Finite Group Databasebeta

← #6433745

Subgroups — #6433745

Order 512.

2008 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34634C211yesmaximal, characteristic, complemented
2C2×#2620C211yesmaximal, characteristic
3#30195C218yesmaximal, complemented
4#30191C218yesmaximal, complemented
5#23745C212yesmaximal, complemented
6#23629C211yesmaximal, characteristic, complemented
7#23628C211yesmaximal, characteristic, complemented
8#23506C212yesmaximal, complemented
9#23455C211yesmaximal, characteristic, complemented
10#23454C211yesmaximal, characteristic, complemented
11#20973C212yesmaximal
12#20972C212yesmaximal, complemented
13#20335C211yesmaximal, characteristic, complemented
14C23×M16C2212yes—
15C2×#533C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C2×#419C2212yes—
21C2×#419C2212yes—
22C2×#418C2212yes—
23C2×#418C2212yes—
24#2629C2214yescomplemented
25#2628C2214yescomplemented
26#2627—22no—
27#2626—22no—
28#2620—24no—
29#2620C2214yes—
30#2620C2214yes—
31#2396—24no—
32C2×#396—22no—
33C2×#394—22no—
34#2296C2212yes—
35C2×#391C2211yescharacteristic
36C2×#390C2211yescharacteristic
37#2285C2211yescharacteristic
38C2×#384C2211yescharacteristic
39#2280C2212yes—
40#2100C2218yescomplemented
41#2099C2218yescomplemented
42#2074C2214yescomplemented
43#2073C2214yescomplemented
44#2072C2214yescomplemented
45#2071C2214yescomplemented
46#2042C2218yescomplemented
47#2042C2218yescomplemented
48#2026C2214yescomplemented
49#2025C2214yescomplemented
50#2024C2214yescomplemented
51#2023C2214yescomplemented
52#1895C2218yes—
53#1894C2218yescomplemented
54#1893C2218yes—
55#1892C2218yescomplemented
56#1736C2212yes—
57#1736C2212yes—
58#1736C2212yes—
59#1736C2212yes—
60#1729C2214yescomplemented
61#1729C2214yescomplemented
62#1728C2211yescharacteristic
63#1728C2211yescharacteristic
64#1728C2212yes—
65C23×Q8C2311yescharacteristic
66C23×D4—21no—
67C23×D4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C22×M16—28no—
70C22×M16—44no—
71C22×M16—44no—
72C22×M16C2312yes—
73C22×M16C2312yes—
74C22×M16C2314yes—
75C23×C8C2314yes—
76#533—44no—
77C2×C4⋊Q8C2312yes—
78C2×#125C2312yes—
79C2×#124—22no—
80C2×#124—44no—
81C2×#124C2314yes—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8C2314yes—
86C2×#121—22no—
87C2×#121—28no—
88C2×#121C2314yes—
89C2×C4×Q8—22no—
90C2×#117—44no—
91C2×#117D414yes—
92C2×#117C2312yes—
93C2×#117C2312yes—
94C2×#117C2312yes—
95C2×#117C2312yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4D412yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C22⋊C4—21no—
102C22×C22⋊C4—22no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—22no—
105C22×C22⋊C4D412yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108#419—44no—
109#419—44no—
110#418—44no—
111#418—44no—
112C2×#103C2318yes—
113C2×#103C2318yes—
114C2×#102C2318yes—
115C2×#102C2318yes—
116#400—24no—
117#399—24no—
118#398—24no—
119#398—24no—
120#396—28no—
121#396—28no—
122#396—24no—
123#396—24no—
124#395—24no—
125#394—28no—
126#394—28no—
127#393—24no—
128#391—22no—
129#391C2312yes—
130#391C2312yes—
131#390—22no—
132#390C2312yes—
133#390C2312yes—
134#389C2314yes—
135#389C2314yes—
136#385—24no—
137#384—22no—
138#384C2312yes—
139#384C2314yes—
140#382C2312yes—
141#381—22no—
142#380—22no—
143#380C2314yes—
144#380C2314yes—
145#379—22no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—24no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150C2×#95—22no—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#336C2312yes—
156#336C2312yes—
157#336C2312yes—
158#336C2312yes—
159#336C2314yes—
160#336C2314yes—
161C25—21no—
162C25C2×D411yescharacteristic
163C22×Q8—21no—
164C22×Q8—22no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8C2412yes—
170C22×D4—24no—
171C22×D4—21no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—44no—
177C22×D4—84no—
178C22×D4C2412yes—
179C23×C4—24no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D412yes—
189C23×C4#4118yes—
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C2×M16—28no—
193C2×M16—24no—
194C2×M16—24no—
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—44no—
200C2×M16—44no—
201C2×M16—432no—
202C22×C8—28no—
203C22×C8—44no—
204C22×C8—44no—
205C22×C8C2414yes—
206C22×C8C2414yes—
207C4⋊Q8—44no—
208#125—44no—
209#124—44no—
210#124—44no—
211#124—48no—
212C22⋊Q8—28no—
213C22⋊Q8—28no—
214C22⋊Q8—48no—
215C22⋊Q8—48no—
216#121—28no—
217#121—28no—
218#121—48no—
219C4×Q8—48no—
220#117—24no—
221#117—24no—
222#117—24no—
223#117—24no—
224#117—48no—
225#117—42no—
226#117—42no—
227#117—42no—
228#117—42no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—84no—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2414yes—
