Finite Group Databasebeta

← #358683

Subgroups — #358683

Order 512.

1690 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2C2×#1768C211yesmaximal, characteristic, complemented
3C2×#1763C211yesmaximal, characteristic, complemented
4C2×#1762C211yesmaximal, characteristic
5C2×C4×#103C211yesmaximal, characteristic, complemented
6C2×C4×#102C211yesmaximal, characteristic
7#13797C211yesmaximal, characteristic, complemented
8#11357C212yesmaximal, complemented
9#11077C212yesmaximal, complemented
10#11076C212yesmaximal
11#11009C212yesmaximal
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#97C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2326—24no—
23#2326C418yescomplemented
24C2×#391C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#388—22no—
27C2×#385—22no—
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×#376C2211yescharacteristic
32#1768—24no—
33#1763—24no—
34#1762—24no—
35C4×#103—24no—
36C4×#102—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—22no—
45#1434—22no—
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#124—24no—
52C2×#124—24no—
53C2×#124—22no—
54C2×#124C2×C414yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8C2×C414yes—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4C2×C414yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×C4⋊C8—24no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#97—22no—
98C2×#97—22no—
99C2×#97—22no—
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103#391—22no—
104#391C2×C414yes—
105#390—22no—
106#390C2×C414yes—
107#388—216no—
108#385—216no—
109#384—22no—
110#384C2×C414yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4D414yes—
114C4×C4⋊C4D414yes—
115#376C2×C414yes—
116#376D414yes—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95D412yes—
125C2×#95D412yes—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#336—24no—
131C25—21no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8C22×C412yes—
141C22×Q8#4114yes—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4C22×C412yes—
151C22×D4#4114yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8#4114yes—
180C22×C8#4114yes—
181#124—216no—
182#124—28no—
183#124—28no—
184C4×Q8—28no—
185C4×Q8—28no—
186C4×Q8—44no—
187C4×Q8—44no—
188C4×D4—28no—
189C4×D4—28no—
190C4×D4—44no—
191C4×D4—44no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×D414yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—24no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—82no—
241C2×C42—82no—
242C2×C42C22×C412yes—
243C2×C42C22×C412yes—
244C2×C42C22×C412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252C4⋊C8—216no—
253#103—216no—
254#103—216no—
255#102—216no—
256#102—216no—
257#97—44no—
258C4×C8—44no—
259#95—28no—
260#95—24no—
261#95—24no—
262#95—82no—
263#95—82no—
264#95—82no—
265#95—82no—
266#95C22⋊C418yes—
267#95C22⋊C418yes—
268C24—22no—
269C24—24no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24C22×D411yescharacteristic
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—41no—
295C2×Q8#11714yes—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—41no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4#11714yes—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—48no—
360C22×C4—88no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4—88no—
364C22×C4C2×C22⋊C414yes—
365C22×C4C2×C22⋊C414yes—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4#12112yes—
369C22×C4#12112yes—
370C22×C4#12112yes—
371C22×C4#12112yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C2×#4112yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4111yescharacteristic
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—48no—
413C4⋊C4—88no—
414C22⋊C4—232no—
415C22⋊C4—48no—
416C22⋊C4—88no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—24no—
422C42—24no—
423C42—48no—
424C42—84no—
425C42—84no—
426C42C4×D414yes—
427C42C4×D414yes—
428C42#12414yes—
429C42#12714yes—
430C23—24no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—42no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—82no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—82no—
450C23—82no—
451C23C22×C22⋊C411yescharacteristic
452C23C2×#12111yescharacteristic
453C23C2×#12111yescharacteristic
454C23#52512yes—
455C23#53212yes—
456C23#57612yes—
457C23#57712yes—
458C23#57712yes—
459C23#57812yes—
460Q8—24no—
461Q8—24no—
462Q8—44no—
463Q8—44no—
464Q8—42no—
465Q8—42no—
466D4—24no—
467D4—24no—
468D4—44no—
469D4—44no—
470D4—42no—
471D4—42no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—816no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4#39014yes—
525C2×C4#39014yes—
526C2×C4C2×#11711yescharacteristic
527C2×C4C2×#11711yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×C4×D411yescharacteristic
530C2×C4C4×#4111yescharacteristic
531C2×C4C4×#4111yescharacteristic
532C2×C4#52512yes—
533C2×C4#52512yes—
534C2×C4#52512yes—
535C2×C4#52512yes—
536C2×C4#52512yes—
537C2×C4#52512yes—
538C2×C4C2×#12411yescharacteristic
539C2×C4#52912yes—
540C2×C4#52912yes—
541C2×C4#52912yes—
542C2×C4#52912yes—
543C2×C4C2×#12711yescharacteristic
544C8—44no—
545C8—44no—
546C22—48no—
547C22—44no—
548C22—82no—
549C22—82no—
550C22—84no—
551C22—84no—
552C22—84no—
553C22—84no—
554C22—84no—
555C22#228011yescharacteristic
556C22C2×#39011yescharacteristic
557C22#229711yescharacteristic
558C22#239212yes—
559C22#239212yes—
560C22#289712yes—
561C22#289812yes—
562C22#304111yescharacteristic
563C22#304211yescharacteristic
564C22#304311yescharacteristic
565C22#304411yescharacteristic
566C22#304512yes—
567C22#304612yes—
568C22#311912yes—
569C22#312012yes—
570C22#312412yes—
571C22#312512yes—
572C22#317312yes—
573C22#317512yes—
574C22#317812yes—
575C22#317912yes—
576C4—24no—
577C4—24no—
578C4—44no—
579C4—44no—
580C4—44no—
581C4—82no—
582C4—82no—
583C4—84no—
584C4—84no—
585C4#229011yescharacteristic
586C4#229011yescharacteristic
587C4#229011yescharacteristic
588C4#229011yescharacteristic
589C4#239712yes—
590C4#239812yes—
591C2—84no—
592C2#1384711yescharacteristic
593C2#2074612yes—
594C2#2126211yescharacteristic
595C2#2126311yescharacteristic
596C2#2141212yes—
597C2#2369212yes—
598C2#2369312yes—
599C2#2369412yes—
600C2#2369512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×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×#1768C2semidirect
3. C2×#1763C2semidirect
5. C2×C4×#103C2semidirect
7. #13797C2semidirect
8. #11357C2semidirect
9. #11077C2semidirect
23. #2326C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#17682
C2×#17632
C2×#17622
C2×C4×#1032
C2×C4×#1022
#137972
#113572
#110772
#110762
#110092