Finite Group Databasebeta

← #208374

Subgroups — #208374

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35005C211yesmaximal, characteristic
2#21902C211yesmaximal, characteristic, complemented
3#13802C211yesmaximal, characteristic, complemented
4#9912C211yesmaximal, characteristic, complemented
5#9909C211yesmaximal, characteristic, complemented
6#9715C211yesmaximal, characteristic, complemented
7#9712C211yesmaximal, characteristic, complemented
8#9155C212yesmaximal, complemented
9#9147C212yesmaximal, complemented
10#8220C212yesmaximal, complemented
11C4×#327C212yesmaximal, complemented
12C2×#538—21no—
13#2857C2211yescharacteristic
14#2856C2211yescharacteristic
15#2847C2212yes—
16#2639—21no—
17#2617—21no—
18#2559—22no—
19#2557—22no—
20#2496C412yescomplemented
21#2491C412yescomplemented
22#2428C412yescomplemented
23#2422C412yescomplemented
24#2325—21no—
25C4×#121C2212yes—
26C2×#380C2211yescharacteristic
27C2×C4×C22⋊C4—21no—
28C2×C4×C22⋊C4C2211yescharacteristic
29C22×#95C2211yescharacteristic
30#1923—22no—
31#1923—22no—
32#1797C2211yescharacteristic
33#1797C2211yescharacteristic
34#1779C2212yes—
35C4×C22⋊C8C2212yes—
36C2×#327C2212yes—
37C2×#327C2212yes—
38C2×#327C2212yes—
39C2×#327C2212yes—
40#1436—21no—
41#1436—21no—
42#1434C2211yescharacteristic
43#1434C2211yescharacteristic
44C22⋊#95—21no—
45C4×#95C2211yescharacteristic
46C4×#95C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53#538—44no—
54#538—44no—
55C2×C4⋊Q8C2×C411yescharacteristic, complemented
56C2×#127C2×C411yescharacteristic, complemented
57C2×#126—22no—
58C2×#124—22no—
59C2×#123—22no—
60C2×C22⋊Q8—21no—
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—22no—
64C2×#121C2×C412yes—
65C2×#121C2×C412yes—
66C2×#121C2312yes—
67C2×#120—22no—
68C2×C4×D4—22no—
69C2×C4×D4C2×C412yescomplemented
70C22×C4⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8C2312yes—
83C2×C22⋊C8C2312yes—
84C2×C4×C8—22no—
85#400—22no—
86#397—22no—
87#397—22no—
88#394—22no—
89#394—22no—
90#394—22no—
91#393—22no—
92#392—22no—
93#391—21no—
94#390—21no—
95#390—21no—
96#390C2×C412yescomplemented
97#390C2×C412yescomplemented
98#389—21no—
99#389C2×C412yescomplemented
100#389C2×C412yescomplemented
101#388—22no—
102#388—22no—
103#387—22no—
104#386—22no—
105#386—22no—
106#386—22no—
107#386—22no—
108#386—22no—
109#385—22no—
110#384C2×C411yescharacteristic, complemented
111#384C2×C411yescharacteristic, complemented
112#380—21no—
113#380—22no—
114#380—22no—
115#380C2×C412yes—
116#380C2×C412yes—
117#380C2312yes—
118C4×C4⋊C4C2×C412yescomplemented
119C4×C22⋊C4—22no—
120C4×C22⋊C4—44no—
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4D412yes—
124C4×C22⋊C4D412yes—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95D411yescharacteristic
135C2×#95D411yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140#336—21no—
141#336—21no—
142#336—22no—
143#327—24no—
144#327—24no—
145#327—24no—
146#327—24no—
147C25C2×D411yescharacteristic
148C22×Q8—21no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4C22×C412yes—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—81no—
170C23×C4—81no—
171C23×C4—81no—
172C23×C4—81no—
173C23×C4C22×C412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2411yescharacteristic
181C22×C8—22no—
182C22×C8—22no—
183C4⋊Q8—81no—
184#127—81no—
185#126—48no—
186#124—48no—
187#123—48no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190#121—22no—
191#121—22no—
192#121—22no—
193#121—22no—
194#121—42no—
195#121—42no—
196#121—42no—
197#121—42no—
198#121—48no—
199#121—82no—
200#121—82no—
201#120—44no—
202#120—44no—
203C4×D4—44no—
204C4×D4—44no—
205C4×D4—82no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—81no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4C22⋊C412yes—
236C2×C22⋊C4C22⋊C412yes—
237C2×C22⋊C4C22⋊C412yes—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4D814yes—
240C2×C22⋊C4SD1614yes—
241C2×C42—22no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42—42no—
246C2×C42—81no—
247C2×C42—81no—
248C2×C42—81no—
249C2×C42—81no—
250C2×C42C22×C412yes—
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C4⋊C8—81no—
257C4⋊C8—81no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—82no—
263C22⋊C8—82no—
264C4×C8—82no—
