Finite Group Databasebeta

← #6470965

Subgroups — #6470965

Order 512.

1864 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2362C211yesmaximal, characteristic
3#33146C212yesmaximal, complemented
4#32957C212yesmaximal, complemented
5#29968C212yesmaximal, complemented
6#29951C212yesmaximal, complemented
7#26429C218yesmaximal, complemented
8#26355C218yesmaximal, complemented
9#21934C211yesmaximal, characteristic
10#21765C212yesmaximal
11#20846C212yesmaximal
12C2×#543—21no—
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C2×#477C2214yescomplemented
17C2×#428C2214yes—
18C2×#407C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2659—21no—
22#2659—21no—
23#2549—24no—
24#2440C2214yes—
25#2440C2214yes—
26#2362—24no—
27#2362C2218yes—
28#2315—21no—
29C2×#391C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#388—21no—
32C2×#388—21no—
33C2×#387—21no—
34C2×#380C2212yes—
35C2×C4×C4⋊C4—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#2100C2214yescomplemented
39#2096C2214yescomplemented
40#2088C2218yescomplemented
41#2066C2218yescomplemented
42#2058C2214yescomplemented
43#2024C2214yescomplemented
44#1988C2214yescomplemented
45#1947C2214yescomplemented
46#1918C2218yescomplemented
47#1894C2218yes—
48#1872C2214yes—
49#1872C2214yes—
50#1863C2218yescomplemented
51#1854C2218yes—
52#1853C2214yes—
53#1852C2214yes—
54#1851C2218yes—
55#1823C2214yes—
56#1803C2214yescomplemented
57#1797C2212yes—
58#1797C2212yes—
59#1787C2214yes—
60#1736C2214yes—
61C23×Q8C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C22×M16C2314yes—
64#543—48no—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—24no—
72C2×C22⋊Q8—24no—
73C2×C22⋊Q8Q818yescomplemented
74C2×C22⋊Q8C2314yes—
75C2×C4×Q8—22no—
76C2×C4×Q8—44no—
77C2×C4×Q8C2314yes—
78C2×#117—22no—
79C2×#117—44no—
80C2×#117C2314yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—21no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—21no—
95C22×C42—21no—
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#477—48no—
100#428—48no—
101C2×#107C2318yes—
102C2×#106C2318yes—
103C2×C4⋊C8C2314yes—
104C2×C4⋊C8C2314yes—
105C2×#103C2318yes—
106C2×#103C2318yes—
107#407—28no—
108C2×C22⋊C8C2314yes—
109C2×C22⋊C8C2314yes—
110#398—24no—
111#398—24no—
112#395—24no—
113#395—24no—
114#391—22no—
115#391—22no—
116#391—22no—
117#391C2314yes—
118#389—22no—
119#389—22no—
120#389—22no—
121#389C2314yes—
122#389C2314yes—
123#389C2314yes—
124#388—28no—
125#388—28no—
126#387—28no—
127#385—24no—
128#385—24no—
129#384C2312yes—
130#384C2312yes—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380C2314yes—
136#380C2314yes—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4C2314yes—
140C4×C22⋊C4D414yes—
141C4×C22⋊C4C2314yes—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—22no—
145C2×#95—21no—
146C2×#95—21no—
147C2×#95D411yescharacteristic
148C2×#95D411yescharacteristic
149#340C2318yes—
150#340C2318yes—
151#340C2318yes—
152#340C2318yes—
153#336C2314yes—
154#336C2314yes—
155C25C2×D411yescharacteristic
156C22×Q8—21no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—42no—
161C22×Q8—42no—
162C22×Q8C2×Q814yes—
163C22×Q8C2412yes—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×Q814yes—
179C23×C4#4114yes—
180C23×C4C2411yescharacteristic
181C23×C4C2412yes—
182C2×M16—28no—
183C2×M16—44no—
184C2×M16—44no—
185C22×C8C2418yes—
186C4⋊Q8—44no—
187C4⋊Q8—44no—
188C22⋊Q8—216no—
189C22⋊Q8—28no—
190C22⋊Q8—416no—
191C22⋊Q8—416no—
192C22⋊Q8—48no—
193C22⋊Q8—48no—
194C22⋊Q8C2×Q8116yes—
195C4×Q8—28no—
196C4×Q8—44no—
197C4×Q8—48no—
198#117—28no—
199#117—44no—
200#117—48no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4C2×Q814yes—
226C2×C4⋊C4C2412yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2414yes—
229C2×C4⋊C4C2412yes—
230C2×C4⋊C4C2412yes—
231C2×C22⋊C4—28no—
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—44no—
239C2×C22⋊C4—48no—
240C2×C22⋊C4C2×D414yes—
241C2×C22⋊C4C2×D414yes—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—44no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2412yes—
