Finite Group Databasebeta

← #6644828

Subgroups — #6644828

Order 512.

2050 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35000C211yesmaximal, characteristic
2C2×#2547C211yesmaximal, characteristic, complemented
3#33094C212yesmaximal, complemented
4#32258C214yesmaximal, complemented
5#31318C218yesmaximal, complemented
6#31260C214yesmaximal, complemented
7#29638C212yesmaximal, complemented
8#27600C212yesmaximal, complemented
9#24550C214yesmaximal, complemented
10#24002C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic, complemented
12C2×#534—21no—
13C22×C4⋊Q8C2211yescharacteristic, complemented
14C22×#124—21no—
15C22×#121C2211yescharacteristic
16C22×C4×D4C2211yescharacteristic
17C2×#480C2214yes—
18C22×C22⋊C8C2212yescomplemented
19#2854C2212yes—
20#2847C2212yes—
21#2618—21no—
22#2617—21no—
23#2547—22no—
24#2547—22no—
25#2547C2214yes—
26#2547C2214yescomplemented
27#2543—22no—
28#2534—22no—
29#2486C2212yes—
30#2416C2214yes—
31#2408C2212yes—
32C2×#397—21no—
33C2×#397—21no—
34#2308—21no—
35C2×#391—21no—
36C2×#380C2212yes—
37C2×#380C2211yescharacteristic
38C2×#380C2212yescomplemented
39C2×C4×C4⋊C4C2211yescharacteristic, complemented
40#2075C2218yescomplemented
41#2075C2218yescomplemented
42#2057C2218yescomplemented
43#2033C2218yescomplemented
44#2033C2218yescomplemented
45#2023C2218yescomplemented
46#1925C2218yescomplemented
47#1918C2218yescomplemented
48#1893C2218yescomplemented
49#1874C2214yes—
50#1869C2214yes—
51#1869C2214yescomplemented
52#1852C2214yescomplemented
53#1852C2214yescomplemented
54#1852C2214yescomplemented
55#1823C2214yescomplemented
56#1823C2214yescomplemented
57#1817C2212yescomplemented
58#1808C2218yescomplemented
59#1786C2214yescomplemented
60#1779C2212yes—
61C23×Q8—21no—
62C23×D4—21no—
63C23×D4D411yescharacteristic
64C23×D4C2311yescharacteristic
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C23×C8C2314yes—
68#534—48no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—44no—
71C2×C4⋊Q8C2314yes—
72C2×#124—24no—
73C2×#124—24no—
74C2×#124—48no—
75C2×#123—22no—
76C2×#123—22no—
77C2×#121—24no—
78C2×#121—21no—
79C2×#121—21no—
80C2×#121—44no—
81C2×#121C2312yes—
82C2×#121C2314yes—
83C2×#121C2312yes—
84C2×#120—22no—
85C2×#120—22no—
86C2×C4×D4—44no—
87C2×C4×D4D412yes—
88C2×C4×D4D412yes—
89C2×C4×D4C2312yes—
90C2×C4×D4C2312yes—
91C22×C4⋊C4—21no—
92C22×C4⋊C4D411yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C22⋊C4—21no—
98C22×C22⋊C4—22no—
99C22×C22⋊C4—22no—
100C22×C22⋊C4—21no—
101C22×C22⋊C4D411yescharacteristic
102C22×C22⋊C4D411yescharacteristic
103C22×C42—21no—
104C22×C42C2311yescharacteristic
105#480—48no—
106C2×#107C2318yes—
107C2×#107C2318yes—
108C2×#102C2318yes—
109C2×#102C2318yes—
110C2×C22⋊C8—44no—
111C2×C22⋊C8C2314yes—
112C2×C22⋊C8C2314yes—
113C2×C22⋊C8C2314yes—
114C2×C22⋊C8C2314yes—
115#398—22no—
116#397—22no—
117#397—24no—
118#397—24no—
119#397—24no—
120#397—24no—
121#396—24no—
122#394—22no—
123#394—24no—
124#392—22no—
125#391—24no—
126#391—24no—
127#390C2314yes—
128#389C2314yes—
129#388—24no—
130#386—22no—
131#386—22no—
132#386—24no—
133#386—24no—
134#381—22no—
135#380—21no—
136#380—21no—
137#380—22no—
138#380—22no—
139#380—24no—
140#380C2314yes—
141#380C2318yes—
142#380C2312yes—
143#380C2314yes—
144#380C2312yes—
145#380C2314yes—
146#379—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4C2314yes—
149C2×#95—21no—
150C2×#95—22no—
151C2×#95—22no—
152C2×#95—21no—
153#340C2318yes—
154#340C2318yes—
155#340C2318yes—
156#340C2318yes—
157#336C2318yes—
158C25—21no—
159C25C2×D411yescharacteristic
160C25C2×D411yescharacteristic
161C22×Q8—24no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×Q8—44no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—24no—
168C22×D4—21no—
169C22×D4—42no—
170C22×D4—44no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4D818yes—
177C22×D4C2×D412yes—
178C22×D4C2×D412yes—
179C22×D4C2412yes—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C22×C8—48no—
200C22×C8—44no—
201C22×C8C2418yes—
202C4⋊Q8—44no—
203C4⋊Q8—44no—
204C4⋊Q8—48no—
205#124—416no—
206#124—416no—
207#123—48no—
208#123—48no—
209#121—28no—
210#121—44no—
211#121—44no—
212#121—44no—
213#121—416no—
214#121—48no—
215#120—48no—
216#120—48no—
217C4×D4—44no—
218C4×D4—44no—
219C4×D4—42no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—42no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2414yes—
239C2×C4⋊C4C2414yes—
240C2×C4⋊C4C2414yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4C2×D412yes—
259C2×C22⋊C4C2×D412yes—
