Finite Group Databasebeta

← #450364

Subgroups — #450364

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1853C212yesmaximal, complemented
2C2×#1853C212yesmaximal, complemented
3#13872C211yesmaximal, characteristic
4#13872C211yesmaximal, characteristic
5#11378C212yesmaximal, complemented
6#11257C212yesmaximal, complemented
7#11200C212yesmaximal, complemented
8#11199C212yesmaximal, complemented
9#10964C211yesmaximal, characteristic
10C22×C4×D4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#106C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2212yes—
17C2×#386—21no—
18C2×#386—21no—
19C2×#380C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×#380C2212yes—
22C2×#380C2212yes—
23#1853—28no—
24#1853—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33#1441—21no—
34#1441—21no—
35#1439—21no—
36#1439—21no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C22⋊#95—21no—
40C22⋊#95—21no—
41C23×D4D411yescharacteristic
42C24×C4—21no—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4×D4—24no—
49C2×C4×D4—24no—
50C2×C4×D4—41no—
51C2×C4×D4—41no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—41no—
57C22×C4⋊C4—41no—
58C22×C4⋊C4—41no—
59C22×C4⋊C4—41no—
60C22×C4⋊C4—42no—
61C22×C4⋊C4—42no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—42no—
77C2×#106—42no—
78C2×#106—42no—
79C2×#106D418yescomplemented
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86#386—44no—
87#386—44no—
88#380—24no—
89#380—24no—
90#380—24no—
91#380—24no—
92#380—42no—
93#380—42no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#336—41no—
109#336—41no—
110C25C2×D411yescharacteristic
111C25C2×D411yescharacteristic
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—41no—
117C22×D4—41no—
118C22×D4SD1614yes—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—44no—
138C23×C4—44no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4#4114yes—
152C23×C4#4114yes—
153C23×C4C2411yescharacteristic
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×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×D414yes—
167C22×C8#4114yes—
168C4×D4—48no—
169C4×D4—48no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—48no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D414yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—21no—
228C2×C42—21no—
229#106—48no—
230#106—48no—
231C22⋊C8—216no—
232C22⋊C8—216no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—42no—
237#95—42no—
238#95—44no—
239#95—82no—
240#95—82no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—21no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—22no—
257C24#12312yes—
258C24#12312yes—
259C24C2×SD1612yes—
260C24C2×SD1612yes—
261C24AGL(1,ℤ/8)12yes—
262C24#13712yes—
263C24C22×D411yescharacteristic
264C2×D4—28no—
265C2×D4—28no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—44no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—48no—
325C22×C4—48no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—88no—
353C22×C4—88no—
354C22×C4—88no—
355C22×C4—88no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4#12312yes—
361C22×C4#12312yes—
362C22×C4C2×SD1612yes—
363C22×C4C22×D412yes—
364C22×C4C2×#4112yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4112yes—
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C4H(5,𝔽2)12yes—
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C42−1+412yes—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—44no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—44no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C42—42no—
411C42—42no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—24no—
416C23—22no—
417C23—24no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23#48114yes—
457C23#48314yes—
458C23C2×#12311yescharacteristic
459C23C2×#12311yescharacteristic
460C23#53411yescharacteristic
461C23D4212yes—
462C23#54712yes—
463C23C22×SD1611yescharacteristic
464C23C2×AGL(1,ℤ/8)11yescharacteristic
465C23C2×#13711yescharacteristic
466C23#57511yescharacteristic
467C23#57712yes—
468C23#57712yes—
469D4—416no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—816no—
507C2×C4—816no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
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—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4#52512yes—
531C2×C4#52512yes—
532C2×C4#53412yes—
533C2×C4#53912yes—
534C2×C4#53912yes—
535C2×C4#53912yes—
536C2×C4D4211yescharacteristic
537C2×C4#54812yes—
538C2×C4#54812yes—
539C2×C4#55112yes—
540C2×C4#55112yes—
541C2×C4#55312yes—
542C2×C4#55312yes—
543C2×C4#55911yescharacteristic
544C2×C4H(3,𝔽4)11yescharacteristic
545C2×C4#56311yescharacteristic
546C8—44no—
547C8—44no—
548C22—28no—
549C22—28no—
550C22—24no—
551C22—24no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22#244312yes—
571C22#244312yes—
572C22#248311yescharacteristic
573C22#267911yescharacteristic
574C22#272411yescharacteristic
575C22C2×#48311yescharacteristic
576C22C2×#48111yescharacteristic
577C22#309411yescharacteristic
578C22#309511yescharacteristic
579C22#309912yes—
580C22#309912yes—
581C22#318711yescharacteristic
582C22#319311yescharacteristic
583C22#320011yescharacteristic
584C22#320011yescharacteristic
585C22#326512yes—
586C22#326612yes—
587C22#327512yes—
588C22D4×SD1612yes—
589C22#333512yes—
590C22#334412yes—
591C22#334512yes—
592C22#335112yes—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—84no—
597C4—84no—
598C4—81no—
599C4—162no—
600C4—162no—
601C4#246312yes—
602C4#260012yes—
603C4#272812yes—
604C4#278212yes—
605C2—44no—
606C2—44no—
607C2#1596611yescharacteristic
608C2#2612812yes—
609C2#2613012yes—
610C2#2614312yes—
611C2#2614912yes—
612C2#2760111yescharacteristic
613C2#2760911yescharacteristic
614C2#3105811yescharacteristic
615C2#3106511yescharacteristic
616C2#3245911yescharacteristic
617C2#3246811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1853C2semidirect
2. C2×#1853C2semidirect
5. #11378C2semidirect
6. #11257C2semidirect
7. #11200C2semidirect
8. #11199C2semidirect
79. C2×#106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#18532
#138722
#113782
#112572
#112002
#111992
#109642