Finite Group Databasebeta

← #368277

Subgroups — #368277

Order 512.

1512 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic
5C4×#389C211yesmaximal, characteristic
6#11328C212yesmaximal
7#11293C212yesmaximal
8#10907C212yesmaximal
9#10885C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10832C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×#107C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2211yescharacteristic
17C4×C4⋊Q8—24no—
18C4×C4⋊Q8—24no—
19C2×#391—21no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1987—24no—
32#1987C418yescomplemented
33#1947—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1436—21no—
44#1436—21no—
45C4⋊#95C2211yescharacteristic
46C4⋊#95C2211yescharacteristic
47C4×#95—21no—
48C23×Q8—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8D414yes—
56C2×C4⋊Q8Q814yes—
57C2×C4×Q8—28no—
58C2×C4×Q8—44no—
59C2×C4×Q8—44no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—22no—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8C2×C414yes—
91#391—28no—
92#389—28no—
93#389—24no—
94#389—24no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389Q814yescomplemented
99#384—22no—
100#384D414yes—
101C4×C4⋊C4—28no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—44no—
108C4×C4⋊C4C2×C414yes—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C43—22no—
120C43—22no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—42no—
126#340C2×C418yes—
127#340C2×C418yes—
128#336—24no—
129#336—42no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8—24no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—42no—
136C22×Q8—42no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8C22×C414yes—
164C4⋊Q8—28no—
165C4⋊Q8—28no—
166C4⋊Q8—28no—
167C4⋊Q8C4⋊C4116yescomplemented
168C4×Q8—432no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C22×C414yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×Q812yes—
219C2×C4⋊C4C2×Q814yes—
220C2×C4⋊C4C2×Q812yes—
221C2×C4⋊C4#4112yes—
222C2×C4⋊C4#4112yes—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—28no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—41no—
245C2×C42—41no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42Q1614yes—
254C2×C42C22×C412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×Q812yes—
258C2×C42C2×Q812yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261C2×C42#4112yes—
262#107—48no—
263C4⋊C8—28no—
264C4⋊C8—44no—
265#95—28no—
266#95—28no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—42no—
271#95—42no—
272#95—42no—
273#95—42no—
274#95—42no—
275C24C22×D411yescharacteristic
276C2×Q8—216no—
277C2×Q8—48no—
278C2×Q8—48no—
279C2×Q8—48no—
280C2×Q8—48no—
281C2×Q8—48no—
282C2×Q8—48no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4#12012yes—
354C22×C4#12012yes—
355C22×C4C22⋊Q812yes—
356C22×C4C22⋊Q812yes—
357C22×C4C22⋊Q812yes—
358C22×C4C22⋊Q812yes—
359C22×C4C2×Q1612yes—
360C22×C4C2×Q1612yes—
361C22×C4C2×Q1612yes—
362C22×C4#13512yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4AGL(1,ℤ/8)12yes—
365C22×C4AGL(1,ℤ/8)12yes—
366C22×C4#13712yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×Q811yescharacteristic
371C22×C4C22×Q811yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C42−1+411yescharacteristic
379C22×C42−1+411yescharacteristic
380C22×C42−1+411yescharacteristic
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—216no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—416no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—42no—
422C4⋊C4—48no—
423C4⋊C4C2×C4⋊C418yes—
424C4⋊C4C4×Q814yescomplemented
425C42—28no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—28no—
431C42—44no—
432C42—416no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—48no—
438C42—44no—
439C42—84no—
440C42C2×C4⋊C414yes—
441C42C4×D414yes—
442C42C22⋊Q814yes—
443C42#12314yes—
444C23C2×C4×D411yescharacteristic
445C23C2×#12011yescharacteristic
