Finite Group Databasebeta

← #160666

Subgroups — #160666

Order 512.

964 conjugacy classes of subgroups.

The subgroup lattice has 727 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34327C211yesmaximal, characteristic
2#22785C211yesmaximal, characteristic, complemented
3#22008C211yesmaximal, characteristic, complemented
4#7600C211yesmaximal, characteristic, complemented
5#7599C211yesmaximal, characteristic, complemented
6#7597C211yesmaximal, characteristic, complemented
7#7595C211yesmaximal, characteristic, complemented
8#7589C211yesmaximal, characteristic, complemented
9#7584C211yesmaximal, characteristic, complemented
10#7582C211yesmaximal, characteristic, complemented
11#7581C211yesmaximal, characteristic, complemented
12#7578C211yesmaximal, characteristic, complemented
13#7575C211yesmaximal, characteristic, complemented
14#7567C211yesmaximal, characteristic, complemented
15#7562C211yesmaximal, characteristic, complemented
16C2×#548C411yescharacteristic
17#2372—21no—
18#2368—21no—
19#2367C411yescharacteristic
20#2364—21no—
21#2362C411yescharacteristic
22#2361C411yescharacteristic
23#2356C411yescharacteristic
24#2353—21no—
25#2352C411yescharacteristic
26Q8×C22⋊C4C411yescharacteristic
27#2339C411yescharacteristic
28#2337—21no—
29#2332—21no—
30#2325—21no—
31C4×#124C2211yescharacteristic
32C4×C22⋊Q8C2211yescharacteristic
33C4×#121C2211yescharacteristic
34#2297C2211yescharacteristic
35#2295C2211yescharacteristic
36C2×#391C2211yescharacteristic
37#2275—21no—
38C42×D4C2211yescharacteristic
39#1954—21no—
40#1939—21no—
41#1923—21no—
42#1923—21no—
43#1923—21no—
44#1916—21no—
45#1869—21no—
46#1839C2211yescharacteristic
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C8×C22⋊C4—21no—
55C4×C22⋊C8C2211yescharacteristic
56#1744C2211yescharacteristic
57#1744C2211yescharacteristic
58C2×#336C2211yescharacteristic
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#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×#41C2×C411yescharacteristic
76C23×Q8C2×C411yescharacteristic
77C23×C8—21no—
78#548—24no—
79#548C818yescomplemented
80C2×#124—21no—
81C2×#124C2×C411yescharacteristic
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×#121—21no—
85C2×#121C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×D4—21no—
90C2×C4×D4—21no—
91C2×C4×D4—21no—
92C2×C4×D4C2×C411yescharacteristic
93C2×C4×D4C2×C411yescharacteristic
94C2×C4×D4C2×C411yescharacteristic
95C2×C4×D4D411yescharacteristic
96C2×C4×D4D411yescharacteristic
97C2×#117C2×C411yescharacteristic
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116#391—21no—
117#391—21no—
118#391—21no—
119#391—21no—
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#391C2×C411yescharacteristic
123#391C2×C411yescharacteristic
124#391C2311yescharacteristic
125#391C2311yescharacteristic
126#390C2×C411yescharacteristic
127#390C2311yescharacteristic
128#389C2×C411yescharacteristic
129#389C2311yescharacteristic
130#388—21no—
131#388—21no—
132#388—21no—
133#388—21no—
134#388—21no—
135#388—21no—
136#388—21no—
137#388—21no—
138#388—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#387—21no—
146#386—21no—
147#386—21no—
148#386—21no—
149#386—21no—
150#386—21no—
151#386—21no—
152#386—21no—
153#385—21no—
154#385—21no—
155#385—21no—
156#383—21no—
157#382C2×C411yescharacteristic
