Finite Group Databasebeta

← #6644832

Subgroups — #6644832

Order 512.

1768 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#2563C211yesmaximal, characteristic
3#33099C212yesmaximal, complemented
4#32261C214yesmaximal, complemented
5#31324C218yesmaximal, complemented
6#31263C214yesmaximal, complemented
7#29642C212yesmaximal, complemented
8#27603C212yesmaximal, complemented
9#24552C214yesmaximal, complemented
10#24006C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×#484C2214yes—
16C22×C22⋊C8C2212yes—
17#2855C2212yes—
18#2848C2212yes—
19#2627—21no—
20#2626—21no—
21#2579—22no—
22#2564—22no—
23#2563—22no—
24#2563—22no—
25#2563C2214yes—
26#2563C2214yes—
27#2487C2212yes—
28#2417C2214yes—
29#2409C2212yes—
30C2×#393—21no—
31C2×#393—21no—
32#2315—21no—
33#2308—21no—
34C2×#390—21no—
35C2×#382—21no—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40#2076C2218yescomplemented
41#2076C2218yescomplemented
42#2057C2218yescomplemented
43#2036C2218yescomplemented
44#2036C2218yescomplemented
45#2024C2218yescomplemented
46#1928C2218yescomplemented
47#1921C2218yescomplemented
48#1892C2218yescomplemented
49#1872C2214yes—
50#1869C2214yes—
51#1869C2214yes—
52#1852C2214yes—
53#1852C2214yescomplemented
54#1852C2214yescomplemented
55#1824C2214yes—
56#1824C2214yescomplemented
57#1817C2212yes—
58#1808C2218yescomplemented
59#1787C2214yescomplemented
60#1779C2212yes—
61C23×Q8—21no—
62C23×Q8C2311yescharacteristic
63C23×D4D411yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C23×C8C2314yes—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—44no—
69C2×C4⋊Q8C2314yes—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—44no—
73C2×C22⋊Q8C2312yes—
74C2×C22⋊Q8C2314yes—
75C2×C22⋊Q8C2312yes—
76C2×C4×D4—44no—
77C2×C4×D4D412yes—
78C2×C4×D4D412yes—
79C2×C4×D4C2312yes—
80C2×C4×D4C2312yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96#484—48no—
97C2×#107C2318yes—
98C2×#107C2318yes—
99C2×#103C2318yes—
100C2×#103C2318yes—
101C2×C22⋊C8—44no—
102C2×C22⋊C8C2314yes—
103C2×C22⋊C8C2314yes—
104C2×C22⋊C8C2314yes—
105C2×C22⋊C8C2314yes—
106#400—22no—
107#399—22no—
108#398—24no—
109#396—24no—
110#395—22no—
111#393—24no—
112#393—24no—
113#393—24no—
114#393—24no—
115#393—22no—
116#391C2314yes—
117#390—24no—
118#390—24no—
119#389C2314yes—
120#387—24no—
121#386—24no—
122#386—22no—
123#386—22no—
124#385—24no—
125#385—22no—
126#385—22no—
127#385—22no—
128#385—22no—
129#382—24no—
130#382—24no—
131#381—22no—
132#380—22no—
133#380—22no—
134#380—24no—
135#380—21no—
136#380—21no—
137#380—24no—
138#380C2318yes—
139#380C2314yes—
140#380C2314yes—
141#380C2314yes—
142#380C2312yes—
143#380C2312yes—
144#379—22no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4C2314yes—
147C2×#95—21no—
148C2×#95—21no—
149C2×#95—22no—
150C2×#95—22no—
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#336C2318yes—
156C25C2×D411yescharacteristic
157C25C2×D411yescharacteristic
158C22×Q8—21no—
159C22×Q8—24no—
160C22×Q8—42no—
161C22×Q8—44no—
162C22×Q8—44no—
163C22×Q8—84no—
164C22×Q8C2412yes—
165C22×D4—42no—
166C22×D4—42no—
167C22×D4Q1618yes—
168C22×D4C2×D412yes—
169C22×D4C2×D412yes—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C22×C8—48no—
191C22×C8—44no—
192C22×C8C2418yes—
193C4⋊Q8—44no—
194C4⋊Q8—44no—
195C4⋊Q8—48no—
196C22⋊Q8—28no—
197C22⋊Q8—44no—
198C22⋊Q8—44no—
199C22⋊Q8—44no—
200C22⋊Q8—48no—
201C4×D4—44no—
202C4×D4—44no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—42no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2412yes—
226C2×C4⋊C4C2414yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2414yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—84no—
243C2×C22⋊C4—84no—
244C2×C22⋊C4C2×D412yes—
245C2×C22⋊C4C2×D412yes—
246C2×C22⋊C4C2×D412yes—
247C2×C22⋊C4C2×D412yes—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—44no—
251C2×C42C2×D412yes—
