Finite Group Databasebeta

← #183236

Subgroups — #183236

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 661 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34203C211yesmaximal, characteristic
2#20379C211yesmaximal, characteristic
3C2×#1511C211yesmaximal, characteristic
4C22×#328C211yesmaximal, characteristic
5#13799C211yesmaximal, characteristic
6#8221C212yesmaximal, complemented
7#8219C212yesmaximal, complemented
8#8217C212yesmaximal, complemented
9C4×#328C212yesmaximal, complemented
10#8024C211yesmaximal, characteristic
11#8020C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C2×#519C411yescharacteristic, complemented
14C22×#117C2211yescharacteristic
15C2×#407C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#405—21no—
18#2362—22no—
19#2360—22no—
20#2344—21no—
21#2344—21no—
22#2342—21no—
23#2342—21no—
24#2323C412yescomplemented
25#2323C412yescomplemented
26#2316C411yescharacteristic, complemented
27#2310C411yescharacteristic, complemented
28C4×C22⋊Q8C2212yes—
29C4×C22⋊Q8C2212yes—
30#2280C2211yescharacteristic
31C2×#381C2211yescharacteristic
32#2274C411yescharacteristic, complemented
33C2×C4×C22⋊C4—21no—
34C22×#95C2211yescharacteristic
35#1750—22no—
36#1749C2212yes—
37C4×C22⋊C8C2212yes—
38#1729—21no—
39#1511C418yescomplemented
40C2×#328C418yescomplemented
41C2×#328C2211yescharacteristic
42C2×#328C2211yescharacteristic
43C2×#328C2211yescharacteristic
44C2×#328C2211yescharacteristic
45C2×#328C2212yes—
46C2×#328C2212yes—
47#1436—22no—
48#1434C2212yes—
49C22⋊#95—21no—
50C4×#95C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1294C2212yes—
55C23×Q8C2×C411yescharacteristic
56C24×C4—21no—
57C24×C4C2311yescharacteristic
58C22×M16—21no—
59C23×C8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2×C414yes—
65C2×C22⋊Q8C2×C414yes—
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×C22⋊Q8C2311yescharacteristic
69C2×C22⋊Q8C2311yescharacteristic
70#519—44no—
71C2×C4×Q8—22no—
72C2×C4×Q8C2×C412yescomplemented
73C2×C4×Q8C2×C412yescomplemented
74C2×#117—21no—
75C2×#117—21no—
76C2×#117—44no—
77C2×#117C2×C411yescharacteristic
78C2×#117C2×C411yescharacteristic
79C2×#117C2×C411yescharacteristic
80C2×#117C2×C411yescharacteristic
81C2×#117C2×C412yes—
82C2×#117D411yescharacteristic
83C2×#117D411yescharacteristic
84C2×#117D412yes—
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4D412yes—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4D411yescharacteristic
91C22×C22⋊C4D411yescharacteristic
92C22×C42—22no—
93C22×C42C2312yes—
94#407—22no—
95#407C2×C414yes—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8C2×C414yes—
100C2×C22⋊C8C2311yescharacteristic
101C2×C22⋊C8C2311yescharacteristic
102C2×C22⋊C8C2312yes—
103#405—44no—
104C2×#97—22no—
105C2×C4×C8—22no—
106#391—22no—
107#391C2×C412yescomplemented
108#391C2×C412yescomplemented
109#389—22no—
110#389—22no—
111#389—22no—
112#389C2×C412yescomplemented
113#389C2×C412yescomplemented
114#388—22no—
115#388—22no—
116#388—22no—
117#388—22no—
118#387—22no—
119#387—22no—
120#387—22no—
121#387—22no—
122#385—22no—
123#385—22no—
124#381—21no—
125#381—22no—
126#381—21no—
127#381—21no—
128#381C2×C414yes—
129#380—21no—
130#380—21no—
131#380C2×C412yes—
132#380C2312yes—
133#379—21no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4C2×C412yescomplemented
138C4×C4⋊C4C2×C412yescomplemented
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145C4×C22⋊C4—44no—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95—22no—
149C2×#95—22no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95C2×C414yes—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156#336—22no—
157#336—22no—
158#328C2×C414yes—
159#328C2×C414yes—
160#328C2×C414yes—
161#328C2×C414yes—
162#328C2×C418yescomplemented
163#328C2×C418yescomplemented
164C25C2×D411yescharacteristic
165C22×Q8—21no—
166C22×Q8—22no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8C4214yescomplemented
170C22×Q8C22×C411yescharacteristic
171C22×Q8C22×C411yescharacteristic
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—22no—
176C23×C4—21no—
177C23×C4—24no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—44no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—82no—
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C412yes—
187C23×C4C22×C412yes—
188C23×C4C2×D412yes—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4#4112yes—
194C23×C4#4112yes—
195C23×C4C2411yescharacteristic
196C2×M16—24no—
197C2×M16—42no—
198C2×M16—42no—
199C22×C8—24no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—42no—
204C22×C8—42no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—24no—
208C22⋊Q8—24no—
209C22⋊Q8—24no—
210C22⋊Q8—24no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8C4214yescomplemented
216C22⋊Q8C4214yescomplemented
217C22⋊Q8C22×C412yes—
218C22⋊Q8C22×C412yes—
219C22⋊Q8C22×C412yes—
220C22⋊Q8C22×C412yes—
221C4×Q8—44no—
222C4×Q8—44no—
223C4×Q8—48no—
224#117—24no—
225#117—24no—
226#117—44no—
227#117—44no—
228#117—42no—
229#117—42no—
