Finite Group Databasebeta

← #399187

Subgroups — #399187

Order 512.

1658 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2093C211yesmaximal, characteristic
3C2×#1880C211yesmaximal, characteristic
4C2×#1880C212yesmaximal
5C2×#1863C212yesmaximal, complemented
6#15148C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic, complemented
8#11346C211yesmaximal, characteristic, complemented
9#11328C212yesmaximal, complemented
10#11066C212yesmaximal
11#10789C211yesmaximal, characteristic, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×#107C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2212yes—
20C2×#399—21no—
21Q8×C4⋊C4—24no—
22Q8×C4⋊C4—24no—
23C2×#390C2211yescharacteristic
24C2×#390C2212yes—
25C2×#389—22no—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#384—21no—
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32#2093—24no—
33#1880—28no—
34#1880—24no—
35#1863—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44#1439—22no—
45#1439—21no—
46C4⋊#95C2211yescharacteristic
47C23×Q8C2311yescharacteristic
48C23×D4C2311yescharacteristic
49C22×SD16—22no—
50C22×SD16—24no—
51C22×SD16—24no—
52C22×SD16—22no—
53C22×SD16Q814yes—
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2311yescharacteristic
57C2×C4×Q8—24no—
58C2×C4×Q8—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—22no—
62C2×C4×Q8—24no—
63C2×C4×Q8—24no—
64C2×C4×Q8—42no—
65C2×C4×Q8—42no—
66C2×C4×Q8D414yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C42—21no—
84C22×C42—22no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C2×#107—24no—
88C2×#107—24no—
89C2×#107—42no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—22no—
94C2×#103D418yes—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—22no—
98C2×#102—24no—
99C2×#102—42no—
100#399—44no—
101#390—24no—
102#390—22no—
103#390—22no—
104#390—42no—
105#389—216no—
106#389—24no—
107#389—24no—
108#389—24no—
109#389—42no—
110#384—28no—
111#384—22no—
112#384D414yes—
113C4×C4⋊C4—28no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95D412yes—
121#340—24no—
122#340—24no—
123#340—22no—
124#340—24no—
125#336—24no—
126#336—24no—
127#336—22no—
128#336—22no—
129C25—21no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8C2×D412yes—
136C22×Q8C2×D414yes—
137C22×Q8C2×Q812yes—
138C22×Q8#4114yes—
139C22×Q8#4112yes—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C2×Q812yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C2×SD16—216no—
161C2×SD16—28no—
162C2×SD16—48no—
163C2×SD16—44no—
164C2×SD16—44no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8C2×D414yes—
181C22×C8C2×Q812yes—
182C4×Q8—216no—
183C4×Q8—28no—
184C4×Q8—28no—
185C4×Q8—48no—
186C4×Q8—48no—
187C4×Q8—48no—
188C4×Q8—48no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D414yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4#4114yes—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4—82no—
241C2×C42—28no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—28no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—44no—
253C2×C42—44no—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256#107—48no—
257#103—216no—
258#103—44no—
259#102—48no—
260#102—44no—
261#95—28no—
262#95—24no—
263#95—44no—
264#95—42no—
265#95—81no—
266C24—22no—
267C24—41no—
268C24—42no—
269C24—42no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—42no—
274C24—42no—
275C24—41no—
276C24—41no—
277C24C22×D411yescharacteristic
278C2×Q8—22no—
279C2×Q8—24no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8C22⋊Q814yes—
292C2×Q8C22⋊Q818yes—
293C2×Q8#12318yes—
294C2×Q8#12314yes—
295C2×Q8#12714yes—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—24no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—41no—
313C2×D4—42no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4#12012yes—
363C22×C4#12012yes—
364C22×C4#13512yes—
365C22×C4AGL(1,ℤ/8)12yes—
