Finite Group Databasebeta

← #6433451

Subgroups — #6433451

Order 512.

2030 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2621C211yesmaximal, characteristic, complemented
2#34065C211yesmaximal, characteristic
3#30124C218yesmaximal, complemented
4#30120C218yesmaximal, complemented
5#23746C211yesmaximal, characteristic, complemented
6#23733C211yesmaximal, characteristic, complemented
7#23507C211yesmaximal, characteristic, complemented
8#23481C212yesmaximal, complemented
9#23432C211yesmaximal, characteristic, complemented
10#23392C212yesmaximal
11#21061C212yesmaximal, complemented
12#20927C212yesmaximal, complemented
13#20683C211yesmaximal, characteristic, complemented
14C22×#137C2212yes—
15C22×C4⋊Q8C2211yescharacteristic
16C2×#525C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C2×C4×#41C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C2×#428C2212yes—
21C2×#417C2212yes—
22C2×#419C2212yes—
23C2×#419C2212yes—
24C2×#419C2212yes—
25C2×#407C2212yes—
26#2621—22no—
27#2621—22no—
28#2621C2218yes—
29C2×#398—22no—
30C2×#397—22no—
31#2324C2214yes—
32#2323C2214yes—
33#2322—22no—
34#2321—22no—
35#2297C2211yescharacteristic
36#2296C2211yescharacteristic
37C2×#391C2212yes—
38#2293—22no—
39#2290—22no—
40C2×#382C2211yescharacteristic
41#2280C2211yescharacteristic
42C4×#117C2211yescharacteristic
43#2101C2218yescomplemented
44#2097C2214yescomplemented
45#2092C2214yescomplemented
46#2043C2218yescomplemented
47#2036C2218yes—
48#2035C2218yes—
49#2016C2214yescomplemented
50#2013C2214yescomplemented
51#2001C2218yes—
52#2000C2218yes—
53#1928C2218yescomplemented
54#1927C2218yescomplemented
55#1878C2218yescomplemented
56#1877C2218yescomplemented
57#1818C2218yes—
58#1736C2212yes—
59#1736C2212yes—
60#1728C2212yes—
61#1728C2211yescharacteristic
62#1728C2211yescharacteristic
63C22×#41C2311yescharacteristic
64C23×Q8C2312yes—
65C24×C4C2311yescharacteristic
66C2×#137—28no—
67C2×#137—44no—
68C2×#137—44no—
69C22×Q16C2314yes—
70C22×SD16C2314yes—
71C22×M16C2312yes—
72C22×M16C2312yes—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—44no—
75C2×C4⋊Q8C2314yes—
76#525—24no—
77#525—24no—
78C2×#123—21no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—44no—
81C2×C22⋊Q8—44no—
82C2×C22⋊Q8C2312yes—
83C2×C22⋊Q8C2318yes—
84C2×#121C2312yes—
85C2×#120—21no—
86C4×#41—44no—
87C2×C4×Q8C2312yes—
88C2×C4×D4—22no—
89C2×C4×D4—22no—
90C2×C4×D4C2312yes—
91C2×#117—21no—
92C2×#117—44no—
93C2×#117D412yes—
94C2×#117D412yes—
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C2×#117C2314yes—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4D412yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—22no—
104C22×C22⋊C4D412yes—
105C22×C42—21no—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108#428—44no—
109C2×#107C2314yes—
110C2×#106C2314yes—
111#419—28no—
112#419—44no—
113#419—44no—
114#417—44no—
115C2×#103C2318yes—
116C2×#103C2314yes—
117C2×#103C2318yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#102C2314yes—
121#407—44no—
122C2×C22⋊C8C2318yes—
123#398—28no—
124#398—28no—
125#397—28no—
126#397—28no—
127#391—22no—
128#391—22no—
129#391C2312yes—
130#391C2318yes—
131#390—24no—
132#390C2312yes—
133#389—24no—
134#389C2314yes—
135#388—24no—
136#388—24no—
137#388—24no—
138#387—24no—
139#386—22no—
140#386—24no—
141#385—22no—
142#385—24no—
143#382—22no—
144#382C2314yes—
145#381—21no—
146#380C2314yes—
147#379—21no—
148C4×C4⋊C4—22no—
149C4×C4⋊C4C2314yes—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—22no—
152C4×C22⋊C4—22no—
153#376—21no—
154C2×#95—22no—
155C2×#95—24no—
156C43—21no—
157#340C2318yes—
158#340C2318yes—
159#336C2314yes—
160#336C2314yes—
161#336C2318yes—
162C25C2×D411yescharacteristic
163C2×#41—22no—
164C2×#41—24no—
165C2×#41—24no—
166C2×#41#4114yes—
167C22×Q8—22no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8C2414yes—
175C22×Q8C2411yescharacteristic
176C22×D4—21no—
177C22×D4—22no—
178C22×D4C2411yescharacteristic
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4C2412yes—
192C23×C4C2411yescharacteristic
193#137—432no—
194C2×Q16—28no—
195C2×Q16—44no—
196C2×Q16—44no—
197C2×SD16—28no—
198C2×SD16—44no—
199C2×SD16—44no—
200C2×M16—24no—
201C2×M16—24no—
202C2×M16—24no—
203C2×M16—24no—
204C2×M16—24no—
205C2×M16—24no—
206C22×C8C2414yes—
207C22×C8C2414yes—
208C4⋊Q8—48no—
209C4⋊Q8—48no—
210#123—28no—
211C22⋊Q8—24no—
212C22⋊Q8—24no—
213C22⋊Q8—48no—
