Finite Group Databasebeta

← #205769

Subgroups — #205769

Order 512.

1315 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2#21938C211yesmaximal, characteristic, complemented
3C2×#1996C211yesmaximal, characteristic, complemented
4C2×#1964C211yesmaximal, characteristic
5#20523C211yesmaximal, characteristic, complemented
6C2×#1520C211yesmaximal, characteristic, complemented
7C2×#1519C211yesmaximal, characteristic
8#9781C218yesmaximal, complemented
9C22×C22⋊Q8C2211yescharacteristic
10C22×#121C2211yescharacteristic
11#3187—22no—
12#3187—22no—
13#3187—22no—
14#3187—22no—
15C2×#481C412yescomplemented
16C2×#483C412yescomplemented
17C2×#465C2214yes—
18C2×#448C412yescomplemented
19C2×#450C412yescomplemented
20C22×#99—22no—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C22⋊C4C2211yescharacteristic
26#1996—22no—
27#1996C2214yescomplemented
28#1964—22no—
29#1964C2214yes—
30#1853—22no—
31#1853—22no—
32#1807C2214yescomplemented
33C8×C22⋊C4—22no—
34C8×C22⋊C4C2214yescomplemented
35C4×C22⋊C8—22no—
36C2×#342—22no—
37#1520—22no—
38#1520C2214yescomplemented
39#1519—22no—
40#1519C2214yes—
41C2×#328—22no—
42C2×#327—22no—
43C23×Q8C2×C411yescharacteristic
44C23×D4—21no—
45C23×D4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×SD16C2×C414yescomplemented
48C23×C8—21no—
49C23×C8C2312yes—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8C2×C412yes—
61C2×C22⋊Q8C2312yes—
62C2×#121—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121C2×C412yes—
73C2×#121C2312yes—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4D412yes—
84C22×C42—21no—
85#483—24no—
86#483—24no—
87#481—24no—
88#481—24no—
89#465—28no—
90#465—28no—
91#450—24no—
92#450—24no—
93#448—24no—
94#448—24no—
95C2×#106—24no—
96C2×#106C2×C414yescomplemented
97C2×#103C2×C414yescomplemented
98C2×#102C2×C414yescomplemented
99C2×#99—28no—
100C2×#99—48no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8C2×C412yes—
105C2×C22⋊C8C2×C412yes—
106C2×C22⋊C8D412yes—
107C2×C22⋊C8D412yes—
108C2×C22⋊C8D412yes—
109C2×C22⋊C8D412yes—
110C2×C22⋊C8D412yes—
111C2×C22⋊C8D412yes—
112C2×C22⋊C8C2312yes—
113C2×C4×C8—24no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380C2312yes—
121#380C2312yes—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124C4×C22⋊C4D412yes—
125C4×C22⋊C4C2312yes—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129#342—28no—
130#340—24no—
131#336—24no—
132#328—28no—
133#327—28no—
134C25—21no—
135C25C2×D411yescharacteristic
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—82no—
142C22×Q8C22×C412yes—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—41no—
151C22×D4—41no—
152C22×D4—41no—
153C22×D4—41no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4—82no—
157C22×D4C22×C412yes—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4—82no—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4#4111yescharacteristic
175C23×C4#4111yescharacteristic
176C23×C4#4111yescharacteristic
177C23×C4#4111yescharacteristic
178C23×C4C2411yescharacteristic
179C2×SD16—28no—
180C2×SD16—84no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—82no—
185C22×C8—82no—
186C22×C8C22×C414yes—
187C22×C8C2×D414yes—
188C22×C8C2×D414yes—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—41no—
198C22⋊Q8—41no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204C22⋊Q8—41no—
205#121—22no—
206#121—22no—
207#121—22no—
208#121—22no—
209#121—22no—
210#121—22no—
211#121—22no—
212#121—22no—
213#121—41no—
214#121—41no—
215#121—41no—
216#121—41no—
217#121—41no—
218#121—41no—
219#121—41no—
220#121—41no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C412yes—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—41no—
265C2×C22⋊C4SD1618yescomplemented
266C2×C22⋊C4C2×D414yes—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—82no—
270#106—28no—
271#106—84no—
272#103—28no—
273#102—28no—
274#99—416no—
275C22⋊C8—22no—
276C22⋊C8—22no—
277C22⋊C8—22no—
278C22⋊C8C22⋊C414yes—
279C22⋊C8C22⋊C414yes—
280C22⋊C8C22⋊C414yes—
281C22⋊C8C22⋊C414yes—
282C22⋊C8C2×D414yes—
283C4×C8—84no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—41no—
289#95—41no—
290C24—22no—
291C24—41no—
292C24—41no—
293C24—44no—
294C24—81no—
295C24—81no—
296C24—82no—
297C24—82no—
298C24C4×D411yescharacteristic
299C24C4×D411yescharacteristic
300C24#12111yescharacteristic
