Finite Group Databasebeta

← #6499275

Subgroups — #6499275

Order 512.

1650 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2C4×#529C211yesmaximal, characteristic
3#33067C212yesmaximal, complemented
4#33066C212yesmaximal, complemented
5#29984C212yesmaximal, complemented
6#29981C212yesmaximal, complemented
7#29278C214yesmaximal, complemented
8#29260C214yesmaximal, complemented
9#29166C214yesmaximal, complemented
10#29162C214yesmaximal
11#21963C212yesmaximal, complemented
12#21890C212yesmaximal
13#20752C211yesmaximal, characteristic, complemented
14C2×#533—21no—
15C2×#529C2211yescharacteristic
16C2×#423C2212yes—
17#2868C2212yes—
18#2862—21no—
19#2859—21no—
20#2859C2212yes—
21#2857—22no—
22#2857C2214yes—
23C4×C4⋊Q8—22no—
24C4×C4⋊Q8C2214yes—
25C4×#125C2212yes—
26C4×#126—22no—
27C4×#124C2212yes—
28C4×C22⋊Q8C2212yes—
29C4×#123—22no—
30C4×#121C2212yes—
31#2298—21no—
32#2295C2211yescharacteristic
33#2293C2211yescharacteristic
34#2290—21no—
35#2285—21no—
36#2285C2212yes—
37#2283C2211yescharacteristic
38C42×Q8C2211yescharacteristic
39C42×D4—21no—
40C42×D4C2211yescharacteristic
41C4×#117—21no—
42C2×C43C2211yescharacteristic
43#2100C2214yes—
44#2100C2214yes—
45#2095C2214yes—
46#2094C2214yes—
47#2058C2214yes—
48#2057C2214yes—
49#2024C2214yes—
50#2023C2214yes—
51#1989C2214yes—
52#1983C2214yes—
53#1949C2218yes—
54#1948C2214yes—
55#1947C2214yes—
56#1919C2218yes—
57#1918C2218yes—
58#1895C2214yes—
59#1894C2214yes—
60#1865C2218yes—
61#1862C2218yes—
62#1831C2212yes—
63#1831C2212yes—
64#1831C2214yes—
65#1809C2218yes—
66#1803C2214yes—
67#1802C2214yes—
68#1728C2214yes—
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×Q8C2311yescharacteristic
86C2×C4×Q8C2312yes—
87C2×C4×D4—21no—
88C2×C4×D4—22no—
89C2×C4×D4C2311yescharacteristic
90C2×C4×D4C2312yes—
91C2×#117—22no—
92C2×#117—21no—
93C2×#117—21no—
94C22×C42—21no—
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42D412yes—
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100C2×#107C2318yes—
101C2×#106C2318yes—
102#423—28no—
103C2×C4⋊C8C2314yes—
104C2×C4⋊C8C2314yes—
105C2×#103C2318yes—
106C2×#102C2318yes—
107#391—24no—
108#391—24no—
109#391C2312yes—
110#390C2312yes—
111#389—24no—
112#389—24no—
113#389—24no—
114#389C2312yes—
115#389C2312yes—
116#389C2318yes—
117#388—22no—
118#388—22no—
119#388—22no—
120#388—22no—
121#387—22no—
122#387—22no—
123#387—22no—
124#387—22no—
125#386—22no—
126#385—22no—
127#384—22no—
128#384—22no—
129#384C2314yes—
130#384C2314yes—
131#383—21no—
132#382—22no—
133#382—22no—
134#382C2312yes—
135#381—21no—
136#380—22no—
137#380—24no—
138#380C2312yes—
139#380C2314yes—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4C2312yes—
149C4×C4⋊C4C2312yes—
150C4×C4⋊C4C2314yes—
151C4×C4⋊C4C2311yescharacteristic
152C4×C4⋊C4C2311yescharacteristic
153C4×C22⋊C4—21no—
154C4×C22⋊C4—22no—
155C4×C22⋊C4—22no—
156C4×C22⋊C4—22no—
157C4×C22⋊C4—22no—
158C4×C22⋊C4—22no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161#376—21no—
162#376—21no—
163C43—21no—
164C43D412yes—
165C43D411yescharacteristic
166C43D411yescharacteristic
167C43C2312yes—
168C43C2311yescharacteristic
169C43C2311yescharacteristic
170#340C2318yes—
171#340C2318yes—
172#340C2318yes—
173#340C2318yes—
174#336C2314yes—
175#336C2314yes—
176#336C2318yes—
177C22×Q8—24no—
178C22×Q8C2411yescharacteristic
179C22×D4—21no—
180C22×D4C2411yescharacteristic
181C23×C4—21no—
182C23×C4—22no—
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2411yescharacteristic
188C2×M16—28no—
189C2×M16—48no—
190C22×C8C2418yes—
191C4⋊Q8—48no—
192C4⋊Q8—48no—
193C4⋊Q8—48no—
194#126—28no—
195#125—22no—
196#125—48no—
197#125C2×Q814yes—
198#124—22no—
199#124C2×Q814yes—
200#123—28no—
201C22⋊Q8—22no—
202C22⋊Q8—416no—
203C22⋊Q8C2×Q814yes—
204#121—22no—
205#121C2×Q814yes—
206C4×Q8—22no—
207C4×Q8—44no—
208C4×Q8—416no—
209C4×Q8C2×Q814yes—
210C4×D4—28no—
211C4×D4—22no—
212C4×D4—44no—
213C4×D4—48no—
214C4×D4C2×Q814yes—
215#117—28no—
216#117—48no—
217#117—44no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4C2412yes—
232C2×C4⋊C4C2412yes—
233C2×C4⋊C4C2418yes—
234C2×C4⋊C4C2411yescharacteristic