446C23C2×C22⋊Q811yescharacteristic
447C23C2×C22⋊Q811yescharacteristic
448C23#52511yescharacteristic
449C23#53611yescharacteristic
450C23#53711yescharacteristic
451C23C22×Q1611yescharacteristic
452C23C2×#13511yescharacteristic
453C23C2×AGL(1,ℤ/8)11yescharacteristic
454C23C2×AGL(1,ℤ/8)11yescharacteristic
455C23C2×AGL(1,ℤ/8)11yescharacteristic
456C23C2×#13711yescharacteristic
457C23#57511yescharacteristic
458C23#57511yescharacteristic
459Q8—432no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4#38514yes—
536C2×C4#38514yes—
537C2×C4C4×Q1614yes—
538C2×C4#44014yes—
539C2×C4#44914yes—
540C2×C4#45114yes—
541C2×C4#47714yes—
542C2×C4#47714yes—
543C2×C4#47914yes—
544C2×C4C22×C4⋊C411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×Q811yescharacteristic
547C2×C4C4×#4111yescharacteristic
548C2×C4#51911yescharacteristic
549C2×C4#52011yescharacteristic
550C2×C4C2×C22⋊Q811yescharacteristic
551C2×C4C2×#12311yescharacteristic
552C2×C4#52911yescharacteristic
553C2×C4#53311yescharacteristic
554C2×C4#54111yescharacteristic
555C2×C4#54211yescharacteristic
556C2×C4#54611yescharacteristic
557C2×C4#54711yescharacteristic
558C2×C4#54811yescharacteristic
559C2×C4#54811yescharacteristic
560C2×C4#54811yescharacteristic
561C2×C4D4×Q811yescharacteristic
562C2×C4D4×Q811yescharacteristic
563C2×C4#55011yescharacteristic
564C2×C4#55611yescharacteristic
565C2×C4#55611yescharacteristic
566C2×C4#55711yescharacteristic
567C2×C4#55711yescharacteristic
568C2×C4#55711yescharacteristic
569C2×C4#55711yescharacteristic
570C2×C4Q8211yescharacteristic
571C2×C4#55911yescharacteristic
572C8—44no—
573C8—82no—
574C22C2×#38511yescharacteristic
575C22C4×C22⋊Q811yescharacteristic
576C22#232311yescharacteristic
577C22#240711yescharacteristic
578C22#240911yescharacteristic
579C22#258111yescharacteristic
580C22#258211yescharacteristic
581C22C2×C4×Q1611yescharacteristic
582C22C2×#44011yescharacteristic
583C22C4×AGL(1,ℤ/8)11yescharacteristic
584C22C4×AGL(1,ℤ/8)11yescharacteristic
585C22C2×#45111yescharacteristic
586C22C2×#44911yescharacteristic
587C22#300311yescharacteristic
588C22#300511yescharacteristic
589C22C2×#47711yescharacteristic
590C22C2×#47711yescharacteristic
591C22C2×#47911yescharacteristic
592C22#307611yescharacteristic
593C22#307711yescharacteristic
594C22#307711yescharacteristic
595C22#307711yescharacteristic
596C22#307911yescharacteristic
597C22#310411yescharacteristic
598C22#310611yescharacteristic
599C22#310911yescharacteristic
600C22#311611yescharacteristic
601C22#323011yescharacteristic
602C22#323311yescharacteristic
603C22#323511yescharacteristic
604C22#323811yescharacteristic
605C22#324711yescharacteristic
606C22#324911yescharacteristic
607C22#325211yescharacteristic
608C22#325311yescharacteristic
609C4—24no—
610C4—24no—
611C4—42no—
612C4—42no—
613C4—44no—
614C4—48no—
615C4—42no—
616C4—82no—
617C4—84no—
618C4—82no—
619C4—82no—
620C4#186318yes—
621C4Q8×C4⋊C411yescharacteristic
622C4#235111yescharacteristic
623C4#236511yescharacteristic
624C4#236711yescharacteristic
625C4#244611yescharacteristic
626C4#244811yescharacteristic
627C4#245111yescharacteristic
628C4#245311yescharacteristic
629C2#1401711yescharacteristic
630C2C2×#186311yescharacteristic
631C2#2087311yescharacteristic
632C2C4×#47711yescharacteristic
633C2#2161411yescharacteristic
634C2#2211311yescharacteristic
635C2#2211711yescharacteristic
636C2#2415211yescharacteristic
637C2#2415511yescharacteristic
638C2#2430011yescharacteristic
639C2#2430211yescharacteristic
640C2#2927511yescharacteristic
641C2#2927711yescharacteristic
642C2#2928911yescharacteristic
643C2#2929011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
32. #1987C4semidirect
98. #389Q8semidirect
167. C4⋊Q8C4⋊C4semidirect
424. C4⋊C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#19872
C2×#19472
C4×#3892
#113282
#112932
#109072
#108852
#108492
#108322