Finite Group Databasebeta

← #399133

Subgroups — #399133

Order 512.

1572 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2071C211yesmaximal, characteristic
3C2×#1880C211yesmaximal, characteristic
4C2×#1880C212yesmaximal
5#15147C211yesmaximal, characteristic
6#11371C211yesmaximal, characteristic, complemented
7#11346C211yesmaximal, characteristic, complemented
8#11292C212yesmaximal, complemented
9#11066C212yesmaximal
10#10789C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C2×#397—21no—
20Q8×C4⋊C4—24no—
21Q8×C4⋊C4—24no—
22C2×#390C2211yescharacteristic
23C2×#390C2212yes—
24C2×#389—22no—
25C2×#389C2212yes—
26C2×#384—21no—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2071—24no—
31#1880—28no—
32#1880—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1440—22no—
43#1439—21no—
44C4⋊#95C2212yes—
45C4⋊#95C2211yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C22×SD16—22no—
49C22×SD16—22no—
50C22×SD16—24no—
51C22×SD16—24no—
52C22×SD16D414yes—
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4×Q8—24no—
57C2×C4×Q8—24no—
58C2×C4×Q8—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8—22no—
61C2×C4×Q8—24no—
62C2×C4×Q8—24no—
63C2×C4×Q8—42no—
64C2×C4×Q8—42no—
65C2×C4×Q8D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—21no—
81C22×C42—21no—
82C22×C42—22no—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85C2×#103—24no—
86C2×#103—22no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—22no—
93C2×#102—24no—
94C2×#102—42no—
95#397—44no—
96#390—24no—
97#390—22no—
98#390—22no—
99#390—42no—
100#389—216no—
101#389—24no—
102#389—24no—
103#384—28no—
104#384—22no—
105#384D414yes—
106C4×C4⋊C4—28no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95D412yes—
115#340—24no—
116#340—24no—
117#340—22no—
118#340—24no—
119#340—42no—
120#336—24no—
121#336—22no—
122#336—22no—
123#336—42no—
124C25—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8C2×D412yes—
131C22×Q8C2×D412yes—
132C22×Q8#4112yes—
133C22×Q8#4114yes—
134C22×Q8#4114yes—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4C2×D412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C2×SD16—28no—
156C2×SD16—216no—
157C2×SD16—44no—
158C2×SD16—44no—
159C2×SD16—48no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8C2×D412yes—
177C4×Q8—216no—
178C4×Q8—28no—
179C4×Q8—28no—
180C4×Q8—48no—
181C4×Q8—48no—
182C4×Q8—48no—
183C4×Q8—48no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4SD1614yescomplemented
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4#4114yes—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C42—28no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—28no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247#103—48no—
248#103—44no—
249#102—48no—
250#102—44no—
251#95—28no—
252#95—24no—
253#95—44no—
254#95—42no—
255#95—81no—
256#95—82no—
257C24—22no—
258C24—41no—
259C24—42no—
260C24—41no—
261C24—42no—
262C24—41no—
263C24—41no—
264C24—42no—
265C24—42no—
266C24—41no—
267C24—41no—
268C24C22×D411yescharacteristic
269C2×Q8—22no—
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8#12114yes—
283C2×Q8#12318yes—
284C2×Q8#12314yes—
285C2×Q8#12414yes—
286C2×Q8#12618yes—
287C2×D4—22no—
288C2×D4—24no—
289C2×D4—22no—
290C2×D4—24no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4#12012yes—
354C22×C4#12012yes—
355C22×C4C2×SD1614yes—
356C22×C4C2×SD1612yes—
357C22×C4#13712yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4112yes—
363C22×C4C2×#4112yes—
364C22×C4C2×#4112yes—
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C42−1+412yes—
368SD16—416no—
369C2×C8—22no—
370C2×C8—24no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C4⋊C4—216no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—28no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—416no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—84no—
405C4⋊C4—88no—
406C4⋊C4#12314yes—
407C22⋊C4—88no—
408C22⋊C4—84no—
409C22⋊C4—88no—
410C22⋊C4—84no—
411C42—216no—
412C42—22no—
413C42—22no—
414C42—24no—
415C42—416no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—42no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—81no—
426C23—82no—
427C23—81no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23—82no—
434C23—82no—
435C23—81no—
436C23—82no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—82no—
443C23C2×#12011yescharacteristic
444C23#52512yes—
445C23D4211yescharacteristic
446C23#54611yescharacteristic
447C23#54812yes—
448C23C22×SD1611yescharacteristic
449C23C2×#13711yescharacteristic
450C23#57512yes—
451C23#57612yes—
452C23#57711yescharacteristic
453C23#57811yescharacteristic
454Q8—24no—
455Q8—28no—
456Q8—28no—
457Q8—24no—
458Q8—24no—
459Q8#39718yescomplemented
460D4—42no—
461D4—44no—
462D4—42no—
463D4—81no—
464D4—82no—
465D4—81no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4#44814yes—
519C2×C4C2×#12111yescharacteristic
520C2×C4C2×#12311yescharacteristic
521C2×C4C2×#12312yes—
522C2×C4C2×#12311yescharacteristic
523C2×C4#52512yes—
524C2×C4C2×#12411yescharacteristic
525C2×C4C2×#12612yes—
526C2×C4#53511yescharacteristic
527C2×C4#53611yescharacteristic
528C2×C4#53912yes—
529C2×C4#53912yes—
530C2×C4D4211yescharacteristic
531C2×C4#54611yescharacteristic
532C2×C4#54712yes—
533C2×C4#54812yes—
534C2×C4#55212yes—
535C2×C4#55312yes—
536C2×C4#55312yes—
537C8—42no—
538C8—82no—
539C8—81no—
540C22—84no—
541C22—84no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—161no—
546C22—161no—
547C22—162no—
548C22—162no—
549C22—161no—
550C22—161no—
551C22#240811yescharacteristic
552C22#243611yescharacteristic
553C22#244312yes—
554C22#248112yes—
555C22#284911yescharacteristic
556C22C2×#44811yescharacteristic
557C22#300511yescharacteristic
558C22#300811yescharacteristic
559C22#301311yescharacteristic
560C22#310312yes—
561C22#311712yes—
562C22#311912yes—
563C22#312412yes—
564C22#326812yes—
565C22#327611yescharacteristic
566C22D4×SD1611yescharacteristic
567C22#329811yescharacteristic
568C22#330912yes—
569C22#332911yescharacteristic
570C22#336212yes—
571C22#337112yes—
572C22#337512yes—
573C22#337712yes—
574C4—28no—
575C4—24no—
576C4—44no—
577C4—42no—
578C4—44no—
579C4—82no—
580C4—84no—
581C4—162no—
582C4—161no—
583C4C2×#39711yescharacteristic
584C4#244312yes—
585C4#248112yes—
586C4#248611yescharacteristic
587C4#250411yescharacteristic
588C4#262311yescharacteristic
589C2—162no—
590C2#1477611yescharacteristic
591C2#2423911yescharacteristic
592C2#2425011yescharacteristic
593C2#2593211yescharacteristic
594C2#2594111yescharacteristic
595C2#2616312yes—
596C2#2617712yes—
597C2#2755412yes—
598C2#2757212yes—
599C2#3300511yescharacteristic
600C2#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
6. #11371C2semidirect
7. #11346C2semidirect
8. #11292C2semidirect
10. #10789C2semidirect
11. #10789C2semidirect
217. C2×C4⋊C4SD16semidirect
459. Q8#397semidirect

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×#20712
C2×#18802
#151472
#113712
#113462
#112922
#110662
#107892