Finite Group Databasebeta

← #196911

Subgroups — #196911

Order 512.

1707 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35102C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic, complemented
3#21790C211yesmaximal, characteristic, complemented
4C2×#1512C211yesmaximal, characteristic, complemented
5C2×#1509C211yesmaximal, characteristic, complemented
6#9155C212yesmaximal, complemented
7#9151C212yesmaximal, complemented
8#9147C212yesmaximal, complemented
9#9143C212yesmaximal, complemented
10#8021C211yesmaximal, characteristic, complemented
11#8017C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C22×C4×D4—21no—
14C22×C4×D4C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#432C2211yescharacteristic
17#2848C2212yes—
18#2847C2212yes—
19#2623—21no—
20#2613—21no—
21#2603—22no—
22#2598—22no—
23#2562C412yescomplemented
24#2540C412yescomplemented
25#2536C412yescomplemented
26#2535C412yescomplemented
27#2349—24no—
28#2349—24no—
29#2321—22no—
30C2×#389—21no—
31C2×#385—22no—
32#2283C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C22×#95C2212yes—
37#1955—22no—
38#1955—22no—
39#1779C2212yes—
40#1779C2212yes—
41#1512—24no—
42#1509—24no—
43C2×#328C2211yescharacteristic
44C2×#328C2211yescharacteristic
45C2×#328C2212yes—
46C2×#327C2211yescharacteristic
47C2×#327C2211yescharacteristic
48C2×#327C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C23×D4D411yescharacteristic
54C24×C4—22no—
55C24×C4C2312yes—
56#529—22no—
57#529—22no—
58#529—22no—
59#529—22no—
60C2×#126—21no—
61C2×#125C2×C411yescharacteristic, complemented
62C2×#124C2×C411yescharacteristic, complemented
63C2×#123—21no—
64C2×C22⋊Q8C2×C414yescomplemented
65C2×C22⋊Q8C2311yescharacteristic
66C2×#121C2×C414yescomplemented
67C2×#121C2311yescharacteristic
68C2×#120—22no—
69C2×C4×Q8C2×C411yescharacteristic, complemented
70C2×C4×D4—28no—
71C2×C4×D4—21no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—48no—
77C2×C4×D4C2×C412yes—
78C2×C4×D4C2×C412yes—
79C2×C4×D4C2×C411yescharacteristic, complemented
80C2×C4×D4D412yes—
81C2×C4×D4D412yes—
82C2×#117—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4D411yescharacteristic
87C22×C22⋊C4—22no—
88C22×C22⋊C4D412yes—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#432—22no—
92#432—22no—
93#432—22no—
94#432—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8C2311yescharacteristic
100C2×C22⋊C8C2311yescharacteristic
101C2×C22⋊C8C2312yes—
102C2×#97—21no—
103C2×#97—21no—
104#400—22no—
105#399—22no—
106#398—22no—
107#397—22no—
108#394—22no—
109#394—22no—
110#392—22no—
111#392—22no—
112#390—24no—
113#389—28no—
114#388—24no—
115#388—24no—
116#386—22no—
117#386—24no—
118#386—24no—
119#385—216no—
120#385—22no—
121#385—24no—
122#383—21no—
123#382—22no—
124#382C2×C412yescomplemented
125#382D414yescomplemented
126#381—21no—
127#380—24no—
128#380—24no—
129#380C2×C414yescomplemented
130#380C2×C414yescomplemented
131#380C2312yes—
132#379—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4C2×C412yescomplemented
135C4×C4⋊C4D414yescomplemented
136C4×C22⋊C4—21no—
137#376—21no—
138C2×#95—22no—
139C2×#95—21no—
140C2×#95—24no—
141C2×#95—24no—
142C2×#95C2311yescharacteristic
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145#328—22no—
146#328—22no—
147#328—24no—
148#327—22no—
149#327—22no—
150#327—24no—
151C25C2×D412yes—
152C22×Q8C22×C411yescharacteristic
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—21no—
156C22×D4—24no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4C22⋊C412yes—
160C22×D4C22⋊C412yes—
161C22×D4C22×C411yescharacteristic
162C23×C4—21no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—28no—
170C23×C4—21no—
171C23×C4—48no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—84no—
175C23×C4—84no—
176C23×C4C22×C414yes—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D412yes—
179C23×C4C2×D414yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2411yescharacteristic
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185#126—42no—
186#126—42no—
187#125—22no—
188#125—22no—
189#124—22no—
190#124—22no—
191#123—42no—
192#123—42no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C22⋊Q8—21no—
197C22⋊Q8—84no—
198#121—21no—
199#121—21no—
200#121—21no—
201#121—21no—
202#121—84no—
203#120—44no—
204#120—44no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×D4—24no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—24no—
217C4×D4—432no—
218C4×D4—42no—
219C4×D4—42no—
220#117—42no—
221#117—42no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—84no—
235C2×C4⋊C4C22⋊C412yes—
236C2×C4⋊C4C22⋊C412yes—