252C2×C42C2412yes—
253#107—48no—
254#107—48no—
255#103—48no—
256#103—48no—
257C22⋊C8—28no—
258C22⋊C8—44no—
259C22⋊C8—44no—
260C22⋊C8—44no—
261#95—24no—
262#95—28no—
263#95—24no—
264#95—24no—
265#95—22no—
266#95—22no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24C2×Q1614yes—
280C24C2×Q1614yes—
281C24C22×D412yes—
282C24C22×D411yescharacteristic
283C24C22×D411yescharacteristic
284C24C22×D411yescharacteristic
285C2×Q8—24no—
286C2×Q8—28no—
287C2×Q8—42no—
288C2×Q8—48no—
289C2×Q8—82no—
290C2×Q8—88no—
291C2×Q8—88no—
292C2×Q8—816no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—416no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4C2×Q1618yes—
299C2×D4C2×Q1618yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—42no—
322C22×C4—44no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4C2×Q1614yes—
343C22×C4C22×D412yes—
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D412yes—
349C22×C4H(5,𝔽2)14yes—
350C22×C4C2511yescharacteristic
351C2×C8—28no—
352C2×C8—28no—
353C2×C8—28no—
354C2×C8—88no—
355C4⋊C4—28no—
356C4⋊C4—24no—
357C4⋊C4—28no—
358C4⋊C4—28no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—42no—
364C4⋊C4—48no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—84no—
368C4⋊C4—82no—
369C4⋊C4—84no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—44no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—84no—
379C22⋊C4—88no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C22⋊C4—84no—
383C42—24no—
384C42—22no—
385C42—22no—
386C42—48no—
387C23—22no—
388C23—22no—
389C23—24no—
390C23—22no—
391C23—22no—
392C23—21no—
393C23—21no—
394C23—22no—
395C23—24no—
396C23—22no—
397C23—22no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—28no—
402C23—28no—
403C23—48no—
404C23—48no—
405C23—82no—
406C23—82no—
407C23—84no—
408C23—84no—
409C23—82no—
410C23—82no—
411C23#53412yes—
412C23C22×Q1614yes—
413C23C22×Q1612yes—
414C23C22×Q1612yes—
415C23C22×Q1612yes—
416C23#57512yes—
417C23C23×D411yescharacteristic
418Q8—48no—
419Q8—816no—
420Q8—84no—
421D4—48no—
422D4—48no—
423C2×C4—28no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—28no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—82no—
455C2×C4—84no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—88no—
460C2×C4—88no—
461C2×C4—84no—
462C2×C4—84no—
463C2×C4—164no—
464C2×C4#53412yes—
465C2×C4C22×Q1614yes—
466C2×C4C2×H(5,𝔽2)12yes—
467C2×C4#58514yes—
468C8—48no—
469C22—24no—
470C22—24no—
471C22—22no—
472C22—22no—
473C22—28no—
474C22—24no—
475C22—24no—
476C22—22no—
477C22—22no—
478C22—24no—
479C22—42no—
480C22—44no—
481C22—42no—
482C22—41no—
483C22—42no—
484C22—41no—
485C22—41no—
486C22—42no—
487C22—41no—
488C22—88no—
489C22—88no—
490C22#318914yes—
491C22C2×#53411yescharacteristic
492C22#348512yes—
493C22C23×Q1611yescharacteristic
494C22C2×#57511yescharacteristic
495C22#358512yes—
496C4—24no—
497C4—22no—
498C4—42no—
499C4—41no—
500C4—42no—
501C4—41no—
502C4—84no—
503C4—84no—
504C4—82no—
505C4—84no—
506C4—82no—
507C4#354514yes—
508C2—24no—
509C2—42no—
510C2—42no—
511C2—44no—
512C2#3711911yescharacteristic
513C2C2×#318911yescharacteristic
514C2#6075311yescharacteristic
515C2#6169812yes—
516C2#6171012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33099C2semidirect
4. #32261C2semidirect
5. #31324C2semidirect
6. #31263C2semidirect
7. #29642C2semidirect
8. #27603C2semidirect
9. #24552C2semidirect
10. #24006C2semidirect
40. #2076C22semidirect
41. #2076C22semidirect
42. #2057C22semidirect
43. #2036C22semidirect
44. #2036C22semidirect
45. #2024C22semidirect
46. #1928C22semidirect
47. #1921C22semidirect
48. #1892C22semidirect
53. #1852C22semidirect
54. #1852C22semidirect
56. #1824C22semidirect
58. #1808C22semidirect
59. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#25632
#330992
#322612
#313242
#312632
#296422
#276032
#245522
#240062
#217382