Finite Group Databasebeta

← #6388671

Subgroups — #6388671

Order 512.

1952 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2985C211yesmaximal, characteristic, complemented
2C2×#2920C211yesmaximal, characteristic
3#36625C211yesmaximal, characteristic
4#23054C212yesmaximal, complemented
5#23050C212yesmaximal, complemented
6#22957C212yesmaximal, complemented
7#22953C212yesmaximal, complemented
8#22604C214yesmaximal, complemented
9#22559C214yesmaximal
10#22106C214yesmaximal, complemented
11#21954C211yesmaximal, characteristic, complemented
12#21821C211yesmaximal, characteristic, complemented
13#21600C214yesmaximal, complemented
14#20714C211yesmaximal, characteristic
15#20537C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#2985—28no—
19C2×#446C2212yescomplemented
20C2×#446C2214yescomplemented
21C2×C8×Q8C2212yescomplemented
22#2920—28no—
23C2×#433C2214yes—
24C2×#432C2211yescharacteristic
25C2×#432C2212yes—
26C2×#432C2211yescharacteristic
27C2×#431C2212yes—
28C2×#431C2211yescharacteristic
29C2×#431C2212yes—
30C2×#431C2211yescharacteristic
31C2×#423C2212yes—
32C2×#407C2212yes—
33C2×C4×M16C2212yes—
34#2857C411yescharacteristic
35#2857C411yescharacteristic
36#2857C411yescharacteristic
37#2857C411yescharacteristic
38#2855C412yes—
39#2855C412yes—
40#2851C412yes—
41#2851C412yes—
42#2848C412yes—
43#2848C412yes—
44C4×C22⋊Q8C2212yes—
45C4×C22⋊Q8C2212yes—
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2212yes—
48#2296C2212yes—
49#2285C2211yescharacteristic
50#2280C2212yes—
51#1985C2214yescomplemented
52#1984C2212yescomplemented
53#1984C2212yescomplemented
54#1940C2214yescomplemented
55#1940C2214yescomplemented
56#1939C2212yescomplemented
57#1939C2212yescomplemented
58#1939C2214yescomplemented
59#1924C2214yes—
60#1924C2214yes—
61#1923C2214yes—
62#1923C2214yes—
63#1917C2214yescomplemented
64#1916C2214yescomplemented
65#1916C2214yescomplemented
66#1916C2214yescomplemented
67#1851C2214yes—
68#1851C2214yes—
69#1818C2214yes—
70#1817C2212yes—
71#1798C2214yes—
72#1797C2212yes—
73#1797C2212yes—
74#1779C2212yes—
75#1771C2214yescomplemented
76#1771C2214yescomplemented
77#1770C2212yescomplemented
78C8×C4⋊C4C2212yescomplemented
79C8×C4⋊C4C2214yescomplemented
80#1753C2214yes—
81#1753C2214yes—
82#1750C2214yes—
83C4×C22⋊C8C2211yescharacteristic
84C4×C22⋊C8C2211yescharacteristic
85C4×C22⋊C8C2211yescharacteristic
86C4×C22⋊C8C2211yescharacteristic
87C24×C4C2×C411yescharacteristic
88C24×C4C2×C411yescharacteristic
89C24×C4C2311yescharacteristic
90C22×M16C2314yes—
91#533—24no—
92#533—24no—
93C2×C4⋊Q8C2×C411yescharacteristic
94C2×C4⋊Q8C2×C411yescharacteristic
95C2×#125C2×C412yes—
96C2×C22⋊Q8C2×C412yes—
97C2×C22⋊Q8C2×C412yes—
98C2×C22⋊Q8C2×C412yes—
99C2×C22⋊Q8C2×C412yes—
100C2×C22⋊Q8C2×C412yes—
101C2×C22⋊Q8C2×C412yes—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2312yes—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C22×C42—24no—
107C22×C42—24no—
108C22×C42C2×C411yescharacteristic
109C22×C42C2×C411yescharacteristic
110C22×C42C2×C411yescharacteristic
111C22×C42C2×C411yescharacteristic
112C22×C42C2×C411yescharacteristic
113C22×C42C2×C411yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#446—28no—
121#446—216no—
122C8×Q8—28no—
123#433—216no—
124#432—24no—
125#432—28no—
126#432D418yes—
127#431—24no—
128#431—24no—
129#431—28no—
130#431—28no—
131#423—28no—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2312yes—
134C2×C4⋊C8C2314yes—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2312yes—
141#407—28no—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C4×M16—28no—
152C2×#97C2312yes—
153C2×#97C2314yes—
154C2×#97C2314yes—
155C2×#97C2312yes—
156C2×C4×C8C2314yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2314yes—
162C2×C4×C8C2314yes—
163#391C2×C412yes—
164#391C2×C412yes—
165#391C2×C412yes—
166#391C2×C412yes—
167#389C2×C412yes—
168#389C2×C412yes—
169#389C2×C412yes—
