Finite Group Databasebeta

← #6631365

Subgroups — #6631365

Order 512.

2124 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C212yesmaximal, complemented
2#32790C211yesmaximal, characteristic, complemented
3#32773C212yesmaximal, complemented
4#32758C211yesmaximal, characteristic, complemented
5#32743C214yesmaximal, complemented
6#32719C214yesmaximal, complemented
7#32707C214yesmaximal, complemented
8#23755C214yesmaximal, complemented
9#23697C214yesmaximal, complemented
10#23567C214yesmaximal, complemented
11#20714C211yesmaximal, characteristic, complemented
12C2×#532C2212yes—
13C2×#525—22no—
14C23×C42C2211yescharacteristic
15C2×C4×D8C2214yescomplemented
16C2×#431C2212yes—
17#2857C2212yes—
18#2857C2212yes—
19#2856C2212yes—
20#2856C2212yes—
21#2854—22no—
22#2854C2214yes—
23#2850—22no—
24#2850—22no—
25#2847—22no—
26#2847C2214yes—
27C4×#121—22no—
28C4×#121—22no—
29C4×#121C2214yes—
30C4×#121C2214yes—
31#2297—22no—
32#2285C2212yes—
33#2100C2214yescomplemented
34#2094C2214yescomplemented
35#2087C2214yescomplemented
36#2084C2214yescomplemented
37#2077C2218yes—
38#2076C2214yes—
39#2071C2214yescomplemented
40#2065C2218yescomplemented
41#2057C2214yescomplemented
42#2054C2214yescomplemented
43#2041C2214yescomplemented
44#2034C2218yes—
45#2034C2218yes—
46#2033C2218yes—
47#2023C2214yescomplemented
48#2011C2214yescomplemented
49#1999C2218yes—
50#1999C2214yes—
51#1817C2211yescharacteristic
52#1817C2211yescharacteristic
53#1797C2214yes—
54#1779C2212yes—
55C4×#106C2214yescomplemented
56C4×#102C2218yescomplemented
57C4×C22⋊C8C2214yes—
58C22×#41—22no—
59C24×C4D411yescharacteristic
60C24×C4D411yescharacteristic
61C24×C4C2311yescharacteristic
62C22×D8C2314yes—
63#532—48no—
64C2×C4⋊Q8C2312yes—
65C2×#127C2312yes—
66C2×#124—22no—
67#525—416no—
68C2×#123—24no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×#121—22no—
72C2×#121—22no—
73C2×#121—22no—
74C2×#121—22no—
75C2×#121—22no—
76C2×#121C2314yes—
77C2×#121C2314yes—
78C2×#121C2314yes—
79C2×#120—24no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4—22no—
83C2×C4×D4C2314yes—
84C2×#117—22no—
85C22×C42—44no—
86C22×C42D411yescharacteristic
87C22×C42D412yes—
88C22×C42D412yes—
89C22×C42D411yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2312yes—
92C22×C42C2312yes—
93C22×C42C2311yescharacteristic
94C4×D8—48no—
95#431—44no—
96C2×#107C2314yes—
97C2×#106C2318yes—
98C2×C4⋊C8C2314yes—
99C2×#102C2318yes—
100C2×#102C2318yes—
101C2×#102C2318yes—
102C2×C22⋊C8C2318yes—
103C2×C22⋊C8C2312yes—
104C2×C22⋊C8C2312yes—
105C2×C4×C8C2314yes—
106#391—22no—
107#391—22no—
108#390—22no—
109#390—22no—
110#390—22no—
111#390—22no—
112#390C2314yes—
113#390C2314yes—
114#389—22no—
115#389—22no—
116#389C2314yes—
117#389C2314yes—
118#388—24no—
119#388—24no—
120#388—24no—
121#388—24no—
122#386—24no—
123#386—24no—
124#386—24no—
125#386—24no—
126#386—24no—
127#386—24no—
128#384C2312yes—
129#384C2312yes—
130#382—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134#380C2314yes—
135#380C2314yes—
136#380C2314yes—
137C4×C4⋊C4—22no—
138C4×C4⋊C4C2314yes—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—24no—
141#340C2314yes—
142#340C2318yes—
143#340C2314yes—
144#340C2318yes—
145#336C2314yes—
146#336C2314yes—
147C25C2×D411yescharacteristic
148C2×#41—48no—
149C2×#41—48no—
150C2×#41—48no—
151C22×Q8—22no—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4C2414yes—
158C23×C4—24no—
159C23×C4—28no—
160C23×C4—24no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4—42no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2×D412yes—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2411yescharacteristic
177C23×C4C2412yes—
178C2×D8—48no—
179C2×D8—44no—
180C22×C8C2414yes—
181C22×C8C2414yes—
182C4⋊Q8—44no—
183#127—44no—
184#124—48no—
185#123—416no—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188#121—44no—
189#121—48no—
190#121—44no—
191#121—48no—
192#121—48no—
193#121—48no—
194#121—48no—
195#121—48no—
196#121—48no—
197#120—416no—
198C4×D4—216no—
199C4×D4—48no—
200C4×D4—416no—
201C4×D4—416no—
202#117—48no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4C2414yes—
211C2×C4⋊C4C2414yes—
212C2×C4⋊C4C2414yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C42—28no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—44no—
