Finite Group Databasebeta

← #196030

Subgroups — #196030

Order 512.

1584 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic, complemented
2C2×#2324C211yesmaximal, characteristic
3#21790C211yesmaximal, characteristic, complemented
4C2×#1464C212yesmaximal, complemented
5#9085C212yesmaximal, complemented
6#9083C212yesmaximal, complemented
7#9081C212yesmaximal, complemented
8#9079C212yesmaximal, complemented
9#7912C212yesmaximal, complemented
10C22×#121C411yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2211yescharacteristic
13#2973—28no—
14C2×#435—24no—
15C2×#432C2212yes—
16C2×#432C2212yes—
17C2×#407C2212yes—
18C22×C22⋊C8C2212yes—
19#2324—24no—
20#2324—24no—
21C2×#390C411yescharacteristic
22C2×#390C411yescharacteristic
23C2×#390C411yescharacteristic
24C2×#389C411yescharacteristic
25C2×#388C411yescharacteristic
26C2×#388C411yescharacteristic
27C2×#386C2211yescharacteristic
28C2×#386C2211yescharacteristic
29C2×#385C2211yescharacteristic
30C2×#385C2211yescharacteristic
31C2×#380C411yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1955—24no—
34#1779C2212yes—
35#1779C2212yes—
36#1464—28no—
37C2×#324C2214yes—
38C2×#323C2214yes—
39#1280C2214yes—
40#1280C2214yes—
41C23×D4C2×C411yescharacteristic
42C23×D4C2×C411yescharacteristic
43C23×D4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C23×C8—22no—
49C2×#121—24no—
50C2×#121—24no—
51C2×#121—24no—
52C2×C4×D4—24no—
53C2×C4×D4—24no—
54C2×C4×D4D414yes—
55C2×C4×D4D414yes—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—24no—
59C2×#117—24no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C4⋊C4C2×C411yescharacteristic
65C22×C4⋊C4C2×C411yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4C2×C411yescharacteristic
68C22×C22⋊C4C2×C411yescharacteristic
69C22×C22⋊C4C2×C411yescharacteristic
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C22⋊C4C2×C411yescharacteristic
75C22×C22⋊C4C2×C411yescharacteristic
76C22×C22⋊C4C2×C411yescharacteristic
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2311yescharacteristic
79C22×C42C2×C411yescharacteristic
80C22×C42C2×C411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42C2311yescharacteristic
83#435—232no—
84#432—24no—
85#432—24no—
86#432—24no—
87#432D418yescomplemented
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90#407—28no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—28no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8C2314yes—
99C2×C22⋊C8C2314yes—
100C2×#97—22no—
101C2×#97—22no—
102#390—24no—
103#390—24no—
104#390—24no—
105#389—24no—
106#388—24no—
107#388—24no—
108#386—22no—
109#386—22no—
110#386—22no—
111#386—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#380—24no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C2×#95C2×C411yescharacteristic
120C2×#95C2×C411yescharacteristic
121C2×#95C2×C411yescharacteristic
122C2×#95C2×C411yescharacteristic
123C2×#95C2311yescharacteristic
124C2×#95C2311yescharacteristic
125#324—28no—
126#323—28no—
127C25C22×C411yescharacteristic
128C25C22×C411yescharacteristic
129C25C22×C411yescharacteristic
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4—81no—
140C22×D4—81no—
141C22×D4—81no—
142C22×D4—81no—
143C22×D4C22⋊C414yes—
144C22×D4M1614yescomplemented
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—24no—
153C23×C4—42no—
154C23×C4—81no—
155C23×C4—81no—
156C23×C4—81no—
157C23×C4—81no—
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2411yescharacteristic
174C2×M16—28no—
175C2×M16—82no—
176C2×M16—82no—
177C22×C8—24no—
178C22×C8—28no—
179C22×C8—24no—
180C22×C8—82no—
181C22×C8—82no—
182#121—232no—
183C4×D4—28no—
184C4×D4—28no—
185C4×D4—28no—
186C4×D4—28no—
187#117—44no—
188#117—44no—
189#117—44no—
190#117—44no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—81no—
206C2×C4⋊C4—81no—
207C2×C4⋊C4C22⋊C414yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—81no—
228C2×C22⋊C4—81no—
229C2×C22⋊C4—81no—
230C2×C22⋊C4—81no—
231C2×C22⋊C4—81no—
232C2×C22⋊C4—81no—
233C2×C22⋊C4C22⋊C414yes—
