Finite Group Databasebeta

← #539278

Subgroups — #539278

Order 512.

3658 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3C2×#2327C211yesmaximal, characteristic
4#14474C214yesmaximal
5#14451C214yesmaximal
6#14097C214yesmaximal
7#14024C212yesmaximal
8#14024C212yesmaximal
9#13942C211yesmaximal, characteristic
10#13931C212yesmaximal
11#13927C214yesmaximal
12#13925C211yesmaximal, characteristic
13#13850C211yesmaximal, characteristic
14Q8×#95C212yesmaximal
15#13786C211yesmaximal, characteristic
16C2×#557—28no—
17C2×#557—28no—
18C2×#557C4116yescomplemented
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125C2214yes—
22C22×#125C2212yes—
23C22×C4×Q8C2212yes—
24C22×C4×Q8C2212yes—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×Q8C2212yes—
27C22×C4×Q8C2212yes—
28C22×C4×Q8C2211yescharacteristic
29#2365—28no—
30#2327—28no—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#391C2212yes—
36C2×#389C2212yes—
37C2×#387C2214yes—
38C2×#387C2214yes—
39C2×#387C2214yes—
40C2×#382C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×#376C2211yescharacteristic
45#1446C2212yes—
46#1446C2212yes—
47#1445C2214yes—
48#1445C2212yes—
49#1445C2212yes—
50#1445C2214yes—
51#1441C2212yes—
52#1441C2212yes—
53#1441C2214yes—
54#1441C2212yes—
55#1441C2214yes—
56#1441C2212yes—
57#1441C2214yes—
58#1441C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72C4⋊#95C2211yescharacteristic
73#1434C2214yes—
74#1434C2214yes—
75#1434C2214yes—
76#1434C2214yes—
77C4×#95C2214yes—
78C4×#95C2212yes—
79C4×#95C2212yes—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×Q8C2312yes—
83#557—2128no—
84C2×C4⋊Q8—28no—
85C2×C4⋊Q8—28no—
86C2×C4⋊Q8C2×C4116yes—
87C2×C4⋊Q8C2×C418yes—
88C2×C4⋊Q8D418yes—
89C2×C4⋊Q8D418yes—
90C2×#125—216no—
91C2×#125—216no—
92C2×#125—28no—
93C2×#125—28no—
94C2×#125C2×C4132yes—
95C2×#125C2×C4116yes—
96C2×C4×Q8—28no—
97C2×C4×Q8—28no—
98C2×C4×Q8—28no—
99C2×C4×Q8—28no—
100C2×C4×Q8—28no—
101C2×C4×Q8—28no—
102C2×C4×Q8—28no—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8—24no—
107C2×C4×Q8—28no—
108C2×C4×Q8—44no—
109C2×C4×Q8—44no—
110C2×C4×Q8C2×C4116yes—
111C2×C4×Q8C2×C4116yes—
112C2×C4×Q8C2×C418yes—
113C2×C4×Q8C2×C418yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2314yes—
133C22×C42C2314yes—
134C22×C42C2314yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#391—28no—
141#391—28no—
142#391—24no—
143#391—24no—
144#391—44no—
145#389—28no—
146#387—216no—
147#387—216no—
148#387—216no—
149#382—24no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153#376—24no—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—22no—
198C22×Q8C22×C414yes—
199C22×Q8C22×C414yes—
200C22×Q8C22×C418yes—
201C22×Q8C2×D418yes—
202C22×Q8C2×D418yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C4⋊Q8—264no—
219C4⋊Q8C22⋊C4164yes—
220#125—2128no—
221#125—264no—
222C4×Q8—264no—
223C4×Q8—264no—
224C4×Q8—232no—
225C4×Q8—232no—
226C4×Q8—432no—
227C4×Q8—432no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C22×C414yes—
261C2×C4⋊C4C22×C4116yes—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C22×C414yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C4116yes—
266C2×C4⋊C4C22×C4116yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C418yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4#4114yes—
274C2×C42—28no—
275C2×C42—28no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—28no—
283C2×C42—24no—
284C2×C42—28no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42C22×C418yes—
297C2×C42C22×C4116yes—
298C2×C42C22×C418yes—
