Finite Group Databasebeta

← #195538

Subgroups — #195538

Order 512.

1135 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic
2#60237C211yesmaximal, characteristic
3#21845C211yesmaximal, characteristic, complemented
4#17561C211yesmaximal, characteristic
5#17496C211yesmaximal, characteristic, complemented
6#9055C218yesmaximal, complemented
7#7898C211yesmaximal, 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#2980—22no—
22#2974—22no—
23#2973—22no—
24C2×C4×SD16—22no—
25#2935C2214yes—
26#2932—22no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#432C2211yescharacteristic
30#2880—22no—
31C22×C4×C8C2211yescharacteristic
32#2283C2211yescharacteristic
33#1995—22no—
34#1986C2214yescomplemented
35#1826—22no—
36#1514—22no—
37#1510—22no—
38#1502C2214yes—
39#1497—22no—
40#1491—22no—
41#1481C2214yescomplemented
42#1279C2214yescomplemented
43#1278C2214yescomplemented
44C22×#41—21no—
45C22×SD16—22no—
46C2×#131—24no—
47C23×C8—21no—
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—
79C22×C42C2311yescharacteristic
80#498—22no—
81#498C2×C414yes—
82#492—22no—
83#492C2×C414yes—
84#487—22no—
85#487C2×C414yes—
86#486—22no—
87#486C2×C414yes—
88#465—28no—
89#465—28no—
90C4×SD16—28no—
91C4×SD16—28no—
92#436—24no—
93#435—28no—
94#435—28no—
95#435—28no—
96C8×D4—24no—
97#432—21no—
98#432—21no—
99#432—21no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432D412yes—
104#432D412yes—
105#431—21no—
106#431—21no—
107#431—21no—
108#431C2312yes—
109C2×#106—22no—
110#423—21no—
111#423D412yes—
112C2×C4⋊C8C2312yes—
113C2×#103—22no—
114C2×#102—22no—
115#408—28no—
116#408—24no—
117#407—24no—
118#407—21no—
119#407D412yes—
120C2×C22⋊C8—24no—
121C2×C22⋊C8—21no—
122C4×M16—21no—
123C4×M16C2312yes—
124C2×#97—21no—
125C2×C4×C8—21no—
126C2×C4×C8—21no—
127C2×C4×C8—21no—
128C2×C4×C8C2×C411yescharacteristic
129C2×C4×C8C2×C411yescharacteristic
130C2×C4×C8C2×C412yes—
131C2×C4×C8C2312yes—
132#383—21no—
133#382—21no—
134#381—21no—
135#380—21no—
136C4×C4⋊C4C2312yes—
137C4×C22⋊C4—21no—
138#376—21no—
139#339—28no—
140#327—28no—
141Q8⋊C8—28no—
142C2×#41—22no—
143C2×#41—22no—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41—84no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×D4—22no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4C22⋊C412yes—
154C23×C4—22no—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4#4111yescharacteristic
158C2×SD16—28no—
159C2×SD16—48no—
160#131—88no—
161C2×M16—28no—
162C2×M16—41no—
163C2×M16C2×D414yes—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—28no—
170C22×C8—44no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8C22⋊C412yes—
180C4⋊Q8C22×C412yes—
181C4⋊Q8C22×C412yes—
182C4⋊Q8C22×C412yes—
183#127—21no—
184#127—21no—
185#127C22⋊C412yes—
186#127C22×C412yes—
187#125—22no—
188#125—22no—
189#125—41no—
190#125—41no—
191#124—22no—
192#124—22no—
193#124—22no—
194#124—22no—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C22⋊Q8—41no—
200C22⋊Q8—41no—
201C22⋊Q8—41no—
202C22⋊Q8—41no—
203#121—22no—
204#121—22no—
205#121—22no—
206#121—22no—
207#121—22no—
208#121—22no—
209#121C22⋊C414yes—
210#121C2×D414yes—
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—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4C22⋊C414yes—
226C4×D4C2×D414yes—
227#117—22no—
228#117—22no—
229#117—41no—
230#117—41no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C22⋊C412yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C42—22no—
245C2×C42—41no—
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42#4111yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42#4111yescharacteristic
257C2×C42C2411yescharacteristic
258#106—28no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—22no—
263C4⋊C8—41no—
264C4⋊C8C2×D414yes—
265#103—28no—
266#102—28no—
267C22⋊C8—28no—
268C22⋊C8—28no—
