Finite Group Databasebeta

← #516666

Subgroups — #516666

Order 512.

3736 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#541C211yesmaximal, characteristic
2C2×#2320C211yesmaximal, characteristic, complemented
3C2×C42×Q8C211yesmaximal, characteristic, complemented
4#14711C212yesmaximal
5#14709C212yesmaximal, complemented
6#14291C212yesmaximal
7#14289C212yesmaximal, complemented
8#14101C214yesmaximal
9#14097C214yesmaximal, complemented
10#14054C212yesmaximal, complemented
11#14024C212yesmaximal, complemented
12#13997C212yesmaximal
13#13969C212yesmaximal
14#13786C211yesmaximal, characteristic, complemented
15#13777C211yesmaximal, characteristic
16C4×#391C211yesmaximal, characteristic, complemented
17#13761C211yesmaximal, characteristic
18C2×#541—28no—
19C2×#541—28no—
20C2×#541C4116yescomplemented
21C22×C4⋊Q8C2211yescharacteristic
22C22×#126C2212yes—
23C22×#126C2212yes—
24C22×#125C2212yes—
25C22×#125C2212yes—
26C22×#125C2211yescharacteristic
27C22×C22⋊Q8C2211yescharacteristic
28C22×C22⋊Q8C2211yescharacteristic
29C22×C4×Q8C2212yes—
30C22×C4×Q8C2211yescharacteristic
31C22×C4×Q8C2211yescharacteristic
32C22×#117C2211yescharacteristic
33#2320—28no—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#389C2212yes—
38C2×#387C2214yes—
39C2×#383C2212yes—
40C2×#382C2212yes—
41C2×#380C2212yes—
42C42×Q8—28no—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2214yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×#376C2211yescharacteristic
50C2×C43C2212yes—
51C2×C43C2211yescharacteristic
52#1445C2214yes—
53#1445C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1437C2212yes—
61#1437C2212yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2211yescharacteristic
65#1437C2212yes—
66#1437C2211yescharacteristic
67#1436C2212yes—
68#1436C2212yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2211yescharacteristic
74#1436C2212yes—
75#1436C2212yes—
76#1436C2212yes—
77#1436C2212yes—
78#1436C2211yescharacteristic
79#1436C2212yes—
80C4⋊#95C2214yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85#1434C2214yes—
86C4×#95C2214yes—
87C4×#95C2212yes—
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C24×C4C2311yescharacteristic
91#541—2128no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8C2×C418yes—
95C2×#126—28no—
96C2×#126—28no—
97C2×#126—28no—
98C2×#126—28no—
99C2×#126C2×C4116yes—
100C2×#126C2×C4116yes—
101C2×#125—24no—
102C2×#125—24no—
103C2×#125—28no—
104C2×#125—28no—
105C2×#125—28no—
106C2×#125—28no—
107C2×#125C2×C418yes—
108C2×#125C2×C4116yes—
109C2×#125C2×C4116yes—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8—24no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8C2×C418yes—
115C2×C22⋊Q8C2×C418yes—
116C2×C4×Q8—28no—
117C2×C4×Q8—28no—
118C2×C4×Q8—24no—
119C2×C4×Q8—24no—
120C2×C4×Q8—28no—
121C2×C4×Q8C2×C418yes—
122C2×C4×Q8C2×C418yes—
123C2×C4×Q8D418yes—
124C2×C4×Q8D418yes—
125C2×#117—24no—
126C2×#117—24no—
127C2×#117C2×C418yes—
128C22×C4⋊C4C2314yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2314yes—
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2311yescharacteristic
146C22×C22⋊C4C2311yescharacteristic
147C22×C22⋊C4C2312yes—
148C22×C22⋊C4C2312yes—
149C22×C22⋊C4C2314yes—
150C22×C22⋊C4C2312yes—
151C22×C42C2314yes—
152C22×C42C2312yes—
153C22×C42C2311yescharacteristic
154C22×C42C2312yes—
155C22×C42C2314yes—
156C22×C42C2311yescharacteristic
157C22×C42C2312yes—
158C22×C42C2312yes—
159C22×C42C2312yes—
160C22×C42C2312yes—
161C22×C42C2311yescharacteristic
162C22×C42C2311yescharacteristic
163C22×C42C2312yes—
164#391—28no—
165#391—28no—
166#391—28no—
167#389—28no—
168#387—216no—
169#383—28no—
170#382—28no—
171#380—28no—
172C4×C4⋊C4—28no—
173C4×C4⋊C4—216no—
174C4×C4⋊C4—24no—
175C4×C4⋊C4—24no—
176C4×C4⋊C4—28no—
177C4×C22⋊C4—28no—
178#376—24no—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C43—24no—
206C43—28no—
207C25C2411yescharacteristic
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—24no—
213C22×Q8—24no—
214C22×Q8C22×C414yes—
215C22×Q8C22×C414yes—
216C22×Q8C2×D414yes—
217C22×Q8C2×D414yes—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4C22×C414yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C4⋊Q8—232no—
240#126—264no—
241#126—264no—
242#125—232no—
243#125—264no—
244#125—264no—
245C22⋊Q8—232no—
246C22⋊Q8—232no—
247C4×Q8—264no—
248C4×Q8—232no—
249C4×Q8C22⋊C4164yescomplemented
250#117—232no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
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—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C22×C414yes—
283C2×C4⋊C4C22×C414yes—
284C2×C4⋊C4C22×C414yes—
285C2×C4⋊C4C22×C418yes—
286C2×C4⋊C4C22×C418yes—
287C2×C4⋊C4C22×C418yes—
288C2×C4⋊C4C22×C418yes—
289C2×C4⋊C4C22×C418yes—
290C2×C4⋊C4C22×C418yes—
291C2×C4⋊C4C22×C418yes—
292C2×C4⋊C4C22×C414yes—
293C2×C4⋊C4C22×C418yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2×D418yes—
