Finite Group Databasebeta

← #6433990

Subgroups — #6433990

Order 512.

1824 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C212yesmaximal
2#30366C214yesmaximal, complemented
3#30326C214yesmaximal, complemented
4#30296C211yesmaximal, characteristic, complemented
5#30250C211yesmaximal, characteristic, complemented
6#30199C212yesmaximal, complemented
7#29743C214yesmaximal
8#23756C214yesmaximal, complemented
9#23571C214yesmaximal, complemented
10C4×#470C214yesmaximal, complemented
11C4×#407C211yesmaximal, characteristic
12C2×#533C2212yes—
13C23×C42C2211yescharacteristic
14C2×#470C2214yes—
15C2×C4×Q16C2214yescomplemented
16C2×#407C2212yes—
17C2×C4×M16C2212yes—
18#2857C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2857C2212yes—
22#2855—22no—
23#2855C2214yes—
24#2851—22no—
25#2851—22no—
26#2848—22no—
27#2848C2214yes—
28C4×C22⋊Q8—22no—
29C4×C22⋊Q8—22no—
30C4×C22⋊Q8C2214yes—
31C4×C22⋊Q8C2214yes—
32#2296—22no—
33#2285C2212yes—
34#2280—22no—
35#2099C2214yescomplemented
36#2096C2214yescomplemented
37#2084C2214yescomplemented
38#2077C2218yes—
39#2072C2214yescomplemented
40#2057C2214yescomplemented
41#2043C2214yescomplemented
42#2036C2218yes—
43#2035C2218yes—
44#2024C2214yescomplemented
45#2016C2214yescomplemented
46#2001C2218yes—
47#1971C2214yescomplemented
48#1944C2214yescomplemented
49#1935C2218yescomplemented
50#1894C2218yes—
51#1809C2214yes—
52C4×#106C2214yescomplemented
53C4×#103C2218yescomplemented
54#1749C2214yes—
55C4×C22⋊C8C2214yes—
56#1728C2211yescharacteristic
57#1728C2211yescharacteristic
58C24×C4D411yescharacteristic
59C24×C4D411yescharacteristic
60C24×C4C2311yescharacteristic
61C22×Q16C2314yes—
62C22×M16C2312yes—
63C22×M16C2312yes—
64#533—48no—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×#125—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8C2314yes—
72C2×C22⋊Q8C2314yes—
73C2×C22⋊Q8C2314yes—
74C2×C4×Q8—22no—
75C2×C4×Q8C2314yes—
76C2×#117—22no—
77C2×#117—22no—
78C22×C42—44no—
79C22×C42D411yescharacteristic
80C22×C42D412yes—
81C22×C42D412yes—
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2312yes—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87#470—216no—
88C4×Q16—48no—
89C2×#107C2314yes—
90C2×#106C2314yes—
91C2×#106C2314yes—
92C2×#103C2318yes—
93C2×#103C2318yes—
94C2×#103C2318yes—
95#407—28no—
96C2×C22⋊C8C2318yes—
97C4×M16—44no—
98C2×#97C2314yes—
99C2×C4×C8C2314yes—
100#391—22no—
101#391—22no—
102#391C2314yes—
103#391C2314yes—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389C2314yes—
111#389C2314yes—
112#387—24no—
113#387—24no—
114#387—24no—
115#387—24no—
116#385—24no—
117#385—24no—
118#385—24no—
119#385—24no—
120#385—24no—
121#385—24no—
122#384C2312yes—
123#384C2312yes—
124#382—22no—
125#381—24no—
126#380—22no—
127#380—22no—
128#380—22no—
129#380—22no—
130#380—22no—
131#380—22no—
132#380—22no—
133#380C2314yes—
134#380C2314yes—
135#380C2314yes—
136#379—24no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4C2314yes—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—24no—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#336C2314yes—
147#336C2314yes—
148C25C2×D411yescharacteristic
149C22×Q8—22no—
150C22×Q8C2414yes—
151C23×C4—28no—
152C23×C4—24no—
153C23×C4—44no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4—42no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2412yes—
170C23×C4C2411yescharacteristic
171C2×Q16—28no—
172C2×Q16—44no—
173C2×Q16—44no—
174C2×M16—24no—
175C2×M16—42no—
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—42no—
179C2×M16#4118yes—
180C22×C8C2414yes—
181C22×C8C2414yes—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184#125—48no—
185C22⋊Q8—216no—
186C22⋊Q8—44no—
187C22⋊Q8—44no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C22⋊Q8—48no—
191C22⋊Q8—48no—
192C4×Q8—216no—
193C4×Q8—48no—
194#117—48no—
195#117—48no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4C2414yes—
