Finite Group Databasebeta

← #6494934

Subgroups — #6494934

Order 512.

1738 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2C4×#529C211yesmaximal, characteristic
3#33198C212yesmaximal, complemented
4#33197C212yesmaximal, complemented
5#29996C212yesmaximal, complemented
6#29994C212yesmaximal, complemented
7#29289C214yesmaximal, complemented
8#28282C214yesmaximal, complemented
9#24846C214yesmaximal, complemented
10#24843C214yesmaximal
11#21963C212yesmaximal, complemented
12#21890C212yesmaximal
13#20791C211yesmaximal, characteristic, complemented
14C2×#533—21no—
15C2×#529C2211yescharacteristic
16C2×#477C2214yes—
17C2×#476C2214yes—
18C2×#423C2212yes—
19C2×#423C2214yes—
20#2868C2212yes—
21#2862—21no—
22#2859—21no—
23#2859C2212yes—
24#2857—22no—
25#2857C2214yes—
26C4×C4⋊Q8—22no—
27C4×C4⋊Q8C2214yes—
28C4×#125C2212yes—
29C4×#126—22no—
30C4×#124C2212yes—
31C4×C22⋊Q8C2212yes—
32C4×#123—22no—
33C4×#121C2212yes—
34#2298—21no—
35#2295C2211yescharacteristic
36#2293C2211yescharacteristic
37#2290—21no—
38#2285—21no—
39#2285C2212yes—
40#2283C2211yescharacteristic
41C42×Q8C2211yescharacteristic
42C42×D4—21no—
43C42×D4C2211yescharacteristic
44C4×#117—21no—
45C2×C43C2211yescharacteristic
46#2088C2218yes—
47#2086C2218yes—
48#2067C2218yes—
49#2066C2218yes—
50#1989C2214yes—
51#1987C2214yes—
52#1987C2214yes—
53#1983C2214yes—
54#1949C2218yes—
55#1948C2214yes—
56#1948C2214yes—
57#1947C2214yes—
58#1947C2214yes—
59#1895C2214yes—
60#1894C2214yes—
61#1842C2212yes—
62#1842C2214yes—
63#1842C2212yes—
64#1809C2218yes—
65#1803C2214yes—
66#1803C2214yes—
67#1802C2214yes—
68#1802C2214yes—
69C22×M16C2314yes—
70#533—48no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8C2314yes—
74#529—24no—
75#529Q818yescomplemented
76C2×#126—21no—
77C2×#125—22no—
78C2×#125C2311yescharacteristic
79C2×#124C2311yescharacteristic
80C2×#123—21no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8C2311yescharacteristic
83C2×#121C2311yescharacteristic
84C2×C4×Q8—24no—
85C2×C4×Q8C2312yes—
86C2×C4×Q8C2311yescharacteristic
87C2×C4×D4—21no—
88C2×C4×D4—22no—
89C2×C4×D4C2312yes—
90C2×C4×D4C2311yescharacteristic
91C2×#117—22no—
92C2×#117—21no—
93C2×#117—21no—
94C22×C42—21no—
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42D412yes—
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100#477—48no—
101#476—48no—
102C2×#107C2318yes—
103C2×#106C2318yes—
104#423—28no—
105#423—48no—
106C2×C4⋊C8C2314yes—
107C2×C4⋊C8C2314yes—
108C2×C4⋊C8C2314yes—
109C2×C4⋊C8C2314yes—
110C2×C4⋊C8C2318yes—
111C2×#103C2318yes—
112C2×#102C2318yes—
113#391—24no—
114#391—24no—
115#391C2312yes—
116#390C2312yes—
117#389—24no—
118#389—24no—
119#389—24no—
120#389C2312yes—
121#389C2312yes—
122#389C2318yes—
123#388—22no—
124#388—22no—
125#388—22no—
126#388—22no—
127#387—22no—
128#387—22no—
129#387—22no—
130#387—22no—
131#386—22no—
132#385—22no—
133#384—22no—
134#384—22no—
135#384C2314yes—
136#384C2314yes—
137#383—21no—
138#382—22no—
139#382—22no—
140#382C2312yes—
141#381—21no—
142#380—22no—
143#380—24no—
144#380C2312yes—
145#380C2314yes—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4C2312yes—
155C4×C4⋊C4C2312yes—
156C4×C4⋊C4C2314yes—
157C4×C4⋊C4C2311yescharacteristic
158C4×C4⋊C4C2311yescharacteristic
159C4×C22⋊C4—21no—
160C4×C22⋊C4—22no—
161C4×C22⋊C4—22no—
162C4×C22⋊C4—22no—
163C4×C22⋊C4—22no—
164C4×C22⋊C4—22no—
165C4×C22⋊C4—21no—
166C4×C22⋊C4—21no—
167#376—21no—
168#376—21no—
169C43—21no—
170C43D412yes—
171C43D411yescharacteristic
172C43D411yescharacteristic
173C43C2312yes—
174C43C2311yescharacteristic
175C43C2311yescharacteristic
176#340C2318yes—
177#340C2318yes—
178#340C2318yes—
179#340C2318yes—
180C22×Q8—24no—
181C22×Q8C2411yescharacteristic
182C22×D4—21no—
183C22×D4C2411yescharacteristic
184C23×C4—21no—
185C23×C4—22no—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C2×M16—28no—
192C2×M16—48no—
193C22×C8C2418yes—
194C4⋊Q8—28no—
195C4⋊Q8—28no—
196C4⋊Q8—48no—
197C4⋊Q8—48no—
198#126—28no—
199#125—22no—
200#125—48no—
201#125C2×Q814yes—
202#124—22no—
203#124C2×Q814yes—
204#123—28no—
205C22⋊Q8—22no—
206C22⋊Q8—416no—
207C22⋊Q8C2×Q814yes—
208#121—22no—
209#121C2×Q814yes—
210C4×Q8—22no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8—416no—
214C4×Q8C2×Q814yes—
215C4×D4—28no—
216C4×D4—22no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—48no—
220C4×D4C2×Q814yes—
