Finite Group Databasebeta

← #201306

Subgroups — #201306

Order 512.

1430 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35019C211yesmaximal, characteristic
2#34358C211yesmaximal, characteristic
3#21752C211yesmaximal, characteristic
4#17578C211yesmaximal, characteristic, complemented
5C2×#1511C211yesmaximal, characteristic, complemented
6#9460C214yesmaximal, complemented
7#9459C214yesmaximal, complemented
8#7942C211yesmaximal, characteristic, complemented
9C2×#1284C211yesmaximal, characteristic, complemented
10C22×C22⋊Q8C2211yescharacteristic
11C22×#121C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C2×#407C2211yescharacteristic
14#2850C414yescomplemented
15#2639—22no—
16#2591—22no—
17#2561—22no—
18#2504C412yescomplemented
19#2503C2212yes—
20#2436C412yescomplemented
21#2435C2212yes—
22#2377—22no—
23#2364—22no—
24#2342—22no—
25D4×C22⋊C4—22no—
26#2333—22no—
27#2325—22no—
28#2313C2212yes—
29#2308—22no—
30#2308C2212yes—
31#2280C2211yescharacteristic
32C2×#381C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1900—22no—
35#1900—22no—
36#1821C2212yes—
37#1783C2212yes—
38#1520—22no—
39#1511—22no—
40#1511C2214yes—
41#1511C2214yes—
42#1508—22no—
43C2×#328—22no—
44#1284C2214yes—
45#1284C2214yes—
46C23×Q8C2×C411yescharacteristic, complemented
47C23×D4D412yes—
48C23×D4C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—21no—
51C22×M16—21no—
52C2×#124C2×C414yescomplemented
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⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8C2×C412yescomplemented
64C2×C22⋊Q8C2312yes—
65C2×#121—21no—
66C2×#121—21no—
67C2×#121—21no—
68C2×#121—21no—
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121—21no—
73C2×#121C2×C412yes—
74C2×#121C2×C412yes—
75C2×#121D412yes—
76C2×#121D412yes—
77C2×C4×D4—24no—
78C2×#117—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C411yescharacteristic, complemented
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4D412yes—
87C22×C42C2×C412yescomplemented
88C2×#100—24no—
89#407—21no—
90#407—21no—
91#407—21no—
92#407—21no—
93#407—21no—
94#407—21no—
95#407C2312yes—
96#407C2312yes—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101#400—24no—
102#397—24no—
103#394—24no—
104#393—24no—
105#391C2×C414yescomplemented
106#390—24no—
107#390C2×C414yescomplemented
108#388—24no—
109#388—24no—
110#388—28no—
111#387—24no—
112#387—24no—
113#386—24no—
114#386—24no—
115#386—24no—
116#386—24no—
117#386—28no—
118#386—24no—
119#386—24no—
120#385—24no—
121#382C2×C414yescomplemented
122#381—24no—
123#381—21no—
124#381—21no—
125#381D412yes—
126#381C2312yes—
127#381C2312yes—
128#380—22no—
129#380—21no—
130#380D412yes—
131#379—24no—
132#379—22no—
133#379—21no—
134C4×C22⋊C4—24no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4C2×C412yescomplemented
140C4×C22⋊C4D412yes—
141C4×C22⋊C4C2312yes—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95D412yes—
145#343—24no—
146#328—28no—
147#328—28no—
148C25C2×D411yescharacteristic
149C25C2×D412yes—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—82no—
156C22×Q8C22×C412yes—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—48no—
166C22×D4—82no—
167C22×D4C22⋊C414yes—
168C22×D4C22⋊C414yes—
169C22×D4C22×C412yes—
170C22×D4C2×D412yes—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—82no—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4#4111yescharacteristic
188C23×C4#4111yescharacteristic
189C23×C4#4111yescharacteristic
190C23×C4#4111yescharacteristic
191C23×C4C2411yescharacteristic
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—41no—
195C2×M16—41no—
196C2×M16—82no—
197C2×M16—82no—
198C22×C8—41no—
199C22×C8—41no—
200#124—84no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—41no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C22⋊Q8—41no—
217#121—22no—
218#121—22no—
219#121—22no—
220#121—22no—
221#121—22no—
222#121—22no—
223#121—22no—
224#121—22no—
225#121—22no—
226#121—22no—
227#121—22no—
228#121—22no—
229#121—22no—
230#121—22no—
231#121—22no—
232#121—22no—
233C4×D4—88no—
234#117—82no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4C22×C412yes—
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—82no—
279C2×C22⋊C4—84no—
280C2×C22⋊C4C22⋊C414yes—
281C2×C22⋊C4C22⋊C414yes—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—82no—
286C2×C42C22×C414yes—
287#100—84no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—22no—
291C22⋊C8—22no—
292C22⋊C8—41no—
