Finite Group Databasebeta

← #719406

Subgroups — #719406

Order 512.

2698 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2549C211yesmaximal, characteristic
2C2×#2503C212yesmaximal
3C22×#398C211yesmaximal, characteristic
4#16835C214yesmaximal, complemented
5#15681C214yesmaximal, complemented
6#15509C214yesmaximal, complemented
7#15073C212yesmaximal
8#14937C214yesmaximal, complemented
9#14934C212yesmaximal
10#14339C211yesmaximal, characteristic
11#13972C212yesmaximal
12#13904C211yesmaximal, characteristic
13#13904C211yesmaximal, characteristic
14#13802C211yesmaximal, characteristic
15#13776C211yesmaximal, characteristic
16C22×C22⋊Q8C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20#2549—28no—
21#2503—216no—
22C2×#400C2214yes—
23C2×#400C2214yes—
24C2×#398—28no—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#395C2212yes—
30C2×#391C2212yes—
31C2×#391C2214yes—
32C2×#389C2212yes—
33C2×#385C2212yes—
34C2×#382C2214yes—
35C2×#381C2212yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#381C2211yescharacteristic
40C2×#381C2212yes—
41C2×#381C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C22×#95C2212yes—
48C22×#95C2211yescharacteristic
49#1446C2214yes—
50#1445C2212yes—
51#1445C2214yes—
52#1445C2212yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2214yes—
70C4⋊#95C2214yes—
71#1434C2214yes—
72#1434C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C22⋊#95C2211yescharacteristic
77C4×#95C2212yes—
78C23×Q8C2312yes—
79C24×C4C2312yes—
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—28no—
86C2×C22⋊Q8—44no—
87C22×C4⋊C4—28no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—24no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C42C2314yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118#400—216no—
119#400—216no—
120#398—216no—
121#398—28no—
122#398—28no—
123#398—28no—
124#395—28no—
125#391—216no—
126#391—28no—
127#389—28no—
128#385—28no—
129#382—216no—
130#381—28no—
131#381—28no—
132#381—28no—
133#381—24no—
134#381—28no—
135#381—44no—
136#381D418yes—
137#380—24no—
138#380—24no—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—28no—
142C2×#95—28no—
143C2×#95D418yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2314yes—
175C25C2411yescharacteristic
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—28no—
179C22×Q8—42no—
180C22×Q8—44no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—28no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D414yes—
204C23×C4C2×D418yes—
205C23×C4C2×D414yes—
206C23×C4#4114yes—
207C23×C4#4114yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C22⋊Q8—432no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
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—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—432no—
256C2×C4⋊C4—416no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4C2×D414yes—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4#4114yes—
279C2×C42—28no—
280C2×C42—28no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—28no—
290#95—22no—
291#95—22no—
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—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95C2×D418yes—
314#95C2×D414yes—
315#95C2×D414yes—
316#95C2×D4116yes—
317C24—24no—
318C24—24no—
319C24C22×D412yes—
320C24C22×D412yes—
321C24C2×#4112yes—
322C24C2×#4112yes—
323C24C2×#4112yes—
324C24H(5,𝔽2)12yes—
325C242−1+412yes—
326C24C2511yescharacteristic
327C2×Q8—48no—
328C2×Q8—416no—
329C2×Q8—48no—
330C2×Q8—84no—
331C2×Q8—88no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—216no—
340C22×C4—28no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—216no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—416no—
403C22×C4—416no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—84no—
418C22×C4—84no—
419C22×C4#124116yes—
420C22×C4C22×D414yes—
421C22×C4C22×D414yes—
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D414yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C42−1+418yes—
433C4⋊C4—416no—
434C4⋊C4—416no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C22⋊C4—28no—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—44no—
452C42—416no—
453C42—84no—
454C42—84no—
455C23—24no—
456C23—24no—
457C23—28no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23C2×#12414yes—
466C23#53214yes—
467C23#54214yes—
468C23D4214yes—
469C23#54714yes—
470C23#54814yes—
471C23#55514yes—
472C23C23×D411yescharacteristic
473C23C23×D411yescharacteristic
474C23C22×#4111yescharacteristic
475C23C22×#4111yescharacteristic
476C23C22×#4111yescharacteristic
477C23C2×H(5,𝔽2)11yescharacteristic
478C23C2×2−1+414yes—
479C23C2×2−1+411yescharacteristic
480C23#58514yes—
481Q8—816no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—28no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—416no—
505C2×C4—432no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—416no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—88no—
557C2×C4—816no—
558C2×C4C2×#12414yes—
559C2×C4C2×#12414yes—
560C2×C4#53618yes—
561C22—28no—
562C22—24no—
563C22—28no—
564C22—28no—
565C22—28no—
566C22—28no—
567C22—24no—
568C22—28no—
569C22#249118yes—
570C22C22×#12411yescharacteristic
571C22C2×#53211yescharacteristic
572C22C2×#53612yes—
573C22C2×#54211yescharacteristic
574C22C2×D4211yescharacteristic
575C22C2×#54711yescharacteristic
576C22C2×#54811yescharacteristic
577C22C2×#55511yescharacteristic
578C22#348512yes—
579C22#348914yes—
580C22#349112yes—
581C22#349112yes—
582C22#350912yes—
583C22#350912yes—
584C22#351514yes—
585C22#353514yes—
586C22#356314yes—
587C4—28no—
588C4—24no—
589C4—24no—
590C4—44no—
591C4—44no—
592C4—44no—
593C4—44no—
594C4—44no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—84no—
599C4—84no—
600C4—82no—
601C2—28no—
602C2C2×#249111yescharacteristic
603C2#3532011yescharacteristic
604C2#3545511yescharacteristic
605C2#3602412yes—
606C2#3764712yes—
607C2#3912212yes—
608C2#4139312yes—
609C2#4351412yes—
610C2#4446112yes—
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
4. #16835C2semidirect
5. #15681C2semidirect
6. #15509C2semidirect
8. #14937C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25492
C2×#25032
C22×#3982
#168352
#156812
#155092
#150732
#149372
#149342
#143392
#139722
#139042
#138022
#137762