Finite Group Databasebeta

← #220255

Subgroups — #220255

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3299C211yesmaximal, characteristic
2#22567C212yesmaximal, complemented
3#22513C211yesmaximal, characteristic
4Q8×#99C211yesmaximal, characteristic
5#9784C212yesmaximal, complemented
6#9258C212yesmaximal, complemented
7#9257C212yesmaximal, complemented
8#8608C211yesmaximal, characteristic
9#8380C211yesmaximal, characteristic
10#8379C211yesmaximal, characteristic
11#8160C211yesmaximal, characteristic
12C2×#554C2211yescharacteristic
13C2×D4×Q8C2211yescharacteristic
14#3299—22no—
15#3299—22no—
16#3299—22no—
17#3299—22no—
18C2×#487C411yescharacteristic, complemented
19C2×#484C411yescharacteristic, complemented
20C2×#484C2211yescharacteristic
21C2×#477C411yescharacteristic, complemented
22C2×#467C411yescharacteristic, complemented
23C2×#467C2211yescharacteristic
24C2×#462C411yescharacteristic, complemented
25C2×#451C411yescharacteristic, complemented
26C2×#451C2211yescharacteristic
27C2×C4×Q16C411yescharacteristic, complemented
28C2×C8×D4C2211yescharacteristic
29C22×#103C411yescharacteristic, complemented
30C22×#103C2211yescharacteristic
31#2345C2212yes—
32Q8×C22⋊C4C2212yes—
33#1996—22no—
34#1996—22no—
35#1996C2212yes—
36#1996C2212yes—
37#1935—22no—
38#1927—22no—
39#1865—22no—
40#1824C2212yes—
41#1807C2212yes—
42#1799—21no—
43#1799—21no—
44#1799C2211yescharacteristic
45#1799C2211yescharacteristic
46#1787C2212yes—
47#1763—22no—
48C4×#99—21no—
49C4×#99—21no—
50C4×#99C2211yescharacteristic
51#1752C2212yes—
52#1520C2212yes—
53#1520C2212yes—
54#1459C2211yescharacteristic
55#1289C2212yes—
56#1282C2212yes—
57C23×Q8C2×C411yescharacteristic
58C23×Q8C2311yescharacteristic
59C22×Q16C2×C411yescharacteristic
60C22×Q16C2×C411yescharacteristic
61C22×Q16C2×C411yescharacteristic
62C23×C8C2×C411yescharacteristic
63C23×C8C2311yescharacteristic
64#554—22no—
65#554—22no—
66#554—42no—
67D4×Q8—22no—
68D4×Q8—22no—
69D4×Q8—42no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8C2×C411yescharacteristic
72C2×C4⋊Q8C2×C411yescharacteristic
73C2×#125—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8C2×C411yescharacteristic
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×C22⋊Q8C2311yescharacteristic
81C2×C22⋊Q8C2311yescharacteristic
82C2×C4×Q8—22no—
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×Q8C2×C411yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4C2311yescharacteristic
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90#487—22no—
91#487—22no—
92#484—22no—
93#484—22no—
94#484—22no—
95#484—22no—
96#477—22no—
97#477—22no—
98#467—22no—
99#467—22no—
100#467—22no—
101#467—22no—
102#462—22no—
103#462—22no—
104#451—22no—
105#451—22no—
106#451—22no—
107#451—22no—
108C4×Q16—21no—
109C4×Q16—21no—
110C4×Q16—21no—
111C4×Q16—21no—
112C8×D4—22no—
113C8×D4—22no—
114C2×#107C2×C411yescharacteristic
115C2×#107C2×C411yescharacteristic
116C2×#107C2×C411yescharacteristic
117C2×#106—22no—
118C2×C4⋊C8C2×C411yescharacteristic
119C2×#103—21no—
120C2×#103—21no—
121C2×#103—21no—
122C2×#103—21no—
123C2×#103—22no—
124C2×#103—22no—
125C2×#103C2×C411yescharacteristic
126C2×#103C2×C411yescharacteristic
127C2×#103C2×C411yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#103C2×C411yescharacteristic
