Finite Group Databasebeta

← #193493

Subgroups — #193493

Order 512.

1349 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2989C211yesmaximal, characteristic
2C4×#536C211yesmaximal, characteristic
3#21374C211yesmaximal, characteristic
4C2×#1491C211yesmaximal, characteristic, complemented
5C2×#1487C211yesmaximal, characteristic, complemented
6#8942C212yesmaximal, complemented
7#8941C212yesmaximal, complemented
8#8938C212yesmaximal, complemented
9#8937C212yesmaximal, complemented
10#7903C211yesmaximal, characteristic, complemented
11#7902C211yesmaximal, characteristic, complemented
12C2×#536C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15#2989—24no—
16#2989—24no—
17C2×#446—21no—
18C2×#436—21no—
19C2×#435—21no—
20C2×#435—21no—
21C2×#432C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#423C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29#2327—21no—
30#2326—21no—
31#2323C412yes—
32#2323C412yes—
33#2322—21no—
34#2322—21no—
35#2317C411yescharacteristic
36#2311C411yescharacteristic
37#2310C411yescharacteristic
38C4×C4⋊Q8—21no—
39C4×#124—21no—
40C4×C22⋊Q8C2212yes—
41C4×C22⋊Q8C2212yes—
42C4×#123—21no—
43C4×#123—21no—
44C4×#117C2211yescharacteristic
45#1985—21no—
46#1955—21no—
47#1870—21no—
48#1870—21no—
49C4×C22⋊C8C2212yes—
50C4×C22⋊C8C2212yes—
51#1491—24no—
52#1487—24no—
53C2×Q8⋊C8C2212yes—
54C2×Q8⋊C8C2212yes—
55C2×Q8⋊C8C2212yes—
56C2×Q8⋊C8C2212yes—
57#1279C2212yes—
58#1279C2212yes—
59#1279C2212yes—
60#1279C2212yes—
61C22×#41C2×C411yescharacteristic
62C23×Q8C2×C411yescharacteristic
63C22×M16—21no—
64C22×M16—21no—
65C22×M16—21no—
66C23×C8—21no—
67#536—28no—
68C2×C4⋊Q8—21no—
69C2×#124—21no—
70C2×#123—21no—
71C2×#123—21no—
72C2×C22⋊Q8C2×C412yes—
73C2×C22⋊Q8C2×C412yes—
74C4×#41—22no—
75C4×#41—22no—
76C4×#41—22no—
77C4×#41—22no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8—22no—
81C2×C4×Q8—22no—
82C2×C4×Q8—22no—
83C2×C4×Q8—22no—
84C2×C4×Q8C2×C412yes—
85C2×C4×Q8D411yescharacteristic
86C2×C4×Q8D411yescharacteristic
87C2×C4×Q8C2312yes—
88C2×C4×D4D411yescharacteristic
89C2×C4×D4D411yescharacteristic
90C2×C4×D4D411yescharacteristic
91C2×C4×D4D411yescharacteristic
92C2×C4×D4D411yescharacteristic
93C2×C4×D4D411yescharacteristic
94C2×#117C2×C411yescharacteristic
95C2×#117C2×C411yescharacteristic
96C2×#117C2×C411yescharacteristic
97C2×#117C2×C411yescharacteristic
98C2×#117C2×C411yescharacteristic
99C22×C4⋊C4—21no—
100C22×C4⋊C4C2×C411yescharacteristic
101C22×C42—21no—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105#446—28no—
106#436—28no—
107#435—28no—
108#435—28no—
109#432—22no—
110#432—22no—
111#432—22no—
112#432D414yes—
113#431—22no—
114#431—22no—
115#431—22no—
116#431—22no—
117#423—22no—
118#423—22no—
119#423—22no—
120#423D414yes—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—22no—
125C2×C4⋊C8—22no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2311yescharacteristic
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—22no—
136C2×C22⋊C8—22no—
137C4×M16—22no—
138C4×M16—22no—
139C2×#97—21no—
140C2×#97—21no—
141C2×#97—21no—
142C2×C4×C8—21no—
143C2×C4×C8C2312yes—
144C2×C4×C8C2312yes—
145#391—21no—
146#391—21no—
147#391—21no—
148#391C2×C412yes—
149#391C2×C412yes—
150#390—21no—
151#389—22no—
152#389—22no—
153#389C2×C412yes—
