4112yes—
480C23C2×H(5,𝔽2)12yes—
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×H(5,𝔽2)11yescharacteristic
483C23C2×H(5,𝔽2)14yes—
484C23C2×2−1+412yes—
485C23C2×2−1+412yes—
486Q8—416no—
487Q8—816no—
488D4—432no—
489D4—816no—
490D4—816no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
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—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—832no—
556C2×C4#53514yes—
557C2×C4#54818yes—
558C2×C4#55514yes—
559C22—48no—
560C22—48no—
561C22—48no—
562C22—48no—
563C22—48no—
564C22—48no—
565C22—48no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22C2×#53511yescharacteristic
575C22C2×#54812yes—
576C22C2×#55511yescharacteristic
577C22#348411yescharacteristic
578C22#348412yes—
579C22#348611yescharacteristic
580C22#351112yes—
581C22#351914yes—
582C22#352214yes—
583C22#353212yes—
584C22#353612yes—
585C22#353614yes—
586C22#353612yes—
587C22#353712yes—
588C22#354414yes—
589C22#354411yescharacteristic
590C4—24no—
591C4—28no—
592C4—24no—
593C4—44no—
594C4—44no—
595C4—42no—
596C4—44no—
597C4—44no—
598C4—84no—
599C4—82no—
600C4—84no—
601C4—84no—
602C2—88no—
603C2#3670711yescharacteristic
604C2#3912012yes—
605C2#4493712yes—
606C2#4537812yes—
607C2#4841414yes—
608C2#5200411yescharacteristic
609C2#5831012yes—
610C2#5913511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2567C2semidirect
3. #16520C2semidirect
4. #16514C2semidirect
5. #15833C2semidirect
6. #15761C2semidirect
7. #15673C2semidirect
8. #15316C2semidirect
9. #14883C2semidirect
10. #14865C2semidirect
11. #14454C2semidirect
12. #14392C2semidirect
13. #14176C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25372
#165202
#165142
#158332
#157612
#156732
#153162
#148832
#148652
#144542
#143922
#141762