269C22⋊C8—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—41no—
275#97—22no—
276#97—22no—
277#97—22no—
278#97—41no—
279C4×C8—22no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—22no—
283C4×C8—22no—
284C4×C8—22no—
285C4×C8C22×C412yes—
286C4×C8C22×C412yes—
287#95—22no—
288#95—41no—
289#95—41no—
290#95—41no—
291C24—22no—
292C24#12111yescharacteristic
293#41—48no—
294#41—84no—
295#41—84no—
296#41—84no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—44no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—82no—
305C2×Q8—82no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—44no—
314C2×D4—82no—
315C2×D4—82no—
316C2×D4—84no—
317C2×D4C2×C22⋊C414yes—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—84no—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12011yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4#12111yescharacteristic
341C22×C4#12311yescharacteristic
342C22×C4#12311yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4111yescharacteristic
346SD16—416no—
347M16—28no—
348M16—88no—
349M16—82no—
350C2×C8—24no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—44no—
356C2×C8—44no—
357C2×C8—42no—
358C2×C8—44no—
359C2×C8—42no—
360C2×C8—88no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—44no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4C2×C22⋊C414yes—
378C22⋊C4—24no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—82no—
385C42—42no—
386C42—42no—
387C42—44no—
388C42C2×C22⋊C411yescharacteristic
389C42C2×C22⋊C411yescharacteristic
390C42C4×D411yescharacteristic
391C42C4×D411yescharacteristic
392C42C4×D411yescharacteristic
393C42C4×D411yescharacteristic
394C42#12011yescharacteristic
395C42#12011yescharacteristic
396C42#12111yescharacteristic
397C42#12111yescharacteristic
398C42#12311yescharacteristic
399C42#12311yescharacteristic
400C42C23×C411yescharacteristic
401C42C22×D411yescharacteristic
402C42C22×D411yescharacteristic
403C42C2×#4111yescharacteristic
404C23—22no—
405C23—22no—
406C23—24no—
407C23—41no—
408C23—44no—
409C23—82no—
410C23—82no—
411C23#38611yescharacteristic
412C23#38611yescharacteristic
413C23C2×#12111yescharacteristic
414Q8—44no—
415Q8—44no—
416Q8—44no—
417Q8—82no—
418Q8—82no—
419Q8—82no—
420D4—44no—
421D4—48no—
422D4—44no—
423D4—44no—
424D4—82no—
425D4—82no—
426D4—82no—
427D4—84no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—48no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—84no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4#38611yescharacteristic
457C2×C4#38611yescharacteristic
458C2×C4#38611yescharacteristic
459C2×C4#38611yescharacteristic
460C2×C4#38611yescharacteristic
461C2×C4#38611yescharacteristic
462C2×C4C22×C22⋊C411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×C4×D411yescharacteristic
465C2×C4C2×#12011yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×#12311yescharacteristic
468C2×C4#57714yes—
469C8—24no—
470C8—42no—
471C8—44no—
472C8—44no—
473C8—82no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—44no—
485C22—164no—
486C22#188412yes—
487C22C2×#38611yescharacteristic
488C22#304212yes—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—44no—
499C4—82no—
500C4—82no—
501C4—162no—
502C4#294912yes—
503C4#295712yes—
504C4#300912yes—
505C2—22no—
506C2—22no—
507C2—41no—
508C2—84no—
509C2#2091911yescharacteristic
510C2#2093311yescharacteristic
511C2C2×#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. #21845C2semidirect
5. #17496C2semidirect
6. #9055C2semidirect
7. #7898C2semidirect
8. #7885C2semidirect
13. #3167C4semidirect
15. C2×#498C4semidirect
16. C2×#492C4semidirect
17. C2×#487C4semidirect
18. C2×#486C4semidirect
34. #1986C22semidirect
41. #1481C22semidirect
42. #1279C22semidirect
43. #1278C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31672
#602372
#218452
#175612
#174962
#90552
#78982
#78852