154#389C2×C412yes—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#387—22no—
160#387—22no—
161#387—22no—
162#387—22no—
163#386—21no—
164#386—21no—
165#385—22no—
166#385—22no—
167#385—21no—
168#385—21no—
169#384—21no—
170#382—21no—
171#381—21no—
172#381—21no—
173#380C2×C412yes—
174#380C2×C412yes—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4C2×C412yes—
178C4×C4⋊C4C2312yes—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—22no—
183#376—21no—
184C43—21no—
185#336—22no—
186#336—22no—
187Q8⋊C8—24no—
188Q8⋊C8—24no—
189Q8⋊C8—24no—
190Q8⋊C8—24no—
191C2×#41—22no—
192C2×#41—22no—
193C2×#41—24no—
194C2×#41—42no—
195C22×Q8—24no—
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—42no—
200C22×Q8C22⋊C411yescharacteristic
201C22×Q8C22×C412yes—
202C22×D4C22⋊C411yescharacteristic
203C22×D4C22⋊C411yescharacteristic
204C22×D4C22⋊C411yescharacteristic
205C23×C4—21no—
206C23×C4—21no—
207C23×C4C22×C411yescharacteristic
208C23×C4C22×C411yescharacteristic
209C23×C4C22×C411yescharacteristic
210C23×C4C2×D411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C2×M16—24no—
217C2×M16—24no—
218C2×M16—44no—
219C2×M16—42no—
220C2×M16—82no—
221C2×M16—82no—
222C22×C8—22no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—24no—
228C22×C8—22no—
229C22×C8—82no—
230C4⋊Q8—28no—
231#124—28no—
232#123—28no—
233#123—28no—
234C22⋊Q8—24no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C22⋊Q8M1618yescomplemented
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—24no—
243C4×Q8—24no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—42no—
249C4×Q8—42no—
250C4×Q8—42no—
251C4×Q8—42no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264#117—22no—
265#117—22no—
266#117—24no—
267#117—22no—
268#117—22no—
269#117—22no—
270#117—22no—
271#117—42no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4C22⋊C411yescharacteristic
284C2×C4⋊C4C22⋊C411yescharacteristic
285C2×C4⋊C4C22⋊C411yescharacteristic
286C2×C4⋊C4C22⋊C411yescharacteristic
287C2×C4⋊C4C22⋊C411yescharacteristic
288C2×C4⋊C4C22⋊C411yescharacteristic
289C2×C4⋊C4C22×C412yes—
290C2×C4⋊C4C22×C412yes—
291C2×C4⋊C4C22×C412yes—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4C22⋊C411yescharacteristic
301C2×C22⋊C4C22⋊C411yescharacteristic
302C2×C22⋊C4C22⋊C411yescharacteristic
303C2×C22⋊C4C22⋊C411yescharacteristic
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42C22×C411yescharacteristic
321C2×C42C22×C411yescharacteristic
322C2×C42C22×C411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2×D412yes—
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D411yescharacteristic
331C2×C42#4112yes—
332C2×C42#4112yes—
333C2×C42C2411yescharacteristic
334C4⋊C8—24no—
335C4⋊C8—24no—
336C4⋊C8—24no—
337C4⋊C8—21no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8—42no—
342C4⋊C8C2×D412yes—
343C4⋊C8C2×D412yes—
344C4⋊C8C2×D412yes—
345C4⋊C8C2×D412yes—
346C22⋊C8—24no—
347C22⋊C8—24no—
348C22⋊C8—24no—
349C22⋊C8—44no—
350C22⋊C8—44no—
351C22⋊C8—42no—
352#97—24no—
353#97—24no—
354#97—42no—
355C4×C8—24no—
356C4×C8—22no—
357C4×C8—22no—
358C4×C8—22no—
359C4×C8—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370C24C2×C22⋊C411yescharacteristic
371C24C2×C22⋊C411yescharacteristic
372C24C2×C22⋊C411yescharacteristic
373#41—48no—
374#41—48no—
