Finite Group Databasebeta

← #6434152

Subgroups — #6434152

Order 512.

2768 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic, complemented
3#30040C218yesmaximal, complemented
4#30039C214yesmaximal, complemented
5#30038C214yesmaximal, complemented
6#23630C212yesmaximal, complemented
7#23454C212yesmaximal, complemented
8#23430C212yesmaximal, complemented
9#23428C212yesmaximal, complemented
10#20925C214yesmaximal, complemented
11#20772C211yesmaximal, characteristic, complemented
12C22×AGL(1,ℤ/8)C2214yes—
13C2×#533C2211yescharacteristic
14C2×#532C2211yescharacteristic
15C22×#127C2211yescharacteristic
16C22×C4×D4—22no—
17C22×C4×D4C2212yes—
18C2×#417C2214yes—
19C2×#418C2214yes—
20#2862C2212yescomplemented
21#2861C2214yescomplemented
22#2858C2212yescomplemented
23#2850—22no—
24#2850—22no—
25C4×#127—22no—
26C4×#127—22no—
27C4×#127C2214yes—
28C4×#127C2214yes—
29C4×#124—22no—
30C4×#124—22no—
31#2297C2212yes—
32#2297C2212yes—
33C2×#390—22no—
34C2×#390C2212yes—
35C2×#384C2211yescharacteristic
36C2×C43C2211yescharacteristic
37#2072C2218yescomplemented
38#2071C2218yescomplemented
39#2025C2218yescomplemented
40#2023C2218yescomplemented
41#2014C2218yes—
42#2013C2214yescomplemented
43#2012C2218yescomplemented
44#2011C2218yes—
45#2011C2214yescomplemented
46#1879C22116yescomplemented
47#1877C22116yescomplemented
48#1839C2214yescomplemented
49#1839C2214yescomplemented
50#1736C2214yes—
51#1728C2212yes—
52#1728C2212yes—
53#1728C2212yes—
54C22×#41C2312yes—
55C23×D4—22no—
56C23×D4C2312yes—
57C24×C4—22no—
58C2×AGL(1,ℤ/8)—216no—
59C2×AGL(1,ℤ/8)—48no—
60C2×AGL(1,ℤ/8)—48no—
61C22×SD16C2318yes—
62C22×D8C2318yes—
63C22×M16C2314yes—
64#533—44no—
65#532—44no—
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2312yes—
68C2×#127—28no—
69C2×#127—21no—
70C2×#127—21no—
71C2×#127C2314yes—
72C2×#127C2311yescharacteristic
73C2×#125C2312yes—
74C2×#124—22no—
75C2×#124—22no—
76C2×#124C2312yes—
77C2×C22⋊Q8—22no—
78C2×#121—22no—
79C2×C4×Q8—22no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4—22no—
86C2×C4×D4—48no—
87C2×C4×D4—416no—
88C2×C4×D4C2318yes—
89C2×#117C2312yes—
90C22×C4⋊C4—22no—
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—22no—
94C22×C42D412yes—
95C22×C42D412yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99#418—48no—
100#417—48no—
101C2×#103C2318yes—
102C2×#102C2318yes—
103C2×#102C23116yes—
104#391—22no—
105#391—22no—
106#391C2314yes—
107#391C2314yes—
108#390—24no—
109#390—24no—
110#390—24no—
111#390—24no—
112#390—24no—
113#390—22no—
114#390—22no—
115#390C2314yes—
116#390C2314yes—
117#390C2314yes—
118#390C2314yes—
119#388—24no—
120#388—24no—
121#388—24no—
122#388—24no—
123#386—24no—
124#386—22no—
125#386—22no—
126#384—21no—
127#384—21no—
128#384C2314yes—
129#382—22no—
130#382—22no—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—22no—
134C43D412yes—
135C43D412yes—
136C43C2312yes—
137C43C2312yes—
138#340C23116yes—
139#336C2318yes—
140#336C2318yes—
141#336C2314yes—
142#336C2314yes—
143C25—22no—
144C2×#41—28no—
145C2×#41—28no—
146C2×#41—24no—
147C2×#41—24no—
148C22×Q8—22no—
149C22×Q8C2412yes—
150C22×D4—216no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—22no—
156C22×D4—24no—
157C22×D4—28no—
158C22×D4—28no—
159C22×D4—48no—
160C22×D4—48no—
161C22×D4#4118yes—
162C22×D4C2412yes—
163C22×D4C2414yes—
164C23×C4—22no—
165C23×C4—28no—
166C23×C4—22no—
167C23×C4—48no—
168C23×C4—416no—
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D412yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2411yescharacteristic
174AGL(1,ℤ/8)—464no—
175C2×SD16—216no—
176C2×SD16—48no—
177C2×SD16—48no—
178C2×D8—216no—
179C2×D8—48no—
180C2×D8—48no—
181C2×M16—28no—
182C2×M16—28no—
183C2×M16—28no—
184C22×C8C2418yes—
185C4⋊Q8—44no—
186C4⋊Q8—42no—
187#127—28no—
188#127—28no—
189#127—216no—
190#127—42no—
191#125—44no—
192#124—44no—
193#124—48no—
194#124—48no—
195C22⋊Q8—48no—
196#121—48no—
197C4×Q8—48no—
198C4×D4—416no—
199C4×D4—416no—
200C4×D4—416no—
201C4×D4—416no—
202C4×D4—416no—
203C4×D4—48no—
204C4×D4—416no—
205#117—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—48no—
214C2×C4⋊C4C2414yes—
215C2×C4⋊C4C2414yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—48no—
226C2×C22⋊C4—416no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—28no—
231C2×C42—22no—
