Finite Group Databasebeta

← #6475938

Subgroups — #6475938

Order 512.

2442 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C212yesmaximal, complemented
2#32930C214yesmaximal, complemented
3#30291C211yesmaximal, characteristic, complemented
4#30239C211yesmaximal, characteristic, complemented
5#30185C212yesmaximal, complemented
6#29157C214yesmaximal, complemented
7#28572C214yesmaximal, complemented
8#27856C214yesmaximal, complemented
9#25496C214yesmaximal, complemented
10#22181C214yesmaximal, complemented
11C22×#336C211yesmaximal, characteristic, complemented
12C24×C8C2212yes—
13C2×#532C2212yes—
14C2×#525—22no—
15C23×C42C2211yescharacteristic
16C2×#466C2214yes—
17C2×#466C2214yes—
18C2×#466C2214yes—
19#2857C2212yes—
20#2857C2212yes—
21#2856C2212yes—
22#2856C2212yes—
23#2854—22no—
24#2854C2214yes—
25#2850—22no—
26#2850—22no—
27#2847—22no—
28#2847C2214yes—
29C4×#121—22no—
30C4×#121—22no—
31C4×#121C2214yes—
32C4×#121C2214yes—
33#2297—22no—
34#2285C2212yes—
35#2099C2214yescomplemented
36#2093C2214yescomplemented
37#2083C2214yescomplemented
38#2071C2214yescomplemented
39#2057C2214yescomplemented
40#2041C2214yescomplemented
41#2023C2214yescomplemented
42#2011C2214yescomplemented
43#1925C2218yescomplemented
44#1918C2218yescomplemented
45#1893C2218yes—
46#1893C2218yes—
47#1893C2218yes—
48#1879C2218yescomplemented
49#1862C2218yescomplemented
50#1808C2214yes—
51#1808C2214yes—
52#1808C2214yes—
53C2×#336C2211yescharacteristic
54C2×#336C2214yes—
55C2×#336C2214yes—
56C2×#336C2212yes—
57C2×#336C2211yescharacteristic
58C22×#41—22no—
59C24×C4D411yescharacteristic
60C24×C4D411yescharacteristic
61C24×C4C2311yescharacteristic
62C22×D8C2314yes—
63C23×C8—28no—
64C23×C8—44no—
65C23×C8—44no—
66C23×C8C2314yes—
67C23×C8C2314yes—
68C23×C8C2312yes—
69C23×C8C2312yes—
70#532—48no—
71C2×C4⋊Q8C2312yes—
72C2×#127C2312yes—
73C2×#124—22no—
74#525—416no—
75C2×#123—24no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121—22no—
82C2×#121—22no—
83C2×#121C2314yes—
84C2×#121C2314yes—
85C2×#121C2314yes—
86C2×#120—24no—
87C2×C4×D4—24no—
88C2×C4×D4—24no—
89C2×C4×D4—22no—
90C2×C4×D4C2314yes—
91C2×#117—22no—
92C22×C42—44no—
93C22×C42D411yescharacteristic
94C22×C42D412yes—
95C22×C42D412yes—
96C22×C42D411yescharacteristic
97C22×C42C2312yes—
98C22×C42C2311yescharacteristic
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101#466—216no—
102#466—48no—
103#466—48no—
104C2×#107C2314yes—
105C2×#107C2314yes—
106C2×#107C2314yes—
107C2×#102C2318yes—
108C2×#102C2318yes—
109C2×#102C2318yes—
110#391—22no—
111#391—22no—
112#390—22no—
113#390—22no—
114#390—22no—
115#390—22no—
116#390C2314yes—
117#390C2314yes—
118#389—22no—
119#389—22no—
120#389C2314yes—
121#389C2314yes—
122#388—24no—
123#388—24no—
124#388—24no—
125#388—24no—
126#386—24no—
127#386—24no—
128#386—24no—
129#386—24no—
130#386—24no—
131#386—24no—
132#384C2312yes—
133#384C2312yes—
134#382—22no—
135#380—22no—
136#380—22no—
137#380—22no—
138#380C2314yes—
139#380C2314yes—
140#380C2314yes—
141C4×C4⋊C4—22no—
142C4×C4⋊C4C2314yes—
143C4×C22⋊C4—24no—
144C4×C22⋊C4—24no—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#336C2314yes—
149#336C2314yes—
150#336C2318yes—
151C25C2×D411yescharacteristic
152C2×#41—48no—
153C2×#41—48no—
154C2×#41—48no—
155C22×Q8—22no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4C2414yes—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4D818yescomplemented
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C2×D8—28no—
188C2×D8—44no—
189C2×D8—44no—
190C22×C8—24no—
191C22×C8—28no—
192C22×C8—28no—
193C22×C8—28no—
194C22×C8—28no—
195C22×C8—42no—
196C22×C8—42no—
197C22×C8—44no—
198C22×C8—44no—
199C22×C8—42no—
200C22×C8—42no—
201C22×C8—432no—
202C22×C8#4118yescomplemented
203C22×C8C2414yes—
204C22×C8C2414yes—
205C4⋊Q8—44no—
206#127—44no—
207#124—48no—
208#123—416no—
209C22⋊Q8—48no—
210C22⋊Q8—48no—
211#121—216no—
212#121—216no—
213#121—28no—
214#121—28no—
215#121—48no—
216#121—48no—
217#121—48no—
218#121—48no—
219#121—48no—
220#120—416no—
221C4×D4—48no—
222C4×D4—48no—
223C4×D4—416no—
224C4×D4—416no—
225#117—48no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4C2414yes—
234C2×C4⋊C4C2414yes—
235C2×C4⋊C4C2414yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C42—28no—
248C2×C42—24no—
