Finite Group Databasebeta

← #735184

Subgroups — #735184

Order 512.

2406 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#2565C211yesmaximal, characteristic
3C2×#2642C211yesmaximal, characteristic
4C2×#2370C211yesmaximal, characteristic
5#16464C214yesmaximal
6#15895C214yesmaximal
7#15888C214yesmaximal
8#15237C211yesmaximal, characteristic
9#15188C211yesmaximal, characteristic
10#15188C211yesmaximal, characteristic
11#15184C211yesmaximal, characteristic
12#14871C214yesmaximal
13#14155C211yesmaximal, characteristic
14#14155C211yesmaximal, characteristic
15#14155C211yesmaximal, characteristic
16#14003C211yesmaximal, characteristic
17#14000C211yesmaximal, characteristic
18#13848C211yesmaximal, characteristic
19#13786C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#125C2211yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25#2642—28no—
26#2565—28no—
27#2546—28no—
28C2×#400C2214yes—
29C2×#400C2214yes—
30C2×#398C2212yes—
31C2×#398C2214yes—
32C2×#398C2212yes—
33#2370—28no—
34C2×#391C2211yescharacteristic
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#391C2211yescharacteristic
47C2×#391C2211yescharacteristic
48C2×#389C2212yes—
49C2×#389C2212yes—
50C2×#389C2212yes—
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#387C2214yes—
54C2×#387C2212yes—
55C2×#387C2212yes—
56C2×#384C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×C4×C4⋊C4C2212yes—
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×C4×C4⋊C4C2211yescharacteristic
61C2×C4×C4⋊C4C2211yescharacteristic
62C2×C4×C4⋊C4C2211yescharacteristic
63C2×C4×C4⋊C4C2211yescharacteristic
64C2×C4×C4⋊C4C2211yescharacteristic
65C2×C4×C4⋊C4C2211yescharacteristic
66C2×#376C2211yescharacteristic
67#1445C2214yes—
68#1445C2214yes—
69#1445C2214yes—
70#1444C2212yes—
71#1443C2214yes—
72#1443C2214yes—
73#1442C2214yes—
74#1442C2214yes—
75#1442C2214yes—
76#1442C2212yes—
77#1441C2212yes—
78#1441C2214yes—
79#1441C2214yes—
80#1441C2214yes—
81#1441C2214yes—
82#1441C2214yes—
83#1441C2212yes—
84#1436C2214yes—
85#1436C2214yes—
86C4⋊#95C2214yes—
87C4⋊#95C2214yes—
88C4⋊#95C2214yes—
89C4⋊#95C2211yescharacteristic
90C4⋊#95C2211yescharacteristic
91#1434C2214yes—
92#1434C2214yes—
93C4×#95C2211yescharacteristic
94C4×#95C2211yescharacteristic
95C23×Q8C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C23×Q8C2311yescharacteristic
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—24no—
103C2×C4⋊Q8—24no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8—42no—
106C2×C4⋊Q8—42no—
107C2×#125—24no—
108C2×#125—24no—
109C2×#125D418yes—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×Q8—24no—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2311yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2312yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2312yes—
144C22×C42C2312yes—
145C22×C42C2311yescharacteristic
146#400—216no—
147#400—216no—
148#398—216no—
149#398—28no—
150#398—28no—
151#391—24no—
152#391—24no—
153#391—24no—
154#391—24no—
155#391—24no—
156#391—24no—
157#391—24no—
158#391—24no—
159#391—24no—
160#391—24no—
161#391—24no—
162#391—24no—
163#391—42no—
164#391—42no—
165#389—28no—
166#389—28no—
167#389—28no—
168#389—24no—
169#389—24no—
170#387—216no—
171#387—28no—
172#387—28no—
173#384—24no—
174#382—24no—
175C4×C4⋊C4—24no—
176C4×C4⋊C4—24no—
177C4×C4⋊C4—24no—
178C4×C4⋊C4—24no—
179C4×C4⋊C4—28no—
180C4×C4⋊C4—42no—
181C4×C4⋊C4—42no—
182C4×C4⋊C4D418yes—
183#376—24no—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2312yes—
206C2×#95C2312yes—
207C2×#95C2312yes—
208C2×#95C2312yes—
209C2×#95C2312yes—
210C2×#95C2312yes—
211C2×#95C2314yes—
212C2×#95C2314yes—
213C2×#95C2314yes—
214C2×#95C2314yes—
215C2×#95C2312yes—
216C2×#95C2312yes—
217C2×#95C2311yescharacteristic
218C2×#95C2311yescharacteristic
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8#4114yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257C23×C4C2411yescharacteristic
