Finite Group Databasebeta

← #647299

Subgroups — #647299

Order 512.

2498 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic
2C2×C4×C22⋊Q8C211yesmaximal, characteristic
3#17021C214yesmaximal, complemented
4#17010C211yesmaximal, characteristic, complemented
5#16992C211yesmaximal, characteristic, complemented
6#16890C211yesmaximal, characteristic, complemented
7#16813C211yesmaximal, characteristic, complemented
8#15085C212yesmaximal, complemented
9#15071C212yesmaximal, complemented
10#14826C212yesmaximal
11#14826C212yesmaximal
12#14766C212yesmaximal
13#14758C212yesmaximal
14#14307C212yesmaximal, complemented
15#14302C212yesmaximal, complemented
16#14047C214yesmaximal
17C2×#1434C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2851—28no—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#398C2212yes—
26C2×#395C2212yes—
27C4×C22⋊Q8—28no—
28C2×#391C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#387C2212yes—
33C2×#387C2212yes—
34C2×#387C2212yes—
35C2×#387C2214yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#382C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2214yes—
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C22×#95C2212yes—
48C22×#95C2212yes—
49C22×#95C2212yes—
50#1446C2212yes—
51#1445C2214yes—
52#1445C2212yes—
53#1445C2212yes—
54#1445C2212yes—
55#1444C2212yes—
56#1444C2214yes—
57#1444C2212yes—
58#1443C2214yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2212yes—
62#1442C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2212yes—
67#1441C2214yes—
68#1441C2212yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1434C2211yescharacteristic
77#1434C2211yescharacteristic
78#1434C2212yes—
79#1434C2212yes—
80#1434C2212yes—
81#1434C2214yes—
82C22⋊#95C2214yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2314yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#125—24no—
91C2×#125—44no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—28no—
94C2×C4×Q8—24no—
95C2×C4×Q8—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C42—44no—
117C22×C42D418yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124#400—28no—
125#400—44no—
126#398—44no—
127#395—28no—
128#391—24no—
129#389—24no—
130#389—24no—
131#389—24no—
132#387—28no—
133#387—28no—
134#387—28no—
135#387—48no—
136#385—28no—
137#385—28no—
138#385—28no—
139#382—24no—
140#380—24no—
141#380—24no—
142#380—24no—
143#380—216no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—24no—
146C4×C22⋊C4—28no—
147C2×#95—28no—
148C2×#95—44no—
149C2×#95—44no—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—42no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225#125—416no—
226C22⋊Q8—232no—
227C4×Q8—416no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4#4118yes—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—416no—
283C2×C42—42no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42C2×D414yes—
288C2×C42C2×D414yes—
289C2×C42C2×D414yes—
290C2×C42C2×D414yes—
291#95—24no—
292#95—24no—
293#95—24no—
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—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—42no—
313#95—44no—
314#95—42no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95#4118yes—
325C24—22no—
326C24—22no—
327C24—21no—
328C24—21no—
329C24—22no—
330C24C22×D412yes—
331C24C2×#4114yes—
332C242−1+414yes—
333C242−1+414yes—
334C24C2511yescharacteristic
335C2×Q8—28no—
336C2×Q8—416no—
337C2×Q8—44no—
338C2×Q8—44no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—416no—
420C22×C4—416no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C2×#4118yes—
437C22×C4C2×#4114yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C42−1+412yes—
443C4⋊C4—28no—
444C4⋊C4—216no—
445C4⋊C4—416no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—48no—
452C4⋊C4—44no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—88no—
459C22⋊C4—216no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C42—28no—
464C42—28no—
465C42—44no—
466C42—44no—
467C42—48no—
468C42—48no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—28no—
474C23—28no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—24no—
481C23—24no—
482C23#52514yes—
483C23#53614yes—
484C23#53614yes—
485C23#541116yes—
486C23C23×D411yescharacteristic
487C23C22×#4112yes—
488C23C22×#4112yes—
489C23C2×H(5,𝔽2)12yes—
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×2−1+412yes—
492C23C2×2−1+412yes—
493C23C2×2−1+411yescharacteristic
494C23#58512yes—
495Q8—416no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—416no—
524C2×C4—416no—
525C2×C4—416no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
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—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—832no—
581C2×C4#53214yes—
582C2×C4#53414yes—
583C2×C4#53714yes—
584C22—24no—
585C22—24no—
586C22—24no—
587C22—216no—
588C22—216no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22#263418yes—
593C22C2×#52511yescharacteristic
594C22C2×#53211yescharacteristic
595C22C2×#53411yescharacteristic
596C22C2×#53611yescharacteristic
597C22C2×#53611yescharacteristic
598C22C2×#53711yescharacteristic
599C22#344911yescharacteristic
600C22C2×#54114yes—
601C22#348114yes—
602C22#349514yes—
603C22#351412yes—
604C22#351512yes—
605C22#352012yes—
606C22#352312yes—
607C22#353514yes—
608C22#353612yes—
609C22#353812yes—
610C4—24no—
611C4—24no—
612C4—24no—
613C4—44no—
614C4—42no—
615C4—42no—
616C4—48no—
617C4—42no—
618C4—44no—
619C4—82no—
620C4—82no—
621C4—82no—
622C4—84no—
623C4—82no—
624C4—82no—
625C4—82no—
626C2—28no—
627C2C2×#263411yescharacteristic
628C2#3495711yescharacteristic
629C2#3513011yescharacteristic
630C2#3524111yescharacteristic
631C2#3532611yescharacteristic
632C2#3744211yescharacteristic
633C2#3744711yescharacteristic
634C2#3850714yes—
635C2#3853914yes—
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
3. #17021C2semidirect
4. #17010C2semidirect
5. #16992C2semidirect
6. #16890C2semidirect
7. #16813C2semidirect
8. #15085C2semidirect
9. #15071C2semidirect
14. #14307C2semidirect
15. #14302C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×C4×C22⋊Q82
#170212
#170102
#169922
#168902
#168132
#150852
#150712
#148262
#147662
#147582
#143072
#143022
#140472
C2×#14342