Finite Group Databasebeta

← #516367

Subgroups — #516367

Order 512.

3082 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2313C211yesmaximal, characteristic
2C22×#387C211yesmaximal, characteristic
3C2×#2274C211yesmaximal, characteristic
4#13966C211yesmaximal, characteristic
5#13965C211yesmaximal, characteristic
6#13962C212yesmaximal
7#13962C212yesmaximal
8#13949C214yesmaximal, complemented
9#13930C214yesmaximal, complemented
10#13889C214yesmaximal, complemented
11#13884C214yesmaximal, complemented
12#13881C211yesmaximal, characteristic
13#13880C211yesmaximal, characteristic
14#13782C212yesmaximal
15C4×#385C212yesmaximal
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2313—28no—
21C2×#387—28no—
22C2×#387C2212yes—
23C2×#387C2214yes—
24C2×#387C2212yes—
25C2×#387C2214yes—
26C2×#387C2212yes—
27C2×#387C2212yes—
28C2×#385C2212yes—
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#384C2212yes—
33C2×#382C2212yes—
34C2×#381C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37#2274C4116yescomplemented
38C2×C4×C4⋊C4C2214yes—
39C2×C4×C4⋊C4C2214yes—
40C2×C4×C4⋊C4C2214yes—
41C2×C4×C22⋊C4C2212yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C4×C22⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1444C2214yes—
50#1444C2212yes—
51#1444C2212yes—
52#1442C2214yes—
53#1442C2212yes—
54#1442C2212yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2214yes—
66C4⋊#95C2214yes—
67C4⋊#95C2214yes—
68C4⋊#95C2214yes—
69#1434C2214yes—
70#1434C2214yes—
71#1434C2214yes—
72#1434C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C4×#95C2214yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×#117—28no—
86C2×#117—28no—
87C2×#117C2×C4116yes—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4C2×C418yes—
91C22×C4⋊C4D418yes—
92C22×C4⋊C4Q818yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C42—24no—
111C22×C42—44no—
112C22×C42C2314yes—
113C22×C42C2314yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2314yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119#387—28no—
120#387—28no—
121#387—28no—
122#387—28no—
123#387—216no—
124#387—216no—
125#385—28no—
126#385—28no—
127#385—28no—
128#385—28no—
129#384—44no—
130#382—44no—
131#381—28no—
132#380—24no—
133#380—24no—
134C4×C4⋊C4—48no—
135C4×C4⋊C4C2×C4132yescomplemented
136C4×C4⋊C4C2×C4132yescomplemented
137C4×C22⋊C4—28no—
138C4×C22⋊C4C2×C418yes—
139C4×C22⋊C4C2×C4116yes—
140C4×C22⋊C4C2×C418yes—
141C2×#95—24no—
142C2×#95—24no—
143C2×#95—24no—
144C2×#95—24no—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C25C2411yescharacteristic
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4C22×C414yes—
197C23×C4C22×C414yes—
198C23×C4C22×C414yes—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4C2×Q814yes—
203C23×C4C2×Q814yes—
204C23×C4C2×Q814yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219#117—264no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—416no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C4⋊C4164yescomplemented
250C2×C4⋊C4C22×C4116yes—
251C2×C4⋊C4C22×C418yes—
252C2×C4⋊C4C22×C418yes—
253C2×C4⋊C4C2×D418yes—
254C2×C4⋊C4C2×D418yes—
255C2×C4⋊C4C2×Q818yes—
256C2×C4⋊C4C2×Q818yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C22×C418yes—
267C2×C22⋊C4#4118yes—
268C2×C22⋊C4#4118yes—
269C2×C42—28no—
270C2×C42—28no—
271C2×C42—28no—
272C2×C42—28no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—44no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42—44no—
279C2×C42—42no—
280C2×C42—44no—
281C2×C42—42no—
282C2×C42—416no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42C22×C4116yes—