375C2×Q8—24no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—41no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—42no—
387C2×Q8—42no—
388C2×Q8C2×M1614yes—
389C2×D4—22no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—82no—
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C4×D412yes—
443C22×C4C4×D412yes—
444C22×C4#12012yes—
445C22×C4#12112yes—
446C22×C4#12112yes—
447C22×C4#12312yes—
448C22×C4C2×M1614yes—
449C22×C4C23×C411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452M16—48no—
453M16—84no—
454M16—84no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—84no—
470C4⋊C4—28no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—28no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—82no—
487C4⋊C4—82no—
488C4⋊C4C2×M1614yes—
489C22⋊C4—24no—
490C22⋊C4—28no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—28no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—42no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C42—24no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—42no—
515C42—42no—
516C42—42no—
517C42—42no—
518C42—42no—
519C42—44no—
520C42—82no—
521C42C4×D412yes—
522C42C4×D412yes—
523C42#12012yes—
524C42#12112yes—
525C42#12112yes—
526C42#12312yes—
527C42C22×D411yescharacteristic
528C42C22×D411yescharacteristic
529C42C2×#4111yescharacteristic
530C42C2×#4111yescharacteristic
531C23—21no—
532C23—21no—
533C23—21no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—22no—
543C23—21no—
544C23#38612yes—
545C23#40712yes—
546C23C22×C22⋊C411yescharacteristic
547Q8—24no—
548Q8—24no—
549Q8—44no—
550Q8—48no—
551Q8—42no—
552Q8—42no—
553D4—44no—
554D4—44no—
555D4—44no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—84no—
609C2×C4#38612yes—
610C2×C4#38612yes—
611C2×C4#38612yes—
612C2×C4#40712yes—
613C2×C4C2×C4×D411yescharacteristic
614C2×C4C2×C4×D411yescharacteristic
615C2×C4C2×#12011yescharacteristic
616C2×C4C2×#12111yescharacteristic
617C2×C4C2×#12111yescharacteristic
618C2×C4C2×#12311yescharacteristic
619C2×C4C22×M1612yes—
620C2×C4#56812yes—
621C2×C4#57712yes—
622C2×C4#57812yes—
623C8—44no—
624C8—42no—
625C8—42no—
626C8—42no—
627C8—42no—
628C8—82no—
629C22—24no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—41no—
638C22—41no—
639C22—42no—
640C22—42no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22C2×#38611yescharacteristic
645C22C2×#40711yescharacteristic
646C22#287911yescharacteristic
647C22#289812yes—
648C22#290112yes—
649C4—22no—
650C4—22no—
651C4—21no—
652C4—21no—
653C4—22no—
654C4—22no—
655C4—44no—
656C4—41no—
657C4—44no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—82no—
663C4—82no—
664C4D4×M1612yes—
665C4D4×M1612yes—
666C4#300911yescharacteristic
667C4#301111yescharacteristic
668C4#309712yes—
669C2—42no—
670C2—42no—
671C2—81no—
672C2#1745412yes—
673C2#1750612yes—
674C2#2089811yescharacteristic
675C2#2090311yescharacteristic
676C2#2091311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1491C2semidirect
5. C2×#1487C2semidirect
6. #8942C2semidirect
7. #8941C2semidirect
8. #8938C2semidirect
9. #8937C2semidirect
10. #7903C2semidirect
11. #7902C2semidirect
237. C22⋊Q8M16semidirect

Maximal subgroups

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

subgroupindex
C2×#29892
C4×#5362
#213742
C2×#14912
C2×#14872
#89422
#89412
#89382
#89372
#79032
#79022