Finite Group Databasebeta

← #6355370

Subgroups — #6355370

Order 512.

1892 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60940C211yesmaximal, characteristic, complemented
2C2×#3109C212yesmaximal, complemented
3C2×#2974C211yesmaximal, characteristic, complemented
4#19465C218yesmaximal, complemented
5#19458C218yesmaximal, complemented
6#19283C214yesmaximal, complemented
7#19258C214yesmaximal, complemented
8#17818C211yesmaximal, characteristic, complemented
9#17754C212yesmaximal, complemented
10C2×#532C2212yes—
11C22×C4×D4C2211yescharacteristic
12#3245—22no—
13#3226C2214yescomplemented
14#3205C2214yescomplemented
15#3186—21no—
16#3186—21no—
17#3169C2214yescomplemented
18#3109—22no—
19#3109—22no—
20#3109—24no—
21#3109C2218yes—
22#3109C2214yescomplemented
23#3109C2214yescomplemented
24#3100—24no—
25C2×#481—22no—
26C2×#476C2214yescomplemented
27C2×#459C2214yescomplemented
28C2×#447—22no—
29#2974—24no—
30#2974C2218yescomplemented
31C2×#442—24no—
32C2×#436C2212yescomplemented
33C2×#435—22no—
34C2×C8×D4C2212yescomplemented
35C2×#432C2211yescharacteristic
36C2×#432C2212yes—
37C2×#431C2211yescharacteristic
38C2×#407—21no—
39C22×C22⋊C8—21no—
40#1697C22116yescomplemented
41#1695C2218yescomplemented
42#1687C2218yescomplemented
43#1677C22116yescomplemented
44#1675C2218yescomplemented
45#1667C2218yescomplemented
46#1573C2218yescomplemented
47#1568C2214yescomplemented
48#1558C2214yescomplemented
49#1555C2214yescomplemented
50#1551C2218yescomplemented
51#1551C2218yescomplemented
52C22×#41—22no—
53C23×D4D411yescharacteristic
54C24×C4—21no—
55C22×D8—24no—
56C22×M16—22no—
57C23×C8—22no—
58#532—22no—
59#532—22no—
60#532—22no—
61#532—22no—
62#532—22no—
63#532—22no—
64#532C2314yes—
65#532C2314yes—
66C2×C4⋊Q8C2312yes—
67C2×#127C2312yes—
68C2×#124—22no—
69C2×#121—22no—
70C2×#121—22no—
71C2×C4×D4—21no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—21no—
77C2×C4×D4D411yescharacteristic
78C2×C4×D4D411yescharacteristic
79C2×C4×D4D414yes—
80C2×C4×D4C2314yes—
81C2×C4×D4C2311yescharacteristic
82C2×C4×D4C2311yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C22⋊C4—21no—
85C22×C42C2311yescharacteristic
86#500C2314yescomplemented
87#493C2314yescomplemented
88#486C2318yescomplemented
89#481—28no—
90#481—24no—
91#481—24no—
92#480—24no—
93#480—24no—
94#476—24no—
95#476—24no—
96#476C2318yes—
97#476C2318yes—
98#474—28no—
99#466—24no—
100#466—24no—
101#459—24no—
102#459—24no—
103#459—28no—
104#459C2318yes—
105#459C2318yes—
106#447—28no—
107#447—24no—
108#447—24no—
109#447—24no—
110#447—24no—
111#442—28no—
112#442—28no—
113#442—28no—
114#442—28no—
115C4×D8—24no—
116C4×D8—24no—
117#436—24no—
118#436C2318yes—
119#435—216no—
120C8×D4—24no—
121C8×D4C2318yes—
122#432—22no—
123#432—22no—
124#432—22no—
125#432C2314yes—
126#432C2314yes—
127#432C2314yes—
128#431D414yes—
129#431C2314yes—
130C2×#106—24no—
131C2×C4⋊C8D412yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#102—24no—
135C2×#102—24no—
136#407—28no—
137C2×C22⋊C8—22no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—22no—
140C2×C22⋊C8—22no—
141C2×C22⋊C8—28no—
