Finite Group Databasebeta

← #198797

Subgroups — #198797

Order 512.

1096 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3210C211yesmaximal, characteristic
2#60250C211yesmaximal, characteristic, complemented
3#17545C211yesmaximal, characteristic, complemented
4#17544C211yesmaximal, characteristic
5#17478C211yesmaximal, characteristic
6#9282C218yesmaximal, complemented
7#7965C211yesmaximal, characteristic, complemented
8#7854C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3210—22no—
12#3210—22no—
13#3210C414yescomplemented
14#3210C2214yes—
15C2×#494C411yescharacteristic, complemented
16C2×#492C411yescharacteristic, complemented
17C2×#487C411yescharacteristic, complemented
18C2×#486C411yescharacteristic, complemented
19C2×#464—22no—
20C2×#452—22no—
21#2986—22no—
22#2985C2214yescomplemented
23#2984—22no—
24#2983—21no—
25#2982—21no—
26#2962—22no—
27#2960—22no—
28#2924—22no—
29#2920C2211yescharacteristic
30#2920C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C4×#117C2211yescharacteristic
34#1508—22no—
35#1507—22no—
36#1506C2214yescomplemented
37#1505C2214yes—
38#1503—22no—
39#1502—22no—
40#1496C2214yes—
41#1492—22no—
42#1478—21no—
43#1478—21no—
44#1287C2214yescomplemented
45#1274C2214yescomplemented
46C22×#41—21no—
47C22×Q16—22no—
48C22×SD16—22no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533C2×C412yes—
54#533D412yes—
55#533D412yes—
56#533C2312yes—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532—21no—
63#532C2×C412yes—
64#532C2312yes—
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×#127C2×C411yescharacteristic
69C2×#125—21no—
70C2×#124—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8D412yes—
73C2×#121—21no—
74C2×#121—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8D412yes—
77C2×C4×D4—21no—
78C2×C4×D4—21no—
79C2×#117—21no—
80C2×#117—21no—
81C22×C42C2311yescharacteristic
82#494—22no—
83#494C2×C414yes—
84#492—22no—
85#492C2×C414yes—
86#487—22no—
87#487C2×C414yes—
88#486—22no—
89#486C2×C414yes—
90#464—28no—
91#464—28no—
92#452—28no—
93#452—28no—
94#446—24no—
95#446—24no—
96#446—28no—
97#446—24no—
98C8×Q8—24no—
99C8×Q8—24no—
100C8×Q8—24no—
101#433—28no—
102#432—24no—
103#432—21no—
104#432—21no—
105#432C2312yes—
106#432C2312yes—
107#431—24no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431C2×C412yes—
116#431D412yes—
117#431D411yescharacteristic
118#431D411yescharacteristic
119#431C2312yes—
120#431C2312yes—
121#424—24no—
122#423—21no—
123#423—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×#103—22no—
127C2×#102—22no—
128#407—21no—
129#407D412yes—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132#405—24no—
133C4×M16—21no—
134C4×M16—21no—
135C2×C4×C8C2×C411yescharacteristic
136C2×C4×C8C2×C411yescharacteristic
137C2×C4×C8D412yes—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#376C2312yes—
143C43—21no—
144#328—28no—
145#327—28no—
146D4⋊C8—24no—
147D4⋊C8—24no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—41no—
151C2×#41—41no—
152C2×#41—84no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8C22⋊C412yes—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—41no—
160C23×C4—41no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
164C2×Q16—28no—
165C2×Q16—84no—
166C2×SD16—28no—
167C2×SD16—84no—
168C2×M16—22no—
169C2×M16—41no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8C22⋊C412yes—
181C4⋊Q8C22⋊C412yes—
182C4⋊Q8C22×C412yes—
183C4⋊Q8C22×C412yes—
184C4⋊Q8C22×C412yes—
185#127—21no—
186#127—21no—
187#127—21no—
188#127C22×C412yes—
189#125—22no—
190#125—22no—
191#125—22no—
192#125—22no—
193#124—22no—
194#124—22no—
195#124—41no—
196#124—41no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8C22⋊C414yes—
204C22⋊Q8C2×D414yes—
205#121—22no—
206#121—22no—
207#121—22no—
208#121—22no—
209#121—41no—
210#121—41no—
211#121—41no—
212#121—41no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8C22⋊C414yes—
220C4×Q8C2×D414yes—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—41no—
228C4×D4—41no—
229#117—22no—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—24no—
