Finite Group Databasebeta

← #195523

Subgroups — #195523

Order 512.

1173 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic
2#60236C211yesmaximal, characteristic
3#21844C211yesmaximal, characteristic, complemented
4#17544C211yesmaximal, characteristic
5#17464C211yesmaximal, characteristic, complemented
6#9055C218yesmaximal, complemented
7#7890C211yesmaximal, characteristic, complemented
8#7885C211yesmaximal, characteristic, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11#3167—22no—
12#3167—22no—
13#3167C414yescomplemented
14#3167C2214yes—
15C2×#498C411yescharacteristic, complemented
16C2×#492C411yescharacteristic, complemented
17C2×#487C411yescharacteristic, complemented
18C2×#486C411yescharacteristic, complemented
19C2×#465—22no—
20#2981—22no—
21#2979—22no—
22#2975—21no—
23#2974—22no—
24#2972—21no—
25C2×C4×SD16—22no—
26#2935C2214yes—
27#2931—22no—
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30C2×#431C2211yescharacteristic
31#2880—22no—
32C22×C4×C8C2211yescharacteristic
33C4×#117C2211yescharacteristic
34#1994—22no—
35#1986C2214yescomplemented
36#1826—22no—
37#1507—22no—
38#1505—22no—
39#1502C2214yes—
40#1485—22no—
41#1481C2214yescomplemented
42#1479—21no—
43#1479—21no—
44#1279C2214yescomplemented
45#1278C2214yescomplemented
46C22×#41—21no—
47C22×SD16—22no—
48C2×#131—24no—
49C23×C8—21no—
50#533—21no—
51#533—21no—
52#533—21no—
53#533—21no—
54#533—21no—
55#533—21no—
56#533C2×C412yes—
57#533C2312yes—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532C2×C412yes—
63#532D412yes—
64#532D412yes—
65#532C2312yes—
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×C4⋊Q8C2×C411yescharacteristic
69C2×#127C2×C411yescharacteristic
70C2×#125—21no—
71C2×#124—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×#121—21no—
75C2×#121D412yes—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×D4—21no—
79C2×C4×D4D412yes—
80C2×#117—21no—
81C2×#117—21no—
82C22×C42C2311yescharacteristic
83#498—22no—
84#498C2×C414yes—
85#492—22no—
86#492C2×C414yes—
87#487—22no—
88#487C2×C414yes—
89#486—22no—
90#486C2×C414yes—
91#465—28no—
92#465—28no—
93C4×SD16—28no—
94C4×SD16—28no—
95#436—24no—
96#436—24no—
97#436—24no—
98#435—28no—
99C8×D4—24no—
100C8×D4—24no—
101C8×D4—24no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432—21no—
106#431—21no—
107#431—21no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431D411yescharacteristic
112#431D411yescharacteristic
113#431D412yes—
114#431C2312yes—
115C2×#106—22no—
116#423—21no—
117#423D412yes—
118C2×C4⋊C8C2312yes—
119C2×#103—22no—
120C2×#102—22no—
121#408—28no—
122#408—24no—
123#407—24no—
124#407—21no—
125#407D412yes—
126C2×C22⋊C8—24no—
127C2×C22⋊C8—21no—
128C4×M16—21no—
129C4×M16C2312yes—
130C2×C4×C8—21no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8C2×C411yescharacteristic
135C2×C4×C8C2×C411yescharacteristic
136C2×C4×C8C2×C412yes—
137C2×C4×C8C2312yes—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2312yes—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—21no—
142#376—21no—
143C43—21no—
144#339—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
164C2×SD16—28no—
165C2×SD16—48no—
166#131—88no—
167C2×M16—28no—
168C2×M16—41no—
169C2×M16C2×D414yes—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—28no—
176C22×C8—44no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8C22⋊C412yes—
186C4⋊Q8C22×C412yes—
187C4⋊Q8C22×C412yes—
188C4⋊Q8C22×C412yes—
189#127—21no—
190#127—21no—
191#127C22⋊C412yes—
192#127C22×C412yes—
193#125—22no—
194#125—22no—
195#125—41no—
196#125—41no—
197#124—22no—
198#124—22no—
199#124—22no—
200#124—22no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—22no—
204C22⋊Q8—22no—
205C22⋊Q8—41no—
206C22⋊Q8—41no—
207C22⋊Q8—41no—
208C22⋊Q8—41no—
209#121—22no—
210#121—22no—
211#121—22no—
212#121—22no—
213#121—22no—
214#121—22no—
215#121C22⋊C414yes—
216#121C2×D414yes—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×Q8—41no—
222C4×Q8—41no—
223C4×Q8—41no—
224C4×Q8—41no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4C22⋊C414yes—
232C4×D4C2×D414yes—
233#117—24no—
234#117—22no—
235#117—22no—
236#117—42no—
237#117—41no—
238#117—41no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4C22⋊C412yes—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C42—22no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42C2411yescharacteristic
