Finite Group Databasebeta

← #6475944

Subgroups — #6475944

Order 512.

2366 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C212yesmaximal, complemented
2#32936C214yesmaximal, complemented
3#30293C211yesmaximal, characteristic, complemented
4#30239C211yesmaximal, characteristic, complemented
5#30185C212yesmaximal, complemented
6#29163C214yesmaximal, complemented
7#28578C214yesmaximal, complemented
8#27862C214yesmaximal, complemented
9#25502C214yesmaximal, complemented
10#22184C214yesmaximal, complemented
11C2×#1728C211yesmaximal, characteristic, complemented
12C23×M16C2212yes—
13C2×#532C2212yes—
14C2×#525—22no—
15C23×C42C2211yescharacteristic
16C2×#469C2214yes—
17C2×#469C2214yes—
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#2084C2214yescomplemented
38#2071C2214yescomplemented
39#2058C2214yescomplemented
40#2041C2214yescomplemented
41#2023C2214yescomplemented
42#2011C2214yescomplemented
43#1926C2218yescomplemented
44#1921C2218yescomplemented
45#1895C2218yes—
46#1895C2218yes—
47#1893C2218yes—
48#1879C2218yescomplemented
49#1864C2218yescomplemented
50#1809C2214yes—
51#1809C2214yes—
52#1807C2214yes—
53#1728C2214yes—
54#1728C2214yes—
55C2×#336C2211yescharacteristic
56C2×#336C2212yes—
57C2×#336C2211yescharacteristic
58C22×#41—22no—
59C24×C4D412yes—
60C24×C4C2311yescharacteristic
61C22×D8C2314yes—
62C22×M16—28no—
63C22×M16—44no—
64C22×M16—44no—
65C22×M16C2314yes—
66C22×M16C2314yes—
67C23×C8C2312yes—
68C23×C8C2312yes—
69#532—48no—
70C2×C4⋊Q8C2312yes—
71C2×#127C2312yes—
72C2×#124—22no—
73#525—416no—
74C2×#123—24no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×#121—22no—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121—22no—
82C2×#121C2314yes—
83C2×#121C2314yes—
84C2×#121C2314yes—
85C2×#120—24no—
86C2×C4×D4—24no—
87C2×C4×D4—24no—
88C2×C4×D4—22no—
89C2×C4×D4C2314yes—
90C2×#117—22no—
91C22×C42—44no—
92C22×C42D411yescharacteristic
93C22×C42D412yes—
94C22×C42D411yescharacteristic
95C22×C42D411yescharacteristic
96C22×C42D411yescharacteristic
97C22×C42C2312yes—
98C22×C42C2311yescharacteristic
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101#469—48no—
102#469—48no—
103#466—216no—
104C2×#107C2314yes—
105C2×#106C2314yes—
106C2×#106C2314yes—
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—28no—
167C23×C4—24no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C2×D8—28no—
183C2×D8—44no—
184C2×D8—44no—
185C2×M16—28no—
186C2×M16—28no—
187C2×M16—28no—
188C2×M16—28no—
189C2×M16—44no—
190C2×M16—44no—
191C2×M16—432no—
192C22×C8—24no—
193C22×C8—42no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8—42no—
197C22×C8#4118yescomplemented
198C22×C8C2414yes—
199C22×C8C2414yes—
200C4⋊Q8—44no—
201#127—44no—
202#124—48no—
203#123—416no—
204C22⋊Q8—48no—
205C22⋊Q8—48no—
206#121—216no—
207#121—216no—
208#121—28no—
209#121—28no—
210#121—48no—
211#121—48no—
212#121—48no—
213#121—48no—
214#121—48no—
215#120—416no—
216C4×D4—48no—
217C4×D4—48no—
218C4×D4—416no—
219C4×D4—416no—
220#117—48no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4C2414yes—
229C2×C4⋊C4C2414yes—
230C2×C4⋊C4C2414yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
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×C42—28no—
243C2×C42—24no—
244C2×C42—42no—
245C2×C42—44no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—44no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D412yes—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D412yes—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2412yes—
261C2×C42C2412yes—
262#107—28no—
263#106—44no—
264#106—44no—
265#102—216no—
266#102—48no—
267#102—48no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—24no—
273#95—24no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—24no—
278C24—24no—
279C24—42no—
280C24C22×D411yescharacteristic