249C2×C4⋊C4C2414yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—48no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—84no—
264C2×C22⋊C4C2×D414yes—
265C2×C22⋊C4C2×D414yes—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2412yes—
273C2×C42C2411yescharacteristic
274C2×C42C2411yescharacteristic
275#103—48no—
276#103—48no—
277#102—48no—
278#102—48no—
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—42no—
290C24—42no—
291C24—44no—
292C24—44no—
293C24—42no—
294C24—42no—
295C24—42no—
296C24#12414yes—
297C24C22×D412yes—
298C24C22×D411yescharacteristic
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—24no—
302C2×Q8—22no—
303C2×Q8—44no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—48no—
308C2×Q8—48no—
309C2×Q8—82no—
310C2×Q8—82no—
311C2×D4—24no—
312C2×D4—22no—
313C2×D4—24no—
314C2×D4—22no—
315C2×D4—48no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—88no—
322C2×D4—816no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—216no—
328C22×C4—216no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4#12414yes—
352C22×C4#12418yes—
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C22×D414yes—
356C22×C4C22×D412yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4114yes—
359C22×C4C2×#4114yes—
360C22×C4C2×#4114yes—
361C22×C4H(5,𝔽2)14yes—
362C22×C4H(5,𝔽2)12yes—
363C22×C42−1+412yes—
364C22×C4C2511yescharacteristic
365M16—416no—
366M16—48no—
367M16—48no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—416no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—44no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—48no—
388C4⋊C4—82no—
389C4⋊C4—84no—
390C4⋊C4—82no—
391C4⋊C4—84no—
392C4⋊C4—88no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—416no—
396C22⋊C4—48no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—48no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C22⋊C4—84no—
403C22⋊C4—84no—
404C42—24no—
405C42—24no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—44no—
411C42—44no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—28no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—84no—
427C23—84no—
428C23—84no—
429C23—88no—
430C23—84no—
431C23—88no—
432C23—84no—
433C23C2×#12414yes—
434C23C2×#12412yes—
435C23#53412yes—
436C23#53712yes—
437C23C23×D411yescharacteristic
438Q8—44no—
439Q8—44no—
440Q8—88no—
441Q8—84no—
442D4—44no—
443D4—44no—
444D4—816no—
445D4—84no—
446C2×C4—28no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—216no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—48no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—84no—
478C2×C4—84no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4C2×#12414yes—
482C2×C4C2×#12412yes—
483C2×C4#53412yes—
484C2×C4#53712yes—
485C2×C4#53818yes—
486C2×C4#53918yes—
487C2×C4C22×#4112yes—
488C2×C4C2×H(5,𝔽2)11yescharacteristic
489C2×C4C2×H(5,𝔽2)12yes—
490C2×C4C2×2−1+411yescharacteristic
491C8—44no—
492C8—44no—
493C22—24no—
494C22—22no—
495C22—22no—
496C22—24no—
497C22—24no—
498C22—24no—
499C22—24no—
500C22—44no—
501C22—44no—
502C22—88no—
503C22—82no—
504C22—84no—
505C22—82no—
506C22—82no—
507C22—84no—
508C22—168no—
509C22#263914yes—
510C22C22×#12411yescharacteristic
511C22C2×#53411yescharacteristic
512C22C2×#53711yescharacteristic
513C22#358512yes—
514C22#358612yes—
515C4—24no—
516C4—44no—
517C4—42no—
518C4—42no—
519C4—82no—
520C4—84no—
521C4—82no—
522C4—82no—
523C4—84no—
524C4#263914yes—
525C4C2×#53812yes—
526C4C2×#53912yes—
527C2—24no—
528C2—44no—
529C2—162no—
530C2C2×#263911yescharacteristic
531C2#6061911yescharacteristic
532C2#6062011yescharacteristic
533C2#6075512yes—
534C2#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
1. #34634C2semidirect
3. #30195C2semidirect
4. #30191C2semidirect
5. #23745C2semidirect
6. #23629C2semidirect
7. #23628C2semidirect
8. #23506C2semidirect
9. #23455C2semidirect
10. #23454C2semidirect
12. #20972C2semidirect
13. #20335C2semidirect
24. #2629C22semidirect
25. #2628C22semidirect
40. #2100C22semidirect
41. #2099C22semidirect
42. #2074C22semidirect
43. #2073C22semidirect
44. #2072C22semidirect
45. #2071C22semidirect
46. #2042C22semidirect
47. #2042C22semidirect
48. #2026C22semidirect
49. #2025C22semidirect
50. #2024C22semidirect
51. #2023C22semidirect
53. #1894C22semidirect
55. #1892C22semidirect
60. #1729C22semidirect
61. #1729C22semidirect

Maximal subgroups

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

subgroupindex
#346342
C2×#26202
#301952
#301912
#237452
#236292
#236282
#235062
#234552
#234542
#209732
#209722
#203352