249C2×C42—42no—
250C2×C42—44no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—44no—
254C2×C42—42no—
255C2×C42—44no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2412yes—
263C2×C42C2412yes—
264#107—28no—
265#107—44no—
266#107—44no—
267#102—216no—
268#102—48no—
269#102—48no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—24no—
283C24C2×D814yes—
284C24C22×D411yescharacteristic
285C24C22×D411yescharacteristic
286C24C22×D411yescharacteristic
287#41—832no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×D4—28no—
292C2×D4—24no—
293C2×D4—28no—
294C2×D4—24no—
295C2×D4—44no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—48no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—44no—
309C2×D4—44no—
310C22×C4—216no—
311C22×C4—216no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—416no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—416no—
355C22×C4—48no—
356C22×C4C2×D818yes—
357C22×C4C2×D814yes—
358C22×C4C2×D818yes—
359C22×C4C2×D814yes—
360C22×C4#13514yes—
361C22×C4C22×D412yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D412yes—
368C22×C4C2×#4114yes—
369C22×C4H(5,𝔽2)14yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C42−1+412yes—
372C22×C4C2511yescharacteristic
373D8—88no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—48no—
380C2×C8—416no—
381C2×C8—416no—
382C2×C8—48no—
383C2×C8C2×#4118yes—
384C4⋊C4—28no—
385C4⋊C4—24no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—44no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—48no—
406C42—24no—
407C42—24no—
408C42—24no—
409C42—24no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42—42no—
414C42—48no—
415C23—28no—
416C23—28no—
417C23—28no—
418C23—28no—
419C23—28no—
420C23—28no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—28no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23#53212yes—
442C23#53412yes—
443C23#53712yes—
444C23C22×D812yes—
445C23C22×D812yes—
446C23C22×D812yes—
447C23C2×#13512yes—
448C23#57512yes—
449C23#57512yes—
450C23C23×D411yescharacteristic
451Q8—88no—
452D4—48no—
453D4—48no—
454D4—88no—
455D4—88no—
456D4—816no—
457D4—816no—
458D4—816no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—216no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—44no—
485C2×C4—48no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—816no—
501C2×C4#53212yes—
502C2×C4#53412yes—
503C2×C4#53712yes—
504C2×C4C22×D814yes—
505C2×C4C22×D814yes—
506C2×C4C2×#13514yes—
507C2×C4C22×#4112yes—
508C2×C4C2×H(5,𝔽2)11yescharacteristic
509C2×C4C2×H(5,𝔽2)12yes—
510C2×C4C2×2−1+411yescharacteristic
511C8—28no—
512C8—44no—
513C22—232no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—84no—
533C22—84no—
534C22—84no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22#316914yes—
540C22#318614yes—
541C22#324514yes—
542C22C2×#53211yescharacteristic
543C22C2×#53411yescharacteristic
544C22C2×#53711yescharacteristic
545C22C23×D811yescharacteristic
546C22C22×#13511yescharacteristic
547C22C2×#57511yescharacteristic
548C22C2×#57511yescharacteristic
549C4—24no—
550C4—24no—
551C4—24no—
552C4—22no—
553C4—22no—
554C4—22no—
555C4—42no—
556C4—84no—
557C4—84no—
558C4—84no—
559C4#349012yes—
560C4#349212yes—
561C2—24no—
562C2—24no—
563C2—24no—
564C2—164no—
565C2#3505111yescharacteristic
566C2C2×#316911yescharacteristic
567C2#6068811yescharacteristic
568C2C2×#318611yescharacteristic
569C2#6073411yescharacteristic
570C2C2×#324511yescharacteristic
571C2#6090011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36624C2semidirect
2. #32930C2semidirect
3. #30291C2semidirect
4. #30239C2semidirect
5. #30185C2semidirect
6. #29157C2semidirect
7. #28572C2semidirect
8. #27856C2semidirect
9. #25496C2semidirect
10. #22181C2semidirect
11. C22×#336C2semidirect
35. #2099C22semidirect
36. #2093C22semidirect
37. #2083C22semidirect
38. #2071C22semidirect
39. #2057C22semidirect
40. #2041C22semidirect
41. #2023C22semidirect
42. #2011C22semidirect
43. #1925C22semidirect
44. #1918C22semidirect
48. #1879C22semidirect
49. #1862C22semidirect
172. C23×C4D8semidirect
202. C22×C8#41semidirect

Maximal subgroups

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

subgroupindex
#366242
#329302
#302912
#302392
#301852
#291572
#285722
#278562
#254962
#221812
C22×#3362