Finite Group Databasebeta

← #1010568

Subgroups — #1010568

Order 512.

2126 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2495C212yesmaximal
2#17120C212yesmaximal, complemented
3#17059C212yesmaximal
4#16671C214yesmaximal, complemented
5#16116C212yesmaximal, complemented
6#15472C214yesmaximal
7#15225C212yesmaximal, complemented
8#15150C211yesmaximal, characteristic
9#15137C212yesmaximal, complemented
10#14581C212yesmaximal
11#14130C211yesmaximal, characteristic
12#14045C212yesmaximal, complemented
13#13982C212yesmaximal, complemented
14#13907C212yesmaximal
15C4×#397C211yesmaximal, characteristic
16C23×C4⋊C4C2211yescharacteristic
17#2495—216no—
18C2×#400C2214yes—
19C2×#398C2214yes—
20C2×#397C2211yescharacteristic
21C2×#397C2212yes—
22C2×#397C2214yes—
23C2×#397C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#385C2212yes—
31C2×#384C2211yescharacteristic
32C2×#383C2214yes—
33C2×#382C2212yes—
34C2×#380C2212yes—
35C2×#380C2214yes—
36C2×#380C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41#1445C2212yes—
42#1445C2214yes—
43#1445C2211yescharacteristic
44#1445C2212yes—
45#1445C2214yes—
46#1444C2212yes—
47#1444C2214yes—
48#1444C2214yes—
49#1443C2212yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2211yescharacteristic
53#1442C2212yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2212yes—
62#1439C2211yescharacteristic
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71#1434C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C4×#95C2212yes—
75C4×#95C2211yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C22×C4⋊C4—28no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2314yes—
105#400—216no—
106#398—48no—
107#397—216no—
108#397—24no—
109#397—28no—
110#397—44no—
111#389—28no—
112#389—28no—
113#389—28no—
114#389—28no—
115#389—44no—
116#388—216no—
117#385—44no—
118#384—42no—
119#383—216no—
120#382—28no—
121#380—28no—
122#380—48no—
123#380D418yes—
124C4×C4⋊C4—28no—
125C4×C4⋊C4—42no—
126C4×C22⋊C4—28no—
127C4×C22⋊C4—42no—
128C2×#95C2314yes—
129C2×#95C2311yescharacteristic
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C25C2411yescharacteristic
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—28no—
167C23×C4—42no—
168C23×C4—44no—
169C23×C4—42no—
170C23×C4C2×D414yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2414yes—
182C23×C4C2411yescharacteristic
183C23×C4C2412yes—
184C23×C4C2411yescharacteristic
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—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—416no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C2×D418yes—
225C2×C4⋊C4#4118yes—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4C2×D418yes—
243C2×C22⋊C4#4114yes—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—28no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—28no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—44no—
257#95—24no—
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—22no—
268#95—24no—
269#95—28no—
270#95—24no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—42no—
275#95—44no—
276#95—44no—
277#95—41no—
278#95—42no—
279#95—41no—
280#95—41no—
281#95—42no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—42no—
286#95—44no—
287#95—44no—
288#95C2×D418yes—
289C24—22no—
290C24—21no—
291C24—21no—
292C24—22no—
293C24—21no—
294C24—22no—
295C24—21no—
296C24—22no—
297C24C22×D412yes—
298C24C2×#4112yes—
299C24H(5,𝔽2)12yes—
300C24C2511yescharacteristic
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—28no—
336C22×C4—216no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—48no—
378C22×C4—416no—
379C22×C4—48no—
380C22×C4—44no—
381C22×C4—44no—
382C22×C4—88no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4C22×D412yes—
386C22×C4C22×D418yes—
387C22×C4C22×D412yes—
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C4C2×#4118yes—
391C22×C4C2×#4114yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C4H(5,𝔽2)12yes—
394C22×C42−1+414yes—
395C22×C42−1+412yes—
396C22×C42−1+414yes—
397C4⋊C4—216no—
398C4⋊C4—216no—
399C4⋊C4—44no—
400C4⋊C4—48no—
401C4⋊C4—44no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—88no—
405C4⋊C4—88no—
406C4⋊C4—84no—
407C4⋊C4—88no—
408C4⋊C4—88no—
409C4⋊C4—84no—
410C4⋊C4—88no—
411C22⋊C4—48no—
412C22⋊C4—44no—
413C22⋊C4—48no—
414C22⋊C4—44no—
415C22⋊C4—48no—
416C22⋊C4—416no—
417C42—416no—
418C42—48no—
419C42—48no—
420C42—44no—
421C42—44no—
422C23—22no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—22no—
432C23—24no—
433C23—22no—
434C23—42no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—41no—
442C23—42no—
443C23—42no—
444C23#55014yes—
445C23C23×D411yescharacteristic
446C23C22×#4112yes—
447C23C22×#4111yescharacteristic
448C23C2×H(5,𝔽2)11yescharacteristic
449C23C2×H(5,𝔽2)11yescharacteristic
450C23C2×H(5,𝔽2)11yescharacteristic
451C23C2×2−1+411yescharacteristic
452C23C2×2−1+412yes—
453C23C2×2−1+412yes—
454C23#58512yes—
455C23#58511yescharacteristic
456C2×C4—22no—
457C2×C4—28no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—416no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—816no—
539C2×C4#53714yes—
540C2×C4#54118yes—
541C2×C4#54718yes—
542C2×C4#55114yes—
543C2×C4#56314yes—
544C22—24no—
545C22—28no—
546C22—42no—
547C22—44no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—44no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22C2×#53711yescharacteristic
556C22C2×#54112yes—
557C22C2×#54712yes—
558C22C2×#55011yescharacteristic
559C22C2×#55111yescharacteristic
560C22#348212yes—
561C22#348912yes—
562C22#349611yescharacteristic
563C22#350511yescharacteristic
564C22#351111yescharacteristic
565C22#351312yes—
566C22#351412yes—
567C22#351512yes—
568C22#351812yes—
569C22C2×#56311yescharacteristic
570C22#353711yescharacteristic
571C22#353811yescharacteristic
572C22#354412yes—
573C22#354412yes—
574C22#355111yescharacteristic
575C22#355911yescharacteristic
576C22#356312yes—
577C22#356512yes—
578C4—42no—
579C4—48no—
580C4—42no—
581C4—48no—
582C4—42no—
583C4—44no—
584C4—42no—
585C4—82no—
586C4—84no—
587C4—82no—
588C4—81no—
589C4—82no—
590C4—84no—
591C4#272218yes—
592C2—44no—
593C2#3974211yescharacteristic
594C2#4009612yes—
595C2#4257312yes—
596C2#4298011yescharacteristic
597C2#4489012yes—
598C2#4521411yescharacteristic
599C2#4619011yescharacteristic
600C2C2×#272211yescharacteristic
601C2#5303111yescharacteristic
602C2#5904712yes—
603C2#5937911yescharacteristic
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
2. #17120C2semidirect
4. #16671C2semidirect
5. #16116C2semidirect
7. #15225C2semidirect
9. #15137C2semidirect
12. #14045C2semidirect
13. #13982C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24952
#171202
#170592
#166712
#161162
#154722
#152252
#151502
#151372
#145812
#141302
#140452
#139822
#139072
C4×#3972