158#381—21no—
159#380—21no—
160#380C2×C411yescharacteristic
161#380C2×C411yescharacteristic
162C4×C4⋊C4—21no—
163C4×C4⋊C4—21no—
164C4×C4⋊C4—21no—
165C4×C4⋊C4C2×C411yescharacteristic
166C4×C4⋊C4C2×C411yescharacteristic
167C4×C4⋊C4C2×C411yescharacteristic
168C4×C4⋊C4C2×C411yescharacteristic
169C4×C4⋊C4D411yescharacteristic
170C4×C4⋊C4D411yescharacteristic
171C4×C22⋊C4—21no—
172C4×C22⋊C4—21no—
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4C2×C411yescharacteristic
181C4×C22⋊C4C2×C411yescharacteristic
182C4×C22⋊C4D411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186#376—21no—
187#376—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C43C2311yescharacteristic
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208C2×#41—22no—
209C2×#41—81no—
210C2×#41—81no—
211C2×#41C2×C814yes—
212C22×Q8—21no—
213C22×Q8—22no—
214C22×Q8—81no—
215C22×Q8—81no—
216C22×Q8C2×C814yescomplemented
217C22×Q8C22×C411yescharacteristic
218C22×Q8C22×C411yescharacteristic
219C22×D4—21no—
220C22×D4—21no—
221C22×D4C22⋊C411yescharacteristic
222C22×D4C22×C411yescharacteristic
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4C22×C411yescharacteristic
229C23×C4C22×C411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—42no—
243C22×C8—42no—
244C22×C8—81no—
245#124—28no—
246#124—22no—
247#124C2×C814yescomplemented
248C22⋊Q8—22no—
249C22⋊Q8—28no—
250C22⋊Q8C2×C814yes—
251#121—28no—
252#121—22no—
253#121C2×C814yescomplemented
254C4×Q8—22no—
255C4×Q8—81no—
256C4×Q8—81no—
257C4×Q8C2×C814yescomplemented
258C4×D4—28no—
259C4×D4—42no—
260C4×D4—41no—
261C4×D4—41no—
262C4×D4—44no—
263C4×D4—42no—
264C4×D4—42no—
265C4×D4—81no—
266C4×D4—81no—
267C4×D4C2×C814yes—
268C4×D4M1614yes—
269#117—81no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—81no—
278C2×C4⋊C4C22⋊C411yescharacteristic
279C2×C4⋊C4C22⋊C411yescharacteristic
280C2×C4⋊C4C22⋊C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4C22⋊C411yescharacteristic
301C2×C22⋊C4C22⋊C411yescharacteristic
302C2×C22⋊C4C22⋊C411yescharacteristic
303C2×C22⋊C4C22⋊C411yescharacteristic
304C2×C22⋊C4C22⋊C411yescharacteristic
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—81no—
313C2×C42—81no—
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2411yescharacteristic
327C4⋊C8—42no—
328C4⋊C8—81no—
329C4⋊C8—81no—
330C4⋊C8—81no—
331C22⋊C8—42no—
332C22⋊C8—42no—
333C22⋊C8—81no—
334C22⋊C8—81no—
335C22⋊C8—81no—
336C22⋊C8—81no—
337C22⋊C8—81no—
338C4×C8—42no—
339C4×C8—42no—
340C4×C8—81no—
341C4×C8—81no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95C22⋊C411yescharacteristic
364#95C22⋊C411yescharacteristic
365#95C22⋊C411yescharacteristic
366#95C22⋊C411yescharacteristic
367#95C22⋊C411yescharacteristic
368#95C22⋊C411yescharacteristic
369C24—21no—
370C24C2×C22⋊C411yescharacteristic
371C24C2×C22⋊C411yescharacteristic
372#41—84no—
373#41—84no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—24no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—81no—
382C2×Q8—81no—
383C2×Q8—84no—
384C2×Q8—84no—
385C2×Q8C22×C812yes—
386C2×Q8C22×C812yes—
387C2×D4—24no—
388C2×D4—24no—
389C2×D4—21no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—81no—
397C2×D4—81no—
398C2×D4C22⋊C814yes—
399C2×D4C22×C812yes—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4—41no—
420C22×C4—41no—
