Finite Group Databasebeta

← #627030

Subgroups — #627030

Order 512.

2232 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#2331C211yesmaximal, characteristic, complemented
3#15852C214yesmaximal, complemented
4#15714C214yesmaximal, complemented
5#15649C214yesmaximal
6#15499C214yesmaximal
7#15225C212yesmaximal, complemented
8#15135C212yesmaximal
9#14714C211yesmaximal, characteristic, complemented
10#14687C211yesmaximal, characteristic, complemented
11#14340C211yesmaximal, characteristic
12#14276C211yesmaximal, characteristic, complemented
13#14029C212yesmaximal, complemented
14#13875C212yesmaximal
15#13761C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18#2628—28no—
19C2×#400C2212yes—
20C2×#399C2212yes—
21C2×#399C2214yes—
22C2×#395C2212yes—
23C2×#393C2212yes—
24#2331—28no—
25C2×#391C2214yes—
26C2×#391C2214yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#384C2211yescharacteristic
33C2×#384C2212yes—
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#382C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×#376C2211yescharacteristic
43C2×C43C2211yescharacteristic
44#1445C2214yes—
45#1445C2214yes—
46#1445C2214yes—
47#1444C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1441C2212yes—
51#1441C2212yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1440C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1439C2214yes—
61#1437C2212yes—
62#1437C2214yes—
63#1437C2212yes—
64#1436C2212yes—
65#1436C2214yes—
66#1436C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2214yes—
70C4⋊#95C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2214yes—
74#1434C2214yes—
75C4×#95C2211yescharacteristic
76C4×#95C2211yescharacteristic
77C23×Q8C2312yes—
78C24×C4C2311yescharacteristic
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8D418yes—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117—24no—
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2314yes—
113C22×C42C2311yescharacteristic
114#400—28no—
115#399—28no—
116#399—48no—
117#395—28no—
118#393—28no—
119#391—216no—
120#391—48no—
121#389—28no—
122#389—28no—
123#389—44no—
124#387—216no—
125#387—48no—
126#384—28no—
127#384—24no—
128#382—24no—
129#382—24no—
130#382—44no—
131#380—28no—
132#380—44no—
133C4×C4⋊C4—28no—
134C4×C22⋊C4—24no—
135C4×C22⋊C4Q818yescomplemented
136#376—24no—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C43—24no—
166C25C2411yescharacteristic
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—44no—
170C22×Q8—44no—
171C22×Q8C2×D418yes—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4C2×Q814yes—
179C23×C4C2414yes—
180C23×C4C2411yescharacteristic
181C23×C4C2414yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2411yescharacteristic
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C4⋊Q8—232no—
194#117—416no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D418yes—
234C2×C4⋊C4C2×D418yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4C2×Q818yes—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—28no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42C2×D414yes—
275C2×C42C2×Q814yes—
276C2×C42C2×Q814yes—
277C2×C42#4114yes—
278C2×C42#4114yes—
279C2×C42#4114yes—
280#95—24no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—44no—
296#95—42no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95C2×Q818yes—
308C24—21no—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—22no—
313C24—21no—
314C24—22no—
315C24—21no—
316C24—22no—
317C24—22no—
318C24C22×Q812yes—
319C24C2511yescharacteristic
320C2×Q8—216no—
321C2×Q8—416no—
322C2×Q8—88no—
323C2×Q8—84no—
324C2×Q8—84no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4C22×D418yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×Q812yes—
407C22×C4C22×Q814yes—
408C22×C4C22×Q814yes—
409C22×C4C22×Q812yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4H(5,𝔽2)14yes—
419C22×C4H(5,𝔽2)12yes—
420C22×C42−1+414yes—
421C22×C42−1+412yes—
422C4⋊C4—216no—
423C4⋊C4—216no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—416no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—88no—
433C4⋊C4—88no—
434C4⋊C4—88no—
435C4⋊C4—88no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—48no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C42—28no—
443C42—28no—
444C42—28no—
445C42—48no—
446C42—48no—
447C42—44no—
448C42—44no—
449C42—88no—
450C42#123116yes—
451C23—22no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—41no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23C23×D411yescharacteristic
475C23C23×Q811yescharacteristic
476C23C22×#4111yescharacteristic
477C23C22×#4111yescharacteristic
478C23C22×#4111yescharacteristic
479C23C2×H(5,𝔽2)11yescharacteristic
480C23C2×H(5,𝔽2)12yes—
481C23C2×2−1+411yescharacteristic
482C23C2×2−1+412yes—
483C23#58512yes—
484C23#58512yes—
485Q8—816no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—416no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4C2×#12314yes—
568C2×C4C2×#12314yes—
569C2×C4C2×#12314yes—
570C2×C4#53314yes—
571C2×C4#54218yes—
572C2×C4#54214yes—
573C2×C4D4×Q814yes—
574C2×C4#55014yes—
575C2×C4#55214yes—
576C2×C4#55414yes—
577C22—42no—
578C22—42no—
579C22—44no—
580C22—42no—
581C22—44no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—82no—
587C22—82no—
588C22C22×#12311yescharacteristic
589C22C2×#53311yescharacteristic
590C22C2×#54211yescharacteristic
591C22C2×#54212yes—
592C22#345812yes—
593C22C2×D4×Q811yescharacteristic
594C22C2×#55011yescharacteristic
595C22C2×#55211yescharacteristic
596C22C2×#55411yescharacteristic
597C22#349412yes—
598C22#349512yes—
599C22#349612yes—
600C22#349712yes—
601C22#350312yes—
602C22#350811yescharacteristic
603C22#350911yescharacteristic
604C22#351011yescharacteristic
605C22#351011yescharacteristic
606C22#351012yes—
607C22#354612yes—
608C22#355412yes—
609C22#355912yes—
610C22#356312yes—
611C4—24no—
612C4—24no—
613C4—28no—
614C4—42no—
615C4—44no—
616C4—48no—
617C4—44no—
618C4—81no—
619C4—84no—
620C4—81no—
621C4—82no—
622C4—84no—
623C4—84no—
624C4#242318yes—
625C2—82no—
626C2C2×#242311yescharacteristic
627C2#3528311yescharacteristic
628C2#3534411yescharacteristic
629C2#3554111yescharacteristic
630C2#3567311yescharacteristic
631C2#3732111yescharacteristic
632C2#3739711yescharacteristic
633C2#3787812yes—
634C2#3788412yes—
635C2#3936512yes—
636C2#3938712yes—
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
2. C2×#2331C2semidirect
3. #15852C2semidirect
4. #15714C2semidirect
7. #15225C2semidirect
9. #14714C2semidirect
10. #14687C2semidirect
12. #14276C2semidirect
13. #14029C2semidirect
135. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#23312
#158522
#157142
#156492
#154992
#152252
#151352
#147142
#146872
#143402
#142762
#140292
#138752
#137612