Finite Group Databasebeta

← #381816

Subgroups — #381816

Order 512.

895 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23609C211yesmaximal, characteristic
2C2×#1938C211yesmaximal, characteristic
3#21047C211yesmaximal, characteristic
4#21021C211yesmaximal, characteristic, complemented
5C2×#1771C211yesmaximal, characteristic, complemented
6#11291C211yesmaximal, characteristic, complemented
7#11233C211yesmaximal, characteristic, complemented
8#11226C211yesmaximal, characteristic, complemented
9#11221C211yesmaximal, characteristic, complemented
10#11191C211yesmaximal, characteristic
11#11125C211yesmaximal, characteristic
12#11075C211yesmaximal, characteristic
13#10929C211yesmaximal, characteristic
14#10882C211yesmaximal, characteristic, complemented
15#10860C211yesmaximal, characteristic, complemented
16C22×#108C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C22×#104C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#416C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#408C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×#97C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30#2297C2211yescharacteristic
31#2296C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#2068C412yescomplemented
34#2067C412yescomplemented
35#2066C412yescomplemented
36#2065C412yescomplemented
37#1938—24no—
38#1924—24no—
39#1917—24no—
40#1771—24no—
41C2×#341C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44#1735C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#339C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55#1728C2211yescharacteristic
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
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×C4⋊C4—21no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#108—22no—
76C2×#108—41no—
77C2×#108—41no—
78#427—41no—
79#427—41no—
80C2×#107C2×C412yes—
81C2×#106C2×C412yes—
82#424—22no—
83#424—22no—
84#424—41no—
85#424—41no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8C2×C412yes—
91C2×C4⋊C8C2×C412yes—
92C2×#104—22no—
93C2×#104—22no—
94C2×#104—41no—
95C2×#104—41no—
96C2×#104—41no—
97C2×#104—41no—
98#416—22no—
99#416—22no—
100#416—41no—
101#416—41no—
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#103C2×C412yes—
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×#102C2×C412yes—
108#408—22no—
109#408—22no—
110#408—41no—
111#408—41no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97—41no—
117C2×#97—41no—
118C2×#97D414yes—
119C2×#97D414yes—
120C2×C4×C8—22no—
121C2×C4×C8—41no—
122C2×C4×C8—41no—
123#391C2×C412yes—
124#390C2×C412yes—
125#389C2×C412yes—
126#389C2×C412yes—
127#386—21no—
128#386—21no—
129#385—21no—
130#385—21no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C22⋊C4—22no—
135C2×#95—21no—
136#341—22no—
137#341—41no—
138#341—41no—
139#340C2×C412yes—
140#340C2×C412yes—
141#339—22no—
142#339—22no—
143#339—41no—
144#339—41no—
145#339—41no—
146#339—41no—
147#336—22no—
148#336—22no—
149#336—22no—
150#336—22no—
151#336—22no—
152#336—22no—
153#336—22no—
154#336—22no—
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—
165C2×#41—21no—
166C2×#41—21no—
167C22×Q8C22×C411yescharacteristic
168C22×D4—21no—
169C22×D4C22×C411yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4#4111yescharacteristic
174C23×C4#4111yescharacteristic
175C23×C4C2411yescharacteristic
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—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—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×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8C22×C412yes—
225C22×C8C22×C411yescharacteristic
226C22×C8C22×C411yescharacteristic
227C22×C8C2×D412yes—
228C22×C8C2×D412yes—
229C22×C8C2×D412yes—
230C22×C8C2×D412yes—
231C22×C8#4112yes—
232C22×C8#4112yes—
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—
242#117—21no—
243#117—21no—
244#117—21no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C412yes—
252C2×C4⋊C4C22×C412yes—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
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—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42C22×C412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42#4112yes—
279#108—44no—
280#107—42no—
281#106—42no—
282C4⋊C8—24no—
