Finite Group Databasebeta

← #219917

Subgroups — #219917

Order 512.

1077 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36117C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic, complemented
3C2×#1521C212yesmaximal, complemented
4#13921C211yesmaximal, characteristic, complemented
5#9712C211yesmaximal, characteristic
6#9680C211yesmaximal, characteristic
7#9157C212yesmaximal, complemented
8#9114C212yesmaximal, complemented
9#8572C211yesmaximal, characteristic, complemented
10#8330C212yesmaximal, complemented
11C4×#328C211yesmaximal, characteristic, complemented
12C2×#529C2212yes—
13C2×#431C2212yes—
14#2856C2211yescharacteristic
15#2854C2212yes—
16#2552C411yescharacteristic, complemented
17#2548C411yescharacteristic, complemented
18#2512—21no—
19#2501—21no—
20#2479—22no—
21#2454C412yescomplemented
22#2448C412yescomplemented
23#2425—22no—
24#2366C411yescharacteristic, complemented
25#2361—21no—
26#2354—21no—
27Q8×C4⋊C4C411yescharacteristic, complemented
28C4×C22⋊Q8C2211yescharacteristic
29C2×#383C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#1939—21no—
32#1916—22no—
33#1817C2212yes—
34#1797C2211yescharacteristic
35C8×C4⋊C4—21no—
36C4×C22⋊C8C2211yescharacteristic
37#1521—28no—
38C2×#328C2211yescharacteristic
39C2×#328C2212yes—
40C2×#328C2211yescharacteristic
41C2×#327C2212yes—
42C2×#327C2212yes—
43C2×#327C2212yes—
44C2×#327C2212yes—
45#1443—22no—
46C4⋊#95—21no—
47C4⋊#95—21no—
48C4⋊#95C2211yescharacteristic
49#1434C2212yes—
50C4×#95C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2211yescharacteristic
53#1294C2212yes—
54C2×#127C2×C411yescharacteristic
55#529—24no—
56#529—24no—
57#529—42no—
58#529—42no—
59C2×#126—22no—
60C2×#125C2×C412yes—
61C2×#124C2×C412yes—
62C2×#123—22no—
63C2×#123—22no—
64C2×C22⋊Q8C2311yescharacteristic
65C2×#121—21no—
66C2×#121—21no—
67C2×#121C2312yes—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C412yes—
73C2×#117—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#431—24no—
81#431—42no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2311yescharacteristic
86C2×C22⋊C8C2311yescharacteristic
87C2×C4×C8—22no—
88C2×C4×C8—21no—
89#399—22no—
90#399—22no—
91#398—22no—
92#397—22no—
93#396—22no—
94#394—22no—
95#391C2×C411yescharacteristic
96#390—21no—
97#390—21no—
98#390—21no—
99#390—21no—
100#390—21no—
101#390C2×C412yes—
102#390C2×C412yes—
103#390C2×C412yes—
104#389—21no—
105#389—22no—
106#389C2×C412yes—
107#389C2×C412yes—
108#389C2×C411yescharacteristic
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#387—21no—
114#386—22no—
115#386—22no—
116#385—21no—
117#385—22no—
118#384—21no—
119#384—21no—
120#384C2×C411yescharacteristic
121#383—24no—
122#383—42no—
123#380—22no—
124#380C2311yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C412yes—
130C4×C4⋊C4C2×C411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D412yes—
133C4×C4⋊C4D411yescharacteristic
134C4×C22⋊C4—21no—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—21no—
141C2×#95D412yes—
142C2×#95C2311yescharacteristic
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145#336—21no—
146#336—22no—
147#336—22no—
148#336—21no—
149#328—24no—
150#328—22no—
151#328D414yescomplemented
152#327—24no—
153#327—24no—
154#327—42no—
155#327—42no—
156C22×Q8C22×C411yescharacteristic
157C22×D4—22no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C22×C412yes—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173#127—42no—
174#126—44no—
175#126—44no—
176#125—24no—
177#125—42no—
178#124—24no—
179#124—42no—
180#123—44no—
181#123—44no—
182#123—82no—
183#123—82no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8C2×D412yes—
187C22⋊Q8#4112yes—
188#121—22no—
189#121—22no—
190#121—22no—
191#121—22no—
192#121—42no—
193#121—42no—
194#121—81no—
195#121—81no—
196C4×Q8—24no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×D4—24no—
200C4×D4—42no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—44no—