260C2×C22⋊C4C2×D412yes—
261C2×C22⋊C4C2×D412yes—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—44no—
265C2×C42C2×D412yes—
266C2×C42C2412yes—
267#107—48no—
268#107—48no—
269#102—48no—
270#102—48no—
271C22⋊C8—28no—
272C22⋊C8—44no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275#95—24no—
276#95—22no—
277#95—28no—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—24no—
282#95—24no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24—42no—
293C24—42no—
294C24—42no—
295C24—82no—
296C24—82no—
297C24C2×D814yes—
298C24C2×D814yes—
299C24C22×D411yescharacteristic
300C24C22×D412yes—
301C24C22×D411yescharacteristic
302C24C22×D411yescharacteristic
303C2×Q8—48no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—816no—
307C2×D4—24no—
308C2×D4—28no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—416no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—48no—
322C2×D4—82no—
323C2×D4—88no—
324C2×D4—84no—
325C2×D4—84no—
326C2×D4—816no—
327C2×D4C2×D818yes—
328C2×D4C2×D818yes—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4—82no—
368C22×C4—82no—
369C22×C4C2×D814yes—
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4H(5,𝔽2)14yes—
377C22×C4C2511yescharacteristic
378C2×C8—28no—
379C2×C8—28no—
380C2×C8—28no—
381C2×C8—88no—
382C4⋊C4—28no—
383C4⋊C4—24no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—48no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C4⋊C4—82no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—48no—
402C22⋊C4—416no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—44no—
406C22⋊C4—48no—
407C22⋊C4—84no—
408C22⋊C4—84no—
409C42—24no—
410C42—22no—
411C42—22no—
412C42—48no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—24no—
417C23—22no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—22no—
424C23—21no—
425C23—21no—
426C23—22no—
427C23—28no—
428C23—28no—
429C23—48no—
430C23—48no—
431C23—44no—
432C23—42no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—84no—
439C23—84no—
440C23—82no—
441C23—82no—
442C23—84no—
443C23—84no—
444C23—84no—
445C23—164no—
446C23—164no—
447C23#53412yes—
448C23C22×D812yes—
449C23C22×D814yes—
450C23C22×D812yes—
451C23C22×D812yes—
452C23#57512yes—
453C23C23×D411yescharacteristic
454Q8—816no—
455D4—48no—
456D4—48no—
457D4—48no—
458D4—84no—
459D4—88no—
460D4—88no—
461D4—168no—
462C2×C4—28no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—84no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—88no—
498C2×C4—84no—
499C2×C4—88no—
500C2×C4—164no—
501C2×C4#53412yes—
502C2×C4C22×D814yes—
503C2×C4C2×H(5,𝔽2)12yes—
504C2×C4#58514yes—
505C8—48no—
506C22—24no—
507C22—22no—
508C22—24no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—24no—
513C22—24no—
514C22—28no—
515C22—24no—
516C22—42no—
517C22—44no—
518C22—42no—
519C22—41no—
520C22—42no—
521C22—41no—
522C22—42no—
523C22—41no—
524C22—41no—
525C22—88no—
526C22—88no—
527C22—84no—
528C22—82no—
529C22—82no—
530C22—84no—
531C22—82no—
532C22—168no—
533C22#318614yes—
534C22C2×#53411yescharacteristic
535C22#348512yes—
536C22C23×D811yescharacteristic
537C22C2×#57511yescharacteristic
538C22#358612yes—
539C4—24no—
540C4—22no—
541C4—42no—
542C4—41no—
543C4—42no—
544C4—41no—
545C4—82no—
546C4—84no—
547C4—84no—
548C4—84no—
549C4#354514yes—
550C2—24no—
551C2—42no—
552C2—44no—
553C2—42no—
554C2—162no—
555C2#3711911yescharacteristic
556C2C2×#318611yescharacteristic
557C2#6075411yescharacteristic
558C2#6169412yes—
559C2#6171212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35000C2
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
2. C2×#2547C2semidirect
3. #33094C2semidirect
4. #32258C2semidirect
5. #31318C2semidirect
6. #31260C2semidirect
7. #29638C2semidirect
8. #27600C2semidirect
9. #24550C2semidirect
10. #24002C2semidirect
11. #21738C2semidirect
13. C22×C4⋊Q8C22semidirect
18. C22×C22⋊C8C22semidirect
26. #2547C22semidirect
38. C2×#380C22semidirect
39. C2×C4×C4⋊C4C22semidirect
40. #2075C22semidirect
41. #2075C22semidirect
42. #2057C22semidirect
43. #2033C22semidirect
44. #2033C22semidirect
45. #2023C22semidirect
46. #1925C22semidirect
47. #1918C22semidirect
48. #1893C22semidirect
51. #1869C22semidirect
52. #1852C22semidirect
53. #1852C22semidirect
54. #1852C22semidirect
55. #1823C22semidirect
56. #1823C22semidirect
57. #1817C22semidirect
58. #1808C22semidirect
59. #1786C22semidirect

Maximal subgroups

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

subgroupindex
#350002
C2×#25472
#330942
#322582
#313182
#312602
#296382
#276002
#245502
#240022
#217382