Finite Group Databasebeta

← #192429

Subgroups — #192429

Order 512.

1340 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
1#34218C211yesmaximal, characteristic
2#21941C211yesmaximal, characteristic, complemented
3#21763C211yesmaximal, characteristic
4C2×#1501C211yesmaximal, characteristic
5#13965C211yesmaximal, characteristic, complemented
6#8873C212yesmaximal, complemented
7#8871C212yesmaximal, complemented
8#8868C212yesmaximal, complemented
9#8866C212yesmaximal, complemented
10#8012C211yesmaximal, characteristic, complemented
11C2×#1280C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15#2368—21no—
16#2364—21no—
17#2359—22no—
18#2353—21no—
19#2349—21no—
20#2343—21no—
21#2339—21no—
22#2329C2212yes—
23#2325C2212yes—
24#2318—21no—
25C4×#120—22no—
26C2×#385—22no—
27#2280C2211yescharacteristic
28C2×#380C2211yescharacteristic
29#2275—21no—
30C2×C4×C22⋊C4C2211yescharacteristic
31C22×#95C2211yescharacteristic
32#1893—21no—
33#1892—21no—
34#1874C2212yes—
35#1872C2212yes—
36#1854—22no—
37#1853C2212yes—
38#1852C2212yes—
39#1825—21no—
40#1808—21no—
41#1807—21no—
42#1788—21no—
43#1501C418yescomplemented
44C2×#328C2211yescharacteristic
45C2×#328C2211yescharacteristic
46C2×#327C2211yescharacteristic
47C2×#327C2211yescharacteristic
48#1444C2212yes—
49#1444C2212yes—
50C4⋊#95—21no—
51C4⋊#95—21no—
52#1294C2212yes—
53#1294C2212yes—
54#1280C2212yes—
55#1280C2212yes—
56C22×#41D411yescharacteristic
57C24×C4D412yes—
58C24×C4C2311yescharacteristic
59C23×C8—21no—
60C23×C8—21no—
61#525—22no—
62#525—22no—
63#525C2×C414yes—
64#525Q814yes—
65C2×#123—21no—
66C2×C22⋊Q8D411yescharacteristic
67C2×C22⋊Q8C2311yescharacteristic
68C2×#121D411yescharacteristic
69C2×#121C2311yescharacteristic
70C2×#120—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×#117D411yescharacteristic
77C2×#117D412yes—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C42—21no—
84C2×#107—22no—
85C2×#106—22no—
86C2×#103—22no—
87C2×#102—22no—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8C2×C414yescomplemented
94C2×C22⋊C8C2312yes—
95C2×C22⋊C8C2311yescharacteristic
96C2×C22⋊C8C2311yescharacteristic
97#391—22no—
98#389—22no—
99#388—22no—
100#388—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#385—22no—
110#385—22no—
111#385—22no—
112#385—22no—
113#385—48no—
114#383—21no—
115#382—21no—
116#381—21no—
117#381—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380D411yescharacteristic
124#380D411yescharacteristic
125#380C2312yes—
126#379—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—22no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4C2×C414yescomplemented
138#376—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95—21no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—21no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95D411yescharacteristic
148C2×#95D411yescharacteristic
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152#340—22no—
153#340—22no—
154#340—22no—
155#340—22no—
156#328C2×C414yescomplemented
157#328C2×C414yescomplemented
158#327C2×C414yescomplemented
159#327C2×C414yescomplemented
160C25C2×D411yescharacteristic
161C2×#41—22no—
162C2×#41—82no—
163C2×#41C4⋊C414yescomplemented
164C22×Q8C2×D411yescharacteristic
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4C2×D411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—42no—
178C23×C4—44no—
179C23×C4—82no—
180C23×C4—82no—
181C23×C4C22×C412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×Q812yes—
190C23×C4#4112yes—
191C23×C4#4112yes—
192C23×C4C2411yescharacteristic
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—24no—
196C22×C8—22no—
197C22×C8—42no—
198C22×C8—82no—
199C22×C8—82no—
200#123—24no—
201#123—24no—
202C22⋊Q8—22no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C22⋊Q8C4⋊C414yescomplemented
206C22⋊Q8C22×C412yes—
207C22⋊Q8C2×Q812yes—
208#121—22no—
209#121—21no—
210#121—21no—
211#121C4⋊C414yescomplemented
212#121C22×C412yes—
213#121C2×Q812yes—
214#120—24no—
215#120—24no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—44no—
221C4×D4—44no—
222C4×D4—84no—
223C4×D4—84no—
224#117—22no—
225#117—82no—
226#117C4⋊C414yescomplemented
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4—82no—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D412yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C42—21no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—82no—