142C2×C22⋊C8D411yescharacteristic
143C2×C22⋊C8D411yescharacteristic
144C2×#97—21no—
145C2×#97—22no—
146C2×C4×C8C2312yes—
147#335C2318yescomplemented
148#334C2318yescomplemented
149#333C23116yescomplemented
150#331C23116yescomplemented
151C25—21no—
152C2×#41—24no—
153C2×#41—24no—
154C2×#41—42no—
155C2×#41—42no—
156C2×#41—88no—
157C22×Q8—22no—
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—44no—
164C22×D4—42no—
165C22×D4C2×D411yescharacteristic
166C22×D4C2×D414yes—
167C22×D4C2×D411yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—42no—
176C23×C4—84no—
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4#4111yescharacteristic
180C2×D8—28no—
181C2×D8—28no—
182C2×D8—24no—
183C2×D8—24no—
184C2×D8—416no—
185C2×M16—28no—
186C2×M16—84no—
187C22×C8—22no—
188C22×C8—24no—
189C22×C8—28no—
190C22×C8—84no—
191C22×C8C2×D412yes—
192C22×C8C2×D412yes—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8C2414yes—
197#127—22no—
198#127—22no—
199#127—22no—
200#127C2414yes—
201#124—24no—
202#124—24no—
203#124—42no—
204#124—42no—
205#121—24no—
206#121—24no—
207#121—24no—
208#121—24no—
209#121—42no—
210#121—42no—
211#121—42no—
212#121—42no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—22no—
216C4×D4—22no—
217C4×D4—21no—
218C4×D4—24no—
219C4×D4—21no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—24no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—24no—
227C4×D4—42no—
228C4×D4—42no—
229C4×D4—42no—
230C4×D4—42no—
231C4×D4C2×D412yes—
232C4×D4C2×D412yes—
233C4×D4C2×D414yes—
234C4×D4C2×D414yes—
235C4×D4C2412yes—
236C4×D4C2412yes—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42#4112yes—
257C2×C42#4111yescharacteristic
258C2×C42C2412yes—
259C2×C42C2411yescharacteristic
260#107—24no—
261#107—24no—
262#106—28no—
263#106—28no—
264C4⋊C8—22no—
265C4⋊C8—24no—
266C4⋊C8C2×D418yes—
267C4⋊C8C2414yes—
268C4⋊C8C2418yes—
269#102—28no—
270#102—28no—
271#102—28no—
272#102—28no—
273#102—28no—
274#102—28no—
275C22⋊C8—24no—
276C22⋊C8—24no—
277C22⋊C8—24no—
278C22⋊C8—28no—
279C22⋊C8—28no—
280C22⋊C8C2×D414yes—
281C22⋊C8C2×D414yes—
282#97—24no—
283#97—24no—
284#97—24no—
285C4×C8C2×D414yes—
286C4×C8C2414yes—
287C24—21no—
288C24—21no—
289C24—24no—
290C24—44no—
291C24—44no—
292C24—42no—
293C24—82no—
294C24#12111yescharacteristic
295#41—88no—
296#41—88no—
297#41—88no—
298#41—88no—
299C2×Q8—24no—
300C2×Q8—42no—
301C2×Q8—84no—
302C2×D4—24no—
303C2×D4—28no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—84no—
321C2×D4—84no—
322C2×D4—88no—
323C2×D4C22×D412yes—
324C2×D4C22×D412yes—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—88no—
343C22×C4—88no—
344C22×C4—88no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4#12112yes—
349C22×C4#12111yescharacteristic
350C22×C4#12112yes—
351C22×C4#12112yes—
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4C2×#4111yescharacteristic
360D8—416no—
361D8—416no—
362D8—84no—
363D8—84no—
364M16—88no—
365C2×C8—24no—
366C2×C8—28no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—44no—
370C2×C8—48no—
371C2×C8—88no—
372C2×C8C22×D414yes—
