Finite Group Databasebeta

← #762943

Subgroups — #762943

Order 512.

2350 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2593C212yesmaximal
2#17076C211yesmaximal, characteristic, complemented
3#17075C211yesmaximal, characteristic, complemented
4#17069C212yesmaximal
5#15085C212yesmaximal, complemented
6#14824C212yesmaximal
7#14784C214yesmaximal, complemented
8#14782C214yesmaximal, complemented
9#14238C212yesmaximal, complemented
10#14229C214yesmaximal, complemented
11#13915C214yesmaximal, complemented
12C4×#389C212yesmaximal, complemented
13C4×#390C211yesmaximal, characteristic
14C22×C4×D4C2211yescharacteristic
15#2593—216no—
16C2×#400C2212yes—
17C2×#398C2212yes—
18C2×#398C2212yes—
19C2×#397C2214yes—
20C2×#397C2212yes—
21C2×#394C2214yes—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#388C2214yes—
29C2×#388C2214yes—
30C2×#382C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2214yes—
33C2×C4×C4⋊C4C2212yes—
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2214yes—
37C2×C43C2211yescharacteristic
38#1445C2214yes—
39#1445C2212yes—
40#1445C2212yes—
41#1444C2214yes—
42#1444C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1441C2212yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2212yes—
52#1437C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59C4⋊#95C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2212yes—
62C4⋊#95C2211yescharacteristic
63#1434C2214yes—
64#1434C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C4×#95C2214yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C2×C4×D4—24no—
71C2×C4×D4—42no—
72C2×C4×D4—42no—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C42C2311yescharacteristic
94C22×C42C2312yes—
95C22×C42C2312yes—
96C22×C42C2314yes—
97C22×C42C2314yes—
98C22×C42C2311yescharacteristic
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101#400—28no—
102#398—28no—
103#398—28no—
104#397—216no—
105#397—44no—
106#394—216no—
107#390—24no—
108#390D418yes—
109#389—28no—
110#389—28no—
111#388—216no—
112#388—216no—
113#388—216no—
114#388—216no—
115#382—44no—
116#380—44no—
117C4×C4⋊C4—28no—
118C4×C4⋊C4—216no—
119C4×C4⋊C4—44no—
120C4×C4⋊C4—42no—
121C4×C22⋊C4—216no—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C43—42no—
146C25C2412yes—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—41no—
150C22×D4—41no—
151C22×D4—41no—
152C22×D4—41no—
153C22×D4C2×D414yes—
154C23×C4—24no—
155C23×C4—28no—
156C23×C4—24no—
157C23×C4—42no—
158C23×C4—44no—
159C23×C4C2414yes—
160C23×C4C2414yes—
161C23×C4C2414yes—
162C23×C4C2412yes—
163C23×C4C2412yes—
164C23×C4C2414yes—
165C23×C4C2412yes—
166C23×C4C2411yescharacteristic
167C23×C4C2411yescharacteristic
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2411yescharacteristic
171C4×D4—416no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4#4118yes—
209C2×C4⋊C4#4114yes—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4C2×D4116yes—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—28no—
238C2×C42—28no—
239C2×C42—28no—
240C2×C42—24no—
241C2×C42—28no—
242C2×C42—41no—
243C2×C42—42no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42—42no—
247C2×C42—41no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42C2×D414yes—
252C2×C42#4118yes—
253C2×C42#4114yes—
254#95—28no—
255#95—28no—
256#95—24no—
257#95—28no—
258#95—24no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—44no—
273#95—44no—
274#95—42no—
275#95—44no—
276#95—42no—
277C24—24no—
278C24—24no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—24no—
284C24—24no—
285C24—22no—
286C24—22no—
287C24C22×D414yes—
288C24C2511yescharacteristic
289C2×D4—28no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—44no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—48no—
356C22×C4—416no—
357C22×C4—48no—
358C22×C4—88no—
359C22×C4—84no—
360C22×C4C22×D414yes—
361C22×C4C22×D412yes—
362C22×C4C22×D414yes—
363C22×C4C2×#4118yes—
364C22×C4C2×#4114yes—
365C22×C4C2×#4114yes—
366C22×C4C2×#4114yes—
367C22×C4C2×#4114yes—
368C22×C4C2×#4112yes—
369C22×C4C2×#4112yes—
370C22×C4C2×#4114yes—
371C22×C4C2×#4114yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C42−1+414yes—
375C4⋊C4—216no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—44no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—416no—
383C4⋊C4—416no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C22⋊C4—232no—
389C22⋊C4—48no—
390C22⋊C4—416no—
391C22⋊C4—416no—
392C22⋊C4—416no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C42—216no—
396C42—28no—
397C42—44no—
398C42—44no—
399C42—48no—
400C42—416no—
401C42—48no—
402C42—88no—
403C23—24no—
404C23—28no—
405C23—28no—
406C23—24no—
407C23—24no—
408C23—44no—
409C23—42no—
410C23—42no—
411C23—44no—
412C23—44no—
413C23—42no—
414C23—44no—
415C23—44no—
416C23—42no—
417C23—42no—
418C23—44no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—44no—
426C23C23×D411yescharacteristic
427C23C22×#4111yescharacteristic
428C23C22×#4112yes—
429C23C22×#4112yes—
430C23C22×#4111yescharacteristic
431C23C2×H(5,𝔽2)11yescharacteristic
432C23C2×H(5,𝔽2)11yescharacteristic
433C23C2×2−1+412yes—
434C23#58511yescharacteristic
435C23#58511yescharacteristic
436C23#58512yes—
437D4—416no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—816no—
509C2×C4#52918yes—
510C2×C4#53214yes—
511C2×C4#54718yes—
512C2×C4#55914yes—
513C22—48no—
514C22—48no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—82no—
519C22—82no—
520C22—84no—
521C22—82no—
522C22—84no—
523C22—82no—
524C22C2×#52912yes—
525C22C2×#53211yescharacteristic
526C22#345911yescharacteristic
527C22#346011yescharacteristic
528C22#346112yes—
529C22C2×#54712yes—
530C22D4×#4112yes—
531C22#347512yes—
532C22C2×#55911yescharacteristic
533C22#348012yes—
534C22#348212yes—
535C22#348311yescharacteristic
536C22#349011yescharacteristic
537C22#349212yes—
538C22#349312yes—
539C22#349311yescharacteristic
540C22#349611yescharacteristic
541C22#349611yescharacteristic
542C22#350512yes—
543C22#350712yes—
544C22#354511yescharacteristic
545C22#355011yescharacteristic
546C22#355312yes—
547C4—28no—
548C4—24no—
549C4—28no—
550C4—24no—
551C4—48no—
552C4—42no—
553C4—44no—
554C4—44no—
555C4—82no—
556C4—84no—
557C4—84no—
558C4—84no—
559C2—84no—
560C2#3493212yes—
561C2#3495411yescharacteristic
562C2#3500311yescharacteristic
563C2#3502112yes—
564C2#3761211yescharacteristic
565C2#5982512yes—
566C2#5982711yescharacteristic
567C2#5983411yescharacteristic
568C2#5986111yescharacteristic
569C2#5987712yes—
570C2#6004711yescharacteristic
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
2. #17076C2semidirect
3. #17075C2semidirect
5. #15085C2semidirect
7. #14784C2semidirect
8. #14782C2semidirect
9. #14238C2semidirect
10. #14229C2semidirect
11. #13915C2semidirect
12. C4×#389C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25932
#170762
#170752
#170692
#150852
#148242
#147842
#147822
#142382
#142292
#139152
C4×#3892
C4×#3902