232C2×C42—42no—
233C2×C42C2×D412yes—
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2412yes—
241C2×C42C2411yescharacteristic
242C2×C42C2411yescharacteristic
243#107—44no—
244#106—48no—
245C4⋊C8—44no—
246#102—48no—
247#102—48no—
248#102—48no—
249C22⋊C8—48no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C4×C8—28no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259C24—22no—
260C24—24no—
261C24—24no—
262C24—41no—
263C24—42no—
264C24—41no—
265C24C22×D411yescharacteristic
266C24C22×D412yes—
267#41—832no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×D4—28no—
272C2×D4—24no—
273C2×D4—28no—
274C2×D4—24no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—48no—
280C2×D4—48no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—48no—
284C2×D4—48no—
285C2×D4—48no—
286C2×D4—48no—
287C2×D4—48no—
288C2×D4—44no—
289C2×D4—44no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—416no—
324C22×C4—416no—
325C22×C4—416no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4#13514yes—
330C22×C4AGL(1,ℤ/8)18yes—
331C22×C4C22×D412yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D412yes—
336C22×C4C22×D412yes—
337C22×C4C22×D411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C2511yescharacteristic
340D8—416no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—28no—
344C2×C8—28no—
345C4⋊C4—28no—
346C4⋊C4—24no—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—48no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—48no—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C22⋊C4—48no—
359C22⋊C4—48no—
360C22⋊C4—48no—
361C22⋊C4—48no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C22⋊C4—48no—
366C22⋊C4—48no—
367C42—22no—
368C42—22no—
369C42—24no—
370C42—24no—
371C42—22no—
372C42—24no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—48no—
377C42H(5,𝔽2)14yes—
378C23—22no—
379C23—21no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—21no—
388C23—44no—
389C23—48no—
390C23—44no—
391C23—48no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23C2×#13512yes—
401C23C2×AGL(1,ℤ/8)14yes—
402C23C23×D411yescharacteristic
403Q8—88no—
404D4—48no—
405D4—48no—
406D4—88no—
407D4—88no—
408D4—816no—
409D4—816no—
410D4—816no—
411C2×C4—28no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—216no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—44no—
429C2×C4—48no—
430C2×C4—48no—
431C2×C4—44no—
432C2×C4—48no—
433C2×C4—44no—
434C2×C4—48no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—48no—
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—44no—
447C2×C4—816no—
448C2×C4#53412yes—
449C2×C4#53412yes—
450C2×C4C2×#13514yes—
451C2×C4C2×AGL(1,ℤ/8)14yes—
452C2×C4C2×AGL(1,ℤ/8)14yes—
453C2×C4#57512yes—
454C2×C4C2×H(5,𝔽2)12yes—
455C2×C4#58514yes—
456C8—44no—
457C8—44no—
458C22—24no—
459C22—24no—
460C22—24no—
461C22—22no—
462C22—24no—
463C22—416no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—41no—
469C22—42no—
470C22—41no—
471C22—84no—
472C22—84no—
473C22—84no—
474C22—84no—
475C22—84no—
476C22—84no—
477C22—84no—
478C22C2×#53411yescharacteristic
479C22#344912yes—
480C22C22×#13511yescharacteristic
481C22C22×AGL(1,ℤ/8)12yes—
482C22C2×#57511yescharacteristic
483C4—22no—
484C4—22no—
485C4—24no—
486C4—24no—
487C4—42no—
488C4—41no—
489C4—41no—
490C4—84no—
491C4—84no—
492C4—84no—
493C4#319214yes—
494C4#352914yes—
495C2—42no—
496C2—44no—
497C2—164no—
498C2#3665211yescharacteristic
499C2#6073411yescharacteristic
500C2C2×#319211yescharacteristic
501C2#6094612yes—
502C2#6095112yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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. #32790C2semidirect
3. #32773C2semidirect
4. #32758C2semidirect
5. #32743C2semidirect
6. #32719C2semidirect
7. #32707C2semidirect
8. #23755C2semidirect
9. #23697C2semidirect
10. #23567C2semidirect
11. #20714C2semidirect
15. C2×C4×D8C22semidirect
33. #2100C22semidirect
34. #2094C22semidirect
35. #2087C22semidirect
36. #2084C22semidirect
39. #2071C22semidirect
40. #2065C22semidirect
41. #2057C22semidirect
42. #2054C22semidirect
43. #2041C22semidirect
47. #2023C22semidirect
48. #2011C22semidirect
55. C4×#106C22semidirect
56. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#366242
#327902
#327732
#327582
#327432
#327192
#327072
#237552
#236972
#235672
#207142