Finite Group Databasebeta

← #206039

Subgroups — #206039

Order 512.

1179 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#20525C211yesmaximal, characteristic, complemented
4C2×#1508C211yesmaximal, characteristic, complemented
5C2×#1507C211yesmaximal, characteristic
6#17397C211yesmaximal, characteristic, complemented
7#17396C211yesmaximal, characteristic
8#9816C214yesmaximal, complemented
9#9813C214yesmaximal, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×#121C2211yescharacteristic
12#3187—22no—
13#3187—22no—
14#3187—21no—
15#3187—21no—
16#3187—22no—
17C2×#481C412yescomplemented
18C2×#483C412yescomplemented
19C2×#465C2214yes—
20C2×#448C412yescomplemented
21C2×#450C412yescomplemented
22C2×#407C2211yescharacteristic
23C2×#407C2211yescharacteristic
24C2×#407C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C2×C4×C22⋊C4C2211yescharacteristic
27#1855—22no—
28#1855—22no—
29#1810C2214yescomplemented
30#1753—22no—
31#1752C2212yescomplemented
32C8×C22⋊C4C2212yescomplemented
33#1749—21no—
34C4×C22⋊C8—21no—
35#1508—22no—
36#1508C2212yescomplemented
37#1508C2212yescomplemented
38#1507—22no—
39#1507C2212yes—
40#1507C2212yes—
41C2×#328—22no—
42C2×#327—22no—
43#1472—22no—
44#1472C2214yescomplemented
45#1471—22no—
46#1471C2214yes—
47#1463—22no—
48#1462—22no—
49C23×Q8C2×C411yescharacteristic
50C23×D4—21no—
51C23×D4C2×C411yescharacteristic
52C24×C4C2311yescharacteristic
53C22×SD16C2×C414yescomplemented
54C22×M16—21no—
55C22×M16—21no—
56C22×M16—21no—
57C23×C8C2312yes—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8C2×C412yes—
69C2×C22⋊Q8C2312yes—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121—21no—
74C2×#121—21no—
75C2×#121—21no—
76C2×#121—21no—
77C2×#121—21no—
78C2×#121—21no—
79C2×#121—21no—
80C2×#121C2×C412yes—
81C2×#121C2312yes—
82C22×C4⋊C4—21no—
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C42—21no—
89#483—22no—
90#483—22no—
91#483—24no—
92#481—22no—
93#481—22no—
94#481—24no—
95#465—24no—
96#465—24no—
97#465—24no—
98#465—24no—
99#450—22no—
100#450—22no—
101#450—24no—
102#448—22no—
103#448—22no—
104#448—24no—
105C2×#108—24no—
106C2×#106C2×C414yescomplemented
107C2×#103C2×C414yescomplemented
108C2×#102C2×C414yescomplemented
109#407—21no—
110#407—21no—
111#407—21no—
112#407—21no—
113#407—21no—
114#407—21no—
115#407—21no—
116#407—21no—
117#407D412yes—
118#407D412yes—
119#407C2312yes—
120#407C2312yes—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8C2×C412yes—
127C2×C22⋊C8C2×C412yes—
128C2×C22⋊C8D412yes—
129C2×C22⋊C8D412yes—
130C2×C22⋊C8D411yescharacteristic
131C2×C22⋊C8D411yescharacteristic
132C2×C22⋊C8D412yes—
133C2×C22⋊C8C2312yes—
134C2×#97—22no—
135C2×C4×C8—22no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4D412yes—
139C4×C22⋊C4C2311yescharacteristic
140C4×C22⋊C4C2311yescharacteristic
141C2×#95D412yes—
142#341—24no—
143#336—22no—
144#336—22no—
145#328—24no—
146#328—24no—
147#327—24no—
148#327—24no—
149#324—28no—
150#323—28no—
151C25—21no—
152C25#4111yescharacteristic
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—82no—
159C22×Q8C22×C412yes—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4—82no—
172C22×D4—84no—
173C22×D4C22×C412yes—
174C23×C4—22no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—82no—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4#4111yescharacteristic
190C23×C4#4111yescharacteristic
191C23×C4#4111yescharacteristic
192C23×C4C2411yescharacteristic
193C2×SD16—24no—
194C2×SD16—24no—
195C2×SD16—84no—
196C2×M16—22no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—41no—
201C2×M16—41no—
202C2×M16—82no—
203C2×M16—82no—
204C2×M16—82no—
205C22×C8—22no—
206C22×C8—41no—
207C22×C8—41no—
208C22×C8—82no—
209C22×C8C22×C414yes—
210C22×C8C2×D412yes—
211C22×C8C2×D412yes—
212C22×C8C2×D414yes—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8—22no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—41no—
222C22⋊Q8—41no—
223C22⋊Q8—41no—
224C22⋊Q8—41no—
225C22⋊Q8—41no—
226C22⋊Q8—41no—
227C22⋊Q8—41no—
228C22⋊Q8—41no—
229#121—22no—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—22no—
235#121—22no—
236#121—22no—
237#121—41no—
238#121—41no—
239#121—41no—
240#121—41no—
241#121—41no—
242#121—41no—
243#121—41no—
244#121—41no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C412yes—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—41no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—82no—
283#108—84no—
284#106—24no—
285#106—24no—
286#103—24no—
287#103—24no—
