Finite Group Databasebeta

← #4463985

Subgroups — #4463985

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17263C212yesmaximal, complemented
2#17230C212yesmaximal, complemented
3#17221C212yesmaximal, complemented
4#17197C212yesmaximal, complemented
5#17182C214yesmaximal
6#17130C214yesmaximal
7#15194C212yesmaximal, complemented
8#15183C212yesmaximal
9C4⋊#401C212yesmaximal, complemented
10#15150C211yesmaximal, characteristic
11#15137C212yesmaximal, complemented
12#15135C211yesmaximal, characteristic
13#15133C212yesmaximal, complemented
14#14264C212yesmaximal
15#13844C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C2×#401C2214yes—
18C2×#399C2212yes—
19C2×#398C2214yes—
20C2×#398C2214yes—
21C2×#397C2212yes—
22C2×#397C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1445C2214yes—
35#1445C2212yes—
36#1445C2212yes—
37#1445C2214yes—
38#1445C2212yes—
39#1444C2211yescharacteristic
40#1444C2211yescharacteristic
41#1444C2214yes—
42#1444C2212yes—
43#1444C2214yes—
44#1444C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2211yescharacteristic
50#1442C2211yescharacteristic
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1439C2212yes—
61#1439C2212yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4×#95C2212yes—
75C24×C4C2311yescharacteristic
76C2×#117—44no—
77C2×#117—42no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4C2311yescharacteristic
95C22×C22⋊C4C2311yescharacteristic
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104#401—48no—
105#399—44no—
106#398—48no—
107#398—48no—
108#397—44no—
109#397—44no—
110#389—44no—
111#389—44no—
112#389—44no—
113#389—44no—
114#384—42no—
115#384—42no—
116#380—28no—
117#380—44no—
118C4×C4⋊C4—44no—
119C4×C22⋊C4—24no—
120C4×C22⋊C4—42no—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2312yes—
125C2×#95C2312yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2311yescharacteristic
129C2×#95C2314yes—
130C2×#95C2311yescharacteristic
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2311yescharacteristic
153C25C2411yescharacteristic
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4#4114yes—
159C23×C4C2414yes—
160C23×C4C2412yes—
161C23×C4C2414yes—
162C23×C4C2414yes—
163C23×C4C2414yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2411yescharacteristic
167C23×C4C2411yescharacteristic
168C23×C4C2411yescharacteristic
169C23×C4C2411yescharacteristic
170C23×C4C2411yescharacteristic
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173#117—416no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—41no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247#95—22no—
248#95—22no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—28no—
257#95—42no—
258#95—44no—
259#95—41no—
260#95—41no—
261#95—42no—
262#95—42no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—42no—
269#95—42no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—42no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—22no—
285C24—21no—
286C24—22no—
287C24—21no—
288C24—22no—
289C24C2×#4112yes—
290C24C2511yescharacteristic
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—216no—
318C22×C4—28no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—48no—
371C22×C4C2×#4112yes—
372C22×C4C2×#4112yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C4H(5,𝔽2)14yes—
377C22×C4H(5,𝔽2)12yes—
378C22×C42−1+414yes—
379C22×C42−1+412yes—
380C22×C42−1+414yes—
381C22×C42−1+412yes—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C4⋊C4—88no—
394C4⋊C4—84no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—88no—
403C22⋊C4—88no—
404C42—48no—
405C42—44no—
406C42—48no—
407C42—44no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—41no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—41no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23C22×#4111yescharacteristic
432C23C2×H(5,𝔽2)11yescharacteristic
433C23C2×H(5,𝔽2)12yes—
434C23C2×H(5,𝔽2)11yescharacteristic
435C23C2×H(5,𝔽2)11yescharacteristic
436C23C2×H(5,𝔽2)11yescharacteristic
437C23C2×2−1+412yes—
438C23C2×2−1+411yescharacteristic
439C23C2×2−1+412yes—
440C23C2×2−1+411yescharacteristic
441C23#58512yes—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
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—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—416no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
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—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4#53918yes—
525C2×C4#53914yes—
526C2×C4#56014yes—
527C2×C4#56314yes—
528C2×C4#56318yes—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—44no—
533C22—44no—
534C22—81no—
535C22—82no—
536C22—82no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22C2×#53911yescharacteristic
541C22C2×#53912yes—
542C22#349512yes—
543C22#350811yescharacteristic
544C22#351111yescharacteristic
545C22C2×#56011yescharacteristic
546C22C2×#56312yes—
547C22C2×#56311yescharacteristic
548C22#353611yescharacteristic
549C22#353711yescharacteristic
550C22#353812yes—
551C22#354212yes—
552C22#354412yes—
553C22#354612yes—
554C22#354812yes—
555C22#354811yescharacteristic
556C22#355011yescharacteristic
557C22#355011yescharacteristic
558C22#355111yescharacteristic
559C22#355212yes—
560C22#355512yes—
561C22#355512yes—
562C22#355912yes—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—44no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—84no—
571C4—84no—
572C4—82no—
573C4—84no—
574C4—82no—
575C4—84no—
576C4#283418yes—
577C2—82no—
578C2C2×#283411yescharacteristic
579C2#3731511yescharacteristic
580C2#3733512yes—
581C2#3734712yes—
582C2#3738311yescharacteristic
583C2#5296111yescharacteristic
584C2#5296611yescharacteristic
585C2#5296811yescharacteristic
586C2#5300711yescharacteristic
587C2#5301212yes—
588C2#5312312yes—
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
1. #17263C2semidirect
2. #17230C2semidirect
3. #17221C2semidirect
4. #17197C2semidirect
7. #15194C2semidirect
9. C4⋊#401C2semidirect
11. #15137C2semidirect
13. #15133C2semidirect

Maximal subgroups

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

subgroupindex
#172632
#172302
#172212
#171972
#171822
#171302
#151942
#151832
C4⋊#4012
#151502
#151372
#151352
#151332
#142642
#138442