Finite Group Databasebeta

← #198389

Subgroups — #198389

Order 512.

1608 conjugacy classes of subgroups.

The subgroup lattice has 597 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3208C211yesmaximal, characteristic
2#34210C211yesmaximal, characteristic
3#21803C211yesmaximal, characteristic
4#21332C211yesmaximal, characteristic, complemented
5#17575C211yesmaximal, characteristic, complemented
6#9255C218yesmaximal, complemented
7#7981C211yesmaximal, characteristic, complemented
8#7924C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C22×#127C2211yescharacteristic
11C2×#518—21no—
12#3208—22no—
13#3208—22no—
14#3208—22no—
15#3208—22no—
16C2×#492C412yescomplemented
17C2×#486C412yescomplemented
18C2×#460C414yescomplemented
19C2×#450C2214yes—
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2366—22no—
23#2366—22no—
24D4×C22⋊C4—22no—
25D4×C22⋊C4C2214yes—
26#2328—22no—
27#2309—21no—
28C4×#127—22no—
29#2285C2211yescharacteristic
30#2285C2211yescharacteristic
31C4×#117C2211yescharacteristic
32#1996—22no—
33#1996C2214yes—
34#1982—22no—
35#1957—22no—
36#1786C2214yes—
37#1740—21no—
38#1740—21no—
39#1520C2214yes—
40#1518—22no—
41#1503—22no—
42#1289C2214yes—
43#1282C2214yes—
44C23×D4—21no—
45C23×D4—21no—
46C23×D4C2312yes—
47C22×SD16C2×C418yescomplemented
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56C2×C4⋊Q8C2×C411yescharacteristic, complemented
57C2×C4⋊Q8C2×C411yescharacteristic, complemented
58C2×#127—21no—
59C2×#127—21no—
60C2×#127—21no—
61C2×#127—21no—
62C2×#127—21no—
63C2×#127—21no—
64C2×#127C2×C412yes—
65C2×#127C2×C412yes—
66C2×#127D412yes—
67C2×#127D412yes—
68C2×#127D412yes—
69C2×#127D412yes—
70C2×#125—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8C2312yes—
73#518—44no—
74#518—44no—
75C2×C4×Q8—21no—
76C2×C4×Q8C2×C412yescomplemented
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C2×#117—21no—
82C2×#117—21no—
83C2×#117—21no—
84C2×#117D411yescharacteristic
85C2×#117D411yescharacteristic
86C22×C22⋊C4—24no—
87C22×C42C2311yescharacteristic
88#492—24no—
89#492—24no—
90#486—24no—
91#486—24no—
92#460—28no—
93#460—28no—
94#450—28no—
95#450—28no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2×C412yescomplemented
106C2×#102—28no—
107C2×#102C2×C418yescomplemented
108C2×C22⋊C8C2312yes—
109C2×C22⋊C8C2312yes—
110C2×C4×C8—21no—
111C2×C4×C8C2×C412yescomplemented
112#390—24no—
113#390—24no—
114#390—24no—
115#390—24no—
116#386—28no—
117#386—28no—
118#384—21no—
119#384—21no—
120#382—21no—
121#382D412yes—
122#380—21no—
123#380—21no—
124#380C2312yes—
125#380C2312yes—
126#379—24no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4D412yes—
133C4×C22⋊C4D412yes—
134C4×C22⋊C4C2312yes—
135#376—21no—
136C43—21no—
137#342—24no—
138#342—24no—
139#328—28no—
140C25—22no—
141C22×Q8—21no—
142C22×Q8C22×C412yes—
143C22×D4—22no—
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4—84no—
163C22×D4—84no—
164C22×D4C22×C414yes—
165C22×D4C2×D414yes—
166C22×D4C2×D414yes—
167C23×C4—28no—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2411yescharacteristic
171C2×SD16—216no—
172C2×SD16—88no—
173C22×C8—22no—
174C22×C8C22×C414yes—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183#127—22no—
184#127—22no—
185#127—22no—
186#127—22no—
187#127—22no—
188#127—22no—
189#127—22no—
190#127—22no—
191#127—22no—
192#127—22no—
193#127—22no—
194#127—22no—
195#127C22⋊C414yes—
196#127C22⋊C414yes—
197#127C22⋊C414yes—
198#127C22⋊C414yes—
199#125—41no—
200#125—41no—
201#125—41no—
202#125—41no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—22no—
206C22⋊Q8—22no—
207C22⋊Q8—41no—
208C22⋊Q8—41no—
209C22⋊Q8—41no—
210C22⋊Q8—41no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—41no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×D4—416no—
220C4×D4—416no—
221C4×D4—88no—
222C4×D4—88no—
223#117—22no—
224#117—22no—
225#117—22no—
226#117—41no—
227#117—41no—
228#117—41no—
229#117—41no—
230#117—42no—
231#117—42no—
232#117—42no—
233#117—42no—
234#117SD1614yescomplemented
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4C22×C412yes—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4C2×D412yes—
258C2×C22⋊C4C2×D412yes—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—21no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—41no—
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42#4111yescharacteristic
276C2×C42#4111yescharacteristic
277C2×C42#4111yescharacteristic
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—41no—
281C4⋊C8—41no—
282#102—216no—
283#102—88no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C4×C8—22no—
289C4×C8—22no—
290C4×C8—41no—
