Finite Group Databasebeta

← #206047

Subgroups — #206047

Order 512.

1157 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3187C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#20528C211yesmaximal, characteristic, complemented
4#17578C211yesmaximal, characteristic, complemented
5#17577C211yesmaximal, characteristic
6C2×#1472C211yesmaximal, characteristic, complemented
7C2×#1471C211yesmaximal, characteristic
8#9816C214yesmaximal, complemented
9#9813C214yesmaximal, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×#121C2211yescharacteristic
12#3187—22no—
13#3187—22no—
14#3187—22no—
15#3187—21no—
16#3187—21no—
17C2×#481C412yescomplemented
18C2×#483C412yescomplemented
19C2×#465C2214yes—
20C2×#448C412yescomplemented
21C2×#450C412yescomplemented
22C2×#407C2211yescharacteristic
23C2×#407C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26#2280C2211yescharacteristic
27#1855—22no—
28#1855—22no—
29#1810C2214yescomplemented
30#1752—21no—
31#1752C2212yescomplemented
32C8×C22⋊C4—21no—
33C8×C22⋊C4C2212yescomplemented
34#1750—22no—
35#1520—22no—
36#1519—22no—
37#1511—22no—
38#1510—22no—
39#1508C2214yescomplemented
40#1507C2214yes—
41#1472—22no—
42#1472C2212yescomplemented
43#1472C2212yescomplemented
44#1471—22no—
45#1471C2212yes—
46#1471C2212yes—
47C2×#324—22no—
48C2×#323—22no—
49C23×Q8C2×C411yescharacteristic
50C23×D4—21no—
51C23×D4C2×C411yescharacteristic
52C24×C4C2311yescharacteristic
53C22×SD16C2×C414yes—
54C22×M16—21no—
55C22×M16—21no—
56C23×C8—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—
82C2×#117—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88#483—24no—
89#483—22no—
90#483—22no—
91#481—24no—
92#481—22no—
93#481—22no—
94#465—24no—
95#465—24no—
96#465—24no—
97#465—24no—
98#450—24no—
99#450—22no—
100#450—22no—
101#448—24no—
102#448—22no—
103#448—22no—
104C2×#108—24no—
105C2×#106C2×C414yes—
106C2×#103C2×C414yes—
107C2×#102C2×C414yes—
108#407—21no—
109#407—21no—
110#407—21no—
111#407—21no—
112#407—21no—
113#407—21no—
114#407C2312yes—
115#407C2312yes—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C22⋊C8C2×C412yes—
124C2×C22⋊C8C2×C412yes—
125C2×C22⋊C8D412yes—
126C2×C22⋊C8D412yes—
127C2×C22⋊C8D412yes—
128C2×C22⋊C8D412yes—
129C2×C22⋊C8D412yes—
130C2×C22⋊C8D411yescharacteristic
131C2×C22⋊C8D411yescharacteristic
132C2×C22⋊C8C2311yescharacteristic
133C2×C22⋊C8C2311yescharacteristic
134C2×#97—22no—
135C2×C4×C8—22no—
136#381D412yes—
137#380—21no—
138#380D412yes—
139#379—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4C2312yes—
142#341—24no—
143#336—22no—
144#336—22no—
145#328—28no—
146#327—28no—
147#324—24no—
148#324—24no—
149#323—24no—
150#323—24no—
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—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—82no—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4#4111yescharacteristic
189C23×C4#4111yescharacteristic
190C23×C4#4111yescharacteristic
191C23×C4C2411yescharacteristic
192C2×SD16—24no—
193C2×SD16—24no—
194C2×SD16—84no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—41no—
198C2×M16—41no—
199C2×M16—82no—
200C2×M16—82no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—41no—
206C22×C8—41no—
207C22×C8—82no—
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—
245#117—82no—
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—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
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×C22⋊C4—41no—
280C2×C42—22no—
281#108—84no—
282#106—24no—
283#106—24no—
284#103—24no—
285#103—24no—
286#102—24no—
287#102—24no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—41no—
297C22⋊C8—41no—
298C22⋊C8—41no—
299C22⋊C8—41no—
300C22⋊C8C22⋊C414yes—
301C22⋊C8C22⋊C412yes—
302C22⋊C8C22⋊C412yes—
303C22⋊C8C22⋊C414yes—
304C22⋊C8C22⋊C412yes—
305C22⋊C8C22⋊C412yes—
306C22⋊C8C2×D412yes—
307C22⋊C8C2×D412yes—
308#97—82no—
309C4×C8—82no—
