Finite Group Databasebeta

← #197232

Subgroups — #197232

Order 512.

1449 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic, complemented
4#17575C211yesmaximal, characteristic, complemented
5C2×#1478C211yesmaximal, characteristic
6#9175C218yesmaximal, complemented
7#8095C211yesmaximal, characteristic, complemented
8#7957C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C2×#518—21no—
11C22×C4×D4C2211yescharacteristic
12#3077—22no—
13#3077—22no—
14#3077C414yescomplemented
15#3077C2214yes—
16C2×#478—22no—
17C2×#476C412yescomplemented
18C2×#474C412yescomplemented
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#428—21no—
22C2×#423C2211yescharacteristic
23C2×#418—21no—
24#2349—22no—
25#2349C2214yes—
26D4×C4⋊C4—22no—
27D4×C4⋊C4—22no—
28#2316—21no—
29#2315—22no—
30#2285C2211yescharacteristic
31#2285C2211yescharacteristic
32C2×#380—22no—
33#1520—22no—
34#1520—22no—
35#1518—22no—
36#1518C2214yescomplemented
37#1503—22no—
38#1503C2214yescomplemented
39#1478—22no—
40#1478C2214yes—
41C2×D4⋊C8—22no—
42#1303C2214yescomplemented
43#1286C2214yescomplemented
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C22×M16—21no—
47#533—21no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533C2×C412yescomplemented
54#533C2312yes—
55C2×C4⋊Q8C2×C411yescharacteristic, complemented
56C2×C4⋊Q8C2×C411yescharacteristic, complemented
57C2×#125—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60#518—44no—
61#518—44no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×D4—24no—
65C2×C4×D4—24no—
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C412yes—
73C2×C4×D4C2×C412yes—
74C2×C4×D4D412yes—
75C2×C4×D4D412yes—
76C2×C4×D4D412yes—
77C2×C4×D4C2312yes—
78C2×#117—21no—
79C2×#117—21no—
80C2×#117—21no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4D411yescharacteristic
83C22×C22⋊C4—21no—
84C22×C22⋊C4—24no—
85C22×C42C2311yescharacteristic
86#478—28no—
87#478—28no—
88#476—24no—
89#476C2×C418yescomplemented
90#474—24no—
91#474C2×C418yescomplemented
92#431—21no—
93#431—21no—
94#431—21no—
95#431—21no—
96#431—21no—
97#431—21no—
98#431C2312yes—
99#431C2312yes—
100#428—24no—
101#428—24no—
102C2×#107—22no—
103C2×#106—22no—
104#423—21no—
105#423—21no—
106#423C2×C412yescomplemented
107#423C2312yes—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8C2×C412yescomplemented
112#418—24no—
113#418—24no—
114C2×#102—22no—
115C2×#102—22no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120#389—24no—
121#389—24no—
122#385—28no—
123#385—28no—
124#384—21no—
125#384—21no—
126#382D412yes—
127#382C2312yes—
128#380—24no—
129#380—24no—
130#380—21no—
131#380—21no—
132#380—24no—
133#380D412yes—
134#380D412yes—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4D412yes—
138C4×C4⋊C4C2312yes—
139C2×#95—24no—
140#328—28no—
141#328—28no—
142D4⋊C8—28no—
143C25—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—22no—
150C22×D4—22no—
151C22×D4C22⋊C412yes—
152C22×D4C22⋊C412yes—
153C22×D4SD1614yes—
154C22×D4Q1614yes—
155C22×D4C2×D412yes—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—28no—
164C23×C4—84no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—22no—
173C22×C8—41no—
174C22×C8—41no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8C22×C412yes—
182C4⋊Q8C22×C412yes—
183#125—22no—
184#125—22no—
185#125—41no—
186#125—41no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—41no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C22⋊Q8—41no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4—22no—
215C4×D4—48no—
216C4×D4—416no—
217C4×D4—48no—
218C4×D4C22⋊C414yes—
219C4×D4C22⋊C414yes—
220C4×D4C22×C414yes—
221C4×D4C2×D414yes—
222#117—22no—
223#117—22no—
224#117—41no—
225#117—41no—
226#117—42no—
227#117—42no—
228#117—42no—
229#117—42no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—88no—
247C2×C4⋊C4C22⋊C412yes—
248C2×C4⋊C4C22⋊C412yes—
249C2×C4⋊C4C2×D412yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—416no—
260C2×C22⋊C4—41no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—41no—
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42C2411yescharacteristic
276#107—28no—
277#106—28no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—41no—
284C4⋊C8—41no—
285C4⋊C8C22×C414yes—
286#102—28no—
287#102—28no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—41no—
291C22⋊C8—41no—
292C4×C8—22no—
293C4×C8—22no—
294C4×C8—41no—
295C4×C8—41no—
296#95—28no—
297#95—22no—
298#95—22no—
299#95C2×D414yes—
300#95C2×D414yes—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—24no—
306C24—82no—
307C24—82no—
308C24#12011yescharacteristic
