Finite Group Databasebeta

← #533471

Subgroups — #533471

Order 512.

2686 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#2369C212yesmaximal
3#14587C212yesmaximal
4#14450C212yesmaximal
5#14238C212yesmaximal
6#14217C214yesmaximal
7#14059C214yesmaximal
8#14053C212yesmaximal
9#14046C212yesmaximal
10#14005C212yesmaximal
11#13926C211yesmaximal, characteristic
12#13821C212yesmaximal
13#13821C212yesmaximal
14C4×#391C211yesmaximal, characteristic
15C4×#389C212yesmaximal
16C22×C4⋊Q8C2212yes—
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19#2546C4116yescomplemented
20C2×#398C2212yes—
21C2×#398C2212yes—
22#2369—216no—
23C2×#391C2212yes—
24C2×#391C2214yes—
25C2×#391C2211yescharacteristic
26C2×#391C2212yes—
27C2×#391C2211yescharacteristic
28C2×#389C2214yes—
29C2×#389C2214yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#384C2211yescharacteristic
36C2×C4×C4⋊C4C2214yes—
37C2×C4×C4⋊C4C2214yes—
38C2×C4×C4⋊C4C2214yes—
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×C43C2211yescharacteristic
46#1446C2214yes—
47#1444C2214yes—
48#1442C2214yes—
49#1441C2212yes—
50#1441C2212yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2212yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72#1434C2214yes—
73#1434C2214yes—
74C4×#95C2214yes—
75C4×#95C2212yes—
76C4×#95C2212yes—
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C2×C4⋊Q8—28no—
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8C2×C4116yes—
84C2×C4⋊Q8C2×C418yes—
85C2×C4×Q8—42no—
86C2×C4×Q8—42no—
87C2×C4×Q8C2×C418yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C42C2312yes—
106C22×C42C2314yes—
107C22×C42C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2312yes—
114C22×C42C2314yes—
115C22×C42C2311yescharacteristic
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118#398C2×C4116yes—
119#398C2×C4116yes—
120#391—28no—
121#391—24no—
122#391—216no—
123#391C2×C4116yes—
124#391D418yes—
125#389—216no—
126#389—28no—
127#389—216no—
128#389—44no—
129#389C2×C4116yes—
130#389C2×C4116yes—
131#389Q8116yescomplemented
132#384—42no—
133C4×C4⋊C4—28no—
134C4×C4⋊C4—28no—
135C4×C4⋊C4—216no—
136C4×C4⋊C4—216no—
137C4×C4⋊C4—216no—
138C4×C4⋊C4—42no—
139C4×C4⋊C4—42no—
140C4×C4⋊C4—44no—
141C4×C4⋊C4C2×C418yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C43—42no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8C22×C418yes—
181C22×Q8C22×C414yes—
182C22×Q8C2×D414yes—
183C22×Q8#4114yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C4⋊Q8—264no—
200C4⋊Q8—416no—
201C4×Q8—416no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C22×C418yes—
240C2×C4⋊C4C22×C4116yes—
241C2×C4⋊C4C22×C4116yes—
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C22×C414yes—
244C2×C4⋊C4C22×C414yes—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C418yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×Q8116yes—
249C2×C4⋊C4C2×Q818yes—
250C2×C4⋊C4C2×Q818yes—
251C2×C4⋊C4#4118yes—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—28no—
263C2×C42—28no—
264C2×C42—28no—
265C2×C42—28no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—28no—
273C2×C42—41no—
274C2×C42—42no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42—44no—
278C2×C42—44no—
279C2×C42—42no—
280C2×C42—41no—
281C2×C42—44no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42—42no—
285C2×C42C22×C414yes—
286C2×C42C22×C414yes—
287C2×C42C22×C414yes—
288C2×C42C22×C418yes—
289C2×C42C2×D414yes—
290C2×C42C2×Q818yes—
291#95—28no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—42no—