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C414yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4C22⋊C414yes—
256C2×C22⋊C4C22⋊C414yes—
257C2×C42—21no—
258C2×C42—24no—
259C2×C42—22no—
260C2×C42—44no—
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42#4112yes—
268C2×C42#4112yes—
269C4⋊C8—42no—
270C4⋊C8—42no—
271C22⋊C8—21no—
272C22⋊C8—21no—
273C22⋊C8—22no—
274C22⋊C8—21no—
275C22⋊C8—21no—
276C22⋊C8—22no—
277C22⋊C8—84no—
278C22⋊C8—84no—
279#97—42no—
280#97—42no—
281#95—28no—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—21no—
289#95—22no—
290#95—21no—
291#95—22no—
292#95C2×D412yes—
293#95C2×D414yes—
294#95C2×D412yes—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—21no—
300C24—44no—
301C24—44no—
302C24C2×C22⋊C414yes—
303C24AGL(1,ℤ/8)14yes—
304C24#13714yes—
305C24C22×D411yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—42no—
309C2×D4—28no—
310C2×D4—216no—
311C2×D4—28no—
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—42no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—48no—
324C2×D4—84no—
325C2×D4—84no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—416no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—416no—
356C22×C4—44no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—82no—
362C22×C4—84no—
363C22×C4—84no—
364C22×C4—84no—
365C22×C4—84no—
366C22×C4—88no—
367C22×C4—816no—
368C22×C4—816no—
369C22×C4C2×C22⋊C414yes—
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4#12012yes—
375C22×C4#12012yes—
376C22×C4#12312yes—
377C22×C4#12312yes—
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C2×C8—42no—
385C2×C8—44no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—44no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—84no—
405C4⋊C4—88no—
406C22⋊C4—28no—
407C22⋊C4—28no—
408C22⋊C4—416no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—48no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—84no—
420C22⋊C4—84no—
421C22⋊C4—84no—
422C42—48no—
423C42—42no—
424C42—82no—
425C42C4×D412yes—
426C42C4×D412yes—
427C42C4×D412yes—
428C42C4×D412yes—
429C42#12112yes—
430C42#12112yes—
431C42#12112yes—
432C42#12112yes—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—48no—
446C23—48no—
447C23—44no—
448C23—44no—
449C23—48no—
450C23—44no—
451C23—48no—
452C23—44no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23#41718yes—
458C23C22×C22⋊C411yescharacteristic
459C23C2×#12011yescharacteristic
460C23C2×#12311yescharacteristic
461C23C2×AGL(1,ℤ/8)12yes—
462C23C2×#13712yes—
463Q8—81no—
464Q8—81no—
465D4—216no—
466D4—216no—
467D4—81no—
468D4—81no—
469D4—82no—
470D4—81no—
471D4—81no—
472D4—168no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—48no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—832no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4#38612yes—
522C2×C4#38612yes—
523C2×C4#38612yes—
524C2×C4#38612yes—
525C2×C4C2×C4×D411yescharacteristic
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C2×#12111yescharacteristic
528C2×C4C2×#12111yescharacteristic
529C8—82no—
530C8—81no—
531C8—81no—
532C22—24no—
533C22—24no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—28no—
539C22—28no—
540C22—28no—
541C22—48no—
542C22—48no—
543C22—48no—
544C22—48no—
545C22—44no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—81no—
550C22—84no—
551C22—81no—
552C22—81no—
553C22C2×#38611yescharacteristic
554C22#288312yes—
555C22C2×#41712yes—
556C22#290112yes—
557C22#300311yescharacteristic
558C22#300411yescharacteristic
559C22#309212yes—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—44no—
569C4—44no—
570C4—82no—
571C4—84no—
572C4—81no—
573C4—84no—
574C4—81no—
575C4—81no—
576C2—21no—
577C2—22no—
578C2—21no—
579C2—48no—
580C2—161no—
581C2#1762214yes—
582C2#2090911yescharacteristic
583C2#2098011yescharacteristic
584C2#2098511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35102C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2349C2semidirect
3. #21790C2semidirect
4. C2×#1512C2semidirect
5. C2×#1509C2semidirect
6. #9155C2semidirect
7. #9151C2semidirect
8. #9147C2semidirect
9. #9143C2semidirect
10. #8021C2semidirect
11. #8017C2semidirect
23. #2562C4semidirect
24. #2540C4semidirect
25. #2536C4semidirect
26. #2535C4semidirect
61. C2×#125C2×C4semidirect
62. C2×#124C2×C4semidirect
64. C2×C22⋊Q8C2×C4semidirect
66. C2×#121C2×C4semidirect
69. C2×C4×Q8C2×C4semidirect
79. C2×C4×D4C2×C4semidirect
124. #382C2×C4semidirect
125. #382D4semidirect
129. #380C2×C4semidirect
130. #380C2×C4semidirect
134. C4×C4⋊C4C2×C4semidirect
135. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
#351022
C2×#23492
#217902
C2×#15122
C2×#15092
#91552
#91512
#91472
#91432
#80212
#80172