Finite Group Databasebeta

← #6506460

Subgroups — #6506460

Order 512.

2312 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2434C211yesmaximal, characteristic
3#29112C212yesmaximal, complemented
4#29105C212yesmaximal, complemented
5#27283C218yesmaximal, complemented
6#27277C218yesmaximal, complemented
7#25861C212yesmaximal, complemented
8#25856C212yesmaximal, complemented
9#21768C212yesmaximal
10#20599C212yesmaximal
11#20524C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C2×#491C2214yescomplemented
17C2×#487C2214yescomplemented
18C2×#419C2214yes—
19C2×#419C2214yes—
20C2×#407C2212yes—
21#2857C2212yes—
22#2851C2212yes—
23#2626—22no—
24#2608—22no—
25#2607—22no—
26#2441C2214yes—
27#2439C2214yes—
28#2434—22no—
29#2434—22no—
30#2434C2218yes—
31C2×#395—22no—
32#2329—22no—
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#385—22no—
36#2280C2212yes—
37C2×C4×C22⋊C4C2211yescharacteristic
38#2074C2214yescomplemented
39#2072C2214yescomplemented
40#2043C2218yescomplemented
41#2043C2218yescomplemented
42#2026C2214yescomplemented
43#2024C2214yescomplemented
44#1960C2214yescomplemented
45#1956C2214yescomplemented
46#1935C2218yescomplemented
47#1933C2218yescomplemented
48#1894C2218yes—
49#1873C2214yes—
50#1873C2214yes—
51#1824C2214yes—
52#1787C2214yes—
53#1787C2214yes—
54#1787C2214yes—
55#1787C2218yes—
56#1753C2218yes—
57#1749C2212yes—
58C4×C22⋊C8C2212yes—
59C23×Q8—21no—
60C23×Q8—22no—
61C23×Q8—28no—
62C23×Q8—44no—
63C23×Q8—44no—
64C23×Q8C2314yes—
65C23×Q8C2311yescharacteristic
66C23×Q8C2311yescharacteristic
67C23×Q8C2311yescharacteristic
68C24×C4—21no—
69C24×C4C2311yescharacteristic
70C22×M16C2314yes—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—44no—
75C2×C4⋊Q8—44no—
76C2×C4⋊Q8C2314yes—
77C2×C4⋊Q8C2314yes—
78C2×C4⋊Q8C2312yes—
79C2×#125C2312yes—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8C2314yes—
84C2×#117C2312yes—
85C2×#117C2312yes—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4D412yes—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94#491—48no—
95#487—48no—
96#419—48no—
97#419—48no—
98C2×#103C2318yes—
99C2×#103C2318yes—
100C2×#103C2318yes—
101C2×#103C2318yes—
102C2×#103C2318yes—
103C2×#103C2318yes—
104#407—28no—
105C2×C22⋊C8C2314yes—
106C2×C22⋊C8C2314yes—
107C2×#97C2314yes—
108C2×C4×C8C2314yes—
109#399—24no—
110#396—24no—
111#396—24no—
112#395—28no—
113#395—28no—
114#393—24no—
115#391—22no—
116#391—22no—
117#391—21no—
118#391—21no—
119#391—24no—
120#391C2314yes—
121#391C2314yes—
122#389—22no—
123#389—24no—
124#389C2314yes—
125#389C2314yes—
126#387—24no—
127#387—24no—
128#387—24no—
129#385—24no—
130#385—28no—
131#385—28no—
132#384C2312yes—
133#382C2312yes—
134#381—22no—
135#380—22no—
136#380C2314yes—
137#380C2314yes—
138#379—22no—
139C4×C22⋊C4—22no—
140C4×C22⋊C4—22no—
141C4×C22⋊C4D412yes—
142C4×C22⋊C4D412yes—
143C4×C22⋊C4C2314yes—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95D412yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151C25C2×D411yescharacteristic
152C22×Q8—24no—
153C22×Q8—24no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—24no—
158C22×Q8—22no—
159C22×Q8—28no—
160C22×Q8—28no—
161C22×Q8—28no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—28no—
165C22×Q8—24no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—44no—
169C22×Q8—44no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—44no—
173C22×Q8—44no—
174C22×Q8—464no—
175C22×Q8—48no—
176C22×Q8—48no—
177C22×Q8—44no—
178C22×Q8—48no—
179C22×Q8C2412yes—
180C22×Q8C2412yes—
181C22×Q8C2412yes—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—24no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4C2×D412yes—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4#4118yes—
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C2×M16—28no—
200C2×M16—44no—
201C2×M16—44no—
202C22×C8C2418yes—
203C4⋊Q8—28no—
204C4⋊Q8—28no—
205C4⋊Q8—44no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8—44no—
209C4⋊Q8—44no—
210C4⋊Q8—48no—
211#125—44no—
212C22⋊Q8—28no—
213C22⋊Q8—28no—
214C22⋊Q8—48no—
215C22⋊Q8—48no—
216#117—44no—
217#117—44no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2412yes—