286C2×C42C22×C418yes—
287C2×C42C22×C4116yes—
288C2×C42C22×C418yes—
289#95—24no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95C22×C418yes—
315#95C22×C4116yes—
316#95C22×C418yes—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24C23×C412yes—
322C24C22×D412yes—
323C24C22×Q812yes—
324C24C2×#4114yes—
325C24C2×#4114yes—
326C24C2511yescharacteristic
327C22×C4—216no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—216no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—28no—
346C22×C4—28no—
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—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4C2×C4⋊C4116yes—
414C22×C4C2×C4⋊C4116yes—
415C22×C4C2×C4⋊C4116yes—
416C22×C4C23×C418yes—
417C22×C4C23×C412yes—
418C22×C4C23×C412yes—
419C22×C4C23×C414yes—
420C22×C4C23×C414yes—
421C22×C4C23×C418yes—
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×Q814yes—
427C22×C4C22×Q812yes—
428C22×C4C22×Q812yes—
429C22×C4C22×Q814yes—
430C22×C4C2×#4118yes—
431C22×C4C2×#4118yes—
432C22×C4H(5,𝔽2)14yes—
433C22×C42−1+414yes—
434C4⋊C4—232no—
435C4⋊C4—416no—
436C4⋊C4—416no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—88no—
444C4⋊C4—88no—
445C4⋊C4C2×C4⋊C4132yes—
446C4⋊C4C2×C4⋊C4132yes—
447C22⋊C4—216no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4#117132yes—
451C42—232no—
452C42—48no—
453C42—416no—
454C42—48no—
455C42—416no—
456C42—84no—
457C42—84no—
458C23—24no—
459C23—24no—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23C22×C4⋊C414yes—
471C23C2×#11718yes—
472C23#53214yes—
473C23#53214yes—
474C23#53318yes—
475C23C24×C411yescharacteristic
476C23C23×D411yescharacteristic
477C23C23×Q811yescharacteristic
478C23C22×#4112yes—
479C23C22×#4112yes—
480C23C2×H(5,𝔽2)12yes—
481C23C2×2−1+412yes—
482C23#58512yes—
483C23#58512yes—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—232no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—416no—
517C2×C4—416no—
518C2×C4—416no—
519C2×C4—416no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—88no—
561C2×C4—88no—
562C2×C4C22×C4⋊C418yes—
563C2×C4C22×C4⋊C414yes—
564C2×C4C22×C4⋊C414yes—
565C2×C4C22×C4⋊C418yes—
566C2×C4C2×#117116yes—
567C22—24no—
568C22—24no—
569C22—216no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—216no—
575C22#228518yes—
576C22C23×C4⋊C411yescharacteristic
577C22C22×#11712yes—
578C22C4×H(5,𝔽2)12yes—
579C22C4×2−1+412yes—
580C22C2×#53211yescharacteristic
581C22C2×#53211yescharacteristic
582C22C2×#53312yes—
583C22#348411yescharacteristic
584C22#348511yescharacteristic
585C22#348611yescharacteristic
586C22#348711yescharacteristic
587C22#350112yes—
588C22#350212yes—
589C22#351314yes—
590C22#351514yes—
591C22#352014yes—
592C22#352114yes—
593C4—28no—
594C4—24no—
595C4—24no—
596C4—216no—
597C4—28no—
598C4—44no—
599C4—44no—
600C4—48no—
601C4—82no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—82no—
606C4—82no—
607C2—28no—
608C2C2×#228511yescharacteristic
609C2#3424811yescharacteristic
610C2#3424911yescharacteristic
611C2#3426014yes—
612C2#3539611yescharacteristic
613C2#3539711yescharacteristic
614C2#3540314yes—
615C2#3546811yescharacteristic
616C2#3546911yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #13949C2semidirect
9. #13930C2semidirect
10. #13889C2semidirect
11. #13884C2semidirect
37. #2274C4semidirect
135. C4×C4⋊C4C2×C4semidirect
136. C4×C4⋊C4C2×C4semidirect
249. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23132
C22×#3872
C2×#22742
#139662
#139652
#139622
#139492
#139302
#138892
#138842
#138812
#138802
#137822
C4×#3852