Finite Group Databasebeta

← #411501

Subgroups — #411501

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1970C211yesmaximal, characteristic
2C2×#1746C211yesmaximal, characteristic, complemented
3#14260C212yesmaximal, complemented
4#11328C211yesmaximal, characteristic, complemented
5#11256C211yesmaximal, characteristic, complemented
6#11189C212yesmaximal, complemented
7#11066C212yesmaximal, complemented
8#11022C211yesmaximal, characteristic
9#11022C211yesmaximal, characteristic
10#10927C212yesmaximal, complemented
11#10820C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#390C2212yes—
20C2×#390C2212yes—
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×#376C2211yescharacteristic
24#1970—24no—
25#1746—24no—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1440—22no—
36#1440—22no—
37#1439—22no—
38#1439—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C23×D4C2312yes—
42C22×D8—24no—
43C22×D8—22no—
44C22×D8—22no—
45C22×D8—22no—
46C22×D8—42no—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#107—22no—
65C2×#107—42no—
66C2×#102—24no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#102—42no—
72C2×#102—42no—
73C2×#102—42no—
74C2×#102—42no—
75C2×#97—22no—
76C2×#97—22no—
77C2×#97—22no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8D414yes—
81#390—24no—
82#390—24no—
83#390—24no—
84#390—24no—
85#390—42no—
86#390—42no—
87#389—42no—
88#389—42no—
89#376—22no—
90#376—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95D411yescharacteristic
97C2×#95D411yescharacteristic
98C2×#95D411yescharacteristic
99C2×#95D411yescharacteristic
100#340—22no—
101#340—22no—
102#340—24no—
103#340—42no—
104#340—42no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109C25—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4#4114yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C2×D8—28no—
135C2×D8—48no—
136C2×D8—48no—
137C2×D8—44no—
138C2×D8—44no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8C2×D412yes—
154C22×C8C2×D412yes—
155C22×C8#4112yes—
156C22×C8#4112yes—
157C22×C8#4112yes—
158C22×C8#4112yes—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4#4114yes—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42C2×D412yes—
211C2×C42#4112yes—
212#107—44no—
213#102—48no—
214#102—48no—
215#102—48no—
216#97—28no—
217C4×C8—28no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—44no—
223#95—82no—
224#95—82no—
225#95—82no—
226#95—82no—
227C24—24no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24C22×D411yescharacteristic
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4#12112yes—
323C22×C4#12112yes—
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4112yes—
332C22×C4H(5,𝔽2)12yes—
333C22×C4H(5,𝔽2)12yes—
334D8—416no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8#12314yes—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—88no—
368C22⋊C4—88no—
369C22⋊C4—88no—
370C22⋊C4—88no—
371C22⋊C4—88no—
372C22⋊C4—88no—
373C22⋊C4—88no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—41no—
379C42—42no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—42no—
384C23—44no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23C2×#12111yescharacteristic
420C23#52512yes—
421C23#53211yescharacteristic
422C23#53211yescharacteristic
423C23#53512yes—
424C23#57611yescharacteristic
425C23#57611yescharacteristic
426C23#57611yescharacteristic
427C23#57612yes—
428C23#57712yes—
429C23#57811yescharacteristic
430D4—44no—
431D4—44no—
432D4—44no—
433D4—44no—
434D4—82no—
435D4—82no—
436D4—82no—
437D4—82no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4C2×#12311yescharacteristic
492C2×C4#52511yescharacteristic
493C2×C4#52511yescharacteristic
494C2×C4#52911yescharacteristic
495C2×C4#52912yes—
496C2×C4#52911yescharacteristic
497C2×C4#53211yescharacteristic
498C2×C4#53511yescharacteristic
499C2×C4#53511yescharacteristic
500C2×C4#54012yes—
501C2×C4#55112yes—
502C2×C4#55112yes—
503C2×C4#55112yes—
504C2×C4#55112yes—
505C2×C4#55312yes—
506C2×C4#55312yes—
507C2×C4#55512yes—
508C2×C4#55512yes—
509C8—24no—
510C8—42no—
511C8—42no—
512C8—42no—
513C22—84no—
514C22—84no—
515C22—84no—
516C22—84no—
517C22—84no—
518C22—84no—
519C22—84no—
520C22—162no—
521C22—162no—
522C22—162no—
523C22—162no—
524C22—162no—
525C22—162no—
526C22—162no—
527C22—162no—
528C22#239211yescharacteristic
529C22#260312yes—
530C22#261012yes—
531C22#261911yescharacteristic
532C22#285411yescharacteristic
533C22#305811yescharacteristic
534C22#305811yescharacteristic
535C22#306011yescharacteristic
536C22#306111yescharacteristic
537C22#311812yes—
538C22#311912yes—
539C22#312212yes—
540C22#312412yes—
541C22#317511yescharacteristic
542C22#317511yescharacteristic
543C22#317611yescharacteristic
544C22#317611yescharacteristic
545C22#317912yes—
546C22#317912yes—
547C22#322012yes—
548C22#322012yes—
549C22#322412yes—
550C22#322512yes—
551C4—42no—
552C4—42no—
553C4—84no—
554C4—81no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4—162no—
559C4—162no—
560C4#248211yescharacteristic
561C4#250011yescharacteristic
562C4#252612yes—
563C4#253112yes—
564C4#259811yescharacteristic
565C4#261011yescharacteristic
566C2—164no—
567C2#1512011yescharacteristic
568C2#2370211yescharacteristic
569C2#2370311yescharacteristic
570C2#2950512yes—
571C2#2950712yes—
572C2#2956412yes—
573C2#2956612yes—
574C2#2981011yescharacteristic
575C2#2981311yescharacteristic
576C2#3313211yescharacteristic
577C2#3313811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1746C2semidirect
3. #14260C2semidirect
4. #11328C2semidirect
5. #11256C2semidirect
6. #11189C2semidirect
7. #11066C2semidirect
10. #10927C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19702
C2×#17462
#142602
#113282
#112562
#111892
#110662
#110222
#109272
#108202