Finite Group Databasebeta

← #164117

Subgroups — #164117

Order 512.

677 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35321C211yesmaximal, characteristic
2#22634C211yesmaximal, characteristic
3#22632C211yesmaximal, characteristic
4#7733C211yesmaximal, characteristic, complemented
5#7715C211yesmaximal, characteristic
6#7709C211yesmaximal, characteristic, complemented
7#7701C211yesmaximal, characteristic
8#7693C211yesmaximal, characteristic, complemented
9#7675C211yesmaximal, characteristic
10#7669C211yesmaximal, characteristic, complemented
11#7661C211yesmaximal, characteristic
12#7602C211yesmaximal, characteristic, complemented
13#7590C211yesmaximal, characteristic, complemented
14#7590C211yesmaximal, characteristic, complemented
15#7568C211yesmaximal, characteristic, complemented
16#2863C2211yescharacteristic
17#2863C2211yescharacteristic
18#2855C2211yescharacteristic
19#2855C2211yescharacteristic
20#2647C411yescharacteristic
21#2642C411yescharacteristic
22#2629C411yescharacteristic
23#2621C411yescharacteristic
24#2585—21no—
25#2585—21no—
26#2573—21no—
27#2573—21no—
28#2524—21no—
29#2524—21no—
30#2515—21no—
31#2515—21no—
32#2323C411yescharacteristic
33#2323C411yescharacteristic
34#2320C411yescharacteristic
35#2320C411yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×#125C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1940—21no—
40#1939—21no—
41#1924—21no—
42#1923—21no—
43#1923—21no—
44#1923—21no—
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1831C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1771—21no—
52C8×C4⋊C4—21no—
53#1749C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55#1745C2211yescharacteristic
56#1744C2211yescharacteristic
57#1744C2211yescharacteristic
58#1744C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#125C2×C411yescharacteristic
77C2×C22⋊Q8C2×C411yescharacteristic
78C2×C22⋊Q8C2×C411yescharacteristic
79C2×C4×Q8C2×C411yescharacteristic
80C2×C4×Q8C2×C411yescharacteristic
81C2×#117C2×C411yescharacteristic
82C2×#117C2×C411yescharacteristic
83C22×C42C2311yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×#97—21no—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100#400—21no—
101#400—21no—
102#400—21no—
103#400—21no—
104#400—21no—
105#400—21no—
106#398—21no—
107#398—21no—
108#398—21no—
109#398—21no—
110#398—21no—
111#398—21no—
112#397—21no—
113#397—21no—
114#396—21no—
115#396—21no—
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#391C2×C411yescharacteristic
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#391C2311yescharacteristic
123#391C2311yescharacteristic
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2311yescharacteristic
135#389C2311yescharacteristic
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#385—21no—
159#385—21no—
160#384C2×C411yescharacteristic
161#383—21no—
162#383—21no—
163#383—21no—
164#383—21no—
165#382—21no—
166#382—21no—
167#382C2×C411yescharacteristic
168#381—21no—
169#381—21no—
170#380C2×C411yescharacteristic
171#380C2×C411yescharacteristic
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C22⋊C4D411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186#376C2×C411yescharacteristic
187#376C2×C411yescharacteristic
188C43C2311yescharacteristic
189C43C2311yescharacteristic
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336C2311yescharacteristic
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206C22×Q8C22×C411yescharacteristic
207C22×Q8C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C2×D411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C4⋊Q8—24no—
220#125—24no—
221C22⋊Q8—42no—
222C22⋊Q8—42no—
223C4×Q8—42no—
224C4×Q8—42no—
225#117—24no—
226#117—42no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C22⋊C4C22⋊C411yescharacteristic
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—42no—
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2411yescharacteristic
284C4⋊C8—42no—
285C4⋊C8—42no—
286C4⋊C8—81no—
287C4⋊C8—81no—
288C4⋊C8—81no—
289C4⋊C8—81no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C22⋊C8—81no—
293C22⋊C8—81no—
294#97—42no—
295#97—42no—
296C4×C8—42no—
297C4×C8—42no—
298C4×C8—42no—
299C4×C8—42no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95C22⋊C411yescharacteristic
331#95C22⋊C411yescharacteristic
332#95C22⋊C411yescharacteristic
333#95C22⋊C411yescharacteristic
334#95C22⋊C411yescharacteristic
335#95C22⋊C411yescharacteristic
336#95C22⋊C411yescharacteristic
337#95C22⋊C411yescharacteristic
338C24C2×C22⋊C411yescharacteristic
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
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—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—41no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—81no—
439C22⋊C4—81no—
440C42—22no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—42no—
460C42—42no—
461C42—81no—
462C42—81no—
463C42#13114yes—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23C22×C22⋊C411yescharacteristic
472Q8—82no—
473Q8—82no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4#40812yes—
539C2×C4#40812yes—
540C2×C4#41012yes—
541C2×C4#41312yes—
542C2×C4C2×#13112yes—
543C2×C4#57711yescharacteristic
544C2×C4#57711yescharacteristic
545C2×C4#57811yescharacteristic
546C2×C4#57811yescharacteristic
547C8—42no—
548C8—42no—
549C8—81no—
550C8—81no—
551C8—81no—
552C8—81no—
553C8—81no—
554C8—81no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22C2×#40811yescharacteristic
563C22C2×#41011yescharacteristic
564C22C2×#41311yescharacteristic
565C22#289811yescharacteristic
566C22#289811yescharacteristic
567C22#290111yescharacteristic
568C22#290111yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—22no—
577C4—22no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—81no—
593C2—81no—
594C2#1738311yescharacteristic
595C2#1746911yescharacteristic
596C2#1748411yescharacteristic
597C2#1760511yescharacteristic
598C2#1760511yescharacteristic
599C2#1769011yescharacteristic
600C2#1777811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7733C2semidirect
6. #7709C2semidirect
8. #7693C2semidirect
10. #7669C2semidirect
12. #7602C2semidirect
13. #7590C2semidirect
14. #7590C2semidirect
15. #7568C2semidirect

Maximal subgroups

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

subgroupindex
#353212
#226342
#226322
#77332
#77152
#77092
#77012
#76932
#76752
#76692
#76612
#76022
#75902
#75682