Finite Group Databasebeta

← #759373

Subgroups — #759373

Order 512.

2818 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2446C212yesmaximal, complemented
2C2×C4×C22⋊Q8C212yesmaximal
3#16337C214yesmaximal, complemented
4#15223C211yesmaximal, characteristic
5#15179C212yesmaximal
6#15178C211yesmaximal, characteristic
7#14810C214yesmaximal
8#14753C214yesmaximal
9#14142C212yesmaximal, complemented
10#14046C212yesmaximal, complemented
11#14035C214yesmaximal, complemented
12#13978C212yesmaximal, complemented
13C4×#385C211yesmaximal, characteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C4×Q8C2212yes—
17C23×C42C2211yescharacteristic
18#2446—216no—
19C2×#400C2214yes—
20C2×#398C2214yes—
21C4×C22⋊Q8—216no—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2214yes—
30C2×#387C2214yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#385C2212yes—
34C2×#385C2214yes—
35C2×#382C2211yescharacteristic
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2214yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2214yes—
42C2×C4×C22⋊C4C2211yescharacteristic
43#1445C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2212yes—
61#1436C2212yes—
62#1436C2214yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66#1434C2214yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C23×Q8C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C2×C4⋊Q8—28no—
73C2×C4⋊Q8—42no—
74C2×C22⋊Q8—28no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—28no—
77C2×C4×Q8—28no—
78C2×C4×Q8—28no—
79C2×C4×Q8—44no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2311yescharacteristic
93C22×C22⋊C4C2311yescharacteristic
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C42—28no—
97C22×C42—42no—
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2311yescharacteristic
103C22×C42C2314yes—
104C22×C42C2314yes—
105C22×C42C2312yes—
106#400—216no—
107#398—216no—
108#391—28no—
109#391—28no—
110#391—28no—
111#391D4116yes—
112#389—28no—
113#389—28no—
114#389—216no—
115#389—28no—
116#387—216no—
117#387—216no—
118#387—216no—
119#385—28no—
120#385—216no—
121#382—42no—
122#380—28no—
123C4×C4⋊C4—28no—
124C4×C4⋊C4—28no—
125C4×C4⋊C4—216no—
126C4×C22⋊C4—24no—
127C4×C22⋊C4—216no—
128C4×C22⋊C4—42no—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25C2411yescharacteristic
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8C2×D418yes—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4#4114yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2411yescharacteristic
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C4⋊Q8—416no—
188C22⋊Q8—264no—
189C4×Q8—432no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C2×D418yes—
221C2×C4⋊C4#4118yes—
222C2×C4⋊C4#4118yes—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—28no—
239C2×C42—28no—
240C2×C42—24no—
241C2×C42—28no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—28no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—416no—
252C2×C42—44no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—44no—
256C2×C42C2×D418yes—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—24no—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—24no—
276#95—44no—
277#95—44no—
278#95—42no—
279#95—42no—
280#95—44no—
281#95C2×D4116yes—
282#95C2×D4116yes—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—21no—
287C24—21no—
288C24—22no—
289C24—22no—
290C24C2×#4112yes—
291C242−1+414yes—
292C24C2511yescharacteristic
293C2×Q8—216no—
294C2×Q8—216no—
295C2×Q8—216no—
296C2×Q8—48no—
297C2×Q8—48no—
298C2×Q8—48no—
299C2×Q8—48no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—216no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—216no—
346C22×C4—28no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—48no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—44no—
377C22×C4C22×D418yes—
378C22×C4C22×D414yes—
379C22×C4C22×D418yes—
380C22×C4C22×D414yes—
381C22×C4C22×D414yes—
382C22×C4C2×#4118yes—
383C22×C4C2×#4114yes—
384C22×C4C2×#4114yes—
385C22×C4C2×#4112yes—
386C22×C4C2×#4114yes—
387C22×C4C2×#4112yes—
388C22×C4C2×#4114yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C42−1+414yes—
391C22×C42−1+412yes—
392C4⋊C4—232no—
393C4⋊C4—216no—
394C4⋊C4—216no—
395C4⋊C4—216no—
396C4⋊C4—416no—
397C4⋊C4—48no—
398C4⋊C4—416no—
399C4⋊C4—416no—
400C4⋊C4—416no—
401C4⋊C4—48no—
402C4⋊C4—416no—
403C22⋊C4—28no—
404C22⋊C4—232no—
405C22⋊C4—416no—
406C22⋊C4—48no—
407C22⋊C4—44no—
408C42—216no—
409C42—416no—
410C42—44no—
411C42—44no—
412C42—416no—
413C42—48no—
414C42—44no—
415C42—44no—
416C23—24no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23#54114yes—
437C23C23×D412yes—
438C23C22×#4111yescharacteristic
439C23C22×#4112yes—
440C23C22×#4112yes—
441C23C2×H(5,𝔽2)11yescharacteristic
442C23C2×2−1+411yescharacteristic
443C23C2×2−1+412yes—
444C23C2×2−1+412yes—
445C23#58512yes—
446Q8—432no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—432no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—41no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—416no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4#52518yes—
524C2×C4#53914yes—
525C2×C4D4214yes—
526C2×C4#54718yes—
527C2×C4#54818yes—
528C2×C4#55914yes—
529C22—24no—
530C22—28no—
531C22—42no—
532C22—44no—
533C22—42no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22C2×#52512yes—
539C22#344411yescharacteristic
540C22C2×#53911yescharacteristic
541C22C2×#54111yescharacteristic
542C22#346112yes—
543C22C2×D4211yescharacteristic
544C22C2×#54712yes—
545C22C2×#54812yes—
546C22D4×#4112yes—
547C22#347512yes—
548C22C2×#55911yescharacteristic
549C22#348212yes—
550C22#348612yes—
551C22#348912yes—
552C22#351612yes—
553C22#352012yes—
554C22#352312yes—
555C22#352312yes—
556C22#354112yes—
557C22#355611yescharacteristic
558C22#355711yescharacteristic
559C4—28no—
560C4—42no—
561C4—48no—
562C4—44no—
563C4—44no—
564C4—48no—
565C4—44no—
566C4—48no—
567C4—84no—
568C4—82no—
569C4—82no—
570C4#245518yes—
571C2—44no—
572C2C2×#245511yescharacteristic
573C2#3472012yes—
574C2#3508711yescharacteristic
575C2#3524412yes—
576C2#3752211yescharacteristic
577C2#5900512yes—
578C2#5907112yes—
579C2#5907312yes—
580C2#5959312yes—
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
1. C2×#2446C2semidirect
3. #16337C2semidirect
9. #14142C2semidirect
10. #14046C2semidirect
11. #14035C2semidirect
12. #13978C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24462
C2×C4×C22⋊Q82
#163372
#152232
#151792
#151782
#148102
#147532
#141422
#140462
#140352
#139782
C4×#3852