234#117—42no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4C22⋊C412yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—41no—
251C2×C42—41no—
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42C2411yescharacteristic
264C4⋊C8—28no—
265C4⋊C8—28no—
266C4⋊C8—22no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—41no—
275#103—28no—
276#102—28no—
277C22⋊C8—22no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—41no—
285C22⋊C8C2×D414yes—
286#97—28no—
287#97—22no—
288#97—22no—
289C4×C8—28no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—22no—
293C4×C8—22no—
294C4×C8—41no—
295C4×C8C22×C412yes—
296C4×C8C22×C412yes—
297C4×C8C2×D412yes—
298C4×C8C2×D412yes—
299#95—22no—
300#95—22no—
301C24—41no—
302C24#12111yescharacteristic
303#41—84no—
304#41—84no—
305#41—84no—
306#41—84no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—41no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8C2×C22⋊C414yes—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—44no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—82no—
329C2×D4—84no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—84no—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4C4×D411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12311yescharacteristic
355C22×C4#12311yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359Q16—88no—
360SD16—88no—
361M16—44no—
362M16—82no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—22no—
368C2×C8—44no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—44no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4C2×C22⋊C414yes—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—44no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—42no—
400C42—82no—
401C42—82no—
402C42C2×C22⋊C411yescharacteristic
403C42C2×C22⋊C411yescharacteristic
404C42C4×D411yescharacteristic
405C42C4×D411yescharacteristic
406C42C4×D411yescharacteristic
407C42C4×D411yescharacteristic
408C42#12011yescharacteristic
409C42#12011yescharacteristic
410C42#12111yescharacteristic
411C42#12111yescharacteristic
412C42#12311yescharacteristic
413C42#12311yescharacteristic
414C42C23×C411yescharacteristic
415C42C22×D411yescharacteristic
416C42C22×D411yescharacteristic
417C42C2×#4111yescharacteristic
418C23—22no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23—82no—
423C23—82no—
424C23—82no—
425C23#38611yescharacteristic
426C23#38611yescharacteristic
427C23C2×#12111yescharacteristic
428Q8—44no—
429Q8—44no—
430Q8—44no—
431Q8—44no—
432Q8—42no—
433Q8—42no—
434Q8—82no—
435D4—44no—
436D4—42no—
437D4—42no—
438D4—82no—
439D4—84no—
440D4—84no—
441D4—82no—
442D4—82no—
443D4—82no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—84no—
469C2×C4—84no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4#38611yescharacteristic
475C2×C4#38611yescharacteristic
476C2×C4#38611yescharacteristic
477C2×C4#38611yescharacteristic
478C2×C4#38611yescharacteristic
479C2×C4#38611yescharacteristic
480C2×C4#44312yes—
481C2×C4#44412yes—
482C2×C4C22×C22⋊C411yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C2×#12011yescharacteristic
486C2×C4C2×#12111yescharacteristic
487C2×C4C2×#12311yescharacteristic
488C8—42no—
489C8—42no—
490C8—42no—
491C8—42no—
492C8—44no—
493C8—82no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—164no—
507C22#187612yes—
508C22C2×#38611yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—44no—
518C4—44no—
519C4—84no—
520C4—82no—
521C4#187612yes—
522C4#190512yes—
523C4#191012yes—
524C4C2×#44311yescharacteristic
525C4C2×#44411yescharacteristic
526C2—22no—
527C2—41no—
528C2—41no—
529C2—162no—
530C2C2×#187611yescharacteristic
531C2C2×#190511yescharacteristic
532C2C2×#191011yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
2. #60250C2semidirect
3. #17545C2semidirect
6. #9282C2semidirect
7. #7965C2semidirect
8. #7854C2semidirect
13. #3210C4semidirect
15. C2×#494C4semidirect
16. C2×#492C4semidirect
17. C2×#487C4semidirect
18. C2×#486C4semidirect
22. #2985C22semidirect
36. #1506C22semidirect
44. #1287C22semidirect
45. #1274C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32102
#602502
#175452
#175442
#174782
#92822
#79652
#78542