Finite Group Databasebeta

← #205039

Subgroups — #205039

Order 512.

1205 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35238C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic, complemented
3C2×#1517C211yesmaximal, characteristic, complemented
4C2×#1521C211yesmaximal, characteristic, complemented
5C2×#1514C211yesmaximal, characteristic, complemented
6C2×#1512C211yesmaximal, characteristic, complemented
7#13781C211yesmaximal, characteristic, complemented
8#9716C212yesmaximal, complemented
9#9715C212yesmaximal, complemented
10#9713C212yesmaximal, complemented
11#9712C212yesmaximal, complemented
12C2×#532C2211yescharacteristic
13C2×#529C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2856C2211yescharacteristic
17#2856C2211yescharacteristic
18#2850C2211yescharacteristic
19#2850C2211yescharacteristic
20#2662—22no—
21#2567C411yescharacteristic, complemented
22#2566C411yescharacteristic, complemented
23#2566C411yescharacteristic, complemented
24#2563C411yescharacteristic, complemented
25#2548C411yescharacteristic, complemented
26#2544C411yescharacteristic, complemented
27#2544C411yescharacteristic, complemented
28#2541C411yescharacteristic, complemented
29#2430—22no—
30#2416—22no—
31#2324—22no—
32C2×#383C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1940—22no—
36#1939—22no—
37#1797C2212yes—
38#1797C2212yes—
39#1521—24no—
40#1517—24no—
41#1514—24no—
42#1512—24no—
43C2×#328C2212yes—
44C2×#328C2212yes—
45C2×#327C2212yes—
46C2×#327C2212yes—
47C2×#327C2212yes—
48C2×#327C2212yes—
49C2×#327C2212yes—
50C2×#327C2212yes—
51#1436—22no—
52#1436—22no—
53#1434C2212yes—
54C4×#95C2212yes—
55C22×#41—21no—
56#532—22no—
57#532—22no—
58#532—22no—
59#532—22no—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×#127C2×C411yescharacteristic
62C2×#127C2×C411yescharacteristic
63C2×#127C2×C411yescharacteristic
64#529—22no—
65#529—22no—
66#529—22no—
67#529—22no—
68C2×#126—21no—
69C2×#125C2×C411yescharacteristic
70C2×#124—21no—
71C2×#124C2×C411yescharacteristic
72C2×#124C2×C411yescharacteristic
73C2×#124C2×C411yescharacteristic
74C2×#123—21no—
75C2×C22⋊Q8C2311yescharacteristic
76C2×#121—21no—
77C2×#121—22no—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121C2311yescharacteristic
81C2×#121C2311yescharacteristic
82C2×#121C2311yescharacteristic
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4C2×C411yescharacteristic
89C2×#117—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4D412yes—
92C22×C42—22no—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#432—22no—
97#432—22no—
98#431—22no—
99#431—22no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C22⋊C8C2312yes—
105C2×C22⋊C8C2312yes—
106C2×#97—21no—
107C2×C4×C8—21no—
108#396—22no—
109#396—22no—
110#395—22no—
111#394—22no—
112#394—22no—
113#394—22no—
114#393—22no—
115#392—22no—
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#390—22no—
119#390—22no—
120#390—22no—
121#390C2×C411yescharacteristic
122#390C2×C411yescharacteristic
123#390C2×C411yescharacteristic
124#390C2×C411yescharacteristic
125#390C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#389—22no—
128#388—22no—
129#388—22no—
130#386—22no—
131#386—22no—
132#386—22no—
133#386—22no—
134#385—22no—
135#385—22no—
136#384C2×C411yescharacteristic
137#384C2×C411yescharacteristic
138#383—22no—
139#383D414yescomplemented
140#382—22no—
141#382C2×C411yescharacteristic
142#382C2×C411yescharacteristic
143#382D414yescomplemented
144#380—22no—
145C4×C4⋊C4C2×C411yescharacteristic
146C4×C4⋊C4C2×C411yescharacteristic
147C4×C4⋊C4C2×C411yescharacteristic
148C4×C4⋊C4C2×C411yescharacteristic
149C4×C4⋊C4D412yes—
150C4×C4⋊C4D412yes—
151C2×#95—21no—
152C2×#95—21no—
153C2×#95—21no—
154C2×#95—22no—
155C2×#95—22no—
156C2×#95—22no—
157C2×#95D412yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160#336—22no—
161#336—22no—
162#328—24no—
163#328—24no—
164#327—24no—
165#327—24no—
166#327—24no—
167#327—24no—
168#327—24no—
169#327—24no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—82no—
173C2×#41—82no—
174C22×Q8—21no—
175C22×Q8C22×C411yescharacteristic
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—21no—
183C22×D4C22×C411yescharacteristic
184C22×D4C22×C411yescharacteristic
185C22×D4C22×C411yescharacteristic
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4C2×D412yes—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2411yescharacteristic
196C22×C8—22no—
197C22×C8—22no—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200#127—22no—
201#127—22no—
202#127—81no—
203#127—81no—
204#126—42no—
205#126—42no—
206#125—22no—
207#125—22no—
208#124—22no—
209#124—22no—
210#124—42no—
211#124—42no—
212#124—81no—
213#124—81no—
214#123—42no—
215#123—42no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220#121—21no—
221#121—21no—
222#121—21no—
