Finite Group Databasebeta

← #392999

Subgroups — #392999

Order 512.

2096 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#465C211yesmaximal, characteristic
2C22×#405C211yesmaximal, characteristic, complemented
3C2×#1973C212yesmaximal, complemented
4C2×#1895C211yesmaximal, characteristic
5C2×#1894C211yesmaximal, characteristic, complemented
6C2×#1808C211yesmaximal, characteristic, complemented
7#14258C211yesmaximal, characteristic, complemented
8#14257C211yesmaximal, characteristic
9#10926C212yesmaximal, complemented
10#10925C212yesmaximal, complemented
11#10863C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#465—24no—
19C2×#465—24no—
20C2×#465C418yescomplemented
21C22×#107C2212yes—
22C22×#106C2211yescharacteristic
23C22×#103C2212yes—
24C22×#103C2211yescharacteristic
25C22×#102C2212yes—
26C22×#102C2211yescharacteristic
27C2×#405—28no—
28C2×#405—24no—
29C22×#97C2212yes—
30C22×C4×C8C2212yes—
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35#1973—28no—
36#1895—24no—
37#1894—24no—
38#1808—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42#1441—22no—
43#1440—22no—
44#1439—22no—
45#1438—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C23×D4—21no—
50C23×D4C2311yescharacteristic
51C24×C4C2311yescharacteristic
52C22×SD16—24no—
53C22×SD16—22no—
54C22×SD16—22no—
55C22×SD16—24no—
56C22×SD16C2×C414yes—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16—24no—
60C22×M16—24no—
61C22×M16D414yes—
62C23×C8—24no—
63C23×C8—24no—
64C23×C8C2×C414yes—
65C23×C8D414yes—
66C23×C8D414yes—
67C23×C8C2311yescharacteristic
68C23×C8C2312yes—
69C23×C8C2311yescharacteristic
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8C2×C414yes—
76C2×#121—22no—
77C2×#121—22no—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121C2×C414yes—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117—24no—
85C2×#117—24no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4D412yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4—21no—
93C22×C22⋊C4—21no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4D412yes—
98C22×C42C2312yes—
99#465—232no—
100C2×#107—24no—
101C2×#107—24no—
102C2×#107—24no—
103C2×#106—22no—
104C2×#106—22no—
105C2×#106C2×C414yes—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103C2×C414yes—
112C2×#102—22no—
113C2×#102—22no—
114C2×#102—24no—
115C2×#102—24no—
116C2×#102—24no—
117C2×#102C2×C414yes—
118#405—232no—
119C2×#97—24no—
120C2×#97—24no—
121C2×#97D418yescomplemented
122C2×C4×C8—24no—
123C2×C4×C8—24no—
124C2×C4×C8—24no—
125#391—24no—
126#391—24no—
127#390—24no—
128#390—24no—
129#380—22no—
130#380—22no—
131#380—22no—
132#380—22no—
133C2×#95—21no—
134C2×#95—24no—
135C2×#95—21no—
136C2×#95—24no—
137C2×#95—24no—
138#340—24no—
139#340—24no—
140#340—24no—
141C25—21no—
142C25C2×D411yescharacteristic
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—21no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8C22×C412yes—
153C22×D4—24no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—84no—
166C22×D4C22×C412yes—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4C22×C412yes—
177C23×C4C2×D414yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C2×SD16—28no—
186C2×SD16—216no—
187C2×SD16—44no—
188C2×SD16—48no—
189C2×SD16—44no—
190C2×M16—216no—
191C2×M16—28no—
192C2×M16—28no—
193C2×M16—28no—
194C2×M16—216no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—216no—
202C22×C8—28no—
203C22×C8—216no—
204C22×C8—28no—
205C22×C8C22⋊C4116yescomplemented
206C22×C8C22×C412yes—
207C22×C8C22×C412yes—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C22×C8C2×D414yes—
211C22×C8C2×D414yes—
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214C22×C8C2×D414yes—
215C22×C8#4114yes—
216C22×C8#4114yes—
217C22⋊Q8—28no—
218C22⋊Q8—28no—
219C22⋊Q8—44no—
220C22⋊Q8—44no—
221#121—28no—
222#121—28no—
223#121—44no—
224#121—44no—
225#117—216no—
226#117—28no—
227#117—28no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4C22×C412yes—
256C2×C4⋊C4C22×C412yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4—84no—
269C2×C22⋊C4—84no—
270C2×C22⋊C4—82no—
271C2×C22⋊C4—82no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42C2×D414yes—
277C2×C42#4114yes—
278#107—48no—
279#106—28no—
280#103—28no—
281#103—48no—
282#102—28no—
283#102—48no—
284#97—216no—
285C4×C8—216no—
286#95—48no—
287#95—42no—