301C24#12111yescharacteristic
302C24#12711yescharacteristic
303C24#12711yescharacteristic
304C24C2×SD1614yes—
305C24#13514yes—
306C24C22×D411yescharacteristic
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—44no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—82no—
313C2×Q8—82no—
314C2×Q8—82no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—44no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—48no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—82no—
330C2×D4—82no—
331C2×D4—82no—
332C2×D4—82no—
333C2×D4—82no—
334C2×D4—84no—
335C2×D4—84no—
336C2×D4—82no—
337C2×D4—82no—
338C2×D4—82no—
339C2×D4—82no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—82no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C4×D411yescharacteristic
383C22×C4C4×D411yescharacteristic
384C22×C4C4×D411yescharacteristic
385C22×C4C4×D411yescharacteristic
386C22×C4C4×D411yescharacteristic
387C22×C4C4×D411yescharacteristic
388C22×C4#12111yescharacteristic
389C22×C4#12111yescharacteristic
390C22×C4#12111yescharacteristic
391C22×C4#12111yescharacteristic
392C22×C4#12111yescharacteristic
393C22×C4#12111yescharacteristic
394C22×C4#12411yescharacteristic
395C22×C4#12411yescharacteristic
396C22×C4#12411yescharacteristic
397C22×C4#12411yescharacteristic
398C22×C4#12711yescharacteristic
399C22×C4#12711yescharacteristic
400C22×C4C23×C411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406SD16—88no—
407C2×C8—24no—
408C2×C8—44no—
409C2×C8—44no—
410C2×C8—44no—
411C2×C8—44no—
412C2×C8—82no—
413C2×C8—82no—
414C2×C8C2×C22⋊C418yes—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C22⋊C4—24no—
432C22⋊C4—28no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4C2×SD1618yes—
446C42—44no—
447C42—82no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—44no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—84no—
471C23—84no—
472C23—82no—
473C23—82no—
474C23—168no—
475C23#39011yescharacteristic
476C23#39011yescharacteristic
477C23#39011yescharacteristic
478C23#39011yescharacteristic
479C23C4×SD1614yes—
480C23C4×SD1614yes—
481C23#46514yes—
482C23#46514yes—
483C23#49212yes—
484C23#49212yes—
485C23#49512yes—
486C23#49512yes—
487C23C2×C4×D411yescharacteristic
488C23C2×#12111yescharacteristic
489C23C2×#12711yescharacteristic
490C23C22×SD1612yes—
491C23C2×#13512yes—
492Q8—44no—
493Q8—82no—
494Q8—82no—
495Q8—82no—
496D4—44no—
497D4—82no—
498D4—82no—
499D4—82no—
500D4—82no—
501D4—82no—
502D4—84no—
503D4—84no—
504D4—84no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—88no—
541C2×C4—82no—
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—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—164no—
556C2×C4—164no—
557C2×C4#39011yescharacteristic
558C2×C4#39011yescharacteristic
559C2×C4#39011yescharacteristic
560C2×C4#39011yescharacteristic
561C2×C4C22×C22⋊C411yescharacteristic
562C2×C4C2×C4×D411yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12411yescharacteristic
565C8—28no—
566C8—44no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—42no—
587C22—44no—
588C22—44no—
589C22—42no—
590C22—42no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—164no—
595C22—164no—
596C22—164no—
597C22#196812yes—
598C22#196812yes—
599C22#196812yes—
600C22#196812yes—
601C22C2×#39011yescharacteristic
602C22C2×C4×SD1612yes—
603C22C2×#46512yes—
604C22C2×#49211yescharacteristic
605C22C2×#49511yescharacteristic
606C4—21no—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—48no—
614C4—82no—
615C4—82no—
616C4—82no—
617C4—162no—
618C4—162no—
619C4#288312yes—
620C2—22no—
621C2—21no—
622C2—21no—
623C2—22no—
624C2—22no—
625C2—22no—
626C2—22no—
627C2—22no—
628C2—162no—
629C2#2121311yescharacteristic
630C2C2×#196811yescharacteristic
631C2#2124411yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21938C2semidirect
3. C2×#1996C2semidirect
5. #20523C2semidirect
6. C2×#1520C2semidirect
8. #9781C2semidirect
15. C2×#481C4semidirect
16. C2×#483C4semidirect
18. C2×#448C4semidirect
19. C2×#450C4semidirect
27. #1996C22semidirect
32. #1807C22semidirect
34. C8×C22⋊C4C22semidirect
38. #1520C22semidirect
47. C22×SD16C2×C4semidirect
96. C2×#106C2×C4semidirect
97. C2×#103C2×C4semidirect
98. C2×#102C2×C4semidirect
265. C2×C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
#219382
C2×#19962
C2×#19642
#205232
C2×#15202
C2×#15192
#97812