268#106—28no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—41no—
274C4⋊C8C2×D412yes—
275C4⋊C8C2×D412yes—
276#103—28no—
277#102—28no—
278C22⋊C8—28no—
279C22⋊C8—28no—
280C22⋊C8—21no—
281C22⋊C8—22no—
282C22⋊C8—21no—
283C22⋊C8—22no—
284C22⋊C8—22no—
285C22⋊C8—22no—
286C22⋊C8—41no—
287#97—22no—
288#97—41no—
289C4×C8—22no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—22no—
294C4×C8—21no—
295C4×C8—22no—
296C4×C8—22no—
297C4×C8—22no—
298C4×C8C22×C412yes—
299C4×C8C22×C412yes—
300#95—22no—
301#95—41no—
302C24—22no—
303C24#12111yescharacteristic
304#41—48no—
305#41—84no—
306#41—84no—
307#41—84no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—44no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—42no—
317C2×Q8—82no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—44no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—82no—
329C2×D4—84no—
330C2×D4C2×C22⋊C414yes—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—84no—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4C4×D411yescharacteristic
346C22×C4C4×D411yescharacteristic
347C22×C4C4×D411yescharacteristic
348C22×C4#12011yescharacteristic
349C22×C4#12011yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12111yescharacteristic
355C22×C4#12311yescharacteristic
356C22×C4#12311yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360SD16—416no—
361M16—28no—
362M16—88no—
363M16—82no—
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—24no—
370C2×C8—44no—
371C2×C8—44no—
372C2×C8—42no—
373C2×C8—44no—
374C2×C8—42no—
375C2×C8—88no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4C2×C22⋊C414yes—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—24no—
398C22⋊C4—44no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C42—44no—
402C42—42no—
403C42—42no—
404C42—44no—
405C42—82no—
406C42C2×C22⋊C411yescharacteristic
407C42C2×C22⋊C411yescharacteristic
408C42C4×D411yescharacteristic
409C42C4×D411yescharacteristic
410C42C4×D411yescharacteristic
411C42C4×D411yescharacteristic
412C42#12011yescharacteristic
413C42#12011yescharacteristic
414C42#12111yescharacteristic
415C42#12111yescharacteristic
416C42#12311yescharacteristic
417C42#12311yescharacteristic
418C42C23×C411yescharacteristic
419C42C22×D411yescharacteristic
420C42C22×D411yescharacteristic
421C42C2×#4111yescharacteristic
422C23—22no—
423C23—22no—
424C23—24no—
425C23—41no—
426C23—44no—
427C23—82no—
428C23—82no—
429C23#38611yescharacteristic
430C23#38611yescharacteristic
431C23C2×#12111yescharacteristic
432Q8—44no—
433Q8—44no—
434Q8—42no—
435Q8—42no—
436Q8—82no—
437Q8—82no—
438Q8—82no—
439D4—44no—
440D4—48no—
441D4—44no—
442D4—44no—
443D4—44no—
444D4—42no—
445D4—42no—
446D4—82no—
447D4—84no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—48no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—84no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4#38611yescharacteristic
478C2×C4#38611yescharacteristic
479C2×C4#38611yescharacteristic
480C2×C4#38611yescharacteristic
481C2×C4#38611yescharacteristic
482C2×C4#38611yescharacteristic
483C2×C4#44312yes—
484C2×C4#44412yes—
485C2×C4C22×C22⋊C411yescharacteristic
486C2×C4C2×C4×D411yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4C2×#12011yescharacteristic
489C2×C4C2×#12111yescharacteristic
490C2×C4C2×#12311yescharacteristic
491C2×C4#57714yes—
492C8—24no—
493C8—42no—
494C8—42no—
495C8—42no—
496C8—44no—
497C8—82no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—44no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—164no—
510C22#187612yes—
511C22C2×#38611yescharacteristic
512C22#304212yes—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—44no—
521C4—84no—
522C4—82no—
523C4—82no—
524C4#187612yes—
525C4#188412yes—
526C4#294912yes—
527C4C2×#44311yescharacteristic
528C4C2×#44411yescharacteristic
529C4#300912yes—
530C2—22no—
531C2—22no—
532C2—41no—
533C2—84no—
534C2C2×#187611yescharacteristic
535C2#2093311yescharacteristic
536C2C2×#188411yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21844C2semidirect
5. #17464C2semidirect
6. #9055C2semidirect
7. #7890C2semidirect
8. #7885C2semidirect
13. #3167C4semidirect
15. C2×#498C4semidirect
16. C2×#492C4semidirect
17. C2×#487C4semidirect
18. C2×#486C4semidirect
35. #1986C22semidirect
41. #1481C22semidirect
44. #1279C22semidirect
45. #1278C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31672
#602362
#218442
#175442
#174642
#90552
#78902
#78852