Finite Group Databasebeta

← #417125

Subgroups — #417125

Order 512.

1638 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#489C211yesmaximal, characteristic
2C22×#488C211yesmaximal, characteristic
3C2×#2660C211yesmaximal, characteristic, complemented
4C2×#1992C211yesmaximal, characteristic, complemented
5C2×#1990C211yesmaximal, characteristic, complemented
6#14273C211yesmaximal, characteristic
7#11052C212yesmaximal, complemented
8#11051C212yesmaximal
9#11050C212yesmaximal, complemented
10#11049C212yesmaximal
11#10694C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#124C2211yescharacteristic
15C2×#489—24no—
16C2×#489—24no—
17C2×#489—24no—
18C2×#488—24no—
19C2×#488—24no—
20C2×#488—24no—
21C22×#103C2211yescharacteristic
22C22×#103C2212yes—
23C22×#103C2211yescharacteristic
24C22×#102C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2660—24no—
30#2660—24no—
31C2×#398—22no—
32C2×#395—22no—
33C2×#391C2212yes—
34C2×#390C2212yes—
35C2×#384C2212yes—
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×#376C2211yescharacteristic
39#1992—24no—
40#1990—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47#1439—22no—
48#1439—22no—
49C23×Q8—21no—
50C23×Q8C2311yescharacteristic
51C23×D4C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D414yes—
61C2×#125—22no—
62C2×#125—22no—
63C2×#125—22no—
64C2×#125—22no—
65C2×#125—22no—
66C2×#125D414yes—
67C2×#124—22no—
68C2×#124—22no—
69C2×#124—22no—
70C2×#124—22no—
71C2×#124—22no—
72C2×#124D414yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—24no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—24no—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88#489—232no—
89#488—232no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—24no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—24no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—24no—
99C2×#102—22no—
100C2×#102—22no—
101C2×#102—24no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#102—24no—
105C2×#102—22no—
106C2×#102—22no—
107C2×#102—24no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97D414yes—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8D414yes—
114#398—216no—
115#395—216no—
116#391—24no—
117#391—24no—
118#390—24no—
119#390—24no—
120#384—24no—
121#384—24no—
122#384—22no—
123#384—22no—
124#382—24no—
125#382—24no—
126#376—22no—
127#376—22no—
128C2×#95—24no—
129C2×#95—24no—
130C2×#95D412yes—
131C2×#95D412yes—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#336—24no—
137#336—24no—
138C25—21no—
139C22×Q8—24no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8—82no—
152C22×Q8—82no—
153C22×Q8C2×D412yes—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—21no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4C2×D412yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4C2×D414yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C22×C8C2×D414yes—
190C4⋊Q8—28no—
191C4⋊Q8—28no—
192C4⋊Q8—44no—
193C4⋊Q8—44no—
194#125—28no—
195#125—28no—
196#125—44no—
197#125—44no—
198#124—28no—
199#124—28no—
200#124—44no—
201#124—44no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—216no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4—84no—
238C2×C4⋊C4—84no—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D414yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—48no—
249C2×C22⋊C4—84no—
250C2×C22⋊C4—84no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42#4114yes—
261C2×C42#4114yes—
262C2×C42#4112yes—
263#103—28no—
264#103—28no—
265#103—216no—
266#102—28no—
267#102—28no—
268#102—216no—
269#97—28no—
270C4×C8—28no—
271#95—216no—
272#95—24no—
273#95—24no—
274#95—48no—
275C24—22no—
276C24—41no—
277C24—42no—
278C24—41no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24C22×D411yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—48no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—42no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—88no—
305C2×Q8—84no—
306C2×Q8—84no—
307C2×Q8—88no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4#12012yes—
362C22×C4#12012yes—
363C22×C4#12112yes—
364C22×C4#12112yes—
365C22×C4#12418yes—
366C22×C4#12712yes—
367C22×C4#12712yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4112yes—
375C22×C4C2×#4112yes—
376C22×C42−1+412yes—
377C22×C42−1+412yes—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—24no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C4⋊C4—816no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—816no—
420C42—24no—
421C42—24no—
422C42—24no—
423C42—24no—
424C42#12014yes—
425C42#12014yes—
426C42#12114yes—
427C42#12114yes—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—81no—
433C23—84no—
434C23—84no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—82no—
441C23—84no—
442C23—84no—
443C23—82no—
444C23—82no—
445C23C2×#12011yescharacteristic
446C23C2×#12111yescharacteristic
447C23C2×#12412yes—
448C23C2×#12711yescharacteristic
449C23#53612yes—
450C23#57612yes—
451C23#57712yes—
452C23#57712yes—
453C23#57812yes—
454Q8—42no—
455Q8—42no—
456Q8—44no—
457Q8—816no—
458Q8—82no—
459Q8—81no—
460Q8—81no—
461D4—42no—
462D4—42no—
463D4—44no—
464D4—82no—
465D4—81no—
466D4—81no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—48no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—84no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4#39214yes—
508C2×C4#39214yes—
509C2×C4C2×#12011yescharacteristic
510C2×C4C2×#12011yescharacteristic
511C2×C4C2×#12111yescharacteristic
512C2×C4C2×#12111yescharacteristic
513C2×C4#53211yescharacteristic
514C2×C4#53211yescharacteristic
515C2×C4#53212yes—
516C2×C4#53611yescharacteristic
517C2×C4#53611yescharacteristic
518C2×C4#53612yes—
519C2×C4#54712yes—
520C2×C4#54712yes—
521C2×C4#54812yes—
522C2×C4#54812yes—
523C2×C4#54812yes—
524C2×C4#54812yes—
525C2×C4#55612yes—
526C2×C4#55612yes—
527C8—42no—
528C8—42no—
529C8—44no—
530C22—88no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—164no—
535C22—161no—
536C22—161no—
537C22—161no—
538C22—161no—
539C22C2×#39211yescharacteristic
540C22#243612yes—
541C22#254712yes—
542C22#265811yescharacteristic
543C22#285811yescharacteristic
544C22#301212yes—
545C22#301312yes—
546C22#304512yes—
547C22#304612yes—
548C22#313614yes—
549C22#313714yes—
550C22#315411yescharacteristic
551C22#315411yescharacteristic
552C22#315511yescharacteristic
553C22#315611yescharacteristic
554C22#324314yes—
555C22#324414yes—
556C4—24no—
557C4—24no—
558C4—48no—
559C4—82no—
560C4—84no—
561C4—82no—
562C4—82no—
563C4—162no—
564C4#243211yescharacteristic
565C4#243211yescharacteristic
566C4#243212yes—
567C4#254512yes—
568C4#254511yescharacteristic
569C4#254511yescharacteristic
570C2—162no—
571C2#1524311yescharacteristic
572C2#2342611yescharacteristic
573C2#2342711yescharacteristic
574C2#2591614yes—
575C2#2873014yes—
576C2#3028911yescharacteristic
577C2#3029011yescharacteristic
578C2#3322811yescharacteristic
579C2#3322911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
3. C2×#2660C2semidirect
4. C2×#1992C2semidirect
5. C2×#1990C2semidirect
7. #11052C2semidirect
9. #11050C2semidirect
11. #10694C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4892
C22×#4882
C2×#26602
C2×#19922
C2×#19902
#142732
#110522
#110512
#110502
#110492
#106942