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—82no—
314C2×Q8—82no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4#10318yes—
329C2×D4#10318yes—
330C2×D4C2×C22⋊C414yes—
331C2×D4C2×SD1614yes—
332C2×D4C2×Q1614yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—42no—
343C22×C4—48no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—416no—
347C22×C4—84no—
348C22×C4—84no—
349C22×C4—81no—
350C22×C4—84no—
351C22×C4—84no—
352C22×C4—84no—
353C22×C4—82no—
354C22×C4—88no—
355C22×C4—88no—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4#12011yescharacteristic
363C22×C4#12011yescharacteristic
364C22×C4#12011yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12111yescharacteristic
369C22×C4#12311yescharacteristic
370C22×C4#12311yescharacteristic
371C22×C4C2×SD1612yes—
372C22×C4C2×Q1612yes—
373C22×C4AGL(1,ℤ/8)12yes—
374C22×C4#13712yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378M16—42no—
379M16—42no—
380M16—44no—
381C2×C8—24no—
382C2×C8—44no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—82no—
386C2×C8—82no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C4⋊C4—88no—
410C4⋊C4C2×C22⋊C414yes—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—24no—
414C22⋊C4—24no—
415C22⋊C4—416no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—44no—
425C42C2×C22⋊C411yescharacteristic
426C42C2×C22⋊C411yescharacteristic
427C42C4×D411yescharacteristic
428C42C4×D411yescharacteristic
429C42C4×D411yescharacteristic
430C42C4×D411yescharacteristic
431C42#12111yescharacteristic
432C42#12111yescharacteristic
433C42#12111yescharacteristic
434C42#12111yescharacteristic
435C42#12311yescharacteristic
436C42#12311yescharacteristic
437C42C23×C411yescharacteristic
438C42C22×D411yescharacteristic
439C42C2×#4111yescharacteristic
440C42C2×#4111yescharacteristic
441C23—24no—
442C23—24no—
443C23—24no—
444C23—28no—
445C23—28no—
446C23—41no—
447C23—41no—
448C23—44no—
449C23—44no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—88no—
455C23—88no—
456C23#38611yescharacteristic
457C23#38611yescharacteristic
458C23#45012yes—
459C23#45112yes—
460C23C2×#12011yescharacteristic
461Q8—82no—
462Q8—82no—
463Q8—82no—
464Q8—82no—
465D4—24no—
466D4—24no—
467D4—24no—
468D4—28no—
469D4—28no—
470D4C2×#10318yes—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—48no—
485C2×C4—44no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—1616no—
502C2×C4#38611yescharacteristic
503C2×C4#38611yescharacteristic
504C2×C4#38611yescharacteristic
505C2×C4#38611yescharacteristic
506C2×C4#38611yescharacteristic
507C2×C4#38611yescharacteristic
508C2×C4C2×#10312yes—
509C2×C4C2×#10312yes—
510C2×C4#41712yes—
511C2×C4#41712yes—
512C2×C4#45012yes—
513C2×C4#45112yes—
514C2×C4#48312yes—
515C2×C4#48312yes—
516C2×C4#48412yes—
517C2×C4#48412yes—
518C2×C4C22×C22⋊C411yescharacteristic
519C2×C4C2×C4×D411yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C2×#12111yescharacteristic
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×#12311yescharacteristic
524C2×C4C22×SD1611yescharacteristic
525C2×C4C22×Q1611yescharacteristic
526C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
527C2×C4C2×#13711yescharacteristic
528C8—44no—
529C8—82no—
530C8—82no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—28no—
538C22—42no—
539C22—42no—
540C22—48no—
541C22—48no—
542C22—44no—
543C22—44no—
544C22—88no—
545C22—88no—
546C22#187212yes—
547C22#187212yes—
548C22C2×#38611yescharacteristic
549C22C2×#45011yescharacteristic
550C22C2×#45111yescharacteristic
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—44no—
559C4—84no—
560C4—84no—
561C4—82no—
562C4—82no—
563C4#187212yes—
564C4#187212yes—
565C4#190412yes—
566C4C22×#10311yescharacteristic
567C4C2×#41711yescharacteristic
568C4C2×#48311yescharacteristic
569C4C2×#48411yescharacteristic
570C2—21no—
571C2—21no—
572C2—22no—
573C2—22no—
574C2—48no—
575C2C2×#187211yescharacteristic
576C2C2×#190411yescharacteristic
577C2#2100011yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17575C2semidirect
4. #17575C2semidirect
6. #9175C2semidirect
7. #8095C2semidirect
8. #7957C2semidirect
14. #3077C4semidirect
17. C2×#476C4semidirect
18. C2×#474C4semidirect
36. #1518C22semidirect
38. #1503C22semidirect
42. #1303C22semidirect
43. #1286C22semidirect
53. #533C2×C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
56. C2×C4⋊Q8C2×C4semidirect
89. #476C2×C4semidirect
91. #474C2×C4semidirect
106. #423C2×C4semidirect
111. C2×C4⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30772
#341812
#175752
C2×#14782
#91752
#80952
#79572