Finite Group Databasebeta

← #371478

Subgroups — #371478

Order 512.

869 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
1#21058C211yesmaximal, characteristic
2#21047C211yesmaximal, characteristic
3#20367C211yesmaximal, characteristic, complemented
4#11291C211yesmaximal, characteristic, complemented
5#11221C211yesmaximal, characteristic, complemented
6#11191C211yesmaximal, characteristic
7#11125C211yesmaximal, characteristic
8#10933C211yesmaximal, characteristic
9#10930C211yesmaximal, characteristic
10#10929C211yesmaximal, characteristic
11#10911C211yesmaximal, characteristic, complemented
12#10859C211yesmaximal, characteristic, complemented
13#10819C211yesmaximal, characteristic, complemented
14#10808C211yesmaximal, characteristic, complemented
15#10696C211yesmaximal, characteristic, complemented
16C22×#108C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2297C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1928C412yescomplemented
32#1927C412yescomplemented
33#1926C412yescomplemented
34#1925C412yescomplemented
35#1924—22no—
36#1924—22no—
37#1746—22no—
38#1745—22no—
39C2×#341C2211yescharacteristic
40C2×#341C2211yescharacteristic
41C2×#341C2211yescharacteristic
42#1735C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49#1729C2211yescharacteristic
50#1729C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1728C2211yescharacteristic
55C2×#336C2211yescharacteristic
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C22×#41C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#108—22no—
75C2×#108—22no—
76C2×#108—41no—
77#427—41no—
78#427—41no—
79C2×#107C2×C412yes—
80C2×#106C2×C412yes—
81#424—41no—
82#424—41no—
83C2×C4⋊C8—22no—
84C2×#104—22no—
85C2×#104—22no—
86C2×#104—22no—
87C2×#104—22no—
88C2×#104—41no—
89C2×#104—41no—
90#416—22no—
91#416—22no—
92#416—41no—
93#416—41no—
94C2×#103C2×C411yescharacteristic
95C2×#103C2×C411yescharacteristic
96C2×#103C2×C412yes—
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C412yes—
100#408—22no—
101#408—22no—
102#408—41no—
103#408—41no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C4×M16—22no—
107C4×M16—22no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—21no—
111C2×#97—22no—
112C2×#97—22no—
113C2×#97—41no—
114C2×#97D412yes—
115C2×#97D412yes—
116C2×C4×C8—21no—
117C2×C4×C8—22no—
118C2×C4×C8—22no—
119C2×C4×C8—41no—
120#391C2×C412yes—
121#390C2×C412yes—
122#386—22no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C412yes—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128#376—21no—
129C2×#95—22no—
130C43—21no—
131#341—22no—
132#341—22no—
133#341—41no—
134#340C2×C412yes—
135#340C2×C412yes—
136#339—22no—
137#339—22no—
138#339—41no—
139#339—41no—
140#339—41no—
141#339—41no—
142#336—22no—
143#336—22no—
144#336—22no—
145#336—22no—
146#336—22no—
147#336—22no—
148#336—21no—
149#336—21no—
150#336—22no—
151#336C2×C412yes—
152#336C2×C412yes—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C22×Q8C22×C411yescharacteristic
166C22×D4—21no—
167C22×D4C22×C411yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16#4112yes—
196C2×M16#4112yes—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8C22×C412yes—
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C22×C8C2×D412yes—
225C22×C8C2×D412yes—
226C22×C8#4112yes—
227C22×C8#4112yes—
228C22×C8#4112yes—
229C22×C8#4112yes—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C22×C412yes—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42C22×C412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42#4112yes—
274C2×C42#4112yes—
275#108—44no—
276#107—42no—
277#106—42no—
278C4⋊C8—44no—
279#104—44no—
280#104—44no—
281#103—22no—
282#103—22no—
283#103—42no—
284#102—22no—