299C2×C42C22×C414yes—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42#4114yes—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—28no—
324#95—44no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—42no—
330#95—44no—
331#95—44no—
332#95—44no—
333C24C2511yescharacteristic
334C2×Q8—216no—
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—216no—
340C2×Q8—28no—
341C2×Q8—216no—
342C2×Q8—216no—
343C2×Q8—28no—
344C2×Q8—216no—
345C2×Q8—216no—
346C2×Q8—216no—
347C2×Q8—216no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8C2×C22⋊C4132yes—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
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—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4C23×C414yes—
424C22×C4C23×C414yes—
425C22×C4C23×C414yes—
426C22×C4C23×C412yes—
427C22×C4C23×C412yes—
428C22×C4C23×C418yes—
429C22×C4C23×C414yes—
430C22×C4C23×C412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D418yes—
435C22×C4C22×D418yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4H(5,𝔽2)12yes—
444C22×C42−1+414yes—
445C22×C42−1+414yes—
446C22×C42−1+414yes—
447C22×C42−1+412yes—
448C4⋊C4—28no—
449C4⋊C4—28no—
450C4⋊C4—216no—
451C4⋊C4—232no—
452C4⋊C4—216no—
453C4⋊C4—28no—
454C4⋊C4—232no—
455C4⋊C4—232no—
456C4⋊C4—28no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—44no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4C2×C22⋊C4132yes—
463C4⋊C4C2×C22⋊C4132yes—
464C42—232no—
465C42—216no—
466C42—216no—
467C42—28no—
468C42—416no—
469C42—416no—
470C42—48no—
471C42—48no—
472C42—44no—
473C42C2×C22⋊C4116yes—
474C23C24×C411yescharacteristic
475C23C23×D411yescharacteristic
476C23C23×D411yescharacteristic
477C23C22×#4111yescharacteristic
478C23C22×#4111yescharacteristic
479C23C2×H(5,𝔽2)11yescharacteristic
480C23C2×2−1+412yes—
481C23C2×2−1+412yes—
482C23C2×2−1+411yescharacteristic
483C23C2×2−1+412yes—
484C23#58512yes—
485Q8—232no—
486Q8—264no—
487Q8—416no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
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—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4C22×C22⋊C414yes—
573C2×C4C22×C22⋊C4116yes—
574C2×C4C22×C22⋊C414yes—
575C2×C4C22×C22⋊C414yes—
576C2×C4C4×#4114yes—
577C2×C4#51918yes—
578C2×C4#52014yes—
579C2×C4#54614yes—
580C2×C4#54614yes—
581C2×C4#54714yes—
582C2×C4#54714yes—
583C22C23×C22⋊C411yescharacteristic
584C22C2×C4×#4111yescharacteristic
585C22C2×#51912yes—
586C22C2×#52011yescharacteristic
587C22C4×2−1+412yes—
588C22C2×#54611yescharacteristic
589C22C2×#54611yescharacteristic
590C22C2×#54711yescharacteristic
591C22C2×#54711yescharacteristic
592C22#348612yes—
593C22#348611yescharacteristic
594C22#348611yescharacteristic
595C22#348611yescharacteristic
596C22#348711yescharacteristic
597C22#349712yes—
598C22#350712yes—
599C22#350912yes—
600C22#351512yes—
601C22#352312yes—
602C22#355212yes—
603C22#355412yes—
604C22#355512yes—
605C22#355612yes—
606C4—216no—
607C4—24no—
608C4—28no—
609C4—24no—
610C4—44no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—44no—
615C4—44no—
616C4—84no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4#233918yes—
621C2C2×#233911yescharacteristic
622C2#3417311yescharacteristic
623C2#3425711yescharacteristic
624C2#3432812yes—
625C2#3448812yes—
626C2#4532312yes—
627C2#4532412yes—
628C2#4533411yescharacteristic
629C2#4533511yescharacteristic
630C2#4533611yescharacteristic
631C2#4539411yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
18. C2×#557C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#23652
C2×#23272
#144742
#144512
#140972
#140242
#139422
#139312
#139272
#139252
#138502
Q8×#952
#137862