303#95—44no—
304#95—44no—
305#95—42no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—44no—
312#95C22×C418yes—
313#95C22×C4116yes—
314#95C22×C4116yes—
315#95C2×D418yes—
316#95C2×D418yes—
317#95C2×Q8116yes—
318C24C2511yescharacteristic
319C2×Q8—216no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×Q8—84no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×Q8—84no—
332C2×Q8C4×D4116yes—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4C23×C412yes—
404C22×C4C23×C412yes—
405C22×C4C23×C412yes—
406C22×C4C23×C418yes—
407C22×C4C23×C412yes—
408C22×C4C23×C412yes—
409C22×C4C23×C412yes—
410C22×C4C23×C418yes—
411C22×C4C23×C412yes—
412C22×C4C22×D414yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D414yes—
417C22×C4C22×Q814yes—
418C22×C4C22×Q818yes—
419C22×C4C22×Q818yes—
420C22×C4C22×Q814yes—
421C22×C4C22×Q814yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4118yes—
426C22×C4C2×#4114yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C42−1+414yes—
430C4⋊C4—232no—
431C4⋊C4—232no—
432C4⋊C4—216no—
433C4⋊C4—232no—
434C4⋊C4—216no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—416no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—84no—
444C4⋊C4—88no—
445C4⋊C4—84no—
446C4⋊C4C4×Q8132yes—
447C42—216no—
448C42—28no—
449C42—28no—
450C42—28no—
451C42—216no—
452C42—44no—
453C42—416no—
454C42—416no—
455C42—44no—
456C42—48no—
457C42—48no—
458C42—88no—
459C42—84no—
460C23C24×C411yescharacteristic
461C23C23×D411yescharacteristic
462C23C23×Q812yes—
463C23C22×#4111yescharacteristic
464C23C22×#4112yes—
465C23C2×H(5,𝔽2)11yescharacteristic
466C23C2×H(5,𝔽2)11yescharacteristic
467C23C2×2−1+412yes—
468C23#58511yescharacteristic
469C23#58511yescharacteristic
470C23#58512yes—
471Q8—416no—
472Q8—816no—
473C2×C4—28no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
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—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4C2×C4×D414yes—
561C2×C4C2×C4×D414yes—
562C2×C4C2×C4×D414yes—
563C2×C4C2×C4×Q8116yes—
564C2×C4C2×C4×Q818yes—
565C2×C4#51814yes—
566C2×C4D4×Q818yes—
567C2×C4#55418yes—
568C2×C4#55514yes—
569C2×C4Q8214yes—
570C22C22×C4×D411yescharacteristic
571C22C22×C4×Q812yes—
572C22C2×#51811yescharacteristic
573C22#342811yescharacteristic
574C22C4×2−1+412yes—
575C22C2×D4×Q812yes—
576C22D4×#4112yes—
577C22C2×#55412yes—
578C22C2×#55511yescharacteristic
579C22C2×Q8211yescharacteristic
580C22Q8×#4112yes—
581C22#348212yes—
582C22#349611yescharacteristic
583C22#349611yescharacteristic
584C22#350212yes—
585C22#350412yes—
586C22#350511yescharacteristic
587C22#350511yescharacteristic
588C22#351612yes—
589C22#352412yes—
590C22#354112yes—
591C22#355011yescharacteristic
592C22#355111yescharacteristic
593C4—216no—
594C4—24no—
595C4—24no—
596C4—24no—
597C4—28no—
598C4—44no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—48no—
603C4—48no—
604C4—44no—
605C4—82no—
606C4—84no—
607C4#236918yescomplemented
608C2C2×#236911yescharacteristic
609C2C4×D4×Q812yes—
610C2#3420511yescharacteristic
611C2#3433612yes—
612C2#3441311yescharacteristic
613C2#5983112yes—
614C2#5983211yescharacteristic
615C2#5984111yescharacteristic
616C2#5992212yes—
617C2#5992511yescharacteristic
618C2#5995111yescharacteristic
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
19. #2546C4semidirect
131. #389Q8semidirect
607. C4#2369semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#23692
#145872
#144502
#142382
#142172
#140592
#140532
#140462
#140052
#139262
#138212
C4×#3912
C4×#3892