Finite Group Databasebeta

← #413432

Subgroups — #413432

Order 512.

1332 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2077C212yesmaximal, complemented
2C2×#1954C211yesmaximal, characteristic
3C2×C4×#106C211yesmaximal, characteristic, complemented
4#13905C211yesmaximal, characteristic
5#13905C211yesmaximal, characteristic
6#11200C212yesmaximal, complemented
7#11199C212yesmaximal, complemented
8#11095C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10842C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#2077—28no—
28#1954—24no—
29C4×#106—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38#1445—21no—
39#1445—21no—
40#1438—21no—
41#1438—21no—
42#1437—22no—
43#1437—22no—
44C23×D4D411yescharacteristic
45C24×C4C2312yes—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D412yes—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70C2×#106—22no—
71C2×#106—22no—
72C2×#106—22no—
73C2×#106—22no—
74C2×#106—22no—
75C2×#106D414yescomplemented
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—24no—
83C2×C22⋊C8—42no—
84C2×C22⋊C8—42no—
85C2×C4×C8—22no—
86C2×C4×C8—22no—
87C2×C4×C8D414yes—
88#382—22no—
89#382—22no—
90#382—22no—
91#382—22no—
92#380—24no—
93#380—24no—
94#380—24no—
95#380—24no—
96#380—42no—
97#380—42no—
98#380—42no—
99#380—42no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114#340—24no—
115#340—22no—
116#340—22no—
117#340—24no—
118#340—24no—
119#340—42no—
120#340—42no—
121#340—42no—
122C25C2×D412yes—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—42no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8C2×D412yes—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C22×C8#4112yes—
173C4×D4—44no—
174C4×D4—44no—
175C4×D4—44no—
176C4×D4—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
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—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4#4112yes—
215C2×C4⋊C4#4112yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—82no—
234C2×C42—82no—
235C2×C42C2×D412yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241#106—28no—
242#106—28no—
243C4⋊C8—44no—
244C22⋊C8—48no—
245C22⋊C8—48no—
246C4×C8—28no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—42no—
255#95—42no—
256#95—42no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24#13514yes—
272C24C22×D411yescharacteristic
273C2×D4—28no—
274C2×D4—48no—
275C2×D4—48no—
276C2×D4—44no—
277C2×D4—44no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
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—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
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#12112yes—
352C22×C4#12112yes—
353C22×C4#12312yes—
354C22×C4#12312yes—
355C22×C4#12312yes—
356C22×C4#12312yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C42−1+411yescharacteristic
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8#12114yes—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C22⋊C4—48no—
418C22⋊C4—88no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C42—84no—
423C42—84no—
424C42#12314yes—
425C42#12414yes—
426C42#12614yes—
427C42#12614yes—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23C2×#12111yescharacteristic
464C23C2×#12311yescharacteristic
465C23C2×#12311yescharacteristic
466C23D4211yescharacteristic
467C23#54611yescharacteristic
468C23#54611yescharacteristic
469C23#54711yescharacteristic
470C23C2×#13512yes—
471C23#57512yes—
472C23#57712yes—
473C23#57712yes—
474D4—416no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
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—82no—
531C2×C4—82no—
532C2×C4#39714yes—
533C2×C4#39714yes—
534C2×C4C2×#12111yescharacteristic
535C2×C4C2×#12311yescharacteristic
536C2×C4#52511yescharacteristic
537C2×C4C2×#12411yescharacteristic
538C2×C4C2×#12611yescharacteristic
539C2×C4C2×#12611yescharacteristic
540C2×C4#52911yescharacteristic
541C2×C4#52911yescharacteristic
542C2×C4#53211yescharacteristic
543C2×C4#53211yescharacteristic
544C2×C4#53511yescharacteristic
545C2×C4#53711yescharacteristic
546C2×C4#54011yescharacteristic
547C2×C4#54011yescharacteristic
548C2×C4#54211yescharacteristic
549C2×C4#54211yescharacteristic
550C2×C4#54711yescharacteristic
551C2×C4#55011yescharacteristic
552C2×C4#55011yescharacteristic
553C2×C4#55311yescharacteristic
554C2×C4#55311yescharacteristic
555C2×C4#55511yescharacteristic
556C2×C4#55611yescharacteristic
557C2×C4#55611yescharacteristic
558C2×C4#55911yescharacteristic
559C2×C4#55911yescharacteristic
560C2×C4#55911yescharacteristic
561C2×C4#55911yescharacteristic
562C8—24no—
563C8—42no—
564C8—82no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22C2×#39711yescharacteristic
581C22#244511yescharacteristic
582C22#244511yescharacteristic
583C22#244711yescharacteristic
584C22#244711yescharacteristic
585C22#253511yescharacteristic
586C22#253711yescharacteristic
587C22#305011yescharacteristic
588C22#305411yescharacteristic
589C22#305711yescharacteristic
590C22#305711yescharacteristic
591C22#309112yes—
592C22#309112yes—
593C22#309912yes—
594C22#309912yes—
595C22#327512yes—
596C22#328212yes—
597C22#329612yes—
598C22#330112yes—
599C22#332312yes—
600C22#332612yes—
601C22#333712yes—
602C22#335112yes—
603C4—24no—
604C4—24no—
605C4—82no—
606C4—82no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4—162no—
611C4—162no—
612C4#251211yescharacteristic
613C4#251211yescharacteristic
614C4#251311yescharacteristic
615C4#251311yescharacteristic
616C4#254811yescharacteristic
617C4#259211yescharacteristic
618C4#262411yescharacteristic
619C4#285411yescharacteristic
620C2—84no—
621C2#1514711yescharacteristic
622C2#2363811yescharacteristic
623C2#2364011yescharacteristic
624C2#2623912yes—
625C2#2624012yes—
626C2#2630412yes—
627C2#2630712yes—
628C2#2814411yescharacteristic
629C2#2814711yescharacteristic
630C2#2822411yescharacteristic
631C2#2822911yescharacteristic
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×#2077C2semidirect
3. C2×C4×#106C2semidirect
6. #11200C2semidirect
7. #11199C2semidirect
8. #11095C2semidirect
9. #10850C2semidirect
10. #10850C2semidirect
11. #10842C2semidirect
75. C2×#106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20772
C2×#19542
C2×C4×#1062
#139052
#112002
#111992
#110952
#108502
#108422