Finite Group Databasebeta

← #354568

Subgroups — #354568

Order 512.

1902 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2C2×#1983C212yesmaximal
3C2×#1768C211yesmaximal, characteristic, complemented
4C2×#1763C211yesmaximal, characteristic, complemented
5C2×#1762C211yesmaximal, characteristic
6C2×C4×#103C211yesmaximal, characteristic, complemented
7C2×C4×#102C211yesmaximal, characteristic
8#13750C211yesmaximal, characteristic, complemented
9#10932C212yesmaximal, complemented
10#10931C212yesmaximal
11#10849C212yesmaximal, complemented
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⋊C4C2212yes—
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1983C4116yescomplemented
35#1768—24no—
36#1763—24no—
37#1762—24no—
38C4×#103—24no—
39C4×#102—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#336C2212yes—
45C4×#95—22no—
46C4×#95—22no—
47C23×Q8C2311yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#124—22no—
53C2×#124—24no—
54C2×#124C2×C414yes—
55C2×#124C2×C418yes—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8C2×C414yes—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4C2×C414yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C42—21no—
78C22×C42—22no—
79C22×C42—22no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—24no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8C2×C418yescomplemented
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103C2×C418yescomplemented
93C2×#103C2×C418yescomplemented
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—24no—
98C2×#102C2×C418yes—
99C2×#102C2×C418yes—
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106#391—22no—
107#391C2×C414yes—
108#390—22no—
109#390C2×C414yes—
110#388—216no—
111#385—216no—
112#384—22no—
113#384C2×C414yes—
114C4×C4⋊C4—24no—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C418yescomplemented
120C4×C4⋊C4D414yes—
121C4×C4⋊C4D414yes—
122#376C2×C414yes—
123#376D414yes—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95D412yes—
130C2×#95D412yes—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#336—24no—
136C25—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8C22×C412yes—
146C22×Q8C22×C414yes—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4C22×C412yes—
156C22×D4C22×C414yes—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—24no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8C22×C414yes—
185C22×C8C22×C414yes—
186#124—28no—
187#124—216no—
188#124C42116yescomplemented
189C4×Q8—28no—
190C4×Q8—28no—
191C4×Q8—44no—
192C4×Q8—44no—
193C4×D4—28no—
194C4×D4—28no—
195C4×D4—44no—
196C4×D4—44no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C414yes—
222C2×C4⋊C4C22×C414yes—
223C2×C4⋊C4C2×D414yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4#4114yes—
226C2×C4⋊C4#4114yes—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—28no—
240C2×C42—28no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—82no—
246C2×C42—82no—
247C2×C42—82no—
248C2×C42—82no—
249C2×C42—82no—
250C2×C42—82no—
251C2×C42C22×C412yes—
252C2×C42C22×C412yes—
253C2×C42C22×C414yes—
254C2×C42C22×C412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42#4114yes—
261C2×C42#4112yes—
262C2×C42#4112yes—
263C4⋊C8—216no—
264#103—216no—
265#103—216no—
266#102—216no—
267#102—216no—
268#97—44no—
269C4×C8—44no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—24no—
275#95C22⋊C418yes—
276#95C22⋊C418yes—
277C24—22no—
278C24—24no—
279C24—41no—
280C24—41no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24C22×D411yescharacteristic
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—24no—
297C2×Q8—41no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8C2×C4214yes—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—41no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4C2×C4214yes—
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—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—41no—
