Finite Group Databasebeta

← #6435251

Subgroups — #6435251

Order 512.

1890 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2620C211yesmaximal, characteristic
2#34048C211yesmaximal, characteristic, complemented
3#29935C218yesmaximal, complemented
4#29932C214yesmaximal, complemented
5#29931C214yesmaximal, complemented
6#23732C211yesmaximal, characteristic, complemented
7#23731C211yesmaximal, characteristic, complemented
8#23629C211yesmaximal, characteristic, complemented
9#23628C211yesmaximal, characteristic, complemented
10#23481C212yesmaximal, complemented
11#23480C212yesmaximal
12#21261C212yesmaximal, complemented
13#21171C212yesmaximal, complemented
14#20627C211yesmaximal, characteristic, complemented
15C22×#124C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#423C2212yes—
22C2×#419C2212yes—
23C2×#419C2212yes—
24C2×#418C2212yes—
25C2×#418C2212yes—
26C2×#407C2212yes—
27#2620—24no—
28#2620C2214yes—
29#2620C2214yes—
30C2×#396—22no—
31C2×#394—22no—
32#2314C2212yescomplemented
33#2314C2212yescomplemented
34#2314C2214yescomplemented
35#2313—22no—
36#2312—22no—
37#2298—24no—
38C2×#391C2211yescharacteristic
39C2×#390C2211yescharacteristic
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42C2×#384C2211yescharacteristic
43#2280C2212yes—
44C4×#117C2212yes—
45#2097C2214yescomplemented
46#2096C2212yescomplemented
47#2095C2212yescomplemented
48#2094C2212yescomplemented
49#2093C2212yescomplemented
50#2092C2214yescomplemented
51#2074C2214yescomplemented
52#2073C2214yescomplemented
53#2072C2214yescomplemented
54#2071C2214yescomplemented
55#2036C2218yescomplemented
56#2035C2218yescomplemented
57#2034C2218yes—
58#2033C2218yes—
59#1983C2218yescomplemented
60#1983C2218yescomplemented
61#1948C2218yescomplemented
62#1947C2218yescomplemented
63#1798C2214yescomplemented
64#1798C2214yescomplemented
65#1736C2212yes—
66#1736C2212yes—
67#1728C2211yescharacteristic
68#1728C2211yescharacteristic
69C23×Q8C2311yescharacteristic
70C23×D4—21no—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C22×M16C2312yes—
74C22×M16C2312yes—
75C2×#124—22no—
76C2×#124—28no—
77C2×#124C2314yes—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—44no—
80C2×C22⋊Q8C2314yes—
81C2×#121—22no—
82C2×#121—44no—
83C2×#121C2314yes—
84C2×#117—28no—
85C2×#117D414yes—
86C2×#117C2312yes—
87C2×#117C2312yes—
88C2×#117C2312yes—
89C2×#117C2312yes—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—22no—
92C22×C4⋊C4D412yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C22⋊C4—21no—
96C22×C22⋊C4—22no—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4D412yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102#428—44no—
103#428—44no—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#106C2314yes—
107C2×#106C2314yes—
108#423—28no—
109C2×C4⋊C8C2318yes—
110#419—44no—
111#419—44no—
112#418—44no—
113#418—44no—
114C2×#103C2318yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#102C2318yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120#407—44no—
121C2×C22⋊C8C2318yes—
122#396—28no—
123#396—28no—
124#394—28no—
125#394—28no—
126#391—22no—
127#391C2312yes—
128#391C2312yes—
129#390—22no—
130#390C2312yes—
131#390C2312yes—
132#388—24no—
133#388—24no—
134#388—24no—
135#388—24no—
136#387—24no—
137#387—24no—
138#387—24no—
139#387—24no—
140#384—22no—
141#384C2312yes—
142#384C2311yescharacteristic
143#384C2311yescharacteristic
144#384C2314yes—
145#382C2312yes—
146#382C2312yes—
147#381—22no—
148#380—22no—
149#380—22no—
150#380C2314yes—
151#380C2314yes—
152#379—22no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4C2314yes—
156C4×C4⋊C4C2314yes—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—24no—
159#376—22no—
160C2×#95—22no—
161C43—22no—
162#340C2318yes—
163#340C2318yes—
164#336C2312yes—
165#336C2312yes—
166#336C2312yes—
167#336C2312yes—
168C25—21no—
169C25C2×D411yescharacteristic
170C22×Q8—21no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8C2412yes—
176C22×D4—24no—
177C22×D4—21no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—44no—
183C22×D4—84no—
184C22×D4C2412yes—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—24no—
201C2×M16—24no—
202C2×M16—24no—
203C2×M16—24no—
204C22×C8C2414yes—
205C22×C8C2414yes—
206#124—216no—
207#124—48no—
208C22⋊Q8—48no—
209C22⋊Q8—48no—
210#121—48no—
211#121—48no—
212#117—216no—
213#117—28no—
214#117—28no—
215#117—24no—
216#117—24no—
217#117—24no—
218#117—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—84no—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2412yes—