170#389C2×C412yes—
171#389C2×C412yes—
172#389C2×C412yes—
173#389C2×C412yes—
174#389C2×C412yes—
175#387C2×C414yes—
176#387C2×C414yes—
177#387C2×C414yes—
178#387C2×C414yes—
179#385C2×C414yes—
180#385C2×C414yes—
181#385C2×C414yes—
182#385C2×C414yes—
183#385C2×C414yes—
184#385C2×C414yes—
185#384C2×C411yescharacteristic
186#384C2×C411yescharacteristic
187#382C2×C412yes—
188#381C2×C414yes—
189#380C2×C412yes—
190#380C2×C412yes—
191#380C2×C412yes—
192#380C2×C412yes—
193#380C2×C412yes—
194#380C2×C412yes—
195#380C2×C412yes—
196#380C2×C412yes—
197#380C2×C412yes—
198#380C2×C412yes—
199#379C2×C414yes—
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2312yes—
204C4×C22⋊C4C2314yes—
205C4×C22⋊C4C2314yes—
206#336C2312yes—
207#336C2312yes—
208#336C2312yes—
209#336C2312yes—
210#336C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2314yes—
215#336C2314yes—
216#336C2314yes—
217#336C2314yes—
218C25C22×C411yescharacteristic
219C22×Q8C22×C412yes—
220C22×Q8C22×C412yes—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4C22×C411yescharacteristic
232C23×C4C22×C411yescharacteristic
233C23×C4C22×C411yescharacteristic
234C23×C4C22×C411yescharacteristic
235C23×C4C22×C411yescharacteristic
236C23×C4C22×C411yescharacteristic
237C23×C4C22×C411yescharacteristic
238C23×C4C22×C411yescharacteristic
239C23×C4C22×C411yescharacteristic
240C23×C4C22×C411yescharacteristic
241C23×C4C22×C411yescharacteristic
242C23×C4C22×C411yescharacteristic
243C23×C4C22×C411yescharacteristic
244C23×C4C22×C411yescharacteristic
245C23×C4C22×C411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C2×M16—28no—
250C2×M16—28no—
251C2×M16—44no—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2414yes—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260#125—28no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—28no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—24no—
271#117—24no—
272#117—24no—
273#117—44no—
274C2×C4⋊C4C22×C412yes—
275C2×C4⋊C4C22×C412yes—
276C2×C4⋊C4C22×C414yes—
277C2×C4⋊C4C22×C414yes—
278C2×C4⋊C4C22×C412yes—
279C2×C4⋊C4C22×C412yes—
280C2×C4⋊C4C22×C412yes—
281C2×C4⋊C4C22×C412yes—
282C2×C4⋊C4C22×C414yes—
283C2×C4⋊C4C22×C414yes—
284C2×C4⋊C4C22×C414yes—
285C2×C4⋊C4C22×C412yes—
286C2×C4⋊C4C22×C412yes—
287C2×C4⋊C4C22×C412yes—
288C2×C4⋊C4C22×C412yes—
289C2×C4⋊C4C22×C414yes—
290C2×C22⋊C4C22×C414yes—
291C2×C22⋊C4C22×C414yes—
292C2×C22⋊C4C22×C414yes—
293C2×C22⋊C4C22×C414yes—
294C2×C22⋊C4C22×C414yes—
295C2×C22⋊C4C22×C414yes—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42C22×C411yescharacteristic
313C2×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C22×C411yescharacteristic
319C2×C42C22×C411yescharacteristic
320C2×C42C22×C411yescharacteristic
321C2×C42C22×C411yescharacteristic
322C2×C42C22×C411yescharacteristic
323C2×C42C22×C411yescharacteristic
324C2×C42C2×D414yes—
325C2×C42#4114yes—
326C2×C42C2414yes—
327C2×C42C2414yes—
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332C4⋊C8—28no—
333C4⋊C8—28no—
334C4⋊C8—28no—
335C4⋊C8—24no—
336C4⋊C8—24no—
337C4⋊C8—24no—
338C4⋊C8—44no—
339C4⋊C8—44no—
340C4⋊C8C2×D418yes—
341C22⋊C8—24no—
342C22⋊C8—24no—
343C22⋊C8—24no—
344C22⋊C8—24no—
345C22⋊C8—24no—
346C22⋊C8—28no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349C22⋊C8C2×D418yes—
350#97—28no—
351#97—24no—
352#97—44no—
353#97C2×D418yes—
354C4×C8—28no—
355C4×C8—28no—
356C4×C8—22no—
357C4×C8—22no—
358C4×C8—22no—
359C4×C8—22no—
360C4×C8—44no—
361#95C22×C414yes—
362#95C22×C414yes—
363#95C22×C414yes—
364#95C22×C414yes—
365#95C22×C414yes—
366#95C22×C414yes—
367#95C22×C414yes—
368#95C22×C414yes—
369#95C22×C414yes—
370#95C22×C414yes—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—41no—
375C24—41no—
376C24—41no—
377C24—41no—
378C24C23×C411yescharacteristic
379C24C23×C411yescharacteristic
380C24C23×C411yescharacteristic
381C2×Q8—24no—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—24no—
385C2×Q8—42no—