348C22×C4—41no—
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—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—48no—
367C22×C4—88no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—88no—
371C22×C4C2×C22⋊C414yes—
372C22×C4C2×C22⋊C414yes—
373C22×C4C2×C22⋊C412yes—
374C22×C4C2×C22⋊C412yes—
375C22×C4C4×D414yes—
376C22×C4C4×D414yes—
377C22×C4#12112yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4#12112yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C23×C412yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4112yes—
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4112yes—
390C22×C4C2×#4112yes—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—48no—
420C4⋊C4—88no—
421C4⋊C4C4×D418yes—
422C4⋊C4C4×D418yes—
423C22⋊C4—232no—
424C22⋊C4—48no—
425C22⋊C4—88no—
426C42—22no—
427C42—22no—
428C42—24no—
429C42—48no—
430C42—88no—
431C42—84no—
432C42—84no—
433C42—88no—
434C42C2×C4214yes—
435C42#11714yes—
436C42C4×D418yes—
437C42C4×D414yes—
438C42C4×D414yes—
439C42#12414yes—
440C42#12714yes—
441C23—24no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—42no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—82no—
461C23—82no—
462C23C22×C22⋊C411yescharacteristic
463C23C2×C4×D412yes—
464C23C2×#12111yescharacteristic
465C23C2×#12111yescharacteristic
466C23#52512yes—
467C23#57612yes—
468C23#57712yes—
469C23#57712yes—
470C23#57812yes—
471Q8—24no—
472Q8—24no—
473Q8—44no—
474Q8—42no—
475Q8—42no—
476Q8—44no—
477D4—24no—
478D4—24no—
479D4—44no—
480D4—42no—
481D4—42no—
482D4—44no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—816no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4C4×C22⋊C418yes—
535C2×C4#39014yes—
536C2×C4#39014yes—
537C2×C4C22×C4211yescharacteristic
538C2×C4C2×#11711yescharacteristic
539C2×C4C2×C4×D412yes—
540C2×C4C2×C4×D412yes—
541C2×C4C2×C4×D412yes—
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C4×#4111yescharacteristic
545C2×C4C4×#4111yescharacteristic
546C2×C4C4×#4112yes—
547C2×C4C4×#4112yes—
548C2×C4C2×#12411yescharacteristic
549C2×C4#52912yes—
550C2×C4#52912yes—
551C2×C4C2×#12711yescharacteristic
552C2×C4#53212yes—
553C2×C4#53212yes—
554C2×C4#53212yes—
555C8—44no—
556C8—44no—
557C22—48no—
558C22—44no—
559C22—84no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—82no—
564C22—82no—
565C22—84no—
566C22C2×C4×C22⋊C411yescharacteristic
567C22C2×#39011yescharacteristic
568C22#229711yescharacteristic
569C22C4×#12112yes—
570C22C4×#12112yes—
571C22#289712yes—
572C22#289812yes—
573C22#294612yes—
574C22#294712yes—
575C22#295112yes—
576C22#295212yes—
577C22#304111yescharacteristic
578C22#304211yescharacteristic
579C22#304311yescharacteristic
580C22#304411yescharacteristic
581C22#304512yes—
582C22#304612yes—
583C22#311812yes—
584C22#312112yes—
585C22#312412yes—
586C22#312512yes—
587C4—24no—
588C4—24no—
589C4—28no—
590C4—48no—
591C4—48no—
592C4—44no—
593C4—44no—
594C4—82no—
595C4—82no—
596C4C42×D411yescharacteristic
597C4C42×D411yescharacteristic
598C4#229311yescharacteristic
599C4#229311yescharacteristic
600C4C4×#12412yes—
601C4C4×#12712yes—
602C2—84no—
603C2C4×#39011yescharacteristic
604C2#2041512yes—
605C2#2126211yescharacteristic
606C2#2126311yescharacteristic
607C2#2139712yes—
608C2#2150912yes—
609C2#2151012yes—
610C2#2151112yes—
611C2#2151212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C2×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×#1768C2semidirect
4. C2×#1763C2semidirect
6. C2×C4×#103C2semidirect
8. #13750C2semidirect
9. #10932C2semidirect
11. #10849C2semidirect
23. #2326C4semidirect
34. #1983C4semidirect
87. C2×C4⋊C8C2×C4semidirect
92. C2×#103C2×C4semidirect
93. C2×#103C2×C4semidirect
119. C4×C4⋊C4C2×C4semidirect
188. #124C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#19832
C2×#17682
C2×#17632
C2×#17622
C2×C4×#1032
C2×C4×#1022
#137502
#109322
#109312
#108492