Finite Group Databasebeta

← #362083

Subgroups — #362083

Order 512.

1848 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2349C211yesmaximal, characteristic
2C2×#1799C212yesmaximal
3C2×#1799C211yesmaximal, characteristic
4C2×C4×#99C211yesmaximal, characteristic
5#13908C212yesmaximal, complemented
6#11203C214yesmaximal, complemented
7#10854C212yesmaximal, complemented
8#10761C212yesmaximal
9C22×C4×D4C2212yes—
10C22×C4×D4C2211yescharacteristic
11C22×#99C2212yes—
12C22×#99C2211yescharacteristic
13C22×#99C2211yescharacteristic
14#2349—24no—
15#2349—22no—
16#2349—22no—
17C2×#389C412yescomplemented
18C2×#385C412yescomplemented
19C2×#385C2212yes—
20C2×#382C411yescharacteristic, complemented
21C2×#380C411yescharacteristic, complemented
22C2×#380C2212yes—
23C2×#380C2214yes—
24C2×#380C2211yescharacteristic
25C2×C4×C4⋊C4C411yescharacteristic, complemented
26C2×C4×C22⋊C4C2212yes—
27C22×#95C411yescharacteristic, complemented
28C22×#95C2211yescharacteristic
29#1799—28no—
30#1799—24no—
31C4×#99—24no—
32C2×#342C2214yes—
33C2×#342C2212yes—
34C2×#342C2212yes—
35C2×#342C2214yes—
36#1444—22no—
37#1442—22no—
38#1436—24no—
39C22⋊#95C2214yes—
40C23×D4C2311yescharacteristic
41C24×C4C2×C412yes—
42C24×C4C2×C411yescharacteristic
43C24×C4C2312yes—
44C24×C4C2311yescharacteristic
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—24no—
48C2×C4×D4—24no—
49C2×C4×D4—24no—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—44no—
53C2×C4×D4D414yes—
54C2×C4×D4Q814yes—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C22⋊C4C2×C411yescharacteristic
64C22×C22⋊C4C2314yes—
65C22×C22⋊C4C2312yes—
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2312yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C42—22no—
70C22×C42C2×C412yes—
71C22×C42C2×C411yescharacteristic
72C2×#99—22no—
73C2×#99—22no—
74C2×#99—22no—
75C2×#99—22no—
76C2×#99—24no—
77C2×#99—24no—
78C2×#99—24no—
79C2×#99D414yes—
80C2×#99Q814yes—
81#389—24no—
82#389—24no—
83#385—24no—
84#385—24no—
85#385—24no—
86#385—24no—
87#382—22no—
88#382—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380—24no—
94#380—24no—
95#380—28no—
96#380—44no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—21no—
100C4×C22⋊C4—24no—
101C4×C22⋊C4—24no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—24no—
106C2×#95—22no—
107C2×#95—24no—
108C2×#95—22no—
109C2×#95—24no—
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C412yes—
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C412yes—
114C2×#95C2×C411yescharacteristic
115C2×#95C2×C411yescharacteristic
116C2×#95C2×C412yes—
117C2×#95D414yes—
118C2×#95D412yes—
119C2×#95D412yes—
120#342—28no—
121#342—24no—
122#342—24no—
123#342—44no—
124C25C22×C411yescharacteristic
125C25C2411yescharacteristic
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4C2×D412yes—
131C22×D4C2×D412yes—
132C22×D4C2×Q812yes—
133C22×D4C2×Q812yes—
134C22×D4#4112yes—
135C22×D4#4112yes—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—24no—
148C23×C4—44no—
149C23×C4—42no—
150C23×C4—44no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4—82no—
155C23×C4C4⋊C414yescomplemented
156C23×C4C22×C412yes—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C412yes—
159C23×C4C22×C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D414yes—
166C23×C4C2×D414yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×Q812yes—
171C23×C4#4114yes—
172C4×D4—28no—
173C4×D4—28no—
174C4×D4—28no—
175C4×D4—28no—
176C4×D4—48no—
177C4×D4—48no—
178C4×D4—48no—
179C4×D4—48no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—48no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—84no—
194C2×C4⋊C4—84no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C4⋊C414yescomplemented
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4C4⋊C414yescomplemented
226C2×C22⋊C4C2×D414yes—
227C2×C22⋊C4C2×D412yes—
228C2×C22⋊C4C2×Q814yes—
229C2×C22⋊C4C2×Q812yes—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—42no—
237C2×C42—44no—
238C2×C42—84no—
239C2×C42C4⋊C414yescomplemented
240#99—28no—
241#99—28no—
242#99—48no—
243#95—21no—
244#95—21no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—24no—
250#95—48no—
251#95—48no—
252#95—44no—
253#95—44no—
254#95—84no—
255#95C22⋊C411yescharacteristic
256#95C22⋊C412yes—
257#95C22⋊C411yescharacteristic
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260#95C22⋊C414yes—