285#102—22no—
286#102—42no—
287C22⋊C8—44no—
288C22⋊C8—44no—
289#97—24no—
290#97—24no—
291#97—24no—
292#97—44no—
293C4×C8—24no—
294C4×C8—44no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—44no—
300C24—21no—
301C24C2×#4111yescharacteristic
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315#41—41no—
316#41—41no—
317#41—41no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C42−1+411yescharacteristic
377M16—22no—
378M16—22no—
379M16—22no—
380M16—22no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8#11712yes—
434C2×C8#11712yes—
435C2×C8#12314yes—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C42—22no—
453C42—22no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—82no—
473C42—82no—
474C23—22no—
475C23—22no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23C2×C4×D411yescharacteristic
482C23#52911yescharacteristic
483C23#52911yescharacteristic
484C23#52911yescharacteristic
485C23#52911yescharacteristic
486C23#54011yescharacteristic
487C23#54211yescharacteristic
488Q8—41no—
489Q8—41no—
490Q8—41no—
491Q8—41no—
492D4—41no—
493D4—41no—
494D4—41no—
495D4—41no—
496D4—81no—
497D4—81no—
498D4—81no—
499D4—81no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4C2×#11711yescharacteristic
536C2×C4C4×#4111yescharacteristic
537C2×C4C4×#4111yescharacteristic
538C2×C4C4×#4111yescharacteristic
539C2×C4#51811yescharacteristic
540C2×C4#52011yescharacteristic
541C2×C4C2×#12311yescharacteristic
542C2×C4#52511yescharacteristic
543C2×C4#53211yescharacteristic
544C2×C4#53211yescharacteristic
545C2×C4#53511yescharacteristic
546C2×C4#53711yescharacteristic
547C2×C4#55111yescharacteristic
548C2×C4#55111yescharacteristic
549C2×C4#55111yescharacteristic
550C2×C4#55111yescharacteristic
551C2×C4#55111yescharacteristic
552C2×C4#55211yescharacteristic
553C2×C4#55211yescharacteristic
554C2×C4#55211yescharacteristic
555C2×C4#55311yescharacteristic
556C2×C4#55311yescharacteristic
557C2×C4#55311yescharacteristic
558C2×C4#55511yescharacteristic
559C2×C4#55511yescharacteristic
560C2×C4#55511yescharacteristic
561C2×C4#55511yescharacteristic
562C2×C4#55611yescharacteristic
563C8—22no—
564C8—22no—
565C8—24no—
566C8—42no—
567C8—42no—
568C8—42no—
569C8—42no—
570C8—81no—
571C8—81no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—22no—
580C22—81no—
581C22—81no—
582C22—82no—
583C22—82no—
584C22—81no—
585C22#229311yescharacteristic
586C22C4×#12311yescharacteristic
587C22#232411yescharacteristic
588C22#252611yescharacteristic
589C22#252911yescharacteristic
590C22#253111yescharacteristic
591C22#253211yescharacteristic
592C22#295612yes—
593C22#295712yes—
594C4—44no—
595C4—42no—
596C4—42no—
597C4—82no—
598C4—81no—
599C4—81no—
600C4—82no—
601C4—82no—
602C4—81no—
603C4#234611yescharacteristic
604C4#235311yescharacteristic
605C4#237211yescharacteristic
606C4#237311yescharacteristic
607C4#249811yescharacteristic
608C4#250011yescharacteristic
609C4#260111yescharacteristic
610C4#261011yescharacteristic
611C2—21no—
612C2—21no—
613C2—21no—
614C2—21no—
615C2—161no—
616C2#1407411yescharacteristic
617C2#2129411yescharacteristic
618C2#2129711yescharacteristic
619C2#2159412yes—
620C2#2216712yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20367C2semidirect
4. #11291C2semidirect
5. #11221C2semidirect
11. #10911C2semidirect
12. #10859C2semidirect
13. #10819C2semidirect
14. #10808C2semidirect
15. #10696C2semidirect
31. #1928C4semidirect
32. #1927C4semidirect
33. #1926C4semidirect
34. #1925C4semidirect

Maximal subgroups

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

subgroupindex
#210582
#210472
#203672
#112912
#112212
#111912
#111252
#109332
#109302
#109292
#109112
#108592
#108192
#108082
#106962