291C4×C8—41no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—41no—
297#95C2×D414yes—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24—42no—
305C24—84no—
306C24—84no—
307C24C22×D411yescharacteristic
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—44no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—82no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—84no—
342C2×D4—84no—
343C2×D4—84no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—84no—
349C2×D4C2×C22⋊C418yes—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—416no—
358C22×C4—42no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4C4×D411yescharacteristic
362C22×C4C4×D411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12011yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4#12111yescharacteristic
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4C2×SD1612yes—
374C22×C4#13512yes—
375C22×C4AGL(1,ℤ/8)12yes—
376C22×C4#13712yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383SD16—816no—
384C2×C8—24no—
385C2×C8—44no—
386C2×C8—44no—
387C2×C8—42no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—416no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—88no—
419C22⋊C4—88no—
420C22⋊C4C2×SD1614yes—
421C42—24no—
422C42—44no—
423C42—44no—
424C42—42no—
425C42—42no—
426C42—44no—
427C42—42no—
428C42—82no—
429C42C2×C22⋊C411yescharacteristic
430C42C2×C22⋊C411yescharacteristic
431C42C2×C22⋊C411yescharacteristic
432C42C2×C22⋊C411yescharacteristic
433C42C4×D411yescharacteristic
434C42C4×D411yescharacteristic
435C42C4×D411yescharacteristic
436C42C4×D411yescharacteristic
437C42#12111yescharacteristic
438C42#12111yescharacteristic
439C42#12111yescharacteristic
440C42#12111yescharacteristic
441C42#12311yescharacteristic
442C42#12311yescharacteristic
443C42#12311yescharacteristic
444C42#12311yescharacteristic
445C23—28no—
446C23—41no—
447C23—41no—
448C23—48no—
449C23—48no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—84no—
457C23—84no—
458C23—84no—
459C23—84no—
460C23—164no—
461C23—164no—
462C23—164no—
463C23—164no—
464C23C2×C4×D411yescharacteristic
465C23C2×#12011yescharacteristic
466C23C2×#12111yescharacteristic
467C23C22×SD1611yescharacteristic
468C23C2×#13511yescharacteristic
469C23C2×AGL(1,ℤ/8)11yescharacteristic
470C23C2×#13711yescharacteristic
471Q8—44no—
472Q8—82no—
473Q8—82no—
474Q8—82no—
475D4—216no—
476D4—48no—
477D4—48no—
478D4—48no—
479D4—48no—
480D4—48no—
481D4—48no—
482D4—48no—
483D4—84no—
484D4—84no—
485D4—84no—
486D4—84no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—1616no—
516C2×C4#38611yescharacteristic
517C2×C4#38611yescharacteristic
518C2×C4#38611yescharacteristic
519C2×C4#38611yescharacteristic
520C2×C4#38611yescharacteristic
521C2×C4#38611yescharacteristic
522C2×C4#38611yescharacteristic
523C2×C4#38611yescharacteristic
524C2×C4C4×SD1612yes—
525C2×C4C4×SD1612yes—
526C2×C4#44012yes—
527C2×C4#44012yes—
528C2×C4#45012yes—
529C2×C4#45012yes—
530C2×C4#45212yes—
531C2×C4#45212yes—
532C2×C4#46112yes—
533C2×C4#46112yes—
534C2×C4#46312yes—
535C2×C4#46312yes—
536C2×C4C22×C22⋊C411yescharacteristic
537C2×C4C2×C4×D411yescharacteristic
538C2×C4C2×#12111yescharacteristic
539C2×C4C2×#12311yescharacteristic
540C8—44no—
541C8—82no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—48no—
553C22—48no—
554C22—48no—
555C22—88no—
556C22—88no—
557C22—164no—
558C22—164no—
559C22—164no—
560C22—164no—
561C22C2×#38611yescharacteristic
562C22C2×C4×SD1611yescharacteristic
563C22C2×#44011yescharacteristic
564C22C2×#45011yescharacteristic
565C22C2×#45211yescharacteristic
566C22C2×#46111yescharacteristic
567C22C2×#46311yescharacteristic
568C4—21no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—44no—
575C4—82no—
576C4—84no—
577C4—82no—
578C4—82no—
579C4—162no—
580C4#187712yes—
581C4#187712yes—
582C4#187712yes—
583C4#187712yes—
584C4#190412yes—
585C4#288312yes—
586C2—21no—
587C2—21no—
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—88no—
593C2C2×#187711yescharacteristic
594C2#2098511yescharacteristic
595C2C2×#190411yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21332C2semidirect
5. #17575C2semidirect
6. #9255C2semidirect
7. #7981C2semidirect
8. #7924C2semidirect
16. C2×#492C4semidirect
17. C2×#486C4semidirect
18. C2×#460C4semidirect
47. C22×SD16C2×C4semidirect
56. C2×C4⋊Q8C2×C4semidirect
57. C2×C4⋊Q8C2×C4semidirect
76. C2×C4×Q8C2×C4semidirect
105. C2×C4⋊C8C2×C4semidirect
107. C2×#102C2×C4semidirect
111. C2×C4×C8C2×C4semidirect
234. #117SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#32082
#342102
#218032
#213322
#175752
#92552
#79812
#79242