261#95C22⋊C414yes—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—42no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24C2×C4⋊C412yes—
271C24C4×D412yes—
272C24C4×Q812yes—
273C24C23×C411yescharacteristic
274C24C22×D412yes—
275C24C22×D411yescharacteristic
276C24C22×D411yescharacteristic
277C24C22×Q811yescharacteristic
278C24C22×Q811yescharacteristic
279C24C2×#4112yes—
280C24C2×#4111yescharacteristic
281C24C2×#4111yescharacteristic
282C24H(5,𝔽2)12yes—
283C242−1+412yes—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4#12114yes—
297C2×D4C22⋊Q814yes—
298C2×D4#12514yes—
299C2×D4C4⋊Q814yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—28no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—44no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—88no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C414yes—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C4⋊C412yes—
384C22×C4C2×C4⋊C412yes—
385C22×C4C2×C4⋊C412yes—
386C22×C4C2×C4⋊C412yes—
387C22×C4C2×C4⋊C412yes—
388C22×C4#11712yes—
389C22×C4#11712yes—
390C22×C4C4×D412yes—
391C22×C4C4×Q812yes—
392C22×C4#12114yes—
393C22×C4#12114yes—
394C22×C4C22⋊Q814yes—
395C22×C4C22⋊Q814yes—
396C4⋊C4—28no—
397C4⋊C4—48no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—816no—
403C4⋊C4—88no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—28no—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—48no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C42—24no—
423C42—24no—
424C42—44no—
425C42—44no—
426C42—88no—
427C23—24no—
428C23—22no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—44no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—42no—
450C23—44no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23#38914yes—
460C23C22×C22⋊C411yescharacteristic
461C23C22×C4⋊C411yescharacteristic
462C23C2×#11711yescharacteristic
463C23C2×C4×D411yescharacteristic
464C23C2×C4×Q811yescharacteristic
465C23#51811yescharacteristic
466C23#51911yescharacteristic
467C23C2×#12112yes—
468C23C2×#12111yescharacteristic
469C23C2×C22⋊Q812yes—
470C23C2×C22⋊Q811yescharacteristic
471C23C2×#12511yescharacteristic
472C23C2×C4⋊Q811yescharacteristic
473C23#53712yes—
474C23#54412yes—
475C23D4212yes—
476C23#54612yes—
477C23#54712yes—
478C23#54812yes—
479C23D4×Q812yes—
480C23#55212yes—
481C23#55312yes—
482C23#55412yes—
483D4—28no—
484D4—28no—
485D4—28no—
486D4—28no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—48no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—88no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—88no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—84no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—84no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—84no—
559C2×C4#38012yes—
560C2×C4#38012yes—
561C2×C4#38014yes—
562C2×C4#38914yes—
563C2×C4#41014yes—
564C22—24no—
565C22—22no—
566C22—22no—
567C22—24no—
568C22—21no—
569C22—21no—
570C22—22no—
571C22—22no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—22no—
576C22—24no—
577C22—24no—
578C22—22no—
579C22—22no—
580C22—24no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22C2×#38011yescharacteristic
600C22C2×#38911yescharacteristic
601C22#231611yescharacteristic
602C22D4×C22⋊C411yescharacteristic
603C22Q8×C22⋊C411yescharacteristic
604C22#234411yescharacteristic
605C22#234511yescharacteristic
606C22#241612yes—
607C22#241712yes—
608C22#244012yes—
609C22#244212yes—
610C22C2×#41012yes—
611C22#288312yes—
612C4—24no—
613C4—24no—
614C4—22no—
615C4—22no—
616C4—24no—
617C4—44no—
618C4—42no—
619C4—42no—
620C4—42no—
621C4—84no—
622C4—84no—
623C4—82no—
624C4—82no—
625C4—164no—
626C2—22no—
627C2—22no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—22no—
633C2—22no—
634C2—44no—
635C2#1390711yescharacteristic
636C2#2063211yescharacteristic
637C2#2063311yescharacteristic
638C2#2240512yes—
639C2#2251312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#95 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13908C2semidirect
6. #11203C2semidirect
7. #10854C2semidirect
17. C2×#389C4semidirect
18. C2×#385C4semidirect
20. C2×#382C4semidirect
21. C2×#380C4semidirect
25. C2×C4×C4⋊C4C4semidirect
27. C22×#95C4semidirect
155. C23×C4C4⋊C4semidirect
197. C2×C4⋊C4C4⋊C4semidirect
225. C2×C22⋊C4C4⋊C4semidirect
239. C2×C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23492
C2×#17992
C2×C4×#992
#139082
#112032
#108542
#107612