235C2×C4⋊C4C2412yes—
236C2×C4⋊C4C2414yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2412yes—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—48no—
252C2×C22⋊C4—84no—
253C2×C22⋊C4C2×D414yes—
254C2×C22⋊C4C2×D414yes—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42#4118yes—
263C2×C42C2411yescharacteristic
264C2×C42C2411yescharacteristic
265C2×C42C2412yes—
266#107—44no—
267#107—44no—
268#106—44no—
269#106—44no—
270C4⋊C8—216no—
271#103—48no—
272#103—44no—
273#103—44no—
274#102—48no—
275#102—44no—
276#102—44no—
277C22⋊C8—48no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—24no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—42no—
291C24—42no—
292C24—44no—
293C24—44no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24C22×D412yes—
298C24C22×D411yescharacteristic
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—24no—
302C2×Q8—22no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—48no—
307C2×Q8—48no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—48no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—48no—
317C2×D4—48no—
318C2×D4—88no—
319C2×D4—816no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4#12318yes—
351C22×C4C22×D412yes—
352C22×C4C22×D412yes—
353C22×C4C22×D412yes—
354C22×C4C22×D414yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C2×#4114yes—
357C22×C4H(5,𝔽2)14yes—
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+412yes—
360C22×C4C2511yescharacteristic
361M16—48no—
362M16—48no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—48no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—82no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—88no—
387C22⋊C4—28no—
388C22⋊C4—28no—
389C22⋊C4—48no—
390C22⋊C4—48no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—416no—
394C22⋊C4—84no—
395C22⋊C4—84no—
396C22⋊C4—84no—
397C22⋊C4—84no—
398C42—24no—
399C42—22no—
400C42—22no—
401C42—44no—
402C42—44no—
403C42—84no—
404C42—84no—
405C42#12318yes—
406C42C2×#4118yes—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—84no—
423C23—84no—
424C23—84no—
425C23—88no—
426C23—84no—
427C23—88no—
428C23—84no—
429C23C2×#12312yes—
430C23#53512yes—
431C23#53712yes—
432C23C23×D411yescharacteristic
433Q8—44no—
434Q8—44no—
435Q8—84no—
436D4—44no—
437D4—44no—
438D4—816no—
439D4—84no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—48no—
452C2×C4—44no—
453C2×C4—48no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—84no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4—84no—
474C2×C4C2×#12318yes—
475C2×C4C2×#12312yes—
476C2×C4#53512yes—
477C2×C4#53712yes—
478C2×C4#54018yes—
479C2×C4#54218yes—
480C2×C4C22×#4112yes—
481C2×C4C2×H(5,𝔽2)11yescharacteristic
482C2×C4C2×H(5,𝔽2)12yes—
483C2×C4C2×2−1+411yescharacteristic
484C8—44no—
485C8—44no—
486C22—24no—
487C22—22no—
488C22—22no—
489C22—24no—
490C22—42no—
491C22—44no—
492C22—42no—
493C22—42no—
494C22—44no—
495C22—88no—
496C22—82no—
497C22—84no—
498C22—82no—
499C22—82no—
500C22—84no—
501C22—168no—
502C22#262414yes—
503C22C22×#12311yescharacteristic
504C22C2×#53511yescharacteristic
505C22C2×#53711yescharacteristic
506C22#358512yes—
507C22#358612yes—
508C4—28no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4—82no—
516C4—82no—
517C4—84no—
518C4#262414yes—
519C4C2×#54012yes—
520C4C2×#54212yes—
521C2—24no—
522C2—82no—
523C2—162no—
524C2C2×#262411yescharacteristic
525C2#6052012yes—
526C2#6080611yescharacteristic
527C2#6080711yescharacteristic
528C2#6091912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34048C2semidirect
3. #29935C2semidirect
4. #29932C2semidirect
5. #29931C2semidirect
6. #23732C2semidirect
7. #23731C2semidirect
8. #23629C2semidirect
9. #23628C2semidirect
10. #23481C2semidirect
12. #21261C2semidirect
13. #21171C2semidirect
14. #20627C2semidirect
32. #2314C22semidirect
33. #2314C22semidirect
34. #2314C22semidirect
45. #2097C22semidirect
46. #2096C22semidirect
47. #2095C22semidirect
48. #2094C22semidirect
49. #2093C22semidirect
50. #2092C22semidirect
51. #2074C22semidirect
52. #2073C22semidirect
53. #2072C22semidirect
54. #2071C22semidirect
55. #2036C22semidirect
56. #2035C22semidirect
59. #1983C22semidirect
60. #1983C22semidirect
61. #1948C22semidirect
62. #1947C22semidirect
63. #1798C22semidirect
64. #1798C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26202
#340482
#299352
#299322
#299312
#237322
#237312
#236292
#236282
#234812
#234802
#212612
#211712
#206272