Finite Group Databasebeta

← #6433151

Subgroups — #6433151

Order 512.

2694 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2615C211yesmaximal, characteristic
2#34044C211yesmaximal, characteristic, complemented
3#29899C218yesmaximal, complemented
4#29898C218yesmaximal, complemented
5#23642C214yesmaximal, complemented
6#23480C214yesmaximal, complemented
7#20912C212yesmaximal, complemented
8#20846C212yesmaximal, complemented
9#20580C211yesmaximal, characteristic, complemented
10C23×#41C2211yescharacteristic
11C22×#121C2211yescharacteristic
12C22×#121C2212yes—
13C22×#117C2212yes—
14C2×#428C2214yes—
15C2×#418C2214yes—
16C2×#407C2212yes—
17C2×#407C2214yes—
18#2615—24no—
19#2615C2218yes—
20C2×#393—21no—
21C2×#392—22no—
22C2×#392—21no—
23#2311C2214yescomplemented
24#2311C2214yescomplemented
25#2310—21no—
26#2309—21no—
27#2308—22no—
28#2297—24no—
29#2280C2214yes—
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32#2077C2218yescomplemented
33#2077C2218yescomplemented
34#2076C2218yescomplemented
35#2075C2218yescomplemented
36#2034C2218yescomplemented
37#2034C2218yescomplemented
38#2033C2218yescomplemented
39#2033C2218yescomplemented
40#1875C2218yescomplemented
41#1874C2214yescomplemented
42#1874C2214yescomplemented
43#1854C2218yescomplemented
44#1853C2214yescomplemented
45#1852C2214yescomplemented
46#1780C2218yescomplemented
47#1736C2214yes—
48#1736C2214yes—
49C22×#41—28no—
50C22×#41—48no—
51C22×#41D414yes—
52C22×#41C2314yes—
53C23×Q8D411yescharacteristic
54C23×D4—22no—
55C23×D4—21no—
56C23×D4D412yes—
57C23×D4D411yescharacteristic
58C23×D4C2311yescharacteristic
59C24×C4C2312yes—
60C24×C4C2311yescharacteristic
61C22×M16C2314yes—
62C2×#121—24no—
63C2×#121—22no—
64C2×#121—28no—
65C2×#121—48no—
66C2×#121C2314yes—
67C2×#121C2318yes—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—48no—
71C2×#117C2314yes—
72C2×#117C2314yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4—21no—
82C22×C42—22no—
83#428—48no—
84C2×#107C2318yes—
85C2×#106C2318yes—
86#418—48no—
87C2×#102C2318yes—
88C2×#102C2318yes—
89#407—28no—
90#407—48no—
91C2×C22⋊C8C2314yes—
92C2×C22⋊C8C2314yes—
93C2×C22⋊C8C2318yes—
94C2×C22⋊C8C2318yes—
95#393—28no—
96#392—28no—
97#392—28no—
98#392—28no—
99#391—24no—
100#391—24no—
101#390—24no—
102#390—24no—
103#386—24no—
104#386—24no—
105#386—24no—
106#386—24no—
107#381—24no—
108#380—24no—
109#380—22no—
110#380C2314yes—
111#380C2314yes—
112#380C2314yes—
113#380C2314yes—
114#380C2318yes—
115#380C2314yes—
116#379—24no—
117C4×C22⋊C4—24no—
118C4×C22⋊C4—24no—
119C2×#95—22no—
120C2×#95—21no—
121#340C2318yes—
122#340C2318yes—
123#340C2318yes—
124#340C2318yes—
125C25—21no—
126C25C2×D412yes—
127C25C2×D411yescharacteristic
128C2×#41—216no—
129C2×#41—28no—
130C2×#41—28no—
131C2×#41—48no—
132C2×#41—48no—
133C2×#41—44no—
134C2×#41—44no—
135C2×#41—44no—
136C2×#41—44no—
137C2×#41—464no—
138C22×Q8—24no—
139C22×Q8—44no—
140C22×Q8C2×D414yes—
141C22×D4—28no—
142C22×D4—28no—
143C22×D4—24no—
144C22×D4—24no—
145C22×D4—21no—
146C22×D4—24no—
147C22×D4—28no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—48no—
152C22×D4—44no—
153C22×D4—84no—
154C22×D4—84no—
155C22×D4—84no—
156C22×D4C2×D414yes—
157C22×D4C2×D414yes—
158C22×D4C2×D414yes—
159C22×D4C2412yes—
160C23×C4—28no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—28no—
164C23×C4—28no—
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4C2×D412yes—
170C23×C4C2×D414yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4#4118yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C2×M16—28no—
178C2×M16—28no—
179C2×M16—28no—
180C22×C8C2418yes—
181#121—28no—
182#121—28no—
183#121—48no—
184#121—48no—
185#121—416no—
186#121—48no—
187#117—28no—
188#117—28no—
189#117—44no—
190#117—44no—
191#117—48no—
192#117—48no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4C2414yes—
202C2×C4⋊C4C2414yes—
203C2×C4⋊C4C2412yes—
204C2×C4⋊C4C2414yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—48no—
216C2×C22⋊C4—48no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—84no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—48no—
223#107—48no—
224#106—48no—
225#102—48no—
226#102—48no—
227C22⋊C8—28no—
228C22⋊C8—28no—
229C22⋊C8—48no—
230C22⋊C8—48no—
231#95—28no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237C24—24no—