234C2×C22⋊C4C22⋊C414yes—
235C2×C22⋊C4M1614yescomplemented
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—81no—
241C2×C42—81no—
242C2×C42C2×D414yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42#4114yes—
246C4⋊C8—28no—
247C4⋊C8—44no—
248C22⋊C8—28no—
249C22⋊C8—28no—
250C22⋊C8—24no—
251C22⋊C8—24no—
252C22⋊C8—24no—
253C22⋊C8—82no—
254C22⋊C8—82no—
255C22⋊C8—82no—
256C22⋊C8—82no—
257C22⋊C8C2×D418yes—
258#97—28no—
259#97—44no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268C24—22no—
269C24—22no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—21no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—41no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24C2×C22⋊C412yes—
291C24C2×C22⋊C412yes—
292C24C2×M1612yes—
293C24C2×M1612yes—
294C24C2×M1612yes—
295C24C23×C411yescharacteristic
296C2×D4—28no—
297C2×D4—28no—
298C2×D4—28no—
299C2×D4—28no—
300C2×D4—44no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4—82no—
306C2×D4—82no—
307C2×D4—84no—
308C2×D4—84no—
309C2×D4—84no—
310C2×D4—84no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—28no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—48no—
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—48no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—84no—
372C22×C4—84no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C4×D414yes—
380C22×C4C4×D414yes—
381C22×C4C2×M1612yes—
382C22×C4C2×M1612yes—
383C22×C4C2×M1612yes—
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D412yes—
387C22×C4C2×#4112yes—
388M16—88no—
389C2×C8—28no—
390C2×C8—44no—
391C2×C8—44no—
392C2×C8—44no—
393C2×C8—44no—
394C2×C8—44no—
395C2×C8—88no—
396C4⋊C4—28no—
397C4⋊C4—28no—
398C4⋊C4—44no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—44no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C22⋊C4—28no—
406C22⋊C4—28no—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4—41no—
410C22⋊C4—42no—
411C22⋊C4—41no—
412C22⋊C4—44no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C42—44no—
420C42—82no—
421C42—82no—
422C42#12014yes—
423C42#12114yes—
424C42#12114yes—
425C42#12314yes—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—41no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23#40714yes—
481C23C22×C22⋊C411yescharacteristic
482C23C2×C4×D412yes—
483C23C22×M1611yescharacteristic
484C23C22×M1611yescharacteristic
485C23#56812yes—
486D4—416no—
487D4—88no—
488D4—88no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—416no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
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—84no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4#38614yes—
549C2×C4#38614yes—
550C2×C4C2×#12011yescharacteristic
551C2×C4C2×#12112yes—
552C2×C4C2×#12311yescharacteristic
553C8—48no—
554C8—84no—
555C22—22no—
556C22—22no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—84no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22C2×#38611yescharacteristic
591C22C2×#40711yescharacteristic
592C22#287911yescharacteristic
593C22#288311yescharacteristic
594C22#288311yescharacteristic
595C22#288811yescharacteristic
596C22#288911yescharacteristic
597C22D4×M1612yes—
598C22D4×M1612yes—
599C4—24no—
600C4—24no—
601C4—21no—
602C4—22no—
603C4—21no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—81no—
610C4—81no—
611C4—81no—
612C4—81no—
613C2—22no—
614C2—22no—
615C2—44no—
616C2—82no—
617C2—82no—
618C2#1736711yescharacteristic
619C2#1736811yescharacteristic
620C2#1740711yescharacteristic
621C2#1740811yescharacteristic
622C2#2089811yescharacteristic
623C2#2098511yescharacteristic
624C2#2099211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2973C2semidirect
3. #21790C2semidirect
4. C2×#1464C2semidirect
5. #9085C2semidirect
6. #9083C2semidirect
7. #9081C2semidirect
8. #9079C2semidirect
9. #7912C2semidirect
87. #432D4semidirect
144. C22×D4M16semidirect
235. C2×C22⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#29732
C2×#23242
#217902
C2×#14642
#90852
#90832
#90812
#90792
#79122