210C2×C4⋊C4C2414yes—
211C2×C4⋊C4C2414yes—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C42—28no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—44no—
223C2×C42—44no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—44no—
227C2×C42—42no—
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2412yes—
236C2×C42C2411yescharacteristic
237C2×C42C2411yescharacteristic
238#107—44no—
239#106—28no—
240#106—44no—
241#103—216no—
242#103—48no—
243#103—48no—
244C22⋊C8—216no—
245#97—44no—
246C4×C8—28no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257C24—22no—
258C24—24no—
259C24—41no—
260C24—41no—
261C24—42no—
262C24C22×D411yescharacteristic
263C24C22×D412yes—
264C2×Q8—28no—
265C2×Q8—28no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—44no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—416no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—416no—
311C22×C4#13718yes—
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D412yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D412yes—
318C22×C4C22×D412yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4114yes—
321C22×C4C2511yescharacteristic
322Q16—416no—
323M16—48no—
324M16—48no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8C2×#4118yes—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—48no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—48no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C4⋊C4—48no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C22⋊C4—48no—
353C22⋊C4—48no—
354C42—22no—
355C42—22no—
356C42—24no—
357C42—24no—
358C42—22no—
359C42—24no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—48no—
364C42—48no—
365C42H(5,𝔽2)14yes—
366C23—22no—
367C23—22no—
368C23—21no—
369C23—22no—
370C23—21no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—44no—
377C23—48no—
378C23—44no—
379C23—48no—
380C23—44no—
381C23#53212yes—
382C23C2×#13714yes—
383C23C23×D411yescharacteristic
384Q8—48no—
385Q8—48no—
386Q8—88no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—216no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—44no—
405C2×C4—48no—
406C2×C4—44no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—44no—
410C2×C4—48no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4#53212yes—
428C2×C4#53412yes—
429C2×C4#53412yes—
430C2×C4C2×#13714yes—
431C2×C4C2×#13714yes—
432C2×C4#57514yes—
433C2×C4C22×#4112yes—
434C2×C4C2×H(5,𝔽2)12yes—
435C2×C4#58511yescharacteristic
436C2×C4#58511yescharacteristic
437C8—28no—
438C8—44no—
439C22—22no—
440C22—24no—
441C22—24no—
442C22—24no—
443C22—24no—
444C22—416no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—41no—
450C22—42no—
451C22—41no—
452C22C2×#53211yescharacteristic
453C22C2×#53411yescharacteristic
454C22#344911yescharacteristic
455C22C22×#13712yes—
456C22C2×#57512yes—
457C4—22no—
458C4—24no—
459C4—22no—
460C4—24no—
461C4—42no—
462C4—41no—
463C4—41no—
464C4—84no—
465C4—84no—
466C4—84no—
467C4—84no—
468C4#319514yes—
469C4#345912yes—
470C4#346012yes—
471C2—42no—
472C2—44no—
473C2#3457511yescharacteristic
474C2#6070112yes—
475C2C2×#319511yescharacteristic
476C2#6074111yescharacteristic
477C2#6095212yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC25C2×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
2. #30366C2semidirect
3. #30326C2semidirect
4. #30296C2semidirect
5. #30250C2semidirect
6. #30199C2semidirect
8. #23756C2semidirect
9. #23571C2semidirect
10. C4×#470C2semidirect
15. C2×C4×Q16C22semidirect
35. #2099C22semidirect
36. #2096C22semidirect
37. #2084C22semidirect
39. #2072C22semidirect
40. #2057C22semidirect
41. #2043C22semidirect
44. #2024C22semidirect
45. #2016C22semidirect
47. #1971C22semidirect
48. #1944C22semidirect
49. #1935C22semidirect
52. C4×#106C22semidirect
53. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#366252
#303662
#303262
#302962
#302502
#301992
#297432
#237562
#235712
C4×#4702
C4×#4072