Finite Group Databasebeta

← #616172

Subgroups — #616172

Order 512.

3534 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2868C211yesmaximal, characteristic, complemented
2C2×#2857C211yesmaximal, characteristic
3C2×#2857C211yesmaximal, characteristic
4C2×#2429C212yesmaximal
5C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
6#15188C212yesmaximal, complemented
7#15178C212yesmaximal
8#15137C214yesmaximal, complemented
9#15131C214yesmaximal
10#14714C211yesmaximal, characteristic, complemented
11#14712C211yesmaximal, characteristic, complemented
12#14024C212yesmaximal, complemented
13#14005C214yesmaximal, complemented
14#13876C214yesmaximal
15C42×C22⋊C4C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C4×Q8C2212yes—
23C23×C42C2211yescharacteristic
24#2868—28no—
25#2857—28no—
26#2857—28no—
27#2429—216no—
28C2×#398C2214yes—
29C2×#396C2214yes—
30C4×C4⋊Q8—28no—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#389C2214yes—
37C2×#389C2214yes—
38C2×#389C2214yes—
39C2×#389C2214yes—
40C2×#389C2212yes—
41C2×#389C2214yes—
42C2×#388C2214yes—
43C2×#384C2211yescharacteristic
44C2×#384C2214yes—
45C2×#384C2214yes—
46C2×#384C2211yescharacteristic
47C2×#380C2214yes—
48C2×#380C2214yes—
49C2×C4×C4⋊C4C2214yes—
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C43C2211yescharacteristic
55C2×C43C2211yescharacteristic
56#1444C2218yes—
57#1442C2218yes—
58#1441C2218yes—
59#1441C2218yes—
60#1439C2218yes—
61#1436C2218yes—
62C4⋊#95C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4×#95C2212yes—
67C4×#95C2211yescharacteristic
68C4×#95C2212yes—
69C4×#95C2211yescharacteristic
70C23×Q8C2312yes—
71C23×Q8C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C2×C4⋊Q8—28no—
75C2×C4⋊Q8—28no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—28no—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—44no—
84C2×C4⋊Q8—44no—
85C2×C4⋊Q8D418yes—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—44no—
91C2×C22⋊Q8—44no—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—44no—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2318yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C42—24no—
114C22×C42—24no—
115C22×C42—24no—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#398—216no—
129#396—216no—
130#391—28no—
131#391—28no—
132#391—28no—
133#391—28no—
134#391—44no—
135#389—216no—
136#389—216no—
137#389—216no—
138#389—28no—
139#389—216no—
140#389—216no—
141#388—216no—
142#384—216no—
143#384—216no—
144#384—24no—
145#384D418yes—
146#380—216no—
147#380—216no—
148C4×C4⋊C4—216no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—28no—
151C4×C22⋊C4D418yes—
152C4×C22⋊C4Q8116yescomplemented
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2318yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2318yes—
163C2×#95C2318yes—
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2312yes—
168C43—24no—
169C43—24no—
170C25C2411yescharacteristic
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8C2×D418yes—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4C2×D414yes—
198C23×C4C2×Q818yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C4⋊Q8—232no—
214C4⋊Q8—432no—
215C4⋊Q8—432no—
216C4⋊Q8—416no—
217C22⋊Q8—432no—
218C22⋊Q8—432no—
219C4×Q8—432no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—216no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—216no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—216no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4C2×D418yes—
256C2×C4⋊C4C2×D4116yes—
257C2×C4⋊C4C2×D4116yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4C2×D414yes—
268C2×C22⋊C4C2×D414yes—
269C2×C22⋊C4C2×Q818yes—
270C2×C22⋊C4C2×Q818yes—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—28no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—44no—
298C2×C42—416no—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×Q818yes—
304C2×C42C2×Q818yes—
305C2×C42#4114yes—
306C2×C42#4118yes—
307C2×C42#4114yes—
308#95—216no—
309#95—216no—
310#95—216no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—216no—
318#95—48no—
319#95C2×D414yes—
320#95C2×D414yes—
321#95C2×Q818yes—
322#95C2×Q818yes—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—22no—