373C2×C8C22×D414yes—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4C22×D412yes—
382C4⋊C4C22×D412yes—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—24no—
386C22⋊C4—24no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C42#12112yes—
394C42#12112yes—
395C42#12112yes—
396C42C22×D411yescharacteristic
397C42C22×D411yescharacteristic
398C42C22×D412yes—
399C42C22×D412yes—
400C42C2×#4111yescharacteristic
401C42C2×#4112yes—
402C42C2511yescharacteristic
403C23—24no—
404C23—22no—
405C23—22no—
406C23—41no—
407C23—41no—
408C23—48no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—42no—
413C23—48no—
414C23—42no—
415C23—44no—
416C23—84no—
417C23—84no—
418C23—84no—
419C23—816no—
420C23—84no—
421C23#47214yes—
422C23C2×#12112yes—
423C23C2×#12111yescharacteristic
424Q8—84no—
425Q8—84no—
426D4—24no—
427D4—22no—
428D4—24no—
429D4—48no—
430D4—42no—
431D4—48no—
432D4—48no—
433D4—84no—
434D4—84no—
435D4—84no—
436D4—84no—
437D4—88no—
438D4—88no—
439D4#57514yes—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—88no—
464C2×C4—88no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4C2×#12111yescharacteristic
468C2×C4C2×#12112yes—
469C2×C4C2×#12111yescharacteristic
470C2×C4C2×#12111yescharacteristic
471C2×C4C2×#12112yes—
472C2×C4C2×#12112yes—
473C2×C4C23×D411yescharacteristic
474C2×C4C23×D411yescharacteristic
475C2×C4C22×#4111yescharacteristic
476C8—24no—
477C8—24no—
478C8—44no—
479C8—48no—
480C22—21no—
481C22—22no—
482C22—21no—
483C22—22no—
484C22—24no—
485C22—44no—
486C22—48no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—816no—
493C22—84no—
494C22—84no—
495C22—84no—
496C22—84no—
497C22—168no—
498C22C2×#47214yes—
499C22C2×#47212yes—
500C22C22×#12111yescharacteristic
501C4—21no—
502C4—22no—
503C4—21no—
504C4—22no—
505C4—21no—
506C4—22no—
507C4—21no—
508C4—42no—
509C4—42no—
510C4—84no—
511C4C2×#57512yes—
512C4#358512yes—
513C2—22no—
514C2—24no—
515C2—44no—
516C2—82no—
517C2—164no—
518C2#6042311yescharacteristic
519C2#6044411yescharacteristic
520C2C22×#47211yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #60940C2semidirect
2. C2×#3109C2semidirect
3. C2×#2974C2semidirect
4. #19465C2semidirect
5. #19458C2semidirect
6. #19283C2semidirect
7. #19258C2semidirect
8. #17818C2semidirect
9. #17754C2semidirect
13. #3226C22semidirect
14. #3205C22semidirect
17. #3169C22semidirect
22. #3109C22semidirect
23. #3109C22semidirect
26. C2×#476C22semidirect
27. C2×#459C22semidirect
30. #2974C22semidirect
32. C2×#436C22semidirect
34. C2×C8×D4C22semidirect
40. #1697C22semidirect
41. #1695C22semidirect
42. #1687C22semidirect
43. #1677C22semidirect
44. #1675C22semidirect
45. #1667C22semidirect
46. #1573C22semidirect
47. #1568C22semidirect
48. #1558C22semidirect
49. #1555C22semidirect
50. #1551C22semidirect
51. #1551C22semidirect
86. #500C23semidirect
87. #493C23semidirect
88. #486C23semidirect
147. #335C23semidirect
148. #334C23semidirect
149. #333C23semidirect
150. #331C23semidirect

Maximal subgroups

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

subgroupindex
#609402
C2×#31092
C2×#29742
#194652
#194582
#192832
#192582
#178182
#177542