366C22×C4#13714yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D412yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×Q811yescharacteristic
371C22×C4C2×#4112yes—
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4112yes—
374C22×C4H(5,𝔽2)12yes—
375C22×C42−1+411yescharacteristic
376C22×C42−1+411yescharacteristic
377SD16—416no—
378C2×C8—24no—
379C2×C8—22no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—42no—
395C4⋊C4—28no—
396C4⋊C4—216no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—416no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—48no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—84no—
414C4⋊C4—88no—
415C4⋊C4—88no—
416C4⋊C4#12314yes—
417C22⋊C4—88no—
418C22⋊C4—84no—
419C22⋊C4—88no—
420C22⋊C4—84no—
421C42—216no—
422C42—22no—
423C42—22no—
424C42—24no—
425C42—416no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C23—42no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—81no—
436C23—82no—
437C23—81no—
438C23—82no—
439C23—81no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—81no—
444C23—82no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23—82no—
449C23—81no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23C2×#12011yescharacteristic
454C23#52512yes—
455C23D4212yes—
456C23#54811yescharacteristic
457C23D4×Q811yescharacteristic
458C23C2×#13511yescharacteristic
459C23C2×AGL(1,ℤ/8)11yescharacteristic
460C23C2×#13712yes—
461C23#57612yes—
462C23#57711yescharacteristic
463C23#57811yescharacteristic
464Q8—24no—
465Q8—28no—
466Q8—28no—
467Q8—24no—
468Q8—24no—
469Q8#39918yescomplemented
470D4—42no—
471D4—44no—
472D4—42no—
473D4—81no—
474D4—82no—
475D4—81no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#44914yes—
529C2×C4C2×C22⋊Q812yes—
530C2×C4C2×C22⋊Q811yescharacteristic
531C2×C4C2×#12311yescharacteristic
532C2×C4C2×#12312yes—
533C2×C4C2×#12311yescharacteristic
534C2×C4#52512yes—
535C2×C4C2×#12711yescharacteristic
536C2×C4#53511yescharacteristic
537C2×C4#53611yescharacteristic
538C2×C4#53912yes—
539C2×C4#53912yes—
540C2×C4D4212yes—
541C2×C4#54712yes—
542C2×C4#54712yes—
543C2×C4#54811yescharacteristic
544C2×C4D4×Q811yescharacteristic
545C2×C4#55312yes—
546C2×C4#55412yes—
547C8—42no—
548C8—44no—
549C8—81no—
550C22—82no—
551C22—84no—
552C22—82no—
553C22—84no—
554C22—82no—
555C22—162no—
556C22—162no—
557C22—161no—
558C22—161no—
559C22—161no—
560C22—161no—
561C22#240911yescharacteristic
562C22#243311yescharacteristic
563C22#244312yes—
564C22#244912yes—
565C22#284911yescharacteristic
566C22C2×#44911yescharacteristic
567C22#300411yescharacteristic
568C22#300811yescharacteristic
569C22#301311yescharacteristic
570C22#310812yes—
571C22#311212yes—
572C22#311912yes—
573C22#312412yes—
574C22#326612yes—
575C22#327012yes—
576C22#327412yes—
577C22#328712yes—
578C22#336411yescharacteristic
579C22#336811yescharacteristic
580C22#337512yes—
581C22#338111yescharacteristic
582C22#338512yes—
583C22#338611yescharacteristic
584C4—28no—
585C4—24no—
586C4—44no—
587C4—42no—
588C4—44no—
589C4—84no—
590C4—82no—
591C4—84no—
592C4—161no—
593C4C2×#39911yescharacteristic
594C4#244312yes—
595C4#244912yes—
596C4#248711yescharacteristic
597C4#250111yescharacteristic
598C4#262311yescharacteristic
599C2—162no—
600C2#1477711yescharacteristic
601C2#2432811yescharacteristic
602C2#2434111yescharacteristic
603C2#2575911yescharacteristic
604C2#2576611yescharacteristic
605C2#2616912yes—
606C2#2617912yes—
607C2#2637912yes—
608C2#2639712yes—
609C2#3300311yescharacteristic
610C2#3301211yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
1. C2×Q8×C4⋊C4C2semidirect
5. C2×#1863C2semidirect
7. #11357C2semidirect
8. #11346C2semidirect
9. #11328C2semidirect
11. #10789C2semidirect
469. Q8#399semidirect

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×#20932
C2×#18802
C2×#18632
#151482
#113572
#113462
#113282
#110662
#107892