265#95—24no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—42no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—81no—
281#95—81no—
282#95C22⋊C412yes—
283#95C22⋊C412yes—
284#95C22⋊C412yes—
285#95C22⋊C412yes—
286#95#4112yes—
287#95#4112yes—
288#95#4112yes—
289#95#4112yes—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24C2×C22⋊C412yes—
300C24C2×D812yes—
301C24C2×SD1612yes—
302C24C22×D411yescharacteristic
303C2×Q8—42no—
304C2×Q8—81no—
305C2×Q8—81no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—42no—
320C2×D4—81no—
321C2×D4—81no—
322C2×D4—81no—
323C2×D4—81no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—82no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—48no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—48no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—44no—
388C22×C4—82no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4—82no—
393C22×C4—82no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4C2×C22⋊C412yes—
400C22×C4C2×C22⋊C412yes—
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C412yes—
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×D812yes—
407C22×C4C2×D812yes—
408C22×C4C2×SD1612yes—
409C22×C4C2×SD1612yes—
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C2×#4111yescharacteristic
413C22×C4C2×#4111yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—84no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—44no—
437C22⋊C4—48no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C22⋊C4#10218yes—
451C22⋊C4#10218yes—
452C42—42no—
453C42—41no—
454C42—41no—
455C42—42no—
456C42—41no—
457C42—41no—
458C42—44no—
459C42—84no—
460C42—84no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23C2×#10214yes—
495C23C22×C22⋊C411yescharacteristic
496C23#52511yescharacteristic
497C23#52511yescharacteristic
498C23C22×D811yescharacteristic
499C23C22×SD1611yescharacteristic
500C23#57511yescharacteristic
501C23#57511yescharacteristic
502Q8—162no—
503D4—82no—
504D4—82no—
505D4—82no—
506D4—82no—
507D4—82no—
508D4—82no—
509D4—162no—
510D4—162no—
511D4—164no—
512D4—164no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—816no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—81no—
591C2×C4—164no—
592C2×C4C2×#10214yes—
593C2×C4C2×#10212yes—
594C2×C4C2×#10212yes—
595C2×C4C4×#4111yescharacteristic
596C2×C4C4×#4111yescharacteristic
597C2×C4#53211yescharacteristic
598C2×C4#53211yescharacteristic
599C8—82no—
600C8—82no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—44no—
617C22—44no—
618C22—44no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22#229711yescharacteristic
632C22#288311yescharacteristic
633C22#288311yescharacteristic
634C22C22×#10211yescharacteristic
635C22#289611yescharacteristic
636C22#290111yescharacteristic
637C22#290111yescharacteristic
638C22#310011yescharacteristic
639C22#310011yescharacteristic
640C22#310311yescharacteristic
641C22#310311yescharacteristic
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—24no—
647C4—22no—
648C4—22no—
649C4—44no—
650C4—42no—
651C4—41no—
652C4—41no—
653C4—42no—
654C4—82no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C2—21no—
661C2—21no—
662C2—21no—
663C2—21no—
664C2—44no—
665C2—162no—
666C2#1761311yescharacteristic
667C2#1761311yescharacteristic
668C2#1762311yescharacteristic
669C2#1762911yescharacteristic
670C2#2138611yescharacteristic
671C2#2141911yescharacteristic
672C2#2142211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35005C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21902C2semidirect
3. #13802C2semidirect
4. #9912C2semidirect
5. #9909C2semidirect
6. #9715C2semidirect
7. #9712C2semidirect
8. #9155C2semidirect
9. #9147C2semidirect
10. #8220C2semidirect
11. C4×#327C2semidirect
20. #2496C4semidirect
21. #2491C4semidirect
22. #2428C4semidirect
23. #2422C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
56. C2×#127C2×C4semidirect
69. C2×C4×D4C2×C4semidirect
96. #390C2×C4semidirect
97. #390C2×C4semidirect
99. #389C2×C4semidirect
100. #389C2×C4semidirect
110. #384C2×C4semidirect
111. #384C2×C4semidirect
118. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#350052
#219022
#138022
#99122
#99092
#97152
#97122
#91552
#91472
#82202
C4×#3272