283C4⋊C8—24no—
284C4⋊C8—44no—
285C4⋊C8—44no—
286#104—44no—
287#104—44no—
288#103—22no—
289#103—22no—
290#103—42no—
291#102—22no—
292#102—22no—
293#102—42no—
294C22⋊C8—44no—
295C22⋊C8—44no—
296#97—28no—
297#97—44no—
298C4×C8—44no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304C24—21no—
305C24C2×#4111yescharacteristic
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—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4C4×D412yes—
368C22×C4C4×D412yes—
369C22×C4C23×C411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C4H(5,𝔽2)11yescharacteristic
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C42−1+411yescharacteristic
380C22×C42−1+411yescharacteristic
381C22×C42−1+411yescharacteristic
382C22×C42−1+411yescharacteristic
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
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—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
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—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8C4×D412yes—
440C2×C8C4×D412yes—
441C2×C8#12114yes—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—48no—
451C4⋊C4—82no—
452C4⋊C4—82no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—22no—
466C42—22no—
467C42—22no—
468C42—22no—
469C42—41no—
470C42—41no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C23—22no—
480C23—22no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23C2×C4×D411yescharacteristic
487C23#52511yescharacteristic
488C23#52911yescharacteristic
489C23#54011yescharacteristic
490C23#54111yescharacteristic
491C23#54211yescharacteristic
492C23#54211yescharacteristic
493Q8—41no—
494Q8—41no—
495Q8—41no—
496Q8—41no—
497D4—41no—
498D4—41no—
499D4—41no—
500D4—41no—
501D4—81no—
502D4—81no—
503D4—81no—
504D4—81no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C4×#4111yescharacteristic
544C2×C4#51811yescharacteristic
545C2×C4#51911yescharacteristic
546C2×C4#52011yescharacteristic
547C2×C4#52011yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4#53211yescharacteristic
550C2×C4#53411yescharacteristic
551C2×C4#53511yescharacteristic
552C2×C4#53611yescharacteristic
553C2×C4#53611yescharacteristic
554C2×C4#53711yescharacteristic
555C2×C4#53711yescharacteristic
556C2×C4#53711yescharacteristic
557C2×C4#53711yescharacteristic
558C2×C4#53811yescharacteristic
559C2×C4#53811yescharacteristic
560C2×C4#53911yescharacteristic
561C2×C4#53911yescharacteristic
562C2×C4#53911yescharacteristic
563C2×C4#53911yescharacteristic
564C2×C4#54111yescharacteristic
565C2×C4#54111yescharacteristic
566C2×C4#54211yescharacteristic
567C2×C4#54211yescharacteristic
568C2×C4#54211yescharacteristic
569C2×C4#54211yescharacteristic
570C8—22no—
571C8—22no—
572C8—24no—
573C8—42no—
574C8—42no—
575C8—81no—
576C8—81no—
577C8—81no—
578C8—81no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—81no—
588C22—81no—
589C22—82no—
590C22—82no—
591C22—81no—
592C22C4×#12111yescharacteristic
593C22#232311yescharacteristic
594C22#232411yescharacteristic
595C22#263011yescharacteristic
596C22#263711yescharacteristic
597C22#265011yescharacteristic
598C22#265411yescharacteristic
599C22#295612yes—
600C22#295712yes—
601C4—44no—
602C4—42no—
603C4—42no—
604C4—82no—
605C4—81no—
606C4—81no—
607C4—82no—
608C4—82no—
609C4—81no—
610C4#232311yescharacteristic
611C4#232511yescharacteristic
612C4#233211yescharacteristic
613C4#233411yescharacteristic
614C4#261811yescharacteristic
615C4#262011yescharacteristic
616C4#266211yescharacteristic
617C4#266411yescharacteristic
618C2—21no—
619C2—21no—
620C2—21no—
621C2—21no—
622C2—161no—
623C2#1431011yescharacteristic
624C2#2154711yescharacteristic
625C2#2155111yescharacteristic
626C2#2213212yes—
627C2#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
4. #21021C2semidirect
5. C2×#1771C2semidirect
6. #11291C2semidirect
7. #11233C2semidirect
8. #11226C2semidirect
9. #11221C2semidirect
14. #10882C2semidirect
15. #10860C2semidirect
33. #2068C4semidirect
34. #2067C4semidirect
35. #2066C4semidirect
36. #2065C4semidirect

Maximal subgroups

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

subgroupindex
#236092
C2×#19382
#210472
#210212
C2×#17712
#112912
#112332
#112262
#112212
#111912
#111252
#110752
#109292
#108822
#108602