258C23×C4C2411yescharacteristic
259C4⋊Q8—416no—
260C4⋊Q8—416no—
261C4⋊Q8—416no—
262#125—232no—
263C4×Q8—416no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—42no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D414yes—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D418yes—
322C2×C4⋊C4C2×D418yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4#4114yes—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—28no—
346C2×C42—22no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—44no—
350C2×C42—44no—
351C2×C42C2×D414yes—
352C2×C42C2×D414yes—
353C2×C42C2×D414yes—
354C2×C42C2×D414yes—
355C2×C42#4114yes—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—28no—
364#95—24no—
365#95—28no—
366#95—28no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—28no—
372#95—28no—
373#95—28no—
374#95—28no—
375#95—28no—
376#95—28no—
377#95—28no—
378#95—28no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—44no—
383#95—42no—
384#95—44no—
385#95—42no—
386#95—42no—
387#95—44no—
388#95—44no—
389#95C2×D414yes—
390#95C2×D414yes—
391C24C2511yescharacteristic
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8—44no—
412C2×Q8—44no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4C22×D414yes—
501C22×C4C22×D412yes—
502C22×C4C22×D414yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C2×#4114yes—
513C22×C4C2×#4114yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4C2×#4112yes—
519C22×C4H(5,𝔽2)12yes—
520C22×C42−1+414yes—
521C22×C42−1+414yes—
522C22×C42−1+414yes—
523C22×C42−1+412yes—
524C4⋊C4—216no—
525C4⋊C4—216no—
526C4⋊C4—28no—
527C4⋊C4—28no—
528C4⋊C4—28no—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—416no—
534C4⋊C4—48no—
535C4⋊C4—48no—
536C4⋊C4—48no—
537C4⋊C4—48no—
538C4⋊C4—48no—
539C4⋊C4—48no—
540C4⋊C4—44no—
541C4⋊C4—48no—
542C4⋊C4—48no—
543C4⋊C4—48no—
544C4⋊C4—44no—
545C4⋊C4—84no—
546C42—28no—
547C42—28no—
548C42—48no—
549C42—48no—
550C42—44no—
551C42—44no—
552C42—44no—
553C42—44no—
554C42—84no—
555C42—84no—
556C42#121116yes—
557C23C23×D411yescharacteristic
558C23C23×D411yescharacteristic
559C23C22×#4111yescharacteristic
560C23C22×#4111yescharacteristic
561C23C22×#4111yescharacteristic
562C23C2×H(5,𝔽2)11yescharacteristic
563C23C2×2−1+412yes—
564C23C2×2−1+412yes—
565C23C2×2−1+411yescharacteristic
566C23C2×2−1+412yes—
567C23#58512yes—
568Q8—416no—
569Q8—88no—
570Q8—88no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4C2×#12114yes—
672C2×C4C2×#12114yes—
673C2×C4C2×#12114yes—
674C2×C4#52914yes—
675C2×C4#53614yes—
676C2×C4#53614yes—
677C2×C4#54214yes—
678C2×C4#54614yes—
679C2×C4#54814yes—
680C2×C4#54814yes—
681C2×C4#55014yes—
682C22C22×#12111yescharacteristic
683C22C2×#52911yescharacteristic
684C22C2×#53611yescharacteristic
685C22C2×#53611yescharacteristic
686C22C2×#54211yescharacteristic
687C22C2×#54611yescharacteristic
688C22C2×#54811yescharacteristic
689C22C2×#54811yescharacteristic
690C22C2×#55011yescharacteristic
691C22#348912yes—
692C22#349911yescharacteristic
693C22#349912yes—
694C22#350011yescharacteristic
695C22#350712yes—
696C22#350912yes—
697C22#351512yes—
698C22#351612yes—
699C22#352312yes—
700C22#352312yes—
701C22#353812yes—
702C22#354112yes—
703C22#355512yes—
704C22#355612yes—
705C4—24no—
706C4—24no—
707C4—24no—
708C4—24no—
709C4—44no—
710C4—44no—
711C4—44no—
712C4—44no—
713C4—42no—
714C4—42no—
715C4—44no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4#242618yes—
724C2C2×#242611yescharacteristic
725C2#3531611yescharacteristic
726C2#3562111yescharacteristic
727C2#3587012yes—
728C2#3808111yescharacteristic
729C2#3808511yescharacteristic
730C2#3917011yescharacteristic
731C2#3917311yescharacteristic
732C2#4270112yes—
733C2#4521712yes—
734C2#4534312yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#25652
C2×#26422
C2×#23702
#164642
#158952
#158882
#152372
#151882
#151842
#148712
#141552
#140032
#140002
#138482
#137862