Finite Group Databasebeta

← #198794

Subgroups — #198794

Order 512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3228C211yesmaximal, characteristic
2#60236C211yesmaximal, characteristic
3#17545C211yesmaximal, characteristic, complemented
4#17544C211yesmaximal, characteristic
5#17479C211yesmaximal, characteristic, complemented
6#9279C218yesmaximal, complemented
7#7965C211yesmaximal, characteristic, complemented
8#7854C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3228—22no—
12#3228—22no—
13#3228C414yescomplemented
14#3228C2214yes—
15C2×#500C411yescharacteristic, complemented
16C2×#498C411yescharacteristic, complemented
17C2×#486C411yescharacteristic, complemented
18C2×#486C411yescharacteristic, complemented
19C2×#482—22no—
20C2×#478—22no—
21#2981C2214yes—
22#2979—22no—
23#2975—21no—
24#2974—22no—
25#2972—21no—
26#2935—22no—
27#2931—22no—
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32#2880—22no—
33C4×#117C2211yescharacteristic
34#1508—22no—
35#1507—22no—
36#1506C2214yescomplemented
37#1505C2214yes—
38#1503—22no—
39#1502—22no—
40#1497C2214yescomplemented
41#1493—22no—
42#1479—21no—
43#1479—21no—
44#1287C2214yescomplemented
45#1274C2214yescomplemented
46C22×#41—21no—
47C2×#131—24no—
48#533—21no—
49#533—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533C2×C412yes—
55#533C2312yes—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532C2×C412yes—
61#532D412yes—
62#532D412yes—
63#532C2312yes—
64C2×C4⋊Q8C2×C411yescharacteristic
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×#127C2×C411yescharacteristic
68C2×#125—21no—
69C2×#124—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×#121—21no—
73C2×#121D412yes—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×D4—21no—
77C2×C4×D4D412yes—
78C2×#117—21no—
79C2×#117—21no—
80C22×C42C2311yescharacteristic
81#500—22no—
82#500C2×C414yes—
83#498—22no—
84#498C2×C414yes—
85#486—22no—
86#486—22no—
87#486C2×C414yes—
88#486C2×C414yes—
89#482—28no—
90#482—28no—
91#478—28no—
92#478—28no—
93#436—24no—
94#436—24no—
95#436—24no—
96#435—28no—
97C8×D4—24no—
98C8×D4—24no—
99C8×D4—24no—
100#432—21no—
101#432—21no—
102#432C2312yes—
103#432C2312yes—
104#431—21no—
105#431—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431C2×C412yes—
112#431D412yes—
113#431D411yescharacteristic
114#431D411yescharacteristic
115#431C2312yes—
116#431C2312yes—
117C2×#107—22no—
118C2×#106—22no—
119#423—21no—
120#423—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×#102—22no—
124C2×#102—22no—
125#408—24no—
126#408—28no—
127#407—24no—
128#407—21no—
129#407D412yes—
130C2×C22⋊C8—24no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
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—
146Q8⋊C8—24no—
147Q8⋊C8—24no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—84no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×D4—22no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4C22⋊C412yes—
160C23×C4—22no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
164#131—88no—
165C2×M16—22no—
166C2×M16—28no—
167C2×M16—41no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—28no—
172C4⋊Q8—21no—
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—
184#127—21no—
185#127—21no—
186#127C22⋊C412yes—
187#127C22×C412yes—
188#125—22no—
189#125—22no—
190#125—41no—
191#125—41no—
192#124—22no—
193#124—22no—
194#124—22no—
195#124—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203C22⋊Q8—41no—
204#121—22no—
205#121—22no—
206#121—22no—
207#121—22no—
208#121—22no—
209#121—22no—
210#121C22⋊C414yes—
211#121C2×D414yes—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—41no—
217C4×Q8—41no—
218C4×Q8—41no—
219C4×Q8—41no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—22no—
226C4×D4C22⋊C414yes—
227C4×D4C2×D414yes—
228#117—24no—
229#117—22no—
230#117—22no—
231#117—42no—
232#117—41no—
233#117—41no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4C22⋊C412yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C42—22no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42C2411yescharacteristic
263#107—28no—
264#106—28no—
265C4⋊C8—22no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—41no—
274#102—28no—
275#102—28no—