204#117—44no—
205#117—44no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42C22×C412yes—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C412yes—
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C4⋊C8—44no—
264C4⋊C8—42no—
265C22⋊C8—21no—
266C22⋊C8—22no—
267C22⋊C8—21no—
268C22⋊C8—21no—
269C22⋊C8—22no—
270C22⋊C8C2×D412yes—
271C4×C8—44no—
272C4×C8—42no—
273#95—21no—
274#95—21no—
275#95—22no—
276#95—21no—
277#95—21no—
278#95—22no—
279#95—21no—
280#95—21no—
281#95—22no—
282#95—21no—
283#95—21no—
284#95—22no—
285#95—44no—
286#95—44no—
287#95—42no—
288#95—44no—
289#95—82no—
290#95—81no—
291#95C22⋊C412yes—
292#95C22⋊C412yes—
293#95C22⋊C412yes—
294#95C22⋊C412yes—
295#95C2×D412yes—
296C24—22no—
297C24C22×D411yescharacteristic
298C2×Q8—21no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8C4×D412yes—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—81no—
317C2×D4—81no—
318C22×C4—22no—
319C22×C4—22no—
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—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—82no—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4#13512yes—
374C22×C4#13512yes—
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C4H(5,𝔽2)11yescharacteristic
381C2×C8—22no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—41no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—81no—
409C4⋊C4—81no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—84no—
413C4⋊C4C4×D412yes—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—44no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C42—41no—
440C42—41no—
441C42—44no—
442C42—42no—
443C42—42no—
444C42—84no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23C22×C22⋊C411yescharacteristic
457C23D4211yescharacteristic
458C23#54611yescharacteristic
459C23C2×#13511yescharacteristic
460C23C2×#13511yescharacteristic
461C23#57611yescharacteristic
462C23#57711yescharacteristic
463Q8—42no—
464Q8—42no—
465D4—82no—
466D4—82no—
467D4—82no—
468D4—82no—
469D4—82no—
470D4—82no—
471D4—84no—
472D4—162no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
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—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—88no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4#41012yes—
549C2×C4#41612yes—
550C2×C4#41612yes—
551C2×C4#42112yes—
552C2×C4C2×C4×D411yescharacteristic
553C2×C4#51811yescharacteristic
554C2×C4D4211yescharacteristic
555C2×C4#54611yescharacteristic
556C8—42no—
557C8—82no—
558C8—81no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22D4×C22⋊C411yescharacteristic
579C22C2×#41011yescharacteristic
580C22#288411yescharacteristic
581C22C2×#41611yescharacteristic
582C22#289711yescharacteristic
583C22C2×#42111yescharacteristic
584C22#290111yescharacteristic
585C22#328411yescharacteristic
586C22#328711yescharacteristic
587C22#330411yescharacteristic
588C22#330511yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—22no—
594C4—42no—
595C4—42no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—81no—
606C4—81no—
607C4—162no—
608C4#152114yescomplemented
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—162no—
614C2C2×#152111yescharacteristic
615C2#1760211yescharacteristic
616C2#1760711yescharacteristic
617C2#1761611yescharacteristic
618C2#2240611yescharacteristic
619C2#2241811yescharacteristic
620C2#2243511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22599C2semidirect
3. C2×#1521C2semidirect
4. #13921C2semidirect
7. #9157C2semidirect
8. #9114C2semidirect
9. #8572C2semidirect
10. #8330C2semidirect
11. C4×#328C2semidirect
16. #2552C4semidirect
17. #2548C4semidirect
21. #2454C4semidirect
22. #2448C4semidirect
24. #2366C4semidirect
27. Q8×C4⋊C4C4semidirect
151. #328D4semidirect
608. C4#1521semidirect

Maximal subgroups

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

subgroupindex
#361172
#225992
C2×#15212
#139212
#97122
#96802
#91572
#91142
#85722
#83302
C4×#3282