281C24C22×D411yescharacteristic
282C24C22×D411yescharacteristic
283#41—832no—
284C2×Q8—44no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×D4—28no—
288C2×D4—24no—
289C2×D4—28no—
290C2×D4—24no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4—48no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—48no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—48no—
304C2×D4—44no—
305C2×D4—44no—
306C22×C4—216no—
307C22×C4—216no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—432no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4AGL(1,ℤ/8)18yes—
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C2×#4114yes—
358C22×C4H(5,𝔽2)14yes—
359C22×C4H(5,𝔽2)12yes—
360C22×C42−1+412yes—
361C22×C4C2511yescharacteristic
362D8—88no—
363M16—416no—
364M16—416no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—48no—
371C2×C8—48no—
372C2×C8C2×#4118yes—
373C4⋊C4—28no—
374C4⋊C4—24no—
375C4⋊C4—28no—
376C4⋊C4—24no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C22⋊C4—48no—
385C22⋊C4—48no—
386C22⋊C4—48no—
387C22⋊C4—48no—
388C22⋊C4—48no—
389C22⋊C4—48no—
390C22⋊C4—48no—
391C22⋊C4—48no—
392C22⋊C4—48no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C42—24no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—41no—
400C42—42no—
401C42—41no—
402C42—42no—
403C42—41no—
404C42—41no—
405C42—48no—
406C23—21no—
407C23—21no—
408C23—22no—
409C23—28no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—48no—
419C23—48no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23#53212yes—
430C23#53412yes—
431C23#53712yes—
432C23C2×AGL(1,ℤ/8)14yes—
433C23C23×D411yescharacteristic
434Q8—88no—
435D4—48no—
436D4—48no—
437D4—88no—
438D4—88no—
439D4—816no—
440D4—816no—
441D4—816no—
442C2×C4—28no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—216no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—48no—
468C2×C4—48no—
469C2×C4—48no—
470C2×C4—48no—
471C2×C4—48no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—816no—
482C2×C4#53212yes—
483C2×C4#53412yes—
484C2×C4#53712yes—
485C2×C4C2×AGL(1,ℤ/8)14yes—
486C2×C4C2×AGL(1,ℤ/8)14yes—
487C2×C4#57514yes—
488C2×C4C22×#4112yes—
489C2×C4C2×H(5,𝔽2)11yescharacteristic
490C2×C4C2×H(5,𝔽2)12yes—
491C2×C4C2×2−1+411yescharacteristic
492C8—28no—
493C8—44no—
494C22—22no—
495C22—24no—
496C22—22no—
497C22—24no—
498C22—24no—
499C22—22no—
500C22—22no—
501C22—24no—
502C22—416no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—42no—
508C22—42no—
509C22—84no—
510C22—84no—
511C22—84no—
512C22—84no—
513C22—84no—
514C22—84no—
515C22—84no—
516C22#318414yes—
517C22C2×#53211yescharacteristic
518C22C2×#53411yescharacteristic
519C22C2×#53711yescharacteristic
520C22C22×AGL(1,ℤ/8)12yes—
521C22C2×#57512yes—
522C4—24no—
523C4—24no—
524C4—24no—
525C4—22no—
526C4—22no—
527C4—22no—
528C4—42no—
529C4—84no—
530C4—84no—
531C4—84no—
532C4#349012yes—
533C4#349212yes—
534C2—24no—
535C2—42no—
536C2—42no—
537C2—164no—
538C2#3505111yescharacteristic
539C2#6067611yescharacteristic
540C2C2×#318411yescharacteristic
541C2#6075012yes—
542C2#6091612yes—
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. #32936C2semidirect
3. #30293C2semidirect
4. #30239C2semidirect
5. #30185C2semidirect
6. #29163C2semidirect
7. #28578C2semidirect
8. #27862C2semidirect
9. #25502C2semidirect
10. #22184C2semidirect
11. C2×#1728C2semidirect
35. #2099C22semidirect
36. #2093C22semidirect
37. #2084C22semidirect
38. #2071C22semidirect
39. #2058C22semidirect
40. #2041C22semidirect
41. #2023C22semidirect
42. #2011C22semidirect
43. #1926C22semidirect
44. #1921C22semidirect
48. #1879C22semidirect
49. #1864C22semidirect
197. C22×C8#41semidirect

Maximal subgroups

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

subgroupindex
#366242
#329362
#302932
#302392
#301852
#291632
#285782
#278622
#255022
#221842
C2×#17282