298C2×C4⋊C4#4118yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4C22×C414yes—
317C2×C22⋊C4C22×C414yes—
318C2×C22⋊C4C22×C418yes—
319C2×C22⋊C4C22×C418yes—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—28no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—28no—
328C2×C42—28no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—28no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—24no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—28no—
340C2×C42—24no—
341C2×C42—42no—
342C2×C42—44no—
343C2×C42—42no—
344C2×C42—44no—
345C2×C42—44no—
346C2×C42C22×C414yes—
347C2×C42C22×C418yes—
348C2×C42C22×C414yes—
349C2×C42C22×C418yes—
350C2×C42C2×D414yes—
351C2×C42C2×D414yes—
352C2×C42C2×D418yes—
353C2×C42C2×D418yes—
354C2×C42#4118yes—
355#95—22no—
356#95—22no—
357#95—24no—
358#95—24no—
359#95—28no—
360#95—24no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—24no—
365#95—24no—
366#95—28no—
367#95—28no—
368#95—28no—
369#95—28no—
370#95—28no—
371#95—28no—
372#95—24no—
373#95—24no—
374#95—44no—
375#95—44no—
376#95—44no—
377#95—42no—
378#95—44no—
379#95—44no—
380#95—42no—
381C24—21no—
382C24—21no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24C23×C412yes—
390C24C2511yescharacteristic
391C2×Q8—28no—
392C2×Q8—28no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—216no—
397C2×Q8—216no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8C2×C22⋊C4116yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—28no—
446C22×C4—28no—
447C22×C4—28no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4C23×C412yes—
472C22×C4C23×C412yes—
473C22×C4C23×C414yes—
474C22×C4C23×C414yes—
475C22×C4C23×C412yes—
476C22×C4C23×C412yes—
477C22×C4C23×C412yes—
478C22×C4C23×C414yes—
479C22×C4C23×C414yes—
480C22×C4C23×C412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D412yes—
488C22×C4C22×D414yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C2×#4118yes—
492C22×C4C2×#4114yes—
493C22×C4C2×#4114yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C42−1+414yes—
497C22×C42−1+414yes—
498C4⋊C4—232no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—216no—
507C4⋊C4—216no—
508C4⋊C4—216no—
509C4⋊C4—216no—
510C4⋊C4—48no—
511C4⋊C4—88no—
512C4⋊C4C2×C22⋊C4116yes—
513C4⋊C4C2×C22⋊C4132yes—
514C22⋊C4—28no—
515C22⋊C4—28no—
516C22⋊C4—216no—
517C22⋊C4—216no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—88no—
521C42—28no—
522C42—232no—
523C42—28no—
524C42—216no—
525C42—216no—
526C42—48no—
527C42—416no—
528C42—44no—
529C42—48no—
530C42—48no—
531C42—84no—
532C42C2×C22⋊C4116yes—
533C42C2×C22⋊C4132yes—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—41no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23C24×C411yescharacteristic
556C23C23×D411yescharacteristic
557C23C23×D411yescharacteristic
558C23C22×#4112yes—
559C23C22×#4112yes—
560C23C2×2−1+412yes—
561C23C2×2−1+412yes—
562C23#58512yes—
563C23#58512yes—
564Q8—232no—
565Q8—416no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—416no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4C22×C22⋊C414yes—
650C2×C4C22×C22⋊C414yes—
651C2×C4C22×C22⋊C418yes—
652C2×C4C22×C22⋊C418yes—
653C2×C4C22×C22⋊C414yes—
654C2×C4C4×#4114yes—
655C2×C4C4×#4114yes—
656C2×C4#52518yes—
657C2×C4#53218yes—
658C2×C4#54118yescomplemented
659C22—44no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—44no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22C23×C22⋊C411yescharacteristic
669C22C2×C4×#4111yescharacteristic
670C22C2×C4×#4111yescharacteristic
671C22C4×2−1+412yes—
672C22C4×2−1+412yes—
673C22C2×#52512yes—
674C22C2×#53212yes—
675C22C2×#54112yes—
676C22#345512yes—
677C22#346112yes—
678C22#346112yes—
679C22#348611yescharacteristic
680C22#348711yescharacteristic
681C22#348912yes—
682C22#349712yes—
683C22#349911yescharacteristic
684C22#350011yescharacteristic
685C22#351512yes—
686C22#351612yes—
687C22#352112yes—
688C22#352312yes—
689C4—28no—
690C4—24no—
691C4—24no—
692C4—28no—
693C4—28no—
694C4—44no—
695C4—48no—
696C4—42no—
697C4—44no—
698C4—42no—
699C4—42no—
700C4—44no—
701C4—82no—
702C4—82no—
703C4—84no—
704C4#229718yes—
705C2—82no—
706C2C2×#229711yescharacteristic
707C2C4×#54112yes—
708C2#3425711yescharacteristic
709C2#3425911yescharacteristic
710C2#3428612yes—
711C2#3520512yes—
712C2#3520912yes—
713C2#3522012yes—
714C2#3524412yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2320C2semidirect
3. C2×C42×Q8C2semidirect
5. #14709C2semidirect
7. #14289C2semidirect
9. #14097C2semidirect
10. #14054C2semidirect
11. #14024C2semidirect
14. #13786C2semidirect
16. C4×#391C2semidirect
20. C2×#541C4semidirect
249. C4×Q8C22⋊C4semidirect
658. C2×C4#541semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×#23202
C2×C42×Q82
#147112
#147092
#142912
#142892
#141012
#140972
#140542
#140242
#139972
#139692
#137862
#137772
C4×#3912
#137612