223#121—21no—
224#121—21no—
225#121—21no—
226#121—21no—
227#121—21no—
228#121—21no—
229#121—21no—
230#121—21no—
231#121—21no—
232#121—48no—
233#121—42no—
234#121—42no—
235#121—82no—
236#121—82no—
237#121—82no—
238#121—82no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—42no—
248C4×D4—42no—
249C4×D4—42no—
250C4×D4—42no—
251#117—42no—
252#117—42no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4C22⋊C412yes—
264C2×C4⋊C4C22⋊C412yes—
265C2×C4⋊C4C22⋊C412yes—
266C2×C4⋊C4C22⋊C412yes—
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C42—21no—
289C2×C42—44no—
290C2×C42—81no—
291C2×C42—81no—
292C2×C42—82no—
293C2×C42—82no—
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42#4112yes—
307C2×C42#4112yes—
308C4⋊C8—42no—
309C4⋊C8—42no—
310C4⋊C8—82no—
311C4⋊C8—82no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316#97—42no—
317C4×C8—42no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—44no—
328#95—82no—
329#95—82no—
330#95C22⋊C412yes—
331#95C22⋊C412yes—
332#95C22⋊C412yes—
333#95C22⋊C412yes—
334#95C2×D414yes—
335#95C2×D414yes—
336C24—21no—
337C24—21no—
338C24—21no—
339C24C22×D411yescharacteristic
340#41—84no—
341#41—84no—
342#41—88no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—42no—
348C2×Q8—82no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—44no—
378C2×D4—82no—
379C2×D4—82no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4—82no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4C2×C22⋊C412yes—
422C22×C4C2×C22⋊C412yes—
423C22×C4C2×C22⋊C412yes—
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C412yes—
429C22×C4#12112yes—
430C22×C4#12112yes—
431C22×C4#12112yes—
432C22×C4#12112yes—
433C22×C4C23×C411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C2×#4111yescharacteristic
438C22×C4C2×#4111yescharacteristic
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—84no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C22⋊C4—44no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—44no—
481C22⋊C4—82no—
482C22⋊C4—82no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C22⋊C4—82no—
486C22⋊C4—82no—
487C42—42no—
488C42—42no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—42no—
494C42—88no—
495C42C4×D412yes—
496C42C4×D412yes—
497C42C4×D412yes—
498C42C4×D412yes—
499C42#12412yes—
500C42#12412yes—
501C42#12712yes—
502C42#12712yes—
503C23—21no—
504C23—21no—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23C22×C22⋊C411yescharacteristic
522C23C2×#12111yescharacteristic
523C23C2×#12111yescharacteristic
524C23#57612yes—
525C23#57712yes—
526Q8—81no—
527Q8—81no—
528Q8—84no—
529D4—81no—
530D4—81no—
531D4—81no—
532D4—81no—
533D4—81no—
534D4—81no—
535D4—84no—
536D4—84no—
537D4—81no—
538D4—81no—
539D4—81no—
540D4—81no—
541D4—81no—
542D4—81no—
543D4—82no—
544D4—82no—
545D4—82no—
546D4—164no—
547D4—164no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—84no—
597C2×C4—84no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—82no—
602C2×C4—82no—
603C2×C4—82no—
604C2×C4—82no—
605C2×C4—82no—
606C2×C4—82no—
607C2×C4—82no—
608C2×C4#39012yes—
609C2×C4#39012yes—
610C2×C4#39012yes—
611C2×C4#39012yes—
612C2×C4#42114yes—
613C2×C4C2×C4×D411yescharacteristic
614C2×C4C2×C4×D411yescharacteristic
615C2×C4C2×#12411yescharacteristic
616C2×C4C2×#12711yescharacteristic
617C8—82no—
618C8—82no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—81no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—81no—
642C22C2×#39011yescharacteristic
643C22#288311yescharacteristic
644C22#288411yescharacteristic
645C22#289712yes—
646C22C2×#42112yes—
647C22#304111yescharacteristic
648C22#304311yescharacteristic
649C22#304512yes—
650C4—22no—
651C4—22no—
652C4—24no—
653C4—22no—
654C4—22no—
655C4—21no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—84no—
663C4—84no—
664C4—81no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4—81no—
669C2—21no—
670C2—22no—
671C2—21no—
672C2—161no—
673C2—161no—
674C2—161no—
675C2#1759712yes—
676C2#1760212yes—
677C2#2120911yescharacteristic
678C2#2121411yescharacteristic
679C2#2126311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35238C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21953C2semidirect
3. C2×#1517C2semidirect
4. C2×#1521C2semidirect
5. C2×#1514C2semidirect
6. C2×#1512C2semidirect
7. #13781C2semidirect
8. #9716C2semidirect
9. #9715C2semidirect
10. #9713C2semidirect
11. #9712C2semidirect
21. #2567C4semidirect
22. #2566C4semidirect
23. #2566C4semidirect
24. #2563C4semidirect
25. #2548C4semidirect
26. #2544C4semidirect
27. #2544C4semidirect
28. #2541C4semidirect
139. #383D4semidirect
143. #382D4semidirect

Maximal subgroups

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

subgroupindex
#352382
#219532
C2×#15172
C2×#15212
C2×#15142
C2×#15122
#137812
#97162
#97152
#97132
#97122