235C2×C4⋊C4C2411yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C42—21no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42—44no—
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D412yes—
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×Q814yes—
273C2×C42#4114yes—
274C2×C42C2412yes—
275C2×C42C2412yes—
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278#107—48no—
279#106—48no—
280C4⋊C8—28no—
281C4⋊C8—28no—
282#103—48no—
283#102—48no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—21no—
292#95—22no—
293#95—21no—
294#95—22no—
295#95—28no—
296#95—24no—
297#95—24no—
298C24—21no—
299C24C22×D411yescharacteristic
300C2×Q8—21no—
301C2×Q8—24no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8C22×Q812yes—
306C2×D4—24no—
307C2×D4—21no—
308C2×D4—24no—
309C2×D4—44no—
310C2×D4C22×Q812yes—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—28no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4C22⋊Q814yes—
334C22×C4C22⋊Q814yes—
335C22×C4C22×D412yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D412yes—
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×Q812yes—
345C22×C4C2×#4112yes—
346C22×C4C2511yescharacteristic
347M16—88no—
348C2×C8—28no—
349C2×C8—28no—
350C2×C8—28no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—28no—
358C4⋊C4—24no—
359C4⋊C4—28no—
360C4⋊C4—24no—
361C4⋊C4—44no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—48no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—416no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4C22×Q812yes—
371C4⋊C4C22×Q812yes—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—24no—
375C22⋊C4—24no—
376C22⋊C4—24no—
377C22⋊C4—44no—
378C22⋊C4—48no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C42—24no—
383C42—24no—
384C42—22no—
385C42—24no—
386C42—22no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—24no—
391C42—42no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—48no—
396C42—44no—
397C42—44no—
398C42—44no—
399C42—42no—
400C42—42no—
401C42C22⋊Q814yes—
402C42C22⋊Q814yes—
403C42AGL(1,ℤ/8)14yes—
404C42#13714yes—
405C42C22×Q812yes—
406C42C22×Q812yes—
407C42C2×#4112yes—
408C42C2×#4112yes—
409C23—22no—
410C23—21no—
411C23—21no—
412C23—22no—
413C23—22no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23C2×C22⋊Q812yes—
419C23C23×D411yescharacteristic
420Q8—24no—
421Q8—42no—
422Q8—816no—
423D4—24no—
424D4—48no—
425D4—42no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—28no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—44no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—48no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—48no—
457C2×C4—42no—
458C2×C4—48no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—48no—
463C2×C4—88no—
464C2×C4C2×C22⋊Q812yes—
465C2×C4C2×C22⋊Q812yes—
466C2×C4C2×C22⋊Q812yes—
467C2×C4C2×C22⋊Q812yes—
468C2×C4C2×C22⋊Q812yes—
469C2×C4C2×AGL(1,ℤ/8)14yes—
470C2×C4C2×AGL(1,ℤ/8)12yes—
471C2×C4C2×#13714yes—
472C2×C4C2×#13712yes—
473C2×C4#57514yes—
474C2×C4C23×Q811yescharacteristic
475C2×C4C22×#4111yescharacteristic
476C2×C4#58514yes—
477C8—48no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—82no—
487C22—82no—
488C22C22×C22⋊Q811yescharacteristic
489C22#344912yes—
490C22C22×AGL(1,ℤ/8)11yescharacteristic
491C22C22×#13711yescharacteristic
492C22C2×#57512yes—
493C4—22no—
494C4—24no—
495C4—24no—
496C4—24no—
497C4—22no—
498C4—22no—
499C4—22no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—82no—
505C4—88no—
506C4—82no—
507C4#307914yes—
508C4#350412yes—
509C4#351012yes—
510C2—42no—
511C2—82no—
512C2#3533411yescharacteristic
513C2C2×#307911yescharacteristic
514C2#6048411yescharacteristic
515C2#6094912yes—
516C2#6096612yes—
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. #33067C2semidirect
4. #33066C2semidirect
5. #29984C2semidirect
6. #29981C2semidirect
7. #29278C2semidirect
8. #29260C2semidirect
9. #29166C2semidirect
11. #21963C2semidirect
13. #20752C2semidirect
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
#330672
#330662
#299842
#299812
#292782
#292602
#291662
#291622
#219632
#218902
#207522