270#107—82no—
271#106—82no—
272#103—82no—
273#102—82no—
274C22⋊C8—24no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—42no—
280C22⋊C8C22×C412yes—
281C22⋊C8C22×C412yes—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—21no—
298#95—21no—
299#95—44no—
300#95—44no—
301#95—41no—
302#95—41no—
303#95C22×C412yes—
304#95C22×C412yes—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—21no—
310C24—42no—
311C24C4×D412yes—
312C24#12011yescharacteristic
313C24#12011yescharacteristic
314C24C22⋊Q812yes—
315C24C22×D411yescharacteristic
316#41—84no—
317#41—84no—
318C2×Q8—21no—
319C2×Q8—42no—
320C2×Q8C2×C4⋊C412yes—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—21no—
326C2×D4—42no—
327C2×D4—44no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4C2×C4⋊C412yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—82no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4C2×C4⋊C412yes—
388C22×C4C2×C4⋊C412yes—
389C22×C4C4×D412yes—
390C22×C4C4×D412yes—
391C22×C4C4×D412yes—
392C22×C4#12011yescharacteristic
393C22×C4#12011yescharacteristic
394C22×C4C22⋊Q812yes—
395C22×C4C22⋊Q812yes—
396C22×C4C22⋊Q812yes—
397C22×C4#12312yes—
398C22×C4#12312yes—
399C22×C4#13514yes—
400C22×C4C23×C411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×Q811yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—84no—
414C2×C8—84no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—24no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C4⋊C4—88no—
430C4⋊C4—88no—
431C4⋊C4C2×C4⋊C412yes—
432C4⋊C4C2×C4⋊C412yes—
433C22⋊C4—216no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—24no—
437C22⋊C4—24no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—24no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—82no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C22⋊C4—82no—
451C42—24no—
452C42—44no—
453C42—82no—
454C42—84no—
455C42—82no—
456C42—82no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—22no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—42no—
482C23#38512yes—
483C23#38512yes—
484C23#45312yes—
485C23#45612yes—
486C23C2×C4×D411yescharacteristic
487C23C2×#12011yescharacteristic
488C23C2×C22⋊Q811yescharacteristic
489C23C2×#13512yes—
490C23#57512yes—
491Q8—42no—
492Q8—81no—
493D4—42no—
494D4—48no—
495D4—44no—
496D4—48no—
497D4—42no—
498D4—42no—
499D4—81no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—84no—
550C2×C4—82no—
551C2×C4—88no—
552C2×C4—88no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—164no—
562C2×C4#38512yes—
563C2×C4#38512yes—
564C2×C4C22×C4⋊C411yescharacteristic
565C2×C4C2×C4×D411yescharacteristic
566C2×C4C2×C22⋊Q811yescharacteristic
567C2×C4C2×#12311yescharacteristic
568C8—42no—
569C8—42no—
570C8—82no—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—24no—
580C22—22no—
581C22—22no—
582C22—24no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—44no—
587C22—44no—
588C22—48no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—82no—
595C22—82no—
596C22#185914yes—
597C22C2×#38511yescharacteristic
598C22C4×#13511yescharacteristic
599C22#294311yescharacteristic
600C22#300212yes—
601C22C2×#45311yescharacteristic
602C22C2×#45611yescharacteristic
603C22#302612yes—
604C22#307612yes—
605C4—21no—
606C4—21no—
607C4—22no—
608C4—21no—
609C4—21no—
610C4—22no—
611C4—44no—
612C4—44no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4—44no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—162no—
621C2—21no—
622C2—21no—
623C2—22no—
624C2—24no—
625C2—42no—
626C2—82no—
627C2#1837612yes—
628C2#1837912yes—
629C2C2×#185911yescharacteristic
630C2#2087211yescharacteristic
631C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21941C2semidirect
5. #13965C2semidirect
6. #8873C2semidirect
7. #8871C2semidirect
8. #8868C2semidirect
9. #8866C2semidirect
10. #8012C2semidirect
11. C2×#1280C2semidirect
43. #1501C4semidirect
93. C2×C22⋊C8C2×C4semidirect
137. C4×C22⋊C4C2×C4semidirect
156. #328C2×C4semidirect
157. #328C2×C4semidirect
158. #327C2×C4semidirect
159. #327C2×C4semidirect
163. C2×#41C4⋊C4semidirect
205. C22⋊Q8C4⋊C4semidirect
211. #121C4⋊C4semidirect
226. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342182
#219412
#217632
C2×#15012
#139652
#88732
#88712
#88682
#88662
#80122
C2×#12802