232C2×C42—44no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2412yes—
246C2×C42C2411yescharacteristic
247C2×C42C2412yes—
248C2×C42C2411yescharacteristic
249#103—48no—
250#102—48no—
251#102—416no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256C24—28no—
257C24—24no—
258C24—22no—
259C24—48no—
260C24—48no—
261C24—44no—
262C24—44no—
263C24C22×D411yescharacteristic
264#41—416no—
265#41—416no—
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—22no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×D4—216no—
274C2×D4—216no—
275C2×D4—28no—
276C2×D4—232no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—28no—
280C2×D4—28no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—22no—
284C2×D4—48no—
285C2×D4—464no—
286C2×D4—48no—
287C2×D4—44no—
288C2×D4—44no—
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—416no—
295C2×D4—48no—
296C2×D4—44no—
297C2×D4—48no—
298C2×D4C2×#4118yes—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—416no—
313C22×C4—416no—
314C22×C4—48no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—416no—
320C22×C4—832no—
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C22×D412yes—
324C22×C4C22×D412yes—
325C22×C4C22×D412yes—
326C22×C4C22×D412yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4114yes—
329C22×C4H(5,𝔽2)14yes—
330C22×C4H(5,𝔽2)14yes—
331C22×C4C2511yescharacteristic
332SD16—432no—
333D8—432no—
334M16—416no—
335C2×C8—28no—
336C2×C8—28no—
337C2×C8—28no—
338C4⋊C4—28no—
339C4⋊C4—28no—
340C4⋊C4—48no—
341C4⋊C4—44no—
342C4⋊C4—48no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—48no—
346C22⋊C4—416no—
347C22⋊C4—416no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—44no—
351C22⋊C4—416no—
352C22⋊C4—48no—
353C22⋊C4—48no—
354C22⋊C4—48no—
355C42—22no—
356C42—22no—
357C42—24no—
358C42—24no—
359C42—24no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—24no—
366C42—22no—
367C42—22no—
368C42—42no—
369C42—42no—
370C42—44no—
371C42AGL(1,ℤ/8)18yescomplemented
372C23—22no—
373C23—22no—
374C23—28no—
375C23—432no—
376C23—416no—
377C23—44no—
378C23—44no—
379C23—48no—
380C23—48no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—44no—
385C23—44no—
386C23—88no—
387C23—88no—
388C23—816no—
389C23#52512yes—
390C23#53512yes—
391C23#53512yes—
392C23C23×D411yescharacteristic
393Q8—44no—
394Q8—44no—
395Q8—88no—
396D4—216no—
397D4—432no—
398D4—432no—
399D4—416no—
400D4—48no—
401D4—44no—
402D4—44no—
403D4—88no—
404D4—88no—
405C2×C4—28no—
406C2×C4—216no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—48no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—832no—
432C2×C4—88no—
433C2×C4—88no—
434C2×C4#52512yes—
435C2×C4#53512yes—
436C2×C4#53512yes—
437C2×C4#538116yes—
438C2×C4C2×AGL(1,ℤ/8)14yes—
439C2×C4C2×AGL(1,ℤ/8)14yes—
440C2×C4C2×AGL(1,ℤ/8)14yes—
441C2×C4#57514yes—
442C2×C4C22×#4112yes—
443C2×C4C2×H(5,𝔽2)12yes—
444C2×C4C2×H(5,𝔽2)12yes—
445C8—48no—
446C22—22no—
447C22—22no—
448C22—24no—
449C22—24no—
450C22—48no—
451C22—416no—
452C22—832no—
453C22—84no—
454C22—84no—
455C22—88no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—84no—
460C22—84no—
461C22#263114yes—
462C22C2×#52511yescharacteristic
463C22C2×#53511yescharacteristic
464C22C2×#53511yescharacteristic
465C22C22×AGL(1,ℤ/8)12yes—
466C22C2×#57512yes—
467C4—24no—
468C4—24no—
469C4—24no—
470C4—24no—
471C4—42no—
472C4—42no—
473C4—82no—
474C4—84no—
475C4—84no—
476C4#263114yes—
477C4C2×#53814yes—
478C2—24no—
479C2—88no—
480C2—162no—
481C2C2×#263111yescharacteristic
482C2#6055812yes—
483C2#6079811yescharacteristic
484C2#6080211yescharacteristic
485C2#6080312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36639C2semidirect
2. C2×C4×#127C2semidirect
3. #30040C2semidirect
4. #30039C2semidirect
5. #30038C2semidirect
6. #23630C2semidirect
7. #23454C2semidirect
8. #23430C2semidirect
9. #23428C2semidirect
10. #20925C2semidirect
11. #20772C2semidirect
20. #2862C22semidirect
21. #2861C22semidirect
22. #2858C22semidirect
37. #2072C22semidirect
38. #2071C22semidirect
39. #2025C22semidirect
40. #2023C22semidirect
42. #2013C22semidirect
43. #2012C22semidirect
45. #2011C22semidirect
46. #1879C22semidirect
47. #1877C22semidirect
48. #1839C22semidirect
49. #1839C22semidirect
371. C42AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
#366392
C2×C4×#1272
#300402
#300392
#300382
#236302
#234542
#234302
#234282
#209252
#207722