214C22⋊Q8—48no—
215C22⋊Q8—416no—
216#121—24no—
217#121—24no—
218#120—28no—
219C4×Q8—44no—
220C4×D4—216no—
221C4×D4—44no—
222C4×D4—48no—
223#117—22no—
224#117—24no—
225#117—24no—
226#117—44no—
227#117—44no—
228#117—48no—
229#117—44no—
230#117#4114yes—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—48no—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2414yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4C2×D414yes—
260C2×C22⋊C4C2×D414yes—
261C2×C42—21no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—44no—
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42C2×D411yescharacteristic
272C2×C42C2412yes—
273C2×C42C2412yes—
274#107—44no—
275#106—44no—
276#103—216no—
277#103—44no—
278#103—44no—
279#103—44no—
280#103—48no—
281#102—44no—
282C22⋊C8—48no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290C24—22no—
291C24—21no—
292C24—22no—
293C24—22no—
294C24—44no—
295C24C22×D412yes—
296C24C22×D411yescharacteristic
297#41—216no—
298#41—48no—
299C2×Q8—21no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—48no—
312C2×Q8—48no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8C2×#4112yes—
316C2×D4—24no—
317C2×D4—28no—
318C2×D4—21no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—84no—
324C2×D4C2×#4112yes—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—84no—
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C22×D414yes—
360C22×C4C22×D412yes—
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4112yes—
363C22×C4C2×#4112yes—
364C22×C4H(5,𝔽2)14yes—
365C22×C4H(5,𝔽2)14yes—
366C22×C4C2511yescharacteristic
367Q16—416no—
368SD16—416no—
369M16—48no—
370M16—48no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C4⋊C4—28no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—416no—
387C4⋊C4—416no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—84no—
391C4⋊C4C2×#4114yes—
392C22⋊C4—28no—
393C22⋊C4—28no—
394C22⋊C4—24no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C22⋊C4—416no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—84no—
402C22⋊C4—84no—
403C42—24no—
404C42—22no—
405C42—22no—
406C42—24no—
407C42—24no—
408C42—24no—
409C42—44no—
410C42—42no—
411C42—42no—
412C42—84no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—48no—
421C23—44no—
422C23—44no—
423C23—41no—
424C23—41no—
425C23—84no—
426C23—84no—
427C23#52512yes—
428C23#53412yes—
429C23#53512yes—
430C23C23×D411yescharacteristic
431Q8—24no—
432Q8—42no—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—44no—
437Q8—88no—
438D4—24no—
439D4—48no—
440D4—42no—
441D4—88no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—48no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—48no—
460C2×C4—48no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—84no—
473C2×C4—88no—
474C2×C4—88no—
475C2×C4—88no—
476C2×C4#52512yes—
477C2×C4#53412yes—
478C2×C4#53512yes—
479C2×C4#53818yes—
480C2×C4#54018yes—
481C2×C4C22×#4111yescharacteristic
482C2×C4C22×#4111yescharacteristic
483C2×C4C2×H(5,𝔽2)12yes—
484C2×C4C2×H(5,𝔽2)12yes—
485C8—44no—
486C8—44no—
487C22—24no—
488C22—24no—
489C22—24no—
490C22—42no—
491C22—42no—
492C22—44no—
493C22—44no—
494C22—42no—
495C22—44no—
496C22—42no—
497C22—88no—
498C22—82no—
499C22—82no—
500C22#263514yes—
501C22C2×#52511yescharacteristic
502C22C2×#53411yescharacteristic
503C22C2×#53511yescharacteristic
504C22#358614yes—
505C4—42no—
506C4—42no—
507C4—44no—
508C4—42no—
509C4—42no—
510C4—44no—
511C4—84no—
512C4—82no—
513C4—82no—
514C4—84no—
515C4#263514yes—
516C4C2×#53812yes—
517C4C2×#54012yes—
518C2—24no—
519C2—82no—
520C2—82no—
521C2C2×#263511yescharacteristic
522C2#6056912yes—
523C2#6075412yes—
524C2#6080712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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. C2×#2621C2semidirect
3. #30124C2semidirect
4. #30120C2semidirect
5. #23746C2semidirect
6. #23733C2semidirect
7. #23507C2semidirect
8. #23481C2semidirect
9. #23432C2semidirect
11. #21061C2semidirect
12. #20927C2semidirect
13. #20683C2semidirect
43. #2101C22semidirect
44. #2097C22semidirect
45. #2092C22semidirect
46. #2043C22semidirect
49. #2016C22semidirect
50. #2013C22semidirect
53. #1928C22semidirect
54. #1927C22semidirect
55. #1878C22semidirect
56. #1877C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26212
#340652
#301242
#301202
#237462
#237332
#235072
#234812
#234322
#233922
#210612
#209272
#206832