221#117—28no—
222#117—48no—
223#117—44no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2412yes—
239C2×C4⋊C4C2418yes—
240C2×C4⋊C4C2411yescharacteristic
241C2×C4⋊C4C2411yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C42—21no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42—44no—
264C2×C42—44no—
265C2×C42C2×D412yes—
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×Q814yes—
280C2×C42#4114yes—
281C2×C42C2412yes—
282C2×C42C2411yescharacteristic
283C2×C42C2411yescharacteristic
284C2×C42C2412yes—
285#107—48no—
286#106—48no—
287C4⋊C8—28no—
288C4⋊C8—28no—
289C4⋊C8—28no—
290C4⋊C8—28no—
291C4⋊C8—48no—
292#103—48no—
293#102—48no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—21no—
302#95—22no—
303#95—21no—
304#95—22no—
305#95—28no—
306#95—24no—
307#95—24no—
308C24—21no—
309C24C22×D411yescharacteristic
310C2×Q8—21no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8C22×Q812yes—
317C2×D4—24no—
318C2×D4—21no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—44no—
322C2×D4C22×Q812yes—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—28no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4C22⋊Q814yes—
347C22×C4C22⋊Q814yes—
348C22×C4C22×D412yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D412yes—
357C22×C4C22×Q812yes—
358C22×C4C2×#4112yes—
359C22×C4C2511yescharacteristic
360M16—88no—
361C2×C8—28no—
362C2×C8—28no—
363C2×C8—28no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—28no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—28no—
376C4⋊C4—44no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—44no—
380C4⋊C4—48no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—416no—
384C4⋊C4—48no—
385C4⋊C4C22×Q812yes—
386C4⋊C4C22×Q812yes—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—44no—
393C22⋊C4—48no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C42—24no—
398C42—24no—
399C42—24no—
400C42—24no—
401C42—22no—
402C42—24no—
403C42—22no—
404C42—24no—
405C42—24no—
406C42—22no—
407C42—22no—
408C42—42no—
409C42—42no—
410C42—48no—
411C42—44no—
412C42—42no—
413C42—44no—
414C42—42no—
415C42—44no—
416C42C22⋊Q814yes—
417C42C22⋊Q814yes—
418C42AGL(1,ℤ/8)14yes—
419C42#13714yes—
420C42C22×Q812yes—
421C42C22×Q812yes—
422C42C2×#4112yes—
423C42C2×#4112yes—
424C42H(5,𝔽2)14yes—
425C422−1+414yes—
426C23—22no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—22no—
431C23—42no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23C2×C22⋊Q812yes—
436C23C23×D411yescharacteristic
437Q8—24no—
438Q8—41no—
439Q8—41no—
440Q8—816no—
441D4—24no—
442D4—48no—
443D4—41no—
444D4—41no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—28no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—48no—
471C2×C4—48no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—48no—
481C2×C4—88no—
482C2×C4C2×C22⋊Q812yes—
483C2×C4C2×C22⋊Q812yes—
484C2×C4C2×C22⋊Q812yes—
485C2×C4C2×C22⋊Q812yes—
486C2×C4C2×C22⋊Q812yes—
487C2×C4#53714yes—
488C2×C4#53714yes—
489C2×C4C2×AGL(1,ℤ/8)14yes—
490C2×C4C2×AGL(1,ℤ/8)12yes—
491C2×C4C2×#13712yes—
492C2×C4C2×#13714yes—
493C2×C4#57514yes—
494C2×C4C23×Q811yescharacteristic
495C2×C4C22×#4111yescharacteristic
496C2×C4C2×H(5,𝔽2)12yes—
497C2×C4C2×2−1+412yes—
498C8—48no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—82no—
508C22—82no—
509C22C22×C22⋊Q811yescharacteristic
510C22C2×#53712yes—
511C22C22×AGL(1,ℤ/8)11yescharacteristic
512C22C22×#13711yescharacteristic
513C22C2×#57512yes—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—24no—
519C4—24no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—82no—
525C4—82no—
526C4—88no—
527C4#307914yes—
528C4#325218yes—
529C4#350312yes—
530C4#350811yescharacteristic
531C4#350911yescharacteristic
532C2—42no—
533C2—82no—
534C2#3528011yescharacteristic
535C2C2×#307911yescharacteristic
536C2#6048411yescharacteristic
537C2C2×#325212yes—
538C2#6091612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36645C2semidirect
3. #33198C2semidirect
4. #33197C2semidirect
5. #29996C2semidirect
6. #29994C2semidirect
7. #29289C2semidirect
8. #28282C2semidirect
9. #24846C2semidirect
11. #21963C2semidirect
13. #20791C2semidirect
75. #529Q8semidirect

Maximal subgroups

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

subgroupindex
#366452
C4×#5292
#331982
#331972
#299962
#299942
#292892
#282822
#248462
#248432
#219632
#218902
#207912