Finite Group Databasebeta

← #362107

Subgroups — #362107

Order 512.

1528 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#1939C211yesmaximal, characteristic
3C2×#1805C212yesmaximal
4C2×#1803C211yesmaximal, characteristic
5C2×C4×#103C211yesmaximal, characteristic
6#13911C211yesmaximal, characteristic
7#11227C212yesmaximal
8#11208C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10841C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4×Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20Q8×C4⋊C4—24no—
21Q8×C4⋊C4C418yescomplemented
22C2×#389—22no—
23C2×#389C2212yes—
24C2×#384—21no—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382C2212yes—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1939—24no—
32#1805—28no—
33#1803—24no—
34C4×#103—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1442—22no—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8—42no—
59C2×C4×Q8C2×C418yes—
60C2×C4×Q8C2×C414yes—
61C2×C4×Q8D414yes—
62C2×C4×Q8Q814yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C42—21no—
78C22×C42—22no—
79C22×C42—21no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82C2×C4⋊C8—24no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—42no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103D414yes—
96C2×#103Q814yes—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100#389—24no—
101#389—216no—
102#389C2×C418yes—
103#384—28no—
104#384—22no—
105#384—22no—
106#384—22no—
107#384C2×C414yes—
108#382—24no—
109#382—42no—
110C4×C4⋊C4—28no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4C2×C414yes—
114C4×C4⋊C4D414yes—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95D412yes—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—22no—
126#340—22no—
127#340—42no—
128#340—42no—
129#336—24no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8C22×C412yes—
138C22×Q8C2×D412yes—
139C22×Q8C2×D412yes—
140C22×Q8C2×Q812yes—
141C22×Q8C2×Q812yes—
142C22×Q8#4112yes—
143C22×Q8#4112yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8C2×D412yes—
175C22×C8C2×Q812yes—
176C4×Q8—28no—
177C4×Q8—28no—
178C4×Q8—28no—
179C4×Q8—216no—
180C4×Q8—48no—
181C4×Q8—48no—
182C4×Q8—48no—
183C4×Q8C4⋊C4116yes—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—81no—
228C2×C4⋊C4—81no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—81no—
232C2×C4⋊C4—81no—
233C2×C4⋊C4C22⋊C414yes—
234C2×C4⋊C4C22⋊C414yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×Q812yes—
240C2×C4⋊C4C2×Q812yes—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—28no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42C22×C414yes—
255C2×C42C22×C412yes—
256C2×C42C2×D414yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×Q812yes—
260C2×C42#4112yes—
261C2×C42#4112yes—
262C4⋊C8—48no—
263C4⋊C8—44no—
264#103—28no—
265#103—28no—
266#103—48no—
267C4×C8—44no—
268#95—28no—
269#95—44no—
270#95—44no—
271#95—42no—
272#95—82no—
273#95—82no—
274#95C22⋊C418yes—
275C24C22×D411yescharacteristic
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8C2×C4⋊C414yes—
298C2×Q8C4×D414yes—
299C2×Q8C4×Q814yes—
300C2×Q8#12114yes—
301C2×Q8C22⋊Q814yes—
302C2×Q8#12514yes—
303C2×Q8C4⋊Q814yes—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4C2×C22⋊C414yes—
363C22×C4C2×C22⋊C414yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C412yes—
366C22×C4#12112yes—
367C22×C4#12112yes—
368C22×C4C22⋊Q812yes—
369C22×C4C22⋊Q812yes—
370C22×C4#13512yes—
371C22×C4#13512yes—
372C22×C4#13712yes—
373C22×C4#13712yes—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×Q811yescharacteristic
379C22×C4C22×Q811yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4H(5,𝔽2)11yescharacteristic
385C22×C4H(5,𝔽2)11yescharacteristic
386C22×C42−1+411yescharacteristic
387C22×C42−1+411yescharacteristic
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—216no—
413C4⋊C4—28no—
414C4⋊C4—28no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—416no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—48no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—84no—
434C4⋊C4—88no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4C2×C4⋊C414yes—
438C42—24no—