288#95—42no—
289#95—84no—
290#95—84no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—24no—
295C24—22no—
296C24—22no—
297C24—42no—
298C24—41no—
299C24—41no—
300C24—42no—
301C24—42no—
302C24—44no—
303C24—42no—
304C24C4×D412yes—
305C24#12112yes—
306C24#12712yes—
307C24C22×D411yescharacteristic
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—41no—
313C2×Q8—42no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—41no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—41no—
320C2×Q8—42no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—48no—
331C2×D4—41no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—42no—
338C2×D4—41no—
339C2×D4—42no—
340C2×D4—41no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—88no—
346C2×D4—88no—
347C2×D4—88no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4C2×C22⋊C414yes—
395C22×C4C4×D414yes—
396C22×C4C4×D412yes—
397C22×C4#12118yes—
398C22×C4#12114yes—
399C22×C4#12112yes—
400C22×C4#12414yes—
401C22×C4#12712yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C2×#4112yes—
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4111yescharacteristic
412SD16—416no—
413M16—232no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—232no—
425C2×C8C2×C22⋊C414yes—
426C2×C8C2×C22⋊C414yes—
427C2×C8C4×D418yes—
428C2×C8#12014yes—
429C2×C8#12118yes—
430C2×C8#12118yes—
431C2×C8#12314yes—
432C4⋊C4—216no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—48no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C22⋊C4—216no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—816no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C42—24no—
457C42—24no—
458C42—24no—
459C42—24no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—44no—
479C23—81no—
480C23—81no—
481C23—82no—
482C23—82no—
483C23—81no—
484C23—81no—
485C23—84no—
486C23—84no—
487C23—82no—
488C23—82no—
489C23—84no—
490C23—84no—
491C23#39014yes—
492C23C2×C4×D411yescharacteristic
493C23C2×#12111yescharacteristic
494C23C2×#12112yes—
495C23C2×#12711yescharacteristic
496C23#53212yes—
497C23#57612yes—
498C23#57714yes—
499C23#57812yes—
500Q8—42no—
501Q8—42no—
502Q8—44no—
503Q8—81no—
504Q8—82no—
505Q8—81no—
506D4—42no—
507D4—42no—
508D4—44no—
509D4—816no—
510D4—81no—
511D4—82no—
512D4—81no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4#39014yes—
555C2×C4C22×C22⋊C411yescharacteristic
556C2×C4C2×C4×D412yes—
557C2×C4C2×C4×D411yescharacteristic
558C2×C4C4×#4112yes—
559C2×C4C2×#12011yescharacteristic
560C2×C4C2×#12111yescharacteristic
561C2×C4C2×#12112yes—
562C2×C4C2×#12112yes—
563C2×C4C2×#12311yescharacteristic
564C2×C4#52512yes—
565C2×C4#52511yescharacteristic
566C2×C4#52512yes—
567C2×C4#52511yescharacteristic
568C2×C4C2×#12411yescharacteristic
569C2×C4#52912yes—
570C2×C4#52912yes—
571C2×C4#53212yes—
572C2×C4#53212yes—
573C8—28no—
574C8—24no—
575C8#38618yes—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—48no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—88no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—162no—
589C22—162no—
590C22—162no—
591C22—162no—
592C22C2×#39011yescharacteristic
593C22C4×#12112yes—
594C22#284711yescharacteristic
595C22#285411yescharacteristic
596C22#285612yes—
597C22#294912yes—
598C22#295112yes—
599C22#305712yes—
600C22#306112yes—
601C22#306214yes—
602C22#306314yes—
603C22#315412yes—
604C22#315511yescharacteristic
605C22#315611yescharacteristic
606C22#317814yes—
607C22#317914yes—
608C4—24no—
609C4—48no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—162no—
614C4—162no—
615C4C2×#38611yescharacteristic
616C4#229711yescharacteristic
617C4C4×#12112yes—
618C4#284711yescharacteristic
619C4#285012yes—
620C4#285411yescharacteristic
621C2—44no—
622C2—162no—
623C2#1469811yescharacteristic
624C2#2123011yescharacteristic
625C2#2123111yescharacteristic
626C2#2153114yes—
627C2#3293812yes—
628C2#3313112yes—
629C2#3318712yes—
630C2#3318812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#405C2semidirect
3. C2×#1973C2semidirect
5. C2×#1894C2semidirect
6. C2×#1808C2semidirect
7. #14258C2semidirect
9. #10926C2semidirect
10. #10925C2semidirect
11. #10863C2semidirect
20. C2×#465C4semidirect
121. C2×#97D4semidirect
205. C22×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4652
C22×#4052
C2×#19732
C2×#18952
C2×#18942
C2×#18082
#142582
#142572
#109262
#109252
#108632