421C22×C4—41no—
422C22×C4—41no—
423C22×C4—41no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—81no—
441C22×C4—81no—
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C2×C22⋊C411yescharacteristic
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C22×C812yes—
453C22×C4C22×C812yes—
454C22×C4C2×M1612yes—
455C22×C4C2×M1612yes—
456C22×C4C23×C411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C2×C8—22no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—84no—
484C4⋊C4—21no—
485C4⋊C4—24no—
486C4⋊C4—24no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—81no—
498C4⋊C4—81no—
499C4⋊C4C22⋊C814yes—
500C4⋊C4C22×C812yes—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C22⋊C4C22⋊C814yes—
521C22⋊C4C22⋊C814yes—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—24no—
527C42—22no—
528C42—41no—
529C42—41no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—42no—
538C42—42no—
539C42—81no—
540C42—81no—
541C42—81no—
542C42—81no—
543C42C22×C812yes—
544C42C2×M1612yes—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23C2×C22⋊C812yes—
565C23C2×C22⋊C812yes—
566C23C22×C22⋊C411yescharacteristic
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—84no—
572D4—44no—
573D4—42no—
574D4—42no—
575D4—48no—
576D4—84no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—162no—
644C2×C4—162no—
645C2×C4—162no—
646C2×C4C2×C22⋊C812yes—
647C2×C4C2×C22⋊C812yes—
648C2×C4C2×C22⋊C812yes—
649C2×C4C2×C22⋊C812yes—
650C2×C4#42112yes—
651C2×C4C23×C811yescharacteristic
652C2×C4C22×M1611yescharacteristic
653C2×C4#57511yescharacteristic
654C2×C4#57511yescharacteristic
655C2×C4#57711yescharacteristic
656C2×C4#57811yescharacteristic
657C8—42no—
658C8—81no—
659C8—81no—
660C8—81no—
661C8—81no—
662C8—81no—
663C8—81no—
664C8—81no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22C22×C22⋊C811yescharacteristic
687C22#288311yescharacteristic
688C22#288811yescharacteristic
689C22#289611yescharacteristic
690C22#289811yescharacteristic
691C22C2×#42111yescharacteristic
692C22#290111yescharacteristic
693C4—22no—
694C4—22no—
695C4—22no—
696C4—21no—
697C4—21no—
698C4—21no—
699C4—21no—
700C4—22no—
701C4—22no—
702C4—21no—
703C4—21no—
704C4—42no—
705C4—41no—
706C4—41no—
707C4—41no—
708C4—42no—
709C4—42no—
710C4—41no—
711C4—41no—
712C4—81no—
713C4—81no—
714C4—81no—
715C4—81no—
716C2—42no—
717C2—42no—
718C2—82no—
719C2#1735011yescharacteristic
720C2#1743711yescharacteristic
721C2#1743911yescharacteristic
722C2#1760011yescharacteristic
723C2#1762311yescharacteristic
724C2#1767911yescharacteristic
725C2#1776811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC2×#548C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22785C2semidirect
3. #22008C2semidirect
4. #7600C2semidirect
5. #7599C2semidirect
6. #7597C2semidirect
7. #7595C2semidirect
8. #7589C2semidirect
9. #7584C2semidirect
10. #7582C2semidirect
11. #7581C2semidirect
12. #7578C2semidirect
13. #7575C2semidirect
14. #7567C2semidirect
15. #7562C2semidirect
79. #548C8semidirect
216. C22×Q8C2×C8semidirect
247. #124C2×C8semidirect
253. #121C2×C8semidirect
257. C4×Q8C2×C8semidirect

Maximal subgroups

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

subgroupindex
#343272
#227852
#220082
#76002
#75992
#75972
#75952
#75892
#75842
#75822
#75812
#75782
#75752
#75672
#75622