Finite Group Databasebeta

← #182815

Subgroups — #182815

Order 512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34225C211yesmaximal, characteristic
2#20550C211yesmaximal, characteristic
3#20546C211yesmaximal, characteristic
4C2×#1554C211yesmaximal, characteristic
5C2×#1550C211yesmaximal, characteristic
6#8237C211yesmaximal, characteristic, complemented
7#8236C211yesmaximal, characteristic, complemented
8#8235C211yesmaximal, characteristic, complemented
9#8235C211yesmaximal, characteristic, complemented
10#8233C211yesmaximal, characteristic, complemented
11#8232C211yesmaximal, characteristic, complemented
12C4×#332C211yesmaximal, characteristic, complemented
13C4×#333C211yesmaximal, characteristic, complemented
14#8042C211yesmaximal, characteristic
15#8039C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21#2370—21no—
22#2370—21no—
23#2362—21no—
24#2362—21no—
25#2356—21no—
26#2351—21no—
27#2347—21no—
28#2346—21no—
29#2323C412yes—
30#2323C412yes—
31#2320C412yes—
32C4×C4⋊Q8C2211yescharacteristic
33C4×C4⋊Q8C2211yescharacteristic
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36#2285C2211yescharacteristic
37#2276C412yes—
38C4×#117C2211yescharacteristic
39#1771—21no—
40#1771—21no—
41#1770—21no—
42C8×C4⋊C4—21no—
43#1767C2211yescharacteristic
44#1767C2211yescharacteristic
45#1767C2211yescharacteristic
46#1767C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51#1753—21no—
52#1753—21no—
53#1752—21no—
54C8×C22⋊C4—21no—
55#1554C418yescomplemented
56#1550C418yescomplemented
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71#1296C2211yescharacteristic
72#1296C2211yescharacteristic
73C22×M16—21no—
74C23×C8—21no—
75#533—22no—
76#533—22no—
77#533C2×C414yes—
78#533C2×C414yes—
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×#125—21no—
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8C2×C412yes—
84C2×C4×Q8—21no—
85C2×C4×Q8—21no—
86C2×C4×Q8—21no—
87C2×C4×Q8C2×C412yes—
88C2×C4×Q8C2×C412yes—
89C2×C4×Q8C2×C412yes—
90C2×#117—21no—
91C2×#117—21no—
92C2×#117C2×C411yescharacteristic
93C2×#117C2×C411yescharacteristic
94C22×C42C2311yescharacteristic
95#432—22no—
96#432C2×C414yes—
97#431—22no—
98#431C2×C414yes—
99#423—22no—
100#423C2×C414yes—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8C2×C414yes—
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×#97—21no—
116C2×#97—21no—
117C2×#97—21no—
118C2×#97—21no—
119C2×#97—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125#391—21no—
126#391—21no—
127#391C2×C412yes—
128#391C2×C412yes—
129#389—22no—
130#389—22no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389C2×C412yes—
138#389C2×C412yes—
139#388—22no—
140#388—22no—
141#388—22no—
142#388—22no—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#383—22no—
150#382—21no—
151#382—22no—
152#380—21no—
153#380C2×C412yes—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4—22no—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4C2×C412yes—
165C4×C4⋊C4C2×C412yes—
166C4×C4⋊C4C2×C412yes—
167C4×C4⋊C4D411yescharacteristic
168C4×C4⋊C4D411yescharacteristic
169C4×C4⋊C4D411yescharacteristic
170C4×C4⋊C4D411yescharacteristic
171C4×C22⋊C4D411yescharacteristic
172C4×C22⋊C4D411yescharacteristic
173C4×C22⋊C4D411yescharacteristic
174C4×C22⋊C4D411yescharacteristic
175#376—22no—
176#376C2×C411yescharacteristic
177#376C2×C411yescharacteristic
178C43C2311yescharacteristic
179C43C2311yescharacteristic
180#336—21no—
181#336—21no—
182#336—21no—
183#336—21no—
184#336—21no—
185#336—21no—
186#336—21no—
187#336—21no—
188#333C2×C414yes—
189#333C2×C414yes—
190#333C2×C414yes—
191#333C2×C414yes—
192#332C2×C414yes—
193#332C2×C414yes—
194#332C2×C414yes—
195#332C2×C414yes—
196C22×Q8—21no—
197C22×Q8C22×C412yes—
198C23×C4C22×C411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C2×M16—24no—
202C2×M16—81no—
203C2×M16—81no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—24no—
213C22×C8—81no—
214C22×C8—81no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8C22×C412yes—
220C4⋊Q8C22×C412yes—
221C4⋊Q8C22×C412yes—
222C4⋊Q8C22×C412yes—
223#125—24no—
224#125—24no—
225C22⋊Q8—24no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8C4218yes—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—44no—
233C4×Q8—44no—
234C4×Q8—81no—
235C4×Q8—81no—
236C4×Q8—81no—
237C4×Q8—81no—
238C4×Q8C4218yes—
239#117—24no—
240#117—24no—