230#117—44no—
231#117—44no—
232#117—42no—
233#117—42no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4C4214yescomplemented
248C2×C4⋊C4C22⋊C412yes—
249C2×C4⋊C4C22⋊C412yes—
250C2×C4⋊C4C22⋊C412yes—
251C2×C4⋊C4C22⋊C412yes—
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C412yes—
255C2×C4⋊C4C22×C412yes—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C412yes—
275C2×C22⋊C4C22⋊C412yes—
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—82no—
287C2×C42C22×C412yes—
288C2×C42C22×C412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C22⋊C8—24no—
292C22⋊C8—24no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—22no—
296C22⋊C8C22×C412yes—
297C22⋊C8C22×C412yes—
298C22⋊C8C22×C414yes—
299#97—44no—
300C4×C8—44no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—22no—
313#95—21no—
314#95—21no—
315#95—22no—
316#95—44no—
317#95—44no—
318#95—82no—
319#95C22×C412yes—
320#95C22×C412yes—
321#95C22×C414yes—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—42no—
327C24C2×C22⋊C411yescharacteristic
328C24C2×C22⋊C411yescharacteristic
329C24C4×D412yes—
330C24C4×D412yes—
331C24C22×D411yescharacteristic
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—24no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—48no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—48no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8C2×C4212yes—
344C2×Q8C2×C4212yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—48no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—48no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—82no—
399C22×C4—84no—
400C22×C4—88no—
401C22×C4—88no—
402C22×C4C2×C4212yes—
403C22×C4C2×C4212yes—
404C22×C4C2×C22⋊C412yes—
405C22×C4C2×C22⋊C412yes—
406C22×C4C2×C22⋊C412yes—
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C412yes—
410C22×C4#11712yes—
411C22×C4#11712yes—
412C22×C4C4×D412yes—
413C22×C4C4×D412yes—
414C22×C4C4×D412yes—
415C22×C4C4×D412yes—
416C22×C4C4×D412yes—
417C22×C4C4×D412yes—
418C22×C4C23×C411yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424M16—48no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—48no—
434C2×C8—42no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4C2×C4212yes—
455C4⋊C4C2×C4212yes—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—24no—
466C22⋊C4—42no—
467C22⋊C4—48no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—82no—
473C22⋊C4#9914yes—
474C42—24no—
475C42—24no—
476C42—42no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—44no—
481C42—44no—
482C42—82no—
483C42—82no—
484C42—88no—
485C23—21no—
486C23—21no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23C4×C22⋊C412yes—
506C23C4×C22⋊C412yes—
507C23C2×#9912yes—
508C23#41012yes—
509C23C22×C22⋊C411yescharacteristic
510C23C2×C4×D411yescharacteristic
511C23C2×C4×D411yescharacteristic
512C23#57712yes—
513C23#57812yes—
514Q8—42no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—48no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—84no—
585C2×C4—816no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4C4×C22⋊C412yes—
593C2×C4C4×C22⋊C412yes—
594C2×C4C2×#9914yes—
595C2×C4C22×C4211yescharacteristic
596C2×C4C2×#11711yescharacteristic
597C2×C4C2×C4×D411yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C8—42no—
600C8—42no—
601C8—44no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—22no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—24no—
611C22—22no—
612C22—22no—
613C22—24no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—44no—
618C22—48no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22C4×#9914yes—
623C22C2×C4×C22⋊C411yescharacteristic
624C22C22×#9911yescharacteristic
625C22C2×#41011yescharacteristic
626C22#289812yes—
627C22#290112yes—
628C22#294712yes—
629C22#294911yescharacteristic
630C22#295011yescharacteristic
631C4—21no—
632C4—21no—
633C4—21no—
634C4—22no—
635C4—22no—
636C4—21no—
637C4—24no—
638C4—44no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—84no—
647C4—82no—
648C4—82no—
649C4—82no—
650C2—21no—
651C2—22no—
652C2—21no—
653C2—24no—
654C2—42no—
655C2#1760812yes—
656C2#1760912yes—
657C2C2×C4×#9911yescharacteristic
658C2#2040511yescharacteristic
659C2#2041711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8221C2semidirect
7. #8219C2semidirect
8. #8217C2semidirect
9. C4×#328C2semidirect
13. C2×#519C4semidirect
24. #2323C4semidirect
25. #2323C4semidirect
26. #2316C4semidirect
27. #2310C4semidirect
32. #2274C4semidirect
39. #1511C4semidirect
40. C2×#328C4semidirect
72. C2×C4×Q8C2×C4semidirect
73. C2×C4×Q8C2×C4semidirect
107. #391C2×C4semidirect
108. #391C2×C4semidirect
112. #389C2×C4semidirect
113. #389C2×C4semidirect
137. C4×C4⋊C4C2×C4semidirect
138. C4×C4⋊C4C2×C4semidirect
162. #328C2×C4semidirect
163. #328C2×C4semidirect
169. C22×Q8C42semidirect
215. C22⋊Q8C42semidirect
216. C22⋊Q8C42semidirect
247. C2×C4⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
#342032
#203792
C2×#15112
C22×#3282
#137992
#82212
#82192
#82172
C4×#3282
#80242
#80202