Finite Group Databasebeta

← #625950

Subgroups — #625950

Order 512.

2350 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic
2C2×#2552C211yesmaximal, characteristic
3C2×#2550C211yesmaximal, characteristic
4C2×#2660C211yesmaximal, characteristic
5#15779C214yesmaximal
6#15766C214yesmaximal
7#15698C214yesmaximal
8#15520C214yesmaximal
9#15230C211yesmaximal, characteristic
10#15228C211yesmaximal, characteristic
11#15228C211yesmaximal, characteristic
12#15228C211yesmaximal, characteristic
13#14707C211yesmaximal, characteristic
14#14354C211yesmaximal, characteristic
15#14277C211yesmaximal, characteristic
16#13911C211yesmaximal, characteristic
17#13909C211yesmaximal, characteristic
18#13779C211yesmaximal, characteristic
19#13759C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C22×#125C2211yescharacteristic
24#2859—28no—
25#2660—28no—
26#2552—28no—
27#2550—28no—
28C2×#398C2214yes—
29C2×#398C2212yes—
30C2×#398C2212yes—
31C2×#398C2212yes—
32C2×#395C2214yes—
33C2×#395C2212yes—
34C2×#391C2212yes—
35C2×#389C2214yes—
36C2×#389C2214yes—
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2212yes—
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#384C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#382C2211yescharacteristic
53C2×#382C2211yescharacteristic
54C2×#382C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C4⋊C4C2211yescharacteristic
60C2×#376C2211yescharacteristic
61C2×#376C2211yescharacteristic
62C2×C43C2211yescharacteristic
63#1446C2214yes—
64#1444C2214yes—
65#1444C2214yes—
66#1444C2214yes—
67#1444C2214yes—
68#1444C2214yes—
69#1444C2214yes—
70#1442C2214yes—
71#1442C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2214yes—
76#1441C2214yes—
77#1436C2212yes—
78#1436C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82#1436C2212yes—
83#1436C2214yes—
84#1436C2212yes—
85#1436C2212yes—
86C4⋊#95C2214yes—
87C4⋊#95C2211yescharacteristic
88C4⋊#95C2211yescharacteristic
89C4⋊#95C2211yescharacteristic
90C4⋊#95C2211yescharacteristic
91C4⋊#95C2211yescharacteristic
92C4⋊#95C2211yescharacteristic
93C4×#95C2211yescharacteristic
94C4×#95C2211yescharacteristic
95C23×Q8C2312yes—
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—42no—
99C2×#125—24no—
100C2×#125—24no—
101C2×#125—24no—
102C2×#125—24no—
103C2×#125—24no—
104C2×#125—24no—
105C2×#125—42no—
106C2×#125—42no—
107C2×#125—42no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2314yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142C22×C42C2312yes—
143C22×C42C2312yes—
144C22×C42C2312yes—
145C22×C42C2311yescharacteristic
146C22×C42C2311yescharacteristic
147#398—216no—
148#398—28no—
149#398—28no—
150#398—28no—
151#395—216no—
152#395—28no—
153#391—28no—
154#389—216no—
155#389—216no—
156#389—28no—
157#389—24no—
158#389—24no—
159#389—24no—
160#389—24no—
161#389—28no—
162#384—24no—
163#384—28no—
164#384—24no—
165#384—24no—
166#384—42no—
167#384D418yes—
168#382—24no—
169#382—24no—
170#382—24no—
171#382—24no—
172#382—24no—
173#382—42no—
174#382Q818yescomplemented
175C4×C4⋊C4—24no—
176C4×C4⋊C4—24no—
177C4×C4⋊C4D418yes—
178C4×C4⋊C4Q818yescomplemented
179#376—24no—
180#376Q818yescomplemented
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2311yescharacteristic
207C2×#95C2311yescharacteristic
208C2×#95C2312yes—
209C2×#95C2314yes—
210C2×#95C2312yes—
211C2×#95C2312yes—
212C43—24no—
213C22×Q8—24no—
214C22×Q8—24no—
215C22×Q8—42no—
216C22×Q8—44no—
217C22×Q8—44no—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C4⋊Q8—416no—
238#125—416no—
239#125—416no—
240#125—416no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2×D414yes—
313C2×C4⋊C4C2×D4116yes—
314C2×C4⋊C4C2×Q814yes—
315C2×C4⋊C4C2×Q814yes—
316C2×C4⋊C4C2×Q818yes—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—44no—
332C2×C42—42no—
333C2×C42—42no—
