Finite Group Databasebeta

← #473909

Subgroups — #473909

Order 512.

1218 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1984C211yesmaximal, characteristic, complemented
2C2×#1968C211yesmaximal, characteristic
3#15201C211yesmaximal, characteristic, complemented
4#15157C211yesmaximal, characteristic
5#11378C212yesmaximal, complemented
6#11190C212yesmaximal, complemented
7#11189C212yesmaximal
8#11077C211yesmaximal, characteristic, complemented
9#11076C211yesmaximal, characteristic
10#11023C212yesmaximal
11#10820C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#393—21no—
22C2×#391C2211yescharacteristic
23C2×#391C2212yes—
24C2×#390C2211yescharacteristic
25C2×#390C2212yes—
26#1984—24no—
27#1968—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1446—21no—
35#1442—21no—
36#1442—22no—
37#1439—22no—
38#1439—21no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C23×Q8D411yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×D4C2311yescharacteristic
46C22×SD16—22no—
47C22×SD16—22no—
48C22×SD16—22no—
49C22×SD16—24no—
50C22×SD16—42no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—42no—
57C2×C4×Q8—42no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70C2×#106—22no—
71C2×#106—24no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—42no—
74C2×#103—24no—
75C2×#103—22no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#102—24no—
80C2×#102—22no—
81C2×#102—42no—
82C2×#102—42no—
83C2×#102—42no—
84C2×C4×C8—42no—
85C2×C4×C8D414yes—
86#393—44no—
87#391—22no—
88#391—22no—
89#391—42no—
90#391—42no—
91#390—22no—
92#390—22no—
93#390—42no—
94#390—42no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—21no—
107#340—24no—
108#340—42no—
109#340—42no—
110#340—42no—
111#336—24no—
112#336—24no—
113C25—21no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—42no—
126C22×Q8—42no—
127C22×Q8SD1614yescomplemented
128C22×Q8#4112yes—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4#4112yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C2×SD16—28no—
150C2×SD16—48no—
151C2×SD16—48no—
152C2×SD16—44no—
153C2×SD16—44no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C22×C8#4114yes—
171C4×Q8—44no—
172C4×Q8—44no—
173C4×Q8—48no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C22⋊C4—82no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42C2×D412yes—
219C2×C42#4112yes—
220#106—44no—
221C4⋊C8—44no—
222#103—44no—
223#103—48no—
224#102—44no—
225#102—48no—
226C4×C8—44no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—42no—
231#95—42no—
232#95—44no—
233#95—42no—
234#95—42no—
235#95—81no—
236#95—81no—
237#95—82no—
238#95—82no—
239C24—22no—
240C24—41no—
241C24—42no—
242C24—41no—
243C24—42no—
244C24—41no—
245C24—41no—
246C24—42no—
247C24—41no—
248C24—42no—
249C24—41no—
250C24C22×D411yescharacteristic
251C2×Q8—28no—
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—48no—
257C2×Q8—44no—
258C2×Q8—48no—
259C2×Q8—44no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—42no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4#12112yes—
341C22×C4#12112yes—
342C22×C4C2×SD1614yes—
343C22×C4C2×SD1612yes—
344C22×C4#13512yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4112yes—
349C22×C4C2×#4111yescharacteristic
350C22×C4H(5,𝔽2)12yes—
351C22×C4H(5,𝔽2)12yes—
352C22×C4H(5,𝔽2)12yes—
353C22×C42−1+411yescharacteristic
354C22×C42−1+411yescharacteristic
355SD16—416no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8#12414yes—
373C4⋊C4—44no—
374C4⋊C4—48no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C22⋊C4—88no—
390C22⋊C4—84no—
391C22⋊C4—88no—
392C22⋊C4—84no—
393C42—22no—
394C42—22no—
395C42—24no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—42no—
400C23—42no—
401C23—42no—
402C23—44no—
403C23—44no—
404C23—42no—
405C23—81no—
406C23—81no—
407C23—82no—
408C23—81no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23—81no—
423C23C2×#12111yescharacteristic
424C23#53412yes—
425C23#54811yescharacteristic
426C23#54811yescharacteristic
427C23#55012yes—
428C23C22×SD1611yescharacteristic
429C23C2×#13511yescharacteristic
430C23#57512yes—
431C23#57611yescharacteristic
432C23#57712yes—
433C23#57811yescharacteristic
434Q8—416no—
435Q8—42no—
436Q8—42no—
437Q8—44no—
438Q8—81no—
439Q8—81no—
440Q8—82no—
441D4—42no—
442D4—42no—
443D4—44no—
444D4—81no—
445D4—81no—
446D4—82no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—81no—
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×C4—82no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4#46514yes—
500C2×C4#52511yescharacteristic
501C2×C4C2×#12411yescharacteristic
502C2×C4#53411yescharacteristic
503C2×C4#53511yescharacteristic
504C2×C4#53511yescharacteristic
505C2×C4#53611yescharacteristic
506C2×C4#53812yes—
507C2×C4#53812yes—
508C2×C4#53812yes—
509C2×C4#54212yes—
510C2×C4#54212yes—
511C2×C4#55112yes—
512C2×C4#55112yes—
513C2×C4#55211yescharacteristic
514C2×C4#55211yescharacteristic
515C2×C4#55912yes—
516C2×C4#56212yes—
517C2×C4#56212yes—
518C8—24no—
519C8—42no—
520C8—82no—
521C22—82no—
522C22—82no—
523C22—84no—
524C22—84no—
525C22—82no—
526C22—161no—
527C22—161no—
528C22—161no—
529C22—161no—
530C22—162no—
531C22—162no—
532C22#253611yescharacteristic
533C22#254811yescharacteristic
534C22#261711yescharacteristic
535C22#273012yes—
536C22#273012yes—
537C22C2×#46511yescharacteristic
538C22#305411yescharacteristic
539C22#305711yescharacteristic
540C22#306111yescharacteristic
541C22#318712yes—
542C22#319112yes—
543C22#319812yes—
544C22#320112yes—
545C22#335911yescharacteristic
546C22#336111yescharacteristic
547C22#336211yescharacteristic
548C22#336411yescharacteristic
549C22#337112yes—
550C22#337612yes—
551C22#339212yes—
552C22#339412yes—
553C22#340212yes—
554C22#340312yes—
555C4—24no—
556C4—44no—
557C4—42no—
558C4—82no—
559C4—82no—
560C4—161no—
561C4—161no—
562C4—162no—
563C4—162no—
564C4#256111yescharacteristic
565C4#263111yescharacteristic
566C4#264011yescharacteristic
567C4#275112yes—
568C4#275112yes—
569C4#281611yescharacteristic
570C2—162no—
571C2#1688711yescharacteristic
572C2#2818611yescharacteristic
573C2#2819211yescharacteristic
574C2#2873811yescharacteristic
575C2#2874111yescharacteristic
576C2#2973411yescharacteristic
577C2#2974011yescharacteristic
578C2#3259712yes—
579C2#3259812yes—
580C2#3260012yes—
581C2#3260312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1984C2semidirect
3. #15201C2semidirect
5. #11378C2semidirect
6. #11190C2semidirect
8. #11077C2semidirect
11. #10820C2semidirect
127. C22×Q8SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#19842
C2×#19682
#152012
#151572
#113782
#111902
#111892
#110772
#110762
#110232
#108202