Finite Group Databasebeta

← #4306748

Subgroups — #4306748

Order 512.

2382 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2370C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3#15242C211yesmaximal, characteristic
4#15228C212yesmaximal
5#15202C211yesmaximal, characteristic
6#15151C211yesmaximal, characteristic
7#15133C212yesmaximal
8#15106C212yesmaximal
9#15008C214yesmaximal
10#14987C214yesmaximal
11#14903C212yesmaximal
12#14879C212yesmaximal
13C4×#395C211yesmaximal, characteristic
14#13850C212yesmaximal
15#13818C212yesmaximal
16#13810C211yesmaximal, characteristic
17#13804C212yesmaximal
18C22×#125C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C2×#398C2214yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#395C2211yescharacteristic
25#2370—28no—
26Q8×C4⋊C4—28no—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#384C2212yes—
37C2×#384C2211yescharacteristic
38C2×#382C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×#376C2212yes—
48#1446C2211yescharacteristic
49#1446C2212yes—
50#1445C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1444C2211yescharacteristic
55#1444C2212yes—
56#1444C2214yes—
57#1444C2212yes—
58#1444C2214yes—
59#1444C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2211yescharacteristic
66#1441C2212yes—
67#1441C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2214yes—
77C4⋊#95C2212yes—
78#1434C2212yes—
79#1434C2214yes—
80#1434C2214yes—
81C4×#95C2214yes—
82C4×#95C2214yes—
83C4×#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C2×#125—24no—
87C2×#125—24no—
88C2×#125—24no—
89C2×C4×Q8—28no—
90C2×C4×Q8—24no—
91C2×C4×Q8—28no—
92C2×C4×Q8—28no—
93C2×C4×Q8—44no—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2314yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2314yes—
126#398—44no—
127#398—44no—
128#398—48no—
129#395—24no—
130#389—24no—
131#389—28no—
132#389—24no—
133#389—28no—
134#389—28no—
135#389—216no—
136#389—28no—
137#387—28no—
138#387—28no—
139#384—28no—
140#384—24no—
141#382—28no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4D418yes—
149C4×C4⋊C4Q8116yescomplemented
150#376—44no—
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8#4114yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205#125—232no—
206C4×Q8—232no—
207C4×Q8—432no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×Q818yes—
268C2×C4⋊C4C2×Q818yes—
269C2×C4⋊C4#4114yes—
270C2×C42—24no—
271C2×C42—22no—
272C2×C42—24no—
273C2×C42—24no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—28no—
280C2×C42—22no—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—44no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42C2×D414yes—
291C2×C42C2×D418yes—
292C2×C42C2×Q818yes—
293C2×C42C2×Q8116yes—
294C2×C42#4118yes—
295C2×C42#4114yes—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—22no—
310#95—24no—
311#95—22no—
312#95—24no—
313#95—28no—
314#95—24no—
315#95—28no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—42no—
321#95—44no—
322#95—44no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95C2×D418yes—
327#95C2×Q8116yes—
328C24C2511yescharacteristic
329C2×Q8—28no—
330C2×Q8—28no—
331C2×Q8—216no—
332C2×Q8—28no—
333C2×Q8—216no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4C22×D412yes—
413C22×C4C22×D414yes—
414C22×C4C22×D414yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×Q814yes—
418C22×C4C22×Q818yes—
419C22×C4C22×Q814yes—
420C22×C4C22×Q818yes—
421C22×C4C22×Q814yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4114yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4114yes—
429C22×C4C2×#4114yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)12yes—
434C22×C42−1+414yes—
435C4⋊C4—28no—
436C4⋊C4—216no—
437C4⋊C4—216no—
438C4⋊C4—216no—
439C4⋊C4—28no—
440C4⋊C4—28no—
441C4⋊C4—216no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—416no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—88no—
452C4⋊C4—84no—
453C4⋊C4—88no—
454C4⋊C4—84no—
455C4⋊C4—88no—
456C4⋊C4C4⋊Q8116yes—
457C42—28no—
458C42—216no—
459C42—28no—
460C42—216no—
461C42—216no—
462C42—416no—
463C42—416no—
464C42—44no—
465C42—84no—
466C42—84no—
467C23C23×D411yescharacteristic
468C23C23×Q812yes—
469C23C22×#4112yes—
470C23C22×#4111yescharacteristic
471C23C22×#4111yescharacteristic
472C23C22×#4111yescharacteristic
473C23C2×H(5,𝔽2)11yescharacteristic
474C23C2×H(5,𝔽2)11yescharacteristic
475C23C2×2−1+412yes—
476C23#58512yes—
477C23#58511yescharacteristic
478Q8—232no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
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—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4#52514yes—
573C2×C4#52514yes—
574C2×C4C2×C4⋊Q818yes—
575C2×C4C2×C4⋊Q814yes—
576C2×C4#53318yes—
577C2×C4#53318yes—
578C2×C4#54614yes—
579C2×C4#55218yes—
580C22C2×#52511yescharacteristic
581C22C2×#52511yescharacteristic
582C22C22×C4⋊Q811yescharacteristic
583C22C2×#53312yes—
584C22C2×#53312yes—
585C22C2×#54611yescharacteristic
586C22C2×#55212yes—
587C22Q8×#4112yes—
588C22#348011yescharacteristic
589C22#348311yescharacteristic
590C22#348611yescharacteristic
591C22#348811yescharacteristic
592C22#349212yes—
593C22#349412yes—
594C22#350011yescharacteristic
595C22#350112yes—
596C22#350212yes—
597C22#350312yes—
598C22#350712yes—
599C22#350811yescharacteristic
600C22#351212yes—
601C22#351512yes—
602C22#351711yescharacteristic
603C4—24no—
604C4—28no—
605C4—28no—
606C4—42no—
607C4—44no—
608C4—48no—
609C4—44no—
610C4—42no—
611C4—44no—
612C4—44no—
613C4—42no—
614C4—44no—
615C4—42no—
616C4—84no—
617C4—81no—
618C4—82no—
619C4#239918yes—
620C2C2×#239911yescharacteristic
621C2#3570812yes—
622C2#3571211yescharacteristic
623C2#3593412yes—
624C2#3593711yescharacteristic
625C2#3608311yescharacteristic
626C2#3616311yescharacteristic
627C2#3627611yescharacteristic
628C2#3628512yes—
629C2#3628712yes—
630C2#3641011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
149. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23702
C2×Q8×C4⋊C42
#152422
#152282
#152022
#151512
#151332
#151062
#150082
#149872
#149032
#148792
C4×#3952
#138502
#138182
#138102
#138042