310#95—22no—
311#95C2×D414yes—
312C24—24no—
313C24—22no—
314C24—42no—
315C24—44no—
316C24—41no—
317C24—41no—
318C24—82no—
319C24—81no—
320C24—82no—
321C24C4×D411yescharacteristic
322C24C4×D411yescharacteristic
323C24#12111yescharacteristic
324C24#12111yescharacteristic
325C24#12411yescharacteristic
326C24#12411yescharacteristic
327C24C2×#4111yescharacteristic
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—44no—
331C2×Q8—41no—
332C2×Q8—41no—
333C2×Q8—82no—
334C2×Q8—82no—
335C2×Q8—82no—
336C2×Q8—82no—
337C2×Q8—82no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—44no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—82no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—84no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4—84no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—44no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C4×D411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4#12111yescharacteristic
397C22×C4#12111yescharacteristic
398C22×C4#12111yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12411yescharacteristic
402C22×C4#12411yescharacteristic
403C22×C4#12711yescharacteristic
404C22×C4#12711yescharacteristic
405C22×C4#12711yescharacteristic
406C22×C4#12711yescharacteristic
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C2×#4111yescharacteristic
413SD16—84no—
414SD16—84no—
415M16—44no—
416M16—44no—
417M16—82no—
418M16—82no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—44no—
424C2×C8—44no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—82no—
430C2×C8—82no—
431C2×C8—82no—
432C2×C8—82no—
433C2×C8C2×C22⋊C414yes—
434C2×C8C2×C22⋊C414yes—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—82no—
448C22⋊C4—44no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—82no—
459C42—42no—
460C42—42no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23—44no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—82no—
490C23—88no—
491C23—82no—
492C23—168no—
493C23#39011yescharacteristic
494C23#39011yescharacteristic
495C23#39011yescharacteristic
496C23#39011yescharacteristic
497C23#47114yes—
498C23C2×C4×D411yescharacteristic
499C23C2×#12111yescharacteristic
500C23C2×#12411yescharacteristic
501Q8—44no—
502Q8—82no—
503Q8—82no—
504Q8—82no—
505D4—44no—
506D4—82no—
507D4—82no—
508D4—82no—
509D4—84no—
510D4—84no—
511D4—84no—
512D4—84no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
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—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4#39011yescharacteristic
557C2×C4#39011yescharacteristic
558C2×C4#39011yescharacteristic
559C2×C4#39011yescharacteristic
560C2×C4C22×C22⋊C411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×#12111yescharacteristic
563C2×C4C2×#12711yescharacteristic
564C8—24no—
565C8—24no—
566C8—42no—
567C8—42no—
568C8—82no—
569C8—82no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—164no—
598C22—164no—
599C22—164no—
600C22#197712yes—
601C22#197712yes—
602C22C2×#39011yescharacteristic
603C22#295512yes—
604C22C2×#47112yes—
605C4—21no—
606C4—21no—
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—84no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4#288312yes—
617C2—22no—
618C2—22no—
619C2—22no—
620C2—22no—
621C2—41no—
622C2—41no—
623C2—41no—
624C2—162no—
625C2#2121311yescharacteristic
626C2#2124211yescharacteristic
627C2C2×#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. #20528C2semidirect
4. #17578C2semidirect
6. C2×#1472C2semidirect
8. #9816C2semidirect
9. #9813C2semidirect
17. C2×#481C4semidirect
18. C2×#483C4semidirect
20. C2×#448C4semidirect
21. C2×#450C4semidirect
29. #1810C22semidirect
31. #1752C22semidirect
33. C8×C22⋊C4C22semidirect
39. #1508C22semidirect
42. #1472C22semidirect
43. #1472C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31872
#219482
#205282
#175782
#175772
C2×#14722
C2×#14712
#98162
#98132