Finite Group Databasebeta

← #761649

Subgroups — #761649

Order 512.

2190 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16920C214yesmaximal, complemented
2#15235C211yesmaximal, characteristic, complemented
3#15196C211yesmaximal, characteristic, complemented
4#15181C212yesmaximal
5#15036C214yesmaximal, complemented
6#14986C214yesmaximal
7#14031C214yesmaximal, complemented
8#13974C212yesmaximal, complemented
9#13974C212yesmaximal, complemented
10#13876C212yesmaximal, complemented
11#13875C212yesmaximal, complemented
12C4×#389C212yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic
14C22×C4×D4C2211yescharacteristic
15C2×#390C2212yes—
16C2×#389C2212yes—
17C2×#389C2212yes—
18C2×#388C2214yes—
19C2×#388C2214yes—
20C2×#388C2214yes—
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2214yes—
31C2×C4×C22⋊C4C2212yes—
32C2×C4×C22⋊C4C2212yes—
33C2×C43C2211yescharacteristic
34#1444C2214yes—
35#1443C2214yes—
36#1443C2214yes—
37#1443C2214yes—
38#1443C2214yes—
39#1442C2214yes—
40#1442C2214yes—
41#1442C2214yes—
42#1441C2214yes—
43#1440C2214yes—
44#1440C2214yes—
45#1440C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1438C2214yes—
50#1437C2212yes—
51#1437C2214yes—
52#1437C2212yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2212yes—
65#1434C2214yes—
66C4×#95C2214yes—
67C23×D4C2311yescharacteristic
68C24×C4C2312yes—
69C2×C4×D4—24no—
70C2×C4×D4—42no—
71C2×C4×D4—42no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2312yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2312yes—
93C22×C42C2314yes—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96C22×C42C2312yes—
97C22×C42C2314yes—
98C22×C42C2311yescharacteristic
99#390—28no—
100#389—28no—
101#389—28no—
102#388—216no—
103#388—48no—
104#388—48no—
105#384—42no—
106#382—42no—
107#380—28no—
108#380—28no—
109#380—44no—
110#380—44no—
111C4×C4⋊C4—28no—
112C4×C4⋊C4—48no—
113C4×C4⋊C4—42no—
114C4×C4⋊C4—44no—
115C4×C22⋊C4—28no—
116C4×C22⋊C4—28no—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2312yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C43—42no—
142C25C2412yes—
143C22×D4—24no—
144C22×D4—22no—
145C22×D4—24no—
146C22×D4—41no—
147C22×D4—41no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4C2414yes—
156C23×C4C2414yes—
157C23×C4C2412yes—
158C23×C4C2414yes—
159C23×C4C2412yes—
160C23×C4C2411yescharacteristic
161C23×C4C2412yes—
162C23×C4C2414yes—
163C23×C4C2412yes—
164C23×C4C2412yes—
165C23×C4C2411yescharacteristic
166C23×C4C2411yescharacteristic
167C4×D4—416no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4#4118yes—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4#4118yes—
222C2×C42—24no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—28no—
229C2×C42—28no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—28no—
236C2×C42—41no—
237C2×C42—42no—
238C2×C42—41no—
239C2×C42—44no—
240C2×C42—42no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42#4114yes—
244C2×C42#4118yes—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—42no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24C2×#4114yes—
284C24C2511yescharacteristic
285C2×D4—28no—
286C2×D4—48no—
287C2×D4—48no—
288C2×D4—44no—
289C2×D4—44no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4C2×#4114yes—
362C22×C4C2×#4114yes—
363C22×C4C2×#4118yes—
364C22×C4C2×#4112yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4114yes—
367C22×C4C2×#4114yes—
368C22×C4C2×#4112yes—
369C22×C4C2×#4114yes—
370C22×C4C2×#4114yes—
371C22×C4H(5,𝔽2)14yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C42−1+414yes—
375C4⋊C4—216no—
376C4⋊C4—48no—
377C4⋊C4—416no—
378C4⋊C4—44no—
379C4⋊C4—48no—
380C4⋊C4—416no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—88no—
384C22⋊C4—216no—
385C22⋊C4—48no—
386C22⋊C4—48no—
387C22⋊C4—416no—
388C22⋊C4—416no—
389C22⋊C4—48no—
390C22⋊C4—48no—
391C22⋊C4—88no—
392C22⋊C4—88no—
393C42—28no—
394C42—216no—
395C42—416no—
396C42—48no—
397C42—44no—
398C42—44no—
399C42—44no—
400C42—44no—
401C42—88no—
402C23—24no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23C22×#4111yescharacteristic
438C23C22×#4112yes—
439C23C22×#4112yes—
440C23C22×#4112yes—
441C23C2×H(5,𝔽2)11yescharacteristic
442C23C2×H(5,𝔽2)11yescharacteristic
443C23C2×H(5,𝔽2)12yes—
444C23C2×2−1+412yes—
445C23#58512yes—
446D4—416no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—816no—
519C2×C4#52918yes—
520C2×C4#53814yes—
521C2×C4#54114yes—
522C2×C4#55318yes—
523C2×C4#55318yes—
524C2×C4#55514yes—
525C2×C4#55914yes—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22C2×#52912yes—
542C22#344411yescharacteristic
543C22C2×#53811yescharacteristic
544C22C2×#54111yescharacteristic
545C22#346012yes—
546C22C2×#55312yes—
547C22C2×#55312yes—
548C22C2×#55511yescharacteristic
549C22#347512yes—
550C22#347512yes—
551C22C2×#55911yescharacteristic
552C22#348112yes—
553C22#350712yes—
554C22#351112yes—
555C22#351812yes—
556C22#352012yes—
557C22#352012yes—
558C22#352212yes—
559C22#353711yescharacteristic
560C22#353911yescharacteristic
561C22#354612yes—
562C4—24no—
563C4—28no—
564C4—44no—
565C4—48no—
566C4—42no—
567C4—44no—
568C4—84no—
569C4—82no—
570C4—82no—
571C4—84no—
572C4—84no—
573C4#259318yes—
574C2—84no—
575C2C2×#259311yescharacteristic
576C2#3491712yes—
577C2#3508611yescharacteristic
578C2#3518512yes—
579C2#3751511yescharacteristic
580C2#5954412yes—
581C2#5956612yes—
582C2#5967612yes—
583C2#5967812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #16920C2semidirect
2. #15235C2semidirect
3. #15196C2semidirect
5. #15036C2semidirect
7. #14031C2semidirect
8. #13974C2semidirect
9. #13974C2semidirect
10. #13876C2semidirect
11. #13875C2semidirect
12. C4×#389C2semidirect

Maximal subgroups

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

subgroupindex
#169202
#152352
#151962
#151812
#150362
#149862
#140312
#139742
#138762
#138752
C4×#3892
C4×#3902