241#117—42no—
242#117—42no—
243#117—81no—
244#117—81no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C412yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C412yes—
283C2×C42C22×C412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42#4112yes—
295C2×C42#4112yes—
296C2×C42C2411yescharacteristic
297C4⋊C8—24no—
298C4⋊C8—24no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8C22×C412yes—
308C4⋊C8C22×C412yes—
309C4⋊C8C22×C412yes—
310C4⋊C8C22×C412yes—
311C4⋊C8C22×C412yes—
312C4⋊C8C22×C412yes—
313C4⋊C8C22×C412yes—
314C4⋊C8C22×C412yes—
315C22⋊C8—24no—
316C22⋊C8—24no—
317#97—24no—
318#97—81no—
319#97—81no—
320#97—81no—
321#97—81no—
322C4×C8—24no—
323C4×C8—81no—
324C4×C8—81no—
325C4×C8—81no—
326C4×C8—81no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95C22⋊C411yescharacteristic
338#95C22⋊C411yescharacteristic
339#95C22⋊C411yescharacteristic
340#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345C24C2×C22⋊C411yescharacteristic
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8C2×C4214yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4C2×C4212yes—
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4#11712yes—
383C22×C4C4×D412yes—
384C22×C4C4×D412yes—
385C22×C4C4×D412yes—
386C22×C4C4×D412yes—
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390M16—84no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—84no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—44no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4C2×C4214yes—
444C22⋊C4—28no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—82no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—28no—
467C42—22no—
468C42—22no—
469C42—41no—
470C42—41no—
471C42—44no—
472C42—44no—
473C42—82no—
474C42—82no—
475C42—82no—
476C42C2×C4212yes—
477C42#11712yes—
478C42C4×D412yes—
479C42C4×D412yes—
480C42C4×D412yes—
481C42C4×D412yes—
482C42C23×C411yescharacteristic
483C42C23×C411yescharacteristic
484C42C2×#4111yescharacteristic
485C42C2×#4111yescharacteristic
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23C4×C22⋊C412yes—
493C23C22×C22⋊C411yescharacteristic
494Q8—44no—
495Q8—44no—
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—42no—
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—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4C4×C22⋊C412yes—
553C2×C4C4×C22⋊C412yes—
554C2×C4C4×C22⋊C412yes—
555C2×C4#41312yes—
556C2×C4#41312yes—
557C2×C4C22×C4211yescharacteristic
558C2×C4C2×#11711yescharacteristic
559C2×C4C2×C4×D411yescharacteristic
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×C4×D411yescharacteristic
563C2×C4#57611yescharacteristic
564C2×C4#57711yescharacteristic
565C2×C4#57711yescharacteristic
566C2×C4#57711yescharacteristic
567C2×C4#57711yescharacteristic
568C2×C4#57811yescharacteristic
569C2×C4#57811yescharacteristic
570C2×C4#57811yescharacteristic
571C8—42no—
572C8—42no—
573C8—42no—
574C8—42no—
575C8—42no—
576C8—42no—
577C8—42no—
578C8—42no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22C2×C4×C22⋊C411yescharacteristic
587C22C2×#41311yescharacteristic
588C22C2×#41311yescharacteristic
589C22#289711yescharacteristic
590C22#289811yescharacteristic
591C22#289811yescharacteristic
592C22#289811yescharacteristic
593C4—22no—
594C4—22no—
595C4—21no—
596C4—22no—
597C4—21no—
598C4—22no—
599C4—22no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—44no—
607C4—44no—
608C4—82no—
609C4—82no—
610C4#294611yescharacteristic
611C4#294711yescharacteristic
612C4#294711yescharacteristic
613C4#294711yescharacteristic
614C4#294911yescharacteristic
615C4#295011yescharacteristic
616C4#295111yescharacteristic
617C4#295211yescharacteristic
618C2—42no—
619C2#1777211yescharacteristic
620C2#1777511yescharacteristic
621C2#1777811yescharacteristic
622C2#1778011yescharacteristic
623C2C4×#41311yescharacteristic
624C2#2041511yescharacteristic
625C2#2041711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8237C2semidirect
7. #8236C2semidirect
8. #8235C2semidirect
9. #8235C2semidirect
10. #8233C2semidirect
11. #8232C2semidirect
12. C4×#332C2semidirect
13. C4×#333C2semidirect
55. #1554C4semidirect
56. #1550C4semidirect

Maximal subgroups

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

subgroupindex
#342252
#205502
#205462
C2×#15542
C2×#15502
#82372
#82362
#82352
#82332
#82322
C4×#3322
C4×#3332
#80422
#80392