331C24—22no—
332C24C22×D412yes—
333C24C22×Q814yes—
334C24C2511yescharacteristic
335C2×Q8—216no—
336C2×Q8—216no—
337C2×Q8—216no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—48no—
347C2×Q8—48no—
348C2×Q8—48no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—44no—
413C22×C4—48no—
414C22×C4—44no—
415C22×C4—48no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—44no—
420C22×C4C22×D412yes—
421C22×C4C22×D418yes—
422C22×C4C22×D418yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×Q814yes—
433C22×C4C22×Q814yes—
434C22×C4C22×Q814yes—
435C22×C4C22×Q814yes—
436C22×C4C22×Q814yes—
437C22×C4C22×Q814yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4114yes—
443C22×C4C2×#4112yes—
444C22×C4C2×#4114yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)12yes—
447C22×C42−1+414yes—
448C22×C42−1+414yes—
449C4⋊C4—232no—
450C4⋊C4—216no—
451C4⋊C4—232no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—416no—
455C4⋊C4—416no—
456C4⋊C4—416no—
457C4⋊C4—432no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C22⋊C4—28no—
463C22⋊C4—216no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4C4⋊Q8116yescomplemented
467C42—28no—
468C42—28no—
469C42—28no—
470C42—216no—
471C42—216no—
472C42—216no—
473C42—416no—
474C42—44no—
475C42—48no—
476C42—48no—
477C42—44no—
478C42#121116yes—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—41no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—42no—
496C23—42no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23C2×C4⋊Q814yes—
502C23C23×D411yescharacteristic
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C23×Q812yes—
506C23C22×#4111yescharacteristic
507C23C22×#4111yescharacteristic
508C23C22×#4112yes—
509C23C2×H(5,𝔽2)11yescharacteristic
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×2−1+412yes—
512C23C2×2−1+412yes—
513Q8—432no—
514Q8—816no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—28no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—28no—
544C2×C4—216no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—48no—
570C2×C4—42no—
571C2×C4—416no—
572C2×C4—432no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4C2×#12114yes—
586C2×C4C2×#12118yes—
587C2×C4C2×C4⋊Q814yes—
588C2×C4C2×C4⋊Q814yes—
589C2×C4#53214yes—
590C2×C4#53214yes—
591C2×C4#53318yes—
592C2×C4#53318yes—
593C2×C4#54014yes—
594C2×C4#54118yes—
595C2×C4D4214yes—
596C2×C4D4214yes—
597C2×C4#54614yes—
598C2×C4#54718yes—
599C2×C4#54818yes—
600C2×C4D4×Q818yes—
601C2×C4D4×Q818yes—
602C2×C4#55014yes—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22C22×#12111yescharacteristic
615C22C22×C4⋊Q811yescharacteristic
616C22C2×#53211yescharacteristic
617C22C2×#53211yescharacteristic
618C22C2×#53312yes—
619C22C2×#53312yes—
620C22C2×#54011yescharacteristic
621C22C2×#54112yes—
622C22C2×D4211yescharacteristic
623C22C2×D4211yescharacteristic
624C22C2×#54611yescharacteristic
625C22C2×#54712yes—
626C22C2×#54812yes—
627C22C2×D4×Q812yes—
628C22C2×D4×Q812yes—
629C22C2×#55011yescharacteristic
630C22#349911yescharacteristic
631C22#349911yescharacteristic
632C22#350312yes—
633C22#350312yes—
634C22#350912yes—
635C22#351112yes—
636C22#351112yes—
637C4—24no—
638C4—24no—
639C4—24no—
640C4—28no—
641C4—42no—
642C4—48no—
643C4—44no—
644C4—42no—
645C4—48no—
646C4—44no—
647C4—84no—
648C4—84no—
649C4#242818yes—
650C4#2429116yescomplemented
651C2—44no—
652C2C2×#242811yescharacteristic
653C2C2×#242912yes—
654C2#3528412yes—
655C2#3531911yescharacteristic
656C2D4×C4⋊Q811yescharacteristic
657C2D4×C4⋊Q811yescharacteristic
658C2#3612211yescharacteristic
659C2#3612711yescharacteristic
660C2#3612912yes—
661C2#3613412yes—
662C2#3664511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2868C2semidirect
5. C2×C4×C4⋊Q8C2semidirect
6. #15188C2semidirect
8. #15137C2semidirect
10. #14714C2semidirect
11. #14712C2semidirect
12. #14024C2semidirect
13. #14005C2semidirect
152. C4×C22⋊C4Q8semidirect
466. C22⋊C4C4⋊Q8semidirect
650. C4#2429semidirect

Maximal subgroups

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

subgroupindex
C2×#28682
C2×#28572
C2×#24292
C2×C4×C4⋊Q82
#151882
#151782
#151372
#151312
#147142
#147122
#140242
#140052
#138762
C42×C22⋊C42