293C22⋊C8—41no—
294C22⋊C8—41no—
295C22⋊C8—41no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—28no—
300#95—28no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95C2×D414yes—
307#95C2×D414yes—
308C24—24no—
309C24—22no—
310C24—22no—
311C24—42no—
312C24—44no—
313C24—44no—
314C24—81no—
315C24—82no—
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C24C2×C22⋊C414yes—
319C24#12011yescharacteristic
320C24#12011yescharacteristic
321C24#12311yescharacteristic
322C24#12311yescharacteristic
323C24C22×D411yescharacteristic
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—44no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×Q8—82no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—84no—
348C2×D4—81no—
349C2×D4—82no—
350C2×D4—81no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4—84no—
354C2×D4—84no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—44no—
375C22×C4—81no—
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×C4—82no—
386C22×C4—82no—
387C22×C4—84no—
388C22×C4—82no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—82no—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C414yes—
395C22×C4C4×D411yescharacteristic
396C22×C4C4×D411yescharacteristic
397C22×C4C4×D411yescharacteristic
398C22×C4C4×D411yescharacteristic
399C22×C4C4×D411yescharacteristic
400C22×C4C4×D411yescharacteristic
401C22×C4C4×D411yescharacteristic
402C22×C4C4×D411yescharacteristic
403C22×C4#12011yescharacteristic
404C22×C4#12011yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4#12111yescharacteristic
407C22×C4#12111yescharacteristic
408C22×C4#12111yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12311yescharacteristic
414C22×C4#12311yescharacteristic
415C22×C4C23×C411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421M16—44no—
422M16—44no—
423M16—82no—
424M16—82no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—82no—
428C2×C8—82no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—82no—
442C4⋊C4—84no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—44no—
456C22⋊C4—88no—
457C22⋊C4—82no—
458C22⋊C4—88no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C22⋊C4—82no—
462C22⋊C4—84no—
463C42—44no—
464C42—82no—
465C42—82no—
466C42—84no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—28no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—82no—
497C23—82no—
498C23—88no—
499C23—168no—
500C23#38611yescharacteristic
501C23#38611yescharacteristic
502C23#38611yescharacteristic
503C23#38611yescharacteristic
504C23#45514yes—
505C23C22×C22⋊C411yescharacteristic
506C23C2×#12011yescharacteristic
507C23C2×#12311yescharacteristic
508Q8—82no—
509Q8—82no—
510Q8—82no—
511Q8—82no—
512D4—48no—
513D4—48no—
514D4—82no—
515D4—82no—
516D4—82no—
517D4—82no—
518D4—84no—
519D4—84no—
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—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—88no—
570C2×C4—164no—
571C2×C4—168no—
572C2×C4#38611yescharacteristic
573C2×C4#38611yescharacteristic
574C2×C4#38611yescharacteristic
575C2×C4#38611yescharacteristic
576C2×C4C2×C4×D411yescharacteristic
577C2×C4C2×C4×D411yescharacteristic
578C2×C4C2×#12111yescharacteristic
579C2×C4C2×#12111yescharacteristic
580C8—82no—
581C8—82no—
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—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—88no—
610C22—84no—
611C22—84no—
612C22—164no—
613C22#188512yes—
614C22#188512yes—
615C22C2×#38611yescharacteristic
616C22#288312yes—
617C22C2×#45512yes—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—21no—
626C4—44no—
627C4—84no—
628C4—84no—
629C4—84no—
630C4—82no—
631C4—82no—
632C2—22no—
633C2—22no—
634C2—22no—
635C2—22no—
636C2—41no—
637C2—41no—
638C2—41no—
639C2—84no—
640C2C2×#188511yescharacteristic
641C2#2098411yescharacteristic
642C2#2101211yescharacteristic
subgroupG/H
center, socleC22C2×#386
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
4. #17578C2semidirect
5. C2×#1511C2semidirect
6. #9460C2semidirect
7. #9459C2semidirect
8. #7942C2semidirect
9. C2×#1284C2semidirect
14. #2850C4semidirect
18. #2504C4semidirect
20. #2436C4semidirect
46. C23×Q8C2×C4semidirect
52. C2×#124C2×C4semidirect
63. C2×C22⋊Q8C2×C4semidirect
80. C22×C4⋊C4C2×C4semidirect
87. C22×C42C2×C4semidirect
105. #391C2×C4semidirect
107. #390C2×C4semidirect
121. #382C2×C4semidirect
139. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350192
#343582
#217522
#175782
C2×#15112
#94602
#94592
#79422
C2×#12842