Finite Group Databasebeta

← #222524

Subgroups — #222524

Order 512.

909 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3381C211yesmaximal, characteristic
2#22770C211yesmaximal, characteristic
3#22547C211yesmaximal, characteristic, complemented
4#22487C211yesmaximal, characteristic
5C2×#1582C211yesmaximal, characteristic
6#9828C211yesmaximal, characteristic, complemented
7#9729C211yesmaximal, characteristic, complemented
8#9224C211yesmaximal, characteristic, complemented
9#9219C211yesmaximal, characteristic
10#8981C211yesmaximal, characteristic
11#8974C211yesmaximal, characteristic, complemented
12#8647C211yesmaximal, characteristic, complemented
13#8450C211yesmaximal, characteristic
14#8448C211yesmaximal, characteristic, complemented
15#8201C211yesmaximal, characteristic, complemented
16C2×#557C2211yescharacteristic
17C2×#554C2211yescharacteristic
18#3381—24no—
19#3381C418yescomplemented
20C2×#499—21no—
21C2×#498C2211yescharacteristic
22C2×#479—21no—
23C2×#477C2211yescharacteristic
24C2×#478—21no—
25C2×#474C2211yescharacteristic
26C2×#446C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28C2×C4×SD16—21no—
29C2×C4×SD16C2211yescharacteristic
30C2×C8×D4C2211yescharacteristic
31#2355C2211yescharacteristic
32D4×C4⋊C4C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1987C2211yescharacteristic
36#1981—21no—
37#1981C2211yescharacteristic
38#1960—21no—
39#1956C2211yescharacteristic
40#1956C2211yescharacteristic
41#1933—21no—
42#1919—21no—
43#1892—21no—
44#1877—21no—
45#1872—21no—
46#1834C2211yescharacteristic
47#1824—21no—
48#1805—21no—
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1802C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54#1763—21no—
55C4×#103C2211yescharacteristic
56C4×#102—21no—
57C2×#340—21no—
58#1582—24no—
59C2×#334C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×Q8⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×#322C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1279C2211yescharacteristic
69#1277C2211yescharacteristic
70C22×SD16D411yescharacteristic
71C23×C8—21no—
72#557—22no—
73#557—22no—
74#557—41no—
75#557C2×C414yescomplemented
76#554—22no—
77#554—22no—
78#554—41no—
79#554C2×C414yes—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8C2311yescharacteristic
82C2×C4⋊Q8C2311yescharacteristic
83C2×#125—21no—
84C2×#125—21no—
85C2×#125—41no—
86C2×C22⋊Q8—21no—
87C2×C22⋊Q8—41no—
88C2×C4×Q8—21no—
89C2×C4×Q8—21no—
90C2×C4×Q8C2311yescharacteristic
91C2×C4×Q8C2311yescharacteristic
92C2×C4×D4—21no—
93C2×C4×D4—21no—
94C2×C4×D4C2311yescharacteristic
95C22×C4⋊C4—21no—
96C22×C4⋊C4—21no—
97#499—28no—
98#498—22no—
99#498C2×C414yescomplemented
100#479—28no—
101#478—28no—
102#477—22no—
103#477C2×C414yescomplemented
104#474—22no—
105#474C2×C414yes—
106#446—22no—
107#446—41no—
108C8×Q8—22no—
109C8×Q8C2×C414yescomplemented
110C4×SD16—28no—
111C4×SD16C2×C414yes—
112C4×SD16D414yes—
113C8×D4—22no—
114C8×D4—41no—
115C2×#107—21no—
116C2×#107—21no—
117C2×#106—21no—
118C2×#106—21no—
119C2×#106D411yescharacteristic
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—21no—
129C2×#103—21no—
130C2×#103—21no—
131C2×#103—21no—
132C2×#103—21no—
133C2×#103—21no—
134C2×#103—41no—
135C2×#103D411yescharacteristic
136C2×#103D411yescharacteristic
137C2×#103D411yescharacteristic
138C2×#102—21no—
139C2×#102—21no—
140C2×#102D411yescharacteristic
141C2×C22⋊C8—21no—
142C2×#97—21no—
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148#391—21no—
149#389—21no—
150#387—41no—
151#385—41no—
152#384C2311yescharacteristic
153#384C2311yescharacteristic
154#382—21no—
155#380—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4C2311yescharacteristic
159#340—21no—
160#340—21no—
161#340—21no—
162#340—21no—
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#340—21no—
169#340—41no—
170#340D411yescharacteristic
171#340D411yescharacteristic
172#336—21no—
173#334—22no—
174#333—41no—
175#332—41no—
176Q8⋊C8—22no—
177Q8⋊C8—22no—
178D4⋊C8—22no—
179#322—22no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8C2×D411yescharacteristic
183C22×Q8C2×D411yescharacteristic
184C22×D4C2×D411yescharacteristic
185C23×C4—21no—
186C23×C4—41no—
187C23×C4—41no—
188C2×SD16—22no—
189C2×SD16—22no—
190C2×SD16C22⋊C414yes—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—42no—
197C22×C8—42no—
198C22×C8—81no—
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8C2×D411yescharacteristic
202C22×C8C2×D411yescharacteristic
203C4⋊Q8—24no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—42no—
211C4⋊Q8C22×C412yes—
212C4⋊Q8C22×C412yes—
213#125—24no—
214#125—24no—
215#125—42no—
216#125—42no—
217#125—44no—
218#125—44no—
219C22⋊Q8—24no—
220C22⋊Q8—44no—
221C22⋊Q8—44no—