134C2×#103C2×C411yescharacteristic
135C2×#103D412yes—
136C2×#103D412yes—
137C2×#103D412yes—
138C2×#103D412yes—
139C2×#99—21no—
140C2×#99—21no—
141C2×#99D411yescharacteristic
142C2×#99D411yescharacteristic
143C2×C22⋊C8C2×C411yescharacteristic
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×#97—22no—
147C2×C4×C8C2×C411yescharacteristic
148#391—22no—
149#391—42no—
150#389—22no—
151#387—42no—
152#380—22no—
153#380—22no—
154#380C2312yes—
155#380C2312yes—
156C4×C4⋊C4—22no—
157C4×C22⋊C4—22no—
158C4×C22⋊C4—22no—
159C4×C22⋊C4C2312yes—
160#342—22no—
161#342—22no—
162#342—21no—
163#342—21no—
164#342—22no—
165#342—22no—
166#342D412yes—
167#340—22no—
168#340—22no—
169#340—22no—
170#336—22no—
171#328—42no—
172#328—42no—
173#324—21no—
174#323—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—22no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—41no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—81no—
185C22×Q8—81no—
186C22×Q8C22×C411yescharacteristic
187C22×Q8C22×C411yescharacteristic
188C22×Q8C22×C411yescharacteristic
189C22×Q8C22×C411yescharacteristic
190C22×Q8C2×D412yes—
191C22×Q8C2×D412yes—
192C22×D4C2×D411yescharacteristic
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4C22×C411yescharacteristic
198C23×C4C2411yescharacteristic
199C2×Q16—21no—
200C2×Q16—21no—
201C2×Q16—22no—
202C2×Q16—22no—
203C2×Q16—42no—
204C2×Q16—81no—
205C2×Q16—81no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—22no—
209C22×C8—81no—
210C22×C8—81no—
211C22×C8C22×C411yescharacteristic
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C2×D412yes—
216C22×C8C2×D412yes—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—41no—
220C4⋊Q8—41no—
221C4⋊Q8—42no—
222C4⋊Q8—42no—
223#125—42no—
224#125—42no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—42no—
232C22⋊Q8—42no—
233C22⋊Q8—42no—
234C22⋊Q8—42no—
235C22⋊Q8—42no—
236C22⋊Q8—42no—
237C22⋊Q8—42no—
238C22⋊Q8—42no—
239C22⋊Q8—41no—
240C22⋊Q8—41no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—41no—
246C4×Q8—41no—
247C4×Q8—44no—
248C4×Q8—82no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—42no—
252C4×D4—42no—
253C4×D4—42no—
254C4×D4—42no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C4⋊C4C22×C411yescharacteristic
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4#4111yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D412yes—
289C2×C22⋊C4C2×D411yescharacteristic
290C2×C22⋊C4#4111yescharacteristic
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42C22×C411yescharacteristic
297#107—22no—
298#107—21no—
299#107—21no—
300#107—22no—
301#106—82no—
302C4⋊C8—22no—
303#103—21no—
304#103—21no—
305#103—22no—
306#103—21no—
307#103—21no—
308#103—21no—
309#103—21no—
310#103—21no—
311#103—21no—
312#103—22no—
313#103—44no—
314#103—82no—
315#103C22⋊C411yescharacteristic
316#103C22⋊C411yescharacteristic
317#103C22⋊C411yescharacteristic
318#103C22⋊C411yescharacteristic
319#103C22⋊C412yes—
320#103C22⋊C412yes—
321#103C22⋊C412yes—
322#103C22⋊C412yes—
323#103C22⋊C412yes—
324#103C22⋊C412yes—
325#99—22no—
326#99—22no—
327#99—42no—
328#99—42no—
329C22⋊C8—22no—
330C22⋊C8—22no—
331C22⋊C8—22no—
332C22⋊C8—22no—
333#97—82no—
334C4×C8—21no—
335C4×C8—21no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—42no—
340#95—42no—
341#95—42no—
342#95—42no—
343C24C4×D411yescharacteristic