386C2×Q8—42no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—416no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—416no—
442C22×C4C4×D414yes—
443C22×C4C4×D414yes—
444C22×C4#13114yes—
445C22×C4C23×C411yescharacteristic
446C22×C4C23×C414yes—
447C22×C4C23×C414yes—
448C22×C4C23×C411yescharacteristic
449C22×C4C23×C414yes—
450C22×C4C23×C414yes—
451C22×C4C23×C411yescharacteristic
452C22×C4C23×C411yescharacteristic
453C22×C4C23×C411yescharacteristic
454C22×C4C23×C411yescharacteristic
455C22×C4C23×C411yescharacteristic
456C22×C4C23×C411yescharacteristic
457C22×C4C23×C411yescharacteristic
458C22×C4C23×C411yescharacteristic
459C22×C4C23×C411yescharacteristic
460C22×C4C22×D412yes—
461C22×C4C2×#4112yes—
462C22×C42−1+414yes—
463C22×C4C2511yescharacteristic
464M16—416no—
465C2×C8—24no—
466C2×C8—24no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—24no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—24no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8C22×D418yes—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—28no—
491C4⋊C4—24no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—21no—
516C42—24no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—24no—
521C42—24no—
522C42—24no—
523C42C4×D414yes—
524C42C4×D414yes—
525C42C22×D412yes—
526C42C22×D412yes—
527C42C2×#4112yes—
528C42C2×#4112yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23C2×C4×D412yes—
555C23#51912yes—
556C23C2×#13112yes—
557C23C24×C411yescharacteristic
558Q8—48no—
559Q8—84no—
560C2×C4—28no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—21no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—21no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—21no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4C2×C4×D412yes—
616C2×C4C2×C4×D412yes—
617C2×C4C2×C4×D412yes—
618C2×C4C2×C4×D412yes—
619C2×C4C2×C4×D412yes—
620C2×C4#51912yes—
621C2×C4C2×#13112yes—
622C2×C4C2×#13112yes—
623C2×C4#56812yes—
624C2×C4#56812yes—
625C2×C4#56812yes—
626C2×C4C23×D411yescharacteristic
627C2×C4C22×#4111yescharacteristic
628C2×C4C2×2−1+412yes—
629C2×C4#58512yes—
630C8—28no—
631C8—42no—
632C8—42no—
633C8—44no—
634C8—42no—
635C8—42no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—416no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22C22×C4×D411yescharacteristic
659C22C2×#51911yescharacteristic
660C22#342811yescharacteristic
661C22C22×#13111yescharacteristic
662C22C2×#56811yescharacteristic
663C22C2×#56811yescharacteristic
664C22C2×#56811yescharacteristic
665C4—22no—
666C4—22no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—44no—
674C4—44no—
675C4—41no—
676C4—41no—
677C4—44no—
678C4—44no—
679C4#293314yes—
680C4#348912yes—
681C4#349512yes—
682C2—42no—
683C2—42no—
684C2—42no—
685C2#3420311yescharacteristic
686C2C2×#293311yescharacteristic
687C2#6014111yescharacteristic
688C2#6024911yescharacteristic
689C2#6025011yescharacteristic
690C2#6027911yescharacteristic
691C2#6029511yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2985C2semidirect
4. #23054C2semidirect
5. #23050C2semidirect
6. #22957C2semidirect
7. #22953C2semidirect
8. #22604C2semidirect
10. #22106C2semidirect
11. #21954C2semidirect
12. #21821C2semidirect
13. #21600C2semidirect
15. #20537C2semidirect
19. C2×#446C22semidirect
20. C2×#446C22semidirect
21. C2×C8×Q8C22semidirect
51. #1985C22semidirect
52. #1984C22semidirect
53. #1984C22semidirect
54. #1940C22semidirect
55. #1940C22semidirect
56. #1939C22semidirect
57. #1939C22semidirect
58. #1939C22semidirect
63. #1917C22semidirect
64. #1916C22semidirect
65. #1916C22semidirect
66. #1916C22semidirect
75. #1771C22semidirect
76. #1771C22semidirect
77. #1770C22semidirect
78. C8×C4⋊C4C22semidirect
79. C8×C4⋊C4C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29852
C2×#29202
#366252
#230542
#230502
#229572
#229532
#226042
#225592
#221062
#219542
#218212
#216002
#207142
#205372