222C22⋊Q8—42no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—42no—
230C4×Q8—42no—
231C4×Q8C22×C412yes—
232C4×Q8C22×C412yes—
233C4×Q8C2×D412yes—
234C4×Q8#4112yes—
235C4×D4—24no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—42no—
239C4×D4—42no—
240C4×D4—81no—
241C4×D4C22×C412yes—
242C4×D4C2×D412yes—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—81no—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2×D411yescharacteristic
269C2×C4⋊C4C2×D411yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—41no—
277C2×C42—41no—
278C2×C42C2411yescharacteristic
279#107—24no—
280#107—24no—
281#106—24no—
282#106—24no—
283#106C22⋊C414yes—
284C4⋊C8—24no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—42no—
293C4⋊C8C22×C412yes—
294#103—24no—
295#103—22no—
296#103—22no—
297#103—42no—
298#103—42no—
299#103—81no—
300#103—81no—
301#103—81no—
302#103—81no—
303#103—81no—
304#103—82no—
305#103C22⋊C414yes—
306#102—24no—
307#102—42no—
308#102C22⋊C414yes—
309C22⋊C8—42no—
310#97—42no—
311C4×C8—24no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—81no—
317C4×C8C22×C412yes—
318C4×C8C2×D412yes—
319#95—21no—
320#95—41no—
321#95—41no—
322#95—41no—
323#95—41no—
324C24—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×Q8C2×C22⋊C412yes—
346C2×Q8C4×D412yes—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4C2×C22⋊C412yes—
351C2×D4#13712yes—
352C2×D4#13712yes—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—44no—
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—82no—
372C22×C4—82no—
373C22×C4—161no—
374C22×C4—161no—
375C22×C4—161no—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379SD16—28no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—22no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—84no—
405C2×C8C2×C22⋊C412yes—
406C2×C8C2×C22⋊C412yes—
407C2×C8#13512yes—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—82no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4C2×C22⋊C412yes—
443C4⋊C4C2×C22⋊C412yes—
444C4⋊C4C4×D412yes—
445C4⋊C4#13512yes—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—44no—
449C22⋊C4—82no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—44no—
455C42—41no—
456C42—41no—
457C42—161no—
458C42C23×C411yescharacteristic
459C42C22×D411yescharacteristic
460C42C2×#4111yescharacteristic
461C42H(5,𝔽2)11yescharacteristic
462C23—22no—
463C23—22no—
464C23—22no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23D4211yescharacteristic
470Q8—22no—
471Q8—22no—
472Q8—22no—
473Q8—22no—
474Q8—41no—
475Q8—41no—
476Q8—41no—
477Q8—41no—
478Q8—42no—
479Q8—42no—
480Q8—82no—
481Q8—81no—
482Q8—81no—
483Q8—161no—
484D4—22no—
485D4—22no—
486D4—22no—
487D4#41914yes—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—164no—
528C2×C4—164no—
529C2×C4C22×C22⋊C411yescharacteristic
530C2×C4C2×C4×D411yescharacteristic
531C2×C4#51811yescharacteristic
532C2×C4D4211yescharacteristic
533C2×C4#54611yescharacteristic
534C2×C4#54611yescharacteristic
535C2×C4C2×#13511yescharacteristic
536C2×C4C2×#13511yescharacteristic
537C2×C4C2×#13711yescharacteristic
538C2×C4C2×#13711yescharacteristic
539C2×C4#57711yescharacteristic
540C2×C4#57711yescharacteristic
541C2×C4#57711yescharacteristic
542C2×C4#57811yescharacteristic
543C8—22no—
544C8—22no—
545C8—22no—
546C8—42no—
547C8—41no—
548C8—81no—
549C8—81no—
550C8—81no—
551C8—81no—
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22D4×C22⋊C411yescharacteristic
560C22#326611yescharacteristic
561C22#327011yescharacteristic
562C22#328211yescharacteristic
563C22#328411yescharacteristic
564C22#328811yescharacteristic
565C22#328911yescharacteristic
566C4—21no—
567C4—21no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—42no—
572C4—42no—
573C4—42no—
574C4—81no—
575C4—82no—
576C4—82no—
577C4—161no—
578C4—161no—
579C4C2×#41911yescharacteristic
580C4#289811yescharacteristic
581C4C4×#13511yescharacteristic
582C4#295211yescharacteristic
583C4#328211yescharacteristic
584C4#328411yescharacteristic
585C4#331211yescharacteristic
586C4#332011yescharacteristic
587C2—44no—
588C2#1979011yescharacteristic
589C2#1979711yescharacteristic
590C2#1985311yescharacteristic
591C2#1985711yescharacteristic
592C2#2242511yescharacteristic
593C2#2244911yescharacteristic
594C2#2246411yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22547C2semidirect
6. #9828C2semidirect
7. #9729C2semidirect
8. #9224C2semidirect
11. #8974C2semidirect
12. #8647C2semidirect
14. #8448C2semidirect
15. #8201C2semidirect
19. #3381C4semidirect
75. #557C2×C4semidirect
99. #498C2×C4semidirect
103. #477C2×C4semidirect
109. C8×Q8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#33812
#227702
#225472
#224872
C2×#15822
#98282
#97292
#92242
#92192
#89812
#89742
#86472
#84502
#84482
#82012