334C2×C42—42no—
335C2×C42—42no—
336C2×C42—44no—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42C2×Q814yes—
342C2×C42C2×Q814yes—
343C2×C42C2×Q814yes—
344C2×C42C2×Q814yes—
345C2×C42C2×Q814yes—
346C2×C42C2×Q814yes—
347C2×C42C2×Q814yes—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—28no—
353#95—28no—
354#95—24no—
355#95—28no—
356#95—28no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—44no—
363#95—44no—
364#95—42no—
365#95—42no—
366#95—44no—
367#95—42no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—42no—
372#95—44no—
373#95C2×D414yes—
374#95C2×D414yes—
375#95C2×Q818yes—
376#95C2×Q818yes—
377#95C2×Q818yes—
378#95C2×Q8116yes—
379C24C2511yescharacteristic
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—88no—
383C2×Q8—84no—
384C2×Q8—88no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
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—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×Q814yes—
475C22×C4C22×Q814yes—
476C22×C4C22×Q814yes—
477C22×C4C22×Q814yes—
478C22×C4C22×Q814yes—
479C22×C4C22×Q812yes—
480C22×C4C22×Q812yes—
481C22×C4C22×Q814yes—
482C22×C4C22×Q812yes—
483C22×C4C22×Q814yes—
484C22×C4C22×Q812yes—
485C22×C4C22×Q812yes—
486C22×C4C22×Q812yes—
487C22×C4C22×Q812yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+412yes—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—416no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—416no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—84no—
513C4⋊C4—88no—
514C4⋊C4—84no—
515C4⋊C4—84no—
516C4⋊C4—84no—
517C4⋊C4—88no—
518C42—28no—
519C42—28no—
520C42—28no—
521C42—28no—
522C42—28no—
523C42—48no—
524C42—48no—
525C42—48no—
526C42—84no—
527C42C4⋊Q8116yes—
528C42C4⋊Q8116yes—
529C23C23×D411yescharacteristic
530C23C23×D411yescharacteristic
531C23C23×Q811yescharacteristic
532C23C23×Q811yescharacteristic
533C23C23×Q811yescharacteristic
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×2−1+411yescharacteristic
536C23#58512yes—
537C23#58512yes—
538C23#58512yes—
539C23#58512yes—
540Q8—816no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—42no—
585C2×C4—42no—
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—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4C2×C4⋊Q814yes—
637C2×C4C2×C4⋊Q814yes—
638C2×C4C2×C4⋊Q818yes—
639C2×C4C2×C4⋊Q814yes—
640C2×C4C2×C4⋊Q814yes—
641C2×C4#54414yes—
642C2×C4D4214yes—
643C2×C4D4×Q814yes—
644C2×C4D4×Q814yes—
645C2×C4Q8214yes—
646C22C22×C4⋊Q811yescharacteristic
647C22C22×C4⋊Q811yescharacteristic
648C22#344911yescharacteristic
649C22#344911yescharacteristic
650C22C2×#54411yescharacteristic
651C22#345812yes—
652C22C2×D4211yescharacteristic
653C22C2×D4×Q811yescharacteristic
654C22C2×D4×Q811yescharacteristic
655C22C2×Q8211yescharacteristic
656C22#348512yes—
657C22#348712yes—
658C22#350011yescharacteristic
659C22#350011yescharacteristic
660C22#350412yes—
661C22#350412yes—
662C22#350412yes—
663C22#350412yes—
664C22#350412yes—
665C22#356212yes—
666C22#356212yes—
667C22#356512yes—
668C22#356512yes—
669C4—24no—
670C4—24no—
671C4—28no—
672C4—42no—
673C4—44no—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—44no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4#255118yes—
688C2C2×#255111yescharacteristic
689C2#3533011yescharacteristic
690C2#3533511yescharacteristic
691C2#3564012yes—
692C2#3742912yes—
693C2#3768211yescharacteristic
694C2#3768311yescharacteristic
695C2#3769112yes—
696C2#3961311yescharacteristic
697C2#3961311yescharacteristic
698C2#3962212yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
174. #382Q8semidirect
178. C4×C4⋊C4Q8semidirect
180. #376Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#25522
C2×#25502
C2×#26602
#157792
#157662
#156982
#155202
#152302
#152282
#147072
#143542
#142772
#139112
#139092
#137792
#137592