276C22⋊C8—28no—
277C22⋊C8—28no—
278C22⋊C8—22no—
279C22⋊C8—21no—
280C22⋊C8—22no—
281C22⋊C8—21no—
282C22⋊C8—22no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—41no—
286C22⋊C8C2×D414yes—
287#97—22no—
288#97—22no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—22no—
292C4×C8—22no—
293C4×C8—41no—
294C4×C8C22×C412yes—
295C4×C8C22×C412yes—
296C4×C8C2×D412yes—
297C4×C8C2×D412yes—
298#95—22no—
299#95—22no—
300C24—22no—
301C24#12111yescharacteristic
302#41—48no—
303#41—84no—
304#41—84no—
305#41—84no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—82no—
315C2×Q8—82no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—44no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—82no—
327C2×D4—84no—
328C2×D4C2×C22⋊C414yes—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—84no—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4C4×D411yescharacteristic
346C22×C4#12011yescharacteristic
347C22×C4#12011yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12311yescharacteristic
354C22×C4#12311yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4111yescharacteristic
358M16—44no—
359M16—88no—
360M16—82no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—24no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—44no—
369C2×C8—42no—
370C2×C8—88no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4C2×C22⋊C414yes—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—44no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C42—24no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—82no—
400C42—82no—
401C42C2×C22⋊C411yescharacteristic
402C42C2×C22⋊C411yescharacteristic
403C42C4×D411yescharacteristic
404C42C4×D411yescharacteristic
405C42C4×D411yescharacteristic
406C42C4×D411yescharacteristic
407C42#12011yescharacteristic
408C42#12011yescharacteristic
409C42#12111yescharacteristic
410C42#12111yescharacteristic
411C42#12311yescharacteristic
412C42#12311yescharacteristic
413C42C23×C411yescharacteristic
414C42C22×D411yescharacteristic
415C42C22×D411yescharacteristic
416C42C2×#4111yescharacteristic
417C23—22no—
418C23—24no—
419C23—41no—
420C23—41no—
421C23—44no—
422C23—82no—
423C23—82no—
424C23#38611yescharacteristic
425C23#38611yescharacteristic
426C23C2×#12111yescharacteristic
427Q8—44no—
428Q8—42no—
429Q8—42no—
430Q8—82no—
431Q8—82no—
432Q8—82no—
433Q8—82no—
434D4—44no—
435D4—48no—
436D4—44no—
437D4—44no—
438D4—44no—
439D4—42no—
440D4—42no—
441D4—82no—
442D4—84no—
443C2×C4—22no—
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—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—24no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—48no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—84no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4#38611yescharacteristic
474C2×C4#38611yescharacteristic
475C2×C4#38611yescharacteristic
476C2×C4#38611yescharacteristic
477C2×C4#38611yescharacteristic
478C2×C4#38611yescharacteristic
479C2×C4#44312yes—
480C2×C4#44412yes—
481C2×C4C22×C22⋊C411yescharacteristic
482C2×C4C2×C4×D411yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×#12011yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×#12311yescharacteristic
487C8—42no—
488C8—42no—
489C8—42no—
490C8—42no—
491C8—44no—
492C8—82no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—44no—
502C22—44no—
503C22—44no—
504C22—82no—
505C22—164no—
506C22#187612yes—
507C22C2×#38611yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—44no—
517C4—84no—
518C4—82no—
519C4—82no—
520C4#187612yes—
521C4#190412yes—
522C4#190912yes—
523C4C2×#44311yescharacteristic
524C4C2×#44411yescharacteristic
525C2—22no—
526C2—41no—
527C2—41no—
528C2—84no—
529C2C2×#187611yescharacteristic
530C2C2×#190411yescharacteristic
531C2C2×#190911yescharacteristic
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
3. #17545C2semidirect
5. #17479C2semidirect
6. #9279C2semidirect
7. #7965C2semidirect
8. #7854C2semidirect
13. #3228C4semidirect
15. C2×#500C4semidirect
16. C2×#498C4semidirect
17. C2×#486C4semidirect
18. C2×#486C4semidirect
36. #1506C22semidirect
40. #1497C22semidirect
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×#32282
#602362
#175452
#175442
#174792
#92792
#79652
#78542