238C24—24no—
239C24—24no—
240C24—24no—
241C24—22no—
242C24—41no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—41no—
248C24—42no—
249C24—42no—
250C24—44no—
251C24—42no—
252C24—42no—
253C24—44no—
254C24#12314yes—
255C24C22×D414yes—
256C24C22×D412yes—
257C24C22×D411yescharacteristic
258#41—464no—
259#41—416no—
260#41—416no—
261#41—416no—
262#41—416no—
263C2×Q8—28no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—416no—
267C2×D4—28no—
268C2×D4—28no—
269C2×D4—28no—
270C2×D4—24no—
271C2×D4—28no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—416no—
279C2×D4—432no—
280C2×D4—416no—
281C2×D4—48no—
282C2×D4—42no—
283C2×D4—44no—
284C2×D4—416no—
285C2×D4—84no—
286C2×D4—84no—
287C2×D4—88no—
288C2×D4—88no—
289C2×D4—816no—
290C2×D4—832no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—232no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—416no—
314C22×C4—416no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4#12314yes—
327C22×C4#12318yes—
328C22×C4C22×D414yes—
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4114yes—
332C22×C4C2×#4114yes—
333C22×C4C2×#4114yes—
334C22×C4H(5,𝔽2)18yes—
335C22×C4C2511yescharacteristic
336M16—416no—
337C2×C8—28no—
338C2×C8—28no—
339C2×C8—28no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—28no—
345C4⋊C4—24no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C22⋊C4—416no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C22⋊C4—48no—
354C22⋊C4—84no—
355C22⋊C4—84no—
356C22⋊C4—84no—
357C22⋊C4—84no—
358C22⋊C4—84no—
359C22⋊C4—88no—
360C42—48no—
361C42—48no—
362C23—21no—
363C23—22no—
364C23—21no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—28no—
372C23—24no—
373C23—24no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23—24no—
378C23—24no—
379C23—44no—
380C23—44no—
381C23—48no—
382C23—48no—
383C23—48no—
384C23—42no—
385C23—44no—
386C23—42no—
387C23—42no—
388C23—44no—
389C23—84no—
390C23—84no—
391C23—84no—
392C23—84no—
393C23—84no—
394C23—84no—
395C23—88no—
396C23—84no—
397C23—88no—
398C23—84no—
399C23—84no—
400C23C2×#12314yes—
401C23C2×#12312yes—
402C23#53414yes—
403C23C23×D411yescharacteristic
404Q8—416no—
405D4—416no—
406D4—416no—
407D4—416no—
408D4—48no—
409D4—816no—
410D4—832no—
411D4—84no—
412C2×C4—28no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—216no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—48no—
422C2×C4—416no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—88no—
439C2×C4—84no—
440C2×C4—88no—
441C2×C4—88no—
442C2×C4C2×#12314yes—
443C2×C4C2×#12312yes—
444C2×C4#53414yes—
445C2×C4#538116yes—
446C2×C4C22×#4112yes—
447C2×C4C2×H(5,𝔽2)14yes—
448C8—48no—
449C22—24no—
450C22—24no—
451C22—22no—
452C22—22no—
453C22—24no—
454C22—24no—
455C22—24no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—44no—
461C22—44no—
462C22—44no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22—44no—
467C22—88no—
468C22—816no—
469C22—84no—
470C22—84no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—168no—
475C22#262214yes—
476C22C22×#12311yescharacteristic
477C22C2×#53412yes—
478C22#358511yescharacteristic
479C22#358512yes—
480C22#358611yescharacteristic
481C4—24no—
482C4—44no—
483C4—84no—
484C4—84no—
485C4—84no—
486C4—82no—
487C4#262214yes—
488C4C2×#53814yes—
489C2—24no—
490C2—44no—
491C2—84no—
492C2—162no—
493C2C2×#262211yescharacteristic
494C2#6051811yescharacteristic
495C2#6052011yescharacteristic
496C2#6075312yes—
497C2#6075512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2615C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34044C2semidirect
3. #29899C2semidirect
4. #29898C2semidirect
5. #23642C2semidirect
6. #23480C2semidirect
7. #20912C2semidirect
8. #20846C2semidirect
9. #20580C2semidirect
23. #2311C22semidirect
24. #2311C22semidirect
32. #2077C22semidirect
33. #2077C22semidirect
34. #2076C22semidirect
35. #2075C22semidirect
36. #2034C22semidirect
37. #2034C22semidirect
38. #2033C22semidirect
39. #2033C22semidirect
40. #1875C22semidirect
41. #1874C22semidirect
42. #1874C22semidirect
43. #1854C22semidirect
44. #1853C22semidirect
45. #1852C22semidirect
46. #1780C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26152
#340442
#298992
#298982
#236422
#234802
#209122
#208462
#205802