344C24C22×D411yescharacteristic
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—41no—
360C2×Q8—42no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—42no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8—82no—
371C2×Q8—84no—
372C2×Q8—82no—
373C2×Q8—81no—
374C2×Q8—81no—
375C2×Q8—84no—
376C2×Q8—161no—
377C2×Q8C2×C22⋊C411yescharacteristic
378C2×Q8C2×C22⋊C412yes—
379C2×Q8C2×C22⋊C411yescharacteristic
380C2×D4—22no—
381C2×D4—22no—
382C2×D4#13712yes—
383C2×D4#13712yes—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4—161no—
413C22×C4—161no—
414C22×C4C4×D411yescharacteristic
415C22×C4C4×D411yescharacteristic
416C22×C4C4×D411yescharacteristic
417C22×C4C23×C411yescharacteristic
418C22×C4C22×D412yes—
419C22×C4C2×#4111yescharacteristic
420C22×C4H(5,𝔽2)12yes—
421Q16—42no—
422Q16—42no—
423Q16—82no—
424Q16—82no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—81no—
437C2×C8—81no—
438C2×C8C2×C22⋊C411yescharacteristic
439C2×C8C2×C22⋊C411yescharacteristic
440C2×C8C2×C22⋊C412yes—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—84no—
468C4⋊C4—162no—
469C4⋊C4C2×C22⋊C411yescharacteristic
470C4⋊C4C2×C22⋊C411yescharacteristic
471C4⋊C4C2×C22⋊C412yes—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C42—21no—
487C42—21no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—42no—
493C42—42no—
494C42—84no—
495C42—162no—
496C23—21no—
497C23—21no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23#44112yes—
503C23C2×C4×D411yescharacteristic
504C23D4212yes—
505C23C2×#13711yescharacteristic
506C23C2×#13711yescharacteristic
507C23#57812yes—
508Q8—21no—
509Q8—21no—
510Q8—22no—
511Q8—22no—
512Q8—21no—
513Q8—44no—
514Q8—42no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—41no—
519Q8—82no—
520Q8—81no—
521Q8—82no—
522Q8—81no—
523Q8—81no—
524Q8—82no—
525Q8—162no—
526Q8#41012yes—
527D4—24no—
528D4—24no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—81no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—161no—
581C2×C4—161no—
582C2×C4#44112yes—
583C2×C4C22×C22⋊C411yescharacteristic
584C2×C4#51811yescharacteristic
585C2×C4#54612yes—
586C8—21no—
587C8—21no—
588C8—22no—
589C8—41no—
590C8—41no—
591C8—82no—
592C22—22no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—41no—
603C22—41no—
604C22D4×C22⋊C411yescharacteristic
605C22C2×#44111yescharacteristic
606C22#295011yescharacteristic
607C22#327012yes—
608C22#327012yes—
609C4—22no—
610C4—21no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—42no—
620C4—82no—
621C4—82no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—162no—
627C4—162no—
628C4C2×#41011yescharacteristic
629C4#288311yescharacteristic
630C2—22no—
631C2—22no—
632C2—21no—
633C2—21no—
634C2—42no—
635C2#2240511yescharacteristic
636C2#2244011yescharacteristic
637C2#2245611yescharacteristic
subgroupG/H
center, socleC22D4×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22567C2semidirect
5. #9784C2semidirect
6. #9258C2semidirect
7. #9257C2semidirect
18. C2×#487C4semidirect
19. C2×#484C4semidirect
21. C2×#477C4semidirect
22. C2×#467C4semidirect
24. C2×#462C4semidirect
25. C2×#451C4semidirect
27. C2×C4×Q16C4semidirect
29. C22×#103C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32992
#225672
#225132
Q8×#992
#97842
#92582
#92572
#86082
#83802
#83792
#81602