288#102—24no—
289#102—24no—
290C22⋊C8—22no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—41no—
298C22⋊C8—41no—
299C22⋊C8—41no—
300C22⋊C8—41no—
301C22⋊C8C22⋊C412yes—
302C22⋊C8C22⋊C412yes—
303C22⋊C8C22⋊C414yes—
304C22⋊C8C22⋊C412yes—
305C22⋊C8C22⋊C412yes—
306C22⋊C8C22⋊C414yes—
307C22⋊C8C2×D414yes—
308#97—82no—
309C4×C8—82no—
310#95—22no—
311#95C2×D412yes—
312#95C2×D412yes—
313C24—24no—
314C24—22no—
315C24—42no—
316C24—41no—
317C24—41no—
318C24—82no—
319C24—82no—
320C24—81no—
321C24—82no—
322C24C4×D411yescharacteristic
323C24C4×D411yescharacteristic
324C24#12111yescharacteristic
325C24#12111yescharacteristic
326C24#12411yescharacteristic
327C24#12411yescharacteristic
328C24C2×#4111yescharacteristic
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—44no—
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—82no—
335C2×Q8—82no—
336C2×Q8—82no—
337C2×Q8—82no—
338C2×Q8—82no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—44no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4—84no—
358C2×D4—84no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—44no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4C4×D411yescharacteristic
396C22×C4C4×D411yescharacteristic
397C22×C4C4×D411yescharacteristic
398C22×C4C4×D411yescharacteristic
399C22×C4C4×D411yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12111yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12411yescharacteristic
407C22×C4#12411yescharacteristic
408C22×C4#12711yescharacteristic
409C22×C4#12711yescharacteristic
410C22×C4#12711yescharacteristic
411C22×C4#12711yescharacteristic
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4111yescharacteristic
418SD16—84no—
419SD16—84no—
420M16—44no—
421M16—44no—
422M16—82no—
423M16—82no—
424M16—82no—
425M16—82no—
426C2×C8—24no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—44no—
430C2×C8—44no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—82no—
436C2×C8—82no—
437C2×C8C2×C22⋊C414yes—
438C2×C8C2×C22⋊C414yes—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C22⋊C4—24no—
452C22⋊C4—44no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C42—42no—
463C42—42no—
464C42—82no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—44no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—82no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23—88no—
495C23—82no—
496C23—168no—
497C23#39011yescharacteristic
498C23#39011yescharacteristic
499C23#39011yescharacteristic
500C23#39011yescharacteristic
501C23#47114yes—
502C23C2×C4×D411yescharacteristic
503C23C2×#12111yescharacteristic
504C23C2×#12411yescharacteristic
505Q8—44no—
506Q8—82no—
507Q8—82no—
508Q8—82no—
509D4—44no—
510D4—82no—
511D4—82no—
512D4—82no—
513D4—84no—
514D4—84no—
515D4—84no—
516D4—84no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
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—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—88no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4#39011yescharacteristic
565C2×C4#39011yescharacteristic
566C2×C4#39011yescharacteristic
567C2×C4#39011yescharacteristic
568C2×C4C22×C22⋊C411yescharacteristic
569C2×C4C2×C4×D411yescharacteristic
570C2×C4C2×#12111yescharacteristic
571C2×C4C2×#12711yescharacteristic
572C8—24no—
573C8—24no—
574C8—44no—
575C8—82no—
576C8—82no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—164no—
605C22—164no—
606C22—164no—
607C22#197712yes—
608C22#197712yes—
609C22C2×#39011yescharacteristic
610C22#295612yes—
611C22C2×#47112yes—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—44no—
620C4—82no—
621C4—82no—
622C4—82no—
623C4—82no—
624C4#288312yes—
625C2—22no—
626C2—22no—
627C2—22no—
628C2—22no—
629C2—41no—
630C2—41no—
631C2—41no—
632C2—162no—
633C2#2121311yescharacteristic
634C2#2124111yescharacteristic
635C2C2×#197711yescharacteristic
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. #21948C2semidirect
3. #20525C2semidirect
4. C2×#1508C2semidirect
6. #17397C2semidirect
8. #9816C2semidirect
9. #9813C2semidirect
17. C2×#481C4semidirect
18. C2×#483C4semidirect
20. C2×#448C4semidirect
21. C2×#450C4semidirect
29. #1810C22semidirect
31. #1752C22semidirect
32. C8×C22⋊C4C22semidirect
36. #1508C22semidirect
37. #1508C22semidirect
44. #1472C22semidirect
53. C22×SD16C2×C4semidirect
106. C2×#106C2×C4semidirect
107. C2×#103C2×C4semidirect
108. C2×#102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
#219482
#205252
C2×#15082
C2×#15072
#173972
#173962
#98162
#98132