439C42—24no—
440C42—24no—
441C42—24no—
442C42—216no—
443C42—416no—
444C42—84no—
445C42C2×C4⋊C414yes—
446C42#11714yes—
447C42#12114yes—
448C42C22⋊Q814yes—
449C23C22×C22⋊C411yescharacteristic
450C23C2×#12111yescharacteristic
451C23C2×C22⋊Q811yescharacteristic
452C23D4211yescharacteristic
453C23#54611yescharacteristic
454C23#54811yescharacteristic
455C23D4×Q811yescharacteristic
456C23C2×#13511yescharacteristic
457C23C2×#13511yescharacteristic
458C23C2×#13711yescharacteristic
459C23C2×#13711yescharacteristic
460C23#57611yescharacteristic
461C23#57711yescharacteristic
462C23#57711yescharacteristic
463C23#57811yescharacteristic
464Q8—24no—
465Q8—24no—
466Q8—24no—
467Q8—24no—
468Q8—24no—
469Q8—24no—
470Q8—24no—
471Q8#38918yes—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4#38014yes—
535C2×C4#38014yes—
536C2×C4#41614yes—
537C2×C4#41914yes—
538C2×C4#46414yes—
539C2×C4#46414yes—
540C2×C4#47914yes—
541C2×C4#47914yes—
542C2×C4C22×C4⋊C411yescharacteristic
543C2×C4C2×#11711yescharacteristic
544C2×C4C2×C4×D411yescharacteristic
545C2×C4C2×C4×Q811yescharacteristic
546C2×C4#51811yescharacteristic
547C2×C4#51911yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×#12111yescharacteristic
550C2×C4C2×C22⋊Q811yescharacteristic
551C2×C4C2×C22⋊Q811yescharacteristic
552C2×C4C2×#12511yescharacteristic
553C2×C4C2×C4⋊Q811yescharacteristic
554C2×C4#53711yescharacteristic
555C2×C4#53711yescharacteristic
556C2×C4#54411yescharacteristic
557C2×C4#54411yescharacteristic
558C2×C4D4211yescharacteristic
559C2×C4#54611yescharacteristic
560C2×C4#54711yescharacteristic
561C2×C4#54711yescharacteristic
562C2×C4#54811yescharacteristic
563C2×C4D4×Q811yescharacteristic
564C2×C4#55211yescharacteristic
565C2×C4#55211yescharacteristic
566C2×C4#55311yescharacteristic
567C2×C4#55311yescharacteristic
568C2×C4#55411yescharacteristic
569C2×C4#55411yescharacteristic
570C8—42no—
571C8—42no—
572C8—82no—
573C22C2×#38011yescharacteristic
574C22D4×C22⋊C411yescharacteristic
575C22Q8×C22⋊C411yescharacteristic
576C22#241611yescharacteristic
577C22#241711yescharacteristic
578C22#244011yescharacteristic
579C22#244211yescharacteristic
580C22C2×#41611yescharacteristic
581C22C2×#41911yescharacteristic
582C22#289711yescharacteristic
583C22#289811yescharacteristic
584C22C2×#46411yescharacteristic
585C22C2×#46411yescharacteristic
586C22#304511yescharacteristic
587C22#304611yescharacteristic
588C22C2×#47911yescharacteristic
589C22C2×#47911yescharacteristic
590C22#308311yescharacteristic
591C22#308411yescharacteristic
592C22#327611yescharacteristic
593C22#327811yescharacteristic
594C22#328411yescharacteristic
595C22#328711yescharacteristic
596C22#330411yescharacteristic
597C22#330511yescharacteristic
598C22#332811yescharacteristic
599C22#332911yescharacteristic
600C22#336411yescharacteristic
601C22#336511yescharacteristic
602C22#336911yescharacteristic
603C22#337511yescharacteristic
604C22#338111yescharacteristic
605C22#338311yescharacteristic
606C22#338511yescharacteristic
607C22#338711yescharacteristic
608C4—28no—
609C4—24no—
610C4—44no—
611C4—44no—
612C4—44no—
613C4—84no—
614C4—82no—
615C4—82no—
616C4—162no—
617C4#180518yescomplemented
618C4C2×#38911yescharacteristic
619C4#231611yescharacteristic
620C4#234411yescharacteristic
621C4#234511yescharacteristic
622C4#241611yescharacteristic
623C4#241711yescharacteristic
624C4#244011yescharacteristic
625C4#244211yescharacteristic
626C2#1390711yescharacteristic
627C2C2×#180511yescharacteristic
628C2#2065211yescharacteristic
629C2#2241811yescharacteristic
630C2#2243011yescharacteristic
631C2#2252111yescharacteristic
632C2#2252511yescharacteristic
633C2#2472911yescharacteristic
634C2#2473311yescharacteristic
635C2#2481411yescharacteristic
636C2#2481811yescharacteristic
637C2#2604311yescharacteristic
638C2#2604711yescharacteristic
639C2#2610911yescharacteristic
640C2#2611311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. Q8×C4⋊C4C4semidirect
617. C4#1805semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#19392
C2×#18052
C2×#18032
C2×C4×#1032
#139112
#112272
#112082
#108492
#108412
#107892