Finite Group Databasebeta

← #461768

Subgroups — #461768

Order 512.

1354 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2075C211yesmaximal, characteristic, complemented
2C2×#1869C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic, complemented
4C2×#1807C211yesmaximal, characteristic, complemented
5#15181C212yesmaximal
6#11379C211yesmaximal, characteristic, complemented
7#11328C211yesmaximal, characteristic, complemented
8#11220C212yesmaximal, complemented
9#11200C212yesmaximal, complemented
10#11096C211yesmaximal, characteristic, complemented
11#11095C212yesmaximal, complemented
12C24×C8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C2×#389C2212yes—
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25#2075—24no—
26#1869—24no—
27#1808—24no—
28#1807—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1442—22no—
39#1439—22no—
40#1439—22no—
41#1438—22no—
42C4⋊#95C2212yes—
43C23×D4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8—22no—
47C23×C8—22no—
48C23×C8—24no—
49C23×C8—42no—
50C23×C8D414yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—42no—
58C2×C4×D4—42no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#107—22no—
76C2×#107—42no—
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—42no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—42no—
87#389—42no—
88#389—42no—
89#380—24no—
90#380—24no—
91#380—24no—
92#380—24no—
93#380—42no—
94#380—42no—
95#380—42no—
96#380—42no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104#340—22no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#340—42no—
110#340D414yescomplemented
111#336—22no—
112#336—22no—
113C25C2×D411yescharacteristic
114C25C2×D411yescharacteristic
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—24no—
118C22×D4—22no—
119C22×D4—42no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—28no—
155C22×C8—28no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—44no—
167C22×C8—48no—
168C22×C8—48no—
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8#4112yes—
173C22×C8#4114yes—
174C4×D4—44no—
175C4×D4—44no—
176C4×D4—48no—
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—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—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C2×D414yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C22⋊C4—82no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42—22no—
226#107—44no—
227#107—44no—
228#106—48no—
229C22⋊C8—44no—
230C22⋊C8—44no—
231C22⋊C8—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—82no—
239C24—22no—
240C24—21no—
241C24—21no—
242C24—22no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—21no—
249C24—22no—
250C24—21no—
251C24—22no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—21no—
256C24—21no—
257C24—22no—
258C24#12112yes—
259C24#13512yes—
260C24#13714yes—
261C24C22×D411yescharacteristic
262C2×D4—28no—
263C2×D4—48no—
264C2×D4—48no—
265C2×D4—44no—
266C2×D4—44no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
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#12112yes—
350C22×C4#13512yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4112yes—
355C22×C4C2×#4111yescharacteristic
356C22×C4H(5,𝔽2)12yes—
357C22×C4H(5,𝔽2)12yes—
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C42−1+412yes—
360C22×C42−1+411yescharacteristic
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—416no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8#12114yes—
381C2×C8#12314yes—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—88no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—88no—
403C22⋊C4—88no—
404C22⋊C4—88no—
405C22⋊C4—88no—
406C42—41no—
407C42—41no—
408C42—42no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—24no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—22no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—42no—
427C23—41no—
428C23—42no—
429C23—41no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—41no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—42no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—41no—
449C23—42no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23#47014yes—
455C23C2×#12111yescharacteristic
456C23#53412yes—
457C23D4211yescharacteristic
458C23#54712yes—
459C23#54711yescharacteristic
460C23C2×#13511yescharacteristic
461C23C2×#13511yescharacteristic
462C23C2×#13712yes—
463C23#57611yescharacteristic
464C23#57611yescharacteristic
465C23#57712yes—
466D4—416no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—82no—
507C2×C4—82no—
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×C4C2×#12111yescharacteristic
528C2×C4C2×#12311yescharacteristic
529C2×C4#53411yescharacteristic
530C2×C4#53511yescharacteristic
531C2×C4#53611yescharacteristic
532C2×C4#53712yes—
533C2×C4#53711yescharacteristic
534C2×C4#53812yes—
535C2×C4#54012yes—
536C2×C4#54212yes—
537C2×C4#54212yes—
538C2×C4#54612yes—
539C2×C4#54811yescharacteristic
540C2×C4#55112yes—
541C2×C4#55212yes—
542C2×C4#55311yescharacteristic
543C2×C4#56212yes—
544C2×C4#56312yes—
545C8—24no—
546C8—24no—
547C8—81no—
548C8—81no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—44no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—44no—
558C22—44no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—42no—
563C22—44no—
564C22—42no—
565C22—82no—
566C22—81no—
567C22—81no—
568C22—81no—
569C22—82no—
570C22—81no—
571C22#241611yescharacteristic
572C22#248611yescharacteristic
573C22#261711yescharacteristic
574C22#267112yes—
575C22#271712yes—
576C22#305011yescharacteristic
577C22C2×#47011yescharacteristic
578C22#305811yescharacteristic
579C22#306011yescharacteristic
580C22#319012yes—
581C22#319012yes—
582C22#319812yes—
583C22#319812yes—
584C22#326612yes—
585C22#328011yescharacteristic
586C22#328711yescharacteristic
587C22#333712yes—
588C22#333911yescharacteristic
589C22#334012yes—
590C22#334111yescharacteristic
591C22#334412yes—
592C22#334612yes—
593C22#334612yes—
594C4—42no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—84no—
599C4—162no—
600C4—162no—
601C4—162no—
602C4#241211yescharacteristic
603C4#262011yescharacteristic
604C4#262411yescharacteristic
605C4#270212yes—
606C4#274112yes—
607C4#281411yescharacteristic
608C2—44no—
609C2—82no—
610C2#1640111yescharacteristic
611C2#2477811yescharacteristic
612C2#2478311yescharacteristic
613C2#2765911yescharacteristic
614C2#2766611yescharacteristic
615C2#2973311yescharacteristic
616C2#2973611yescharacteristic
617C2#3056512yes—
618C2#3057012yes—
619C2#3226512yes—
620C2#3226612yes—
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×#2075C2semidirect
3. C2×#1808C2semidirect
4. C2×#1807C2semidirect
6. #11379C2semidirect
7. #11328C2semidirect
8. #11220C2semidirect
9. #11200C2semidirect
10. #11096C2semidirect
11. #11095C2semidirect
110. #340D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
C2×#18692
C2×#18082
C2×#18072
#151812
#113792
#113282
#112202
#112002
#110962
#110952