234C2×C4⋊C4C2414yes—
235C2×C4⋊C4C2414yes—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4C2×D414yes—
244C2×C22⋊C4C2×D414yes—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42C2×D412yes—
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2412yes—
253C2×C42C2412yes—
254#103—216no—
255#103—216no—
256#103—48no—
257#103—48no—
258#103—48no—
259#103—48no—
260C22⋊C8—28no—
261C22⋊C8—28no—
262#97—44no—
263C4×C8—44no—
264#95—28no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95C2×D414yes—
273#95C2×D414yes—
274C24—24no—
275C24—41no—
276C24—41no—
277C24—42no—
278C24#12414yes—
279C24C22×D412yes—
280C24C22×D411yescharacteristic
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—232no—
286C2×Q8—24no—
287C2×Q8—24no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—48no—
294C2×Q8—48no—
295C2×Q8—416no—
296C2×Q8—416no—
297C2×Q8—416no—
298C2×Q8—416no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—416no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—432no—
305C2×Q8—464no—
306C2×Q8—48no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×Q8—44no—
311C2×Q8—82no—
312C2×Q8—82no—
313C2×Q8—88no—
314C2×Q8H(5,𝔽2)14yes—
315C2×Q8H(5,𝔽2)14yes—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—416no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—432no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—48no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4#12414yes—
352C22×C4#12418yes—
353C22×C4#13718yes—
354C22×C4C22×D414yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C2×#4114yes—
360C22×C4C2×#4114yes—
361C22×C4C2×#4114yes—
362C22×C4C2511yescharacteristic
363M16—88no—
364C2×C8—28no—
365C2×C8—28no—
366C2×C8—28no—
367C4⋊C4—24no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—48no—
371C4⋊C4—48no—
372C4⋊C4—416no—
373C4⋊C4—42no—
374C4⋊C4—48no—
375C4⋊C4—44no—
376C4⋊C4—48no—
377C4⋊C4—82no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—82no—
381C22⋊C4—28no—
382C22⋊C4—28no—
383C22⋊C4—48no—
384C22⋊C4—48no—
385C22⋊C4—84no—
386C22⋊C4—82no—
387C22⋊C4—82no—
388C42—24no—
389C42—24no—
390C42—41no—
391C42—41no—
392C42—42no—
393C42—82no—
394C42—82no—
395C23—22no—
396C23—21no—
397C23—21no—
398C23—24no—
399C23—28no—
400C23—22no—
401C23—22no—
402C23—24no—
403C23—44no—
404C23—44no—
405C23—48no—
406C23—48no—
407C23C2×#12414yes—
408C23C2×#12412yes—
409C23C2×#13714yes—
410C23C23×D411yescharacteristic
411Q8—28no—
412Q8—28no—
413Q8—432no—
414Q8—48no—
415Q8—48no—
416Q8—44no—
417Q8—44no—
418Q8—48no—
419Q8—84no—
420Q8—164no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—28no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—44no—
438C2×C4—48no—
439C2×C4—48no—
440C2×C4—48no—
441C2×C4—48no—
442C2×C4—48no—
443C2×C4—432no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4C2×#12414yes—
462C2×C4C2×#12412yes—
463C2×C4C2×#13714yes—
464C2×C4C2×#13714yes—
465C2×C4C22×#4112yes—
466C2×C4C2×H(5,𝔽2)12yes—
467C2×C4C2×H(5,𝔽2)12yes—
468C8—48no—
469C22—22no—
470C22—24no—
471C22—24no—
472C22—22no—
473C22—24no—
474C22—24no—
475C22—24no—
476C22—416no—
477C22—44no—
478C22—44no—
479C22#313914yes—
480C22C22×#12411yescharacteristic
481C22#348411yescharacteristic
482C22#348411yescharacteristic
483C22C22×#13712yes—
484C22#358612yes—
485C4—24no—
486C4—22no—
487C4—24no—
488C4—24no—
489C4—44no—
490C4—44no—
491C4—44no—
492C4—82no—
493C4—84no—
494C4—84no—
495C4—82no—
496C4#349014yes—
497C2—24no—
498C2—44no—
499C2#3536711yescharacteristic
500C2C2×#313911yescharacteristic
501C2#6062011yescharacteristic
502C2#6157412yes—
503C2#6158412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29112C2semidirect
4. #29105C2semidirect
5. #27283C2semidirect
6. #27277C2semidirect
7. #25861C2semidirect
8. #25856C2semidirect
16. C2×#491C22semidirect
17. C2×#487C22semidirect
38. #2074C22semidirect
39. #2072C22semidirect
40. #2043C22semidirect
41. #2043C22semidirect
42. #2026C22semidirect
43. #2024C22semidirect
44. #1960C22semidirect
45. #1956C22semidirect
46. #1935C22semidirect
47. #1933C22semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#24342
#291122
#291052
#272832
#272772
#258612
#258562
#217682
#205992
#205242