253C2×C42C2412yes—
254#107—216no—
255#106—48no—
256C4⋊C8—44no—
257C4⋊C8—44no—
258#103—48no—
259#103—48no—
260C22⋊C8—28no—
261C22⋊C8—28no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95C2×D414yes—
271#95C2×D414yes—
272C24—22no—
273C24—22no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24C22⋊Q814yes—
279C24C22×D412yes—
280C24C22×D411yescharacteristic
281C2×Q8—24no—
282C2×Q8—22no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—48no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—48no—
289C2×Q8—44no—
290C2×Q8—44no—
291C2×Q8—82no—
292C2×Q8—82no—
293C2×Q8—82no—
294C2×Q8—82no—
295C2×Q8C22×Q814yes—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—48no—
312C22×C4—48no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—82no—
339C22×C4—82no—
340C22×C4C22⋊Q814yes—
341C22×C4C22⋊Q818yes—
342C22×C4AGL(1,ℤ/8)14yes—
343C22×C4#13714yes—
344C22×C4C22×D412yes—
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D412yes—
348C22×C4C22×D412yes—
349C22×C4C22×D412yes—
350C22×C4C22×Q812yes—
351C22×C4C22×Q814yes—
352C22×C4C2×#4112yes—
353C22×C4C2×#4114yes—
354C22×C4C2511yescharacteristic
355M16—416no—
356C2×C8—28no—
357C2×C8—28no—
358C2×C8—28no—
359C4⋊C4—24no—
360C4⋊C4—216no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—28no—
365C4⋊C4—24no—
366C4⋊C4—28no—
367C4⋊C4—48no—
368C4⋊C4—42no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—44no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C4⋊C4—82no—
375C4⋊C4—82no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4C22×Q814yes—
381C4⋊C4H(5,𝔽2)14yes—
382C22⋊C4—216no—
383C22⋊C4—28no—
384C22⋊C4—28no—
385C22⋊C4—48no—
386C22⋊C4—48no—
387C22⋊C4—48no—
388C42—24no—
389C42—24no—
390C42—42no—
391C42—42no—
392C42—48no—
393C42—48no—
394C42—48no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—28no—
406C23—44no—
407C23—44no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23C2×C22⋊Q814yes—
413C23C2×C22⋊Q812yes—
414C23C2×AGL(1,ℤ/8)12yes—
415C23C2×#13712yes—
416C23C23×D411yescharacteristic
417Q8—28no—
418Q8—48no—
419Q8—44no—
420Q8—164no—
421Q8—164no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—28no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—44no—
444C2×C4—48no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—42no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—84no—
468C2×C4—164no—
469C2×C4—164no—
470C2×C4C2×C22⋊Q814yes—
471C2×C4C2×C22⋊Q812yes—
472C2×C4C2×AGL(1,ℤ/8)14yes—
473C2×C4C2×#13714yes—
474C2×C4C23×Q811yescharacteristic
475C2×C4C22×#4111yescharacteristic
476C2×C4C2×H(5,𝔽2)12yes—
477C2×C4#58512yes—
478C8—48no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—88no—
493C22#307914yes—
494C22C22×C22⋊Q811yescharacteristic
495C22#348411yescharacteristic
496C22#348511yescharacteristic
497C22C22×AGL(1,ℤ/8)11yescharacteristic
498C22C22×#13711yescharacteristic
499C22#358611yescharacteristic
500C22#358611yescharacteristic
501C4—24no—
502C4—24no—
503C4—24no—
504C4—22no—
505C4—42no—
506C4—44no—
507C4—42no—
508C4—44no—
509C4—44no—
510C4—84no—
511C4—82no—
512C4—84no—
513C4#345912yes—
514C4#350112yes—
515C2—24no—
516C2—44no—
517C2#3498511yescharacteristic
518C2C2×#307911yescharacteristic
519C2#6049111yescharacteristic
520C2#6157311yescharacteristic
521C2#6158511yescharacteristic
522C2#6170411yescharacteristic
523C2#6172311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33146C2semidirect
4. #32957C2semidirect
5. #29968C2semidirect
6. #29951C2semidirect
7. #26429C2semidirect
8. #26355C2semidirect
16. C2×#477C22semidirect
38. #2100C22semidirect
39. #2096C22semidirect
40. #2088C22semidirect
41. #2066C22semidirect
42. #2058C22semidirect
43. #2024C22semidirect
44. #1988C22semidirect
45. #1947C22semidirect
46. #1918C22semidirect
50. #1863C22semidirect
56. #1803C22semidirect
73. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#23622
#331462
#329572
#299682
#299512
#264292
#263552
#219342
#217652
#208462