Finite Group Databasebeta

← #6475478

Subgroups — #6475478

Order 512.

1960 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2C2×#2851C211yesmaximal, characteristic
3#33059C214yesmaximal, complemented
4#30202C214yesmaximal, complemented
5#29329C214yesmaximal, complemented
6#29106C214yesmaximal, complemented
7#27983C212yesmaximal, complemented
8#27966C212yesmaximal, complemented
9#25912C212yesmaximal, complemented
10#25861C212yesmaximal, complemented
11#21765C214yesmaximal
12C4×#407C211yesmaximal, characteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#125C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C23×C42C2211yescharacteristic
17C2×#491C2214yescomplemented
18C2×#407C2212yes—
19C2×C4×M16C2212yes—
20#2857—22no—
21#2857—22no—
22#2857C2212yes—
23#2857C2212yes—
24#2857C2212yes—
25#2857C2212yes—
26#2851—22no—
27#2851—22no—
28#2851C2212yes—
29#2851C2212yes—
30#2851C2212yes—
31#2851C2212yes—
32C2×#391C2212yes—
33C2×#389—22no—
34C2×#389C2212yes—
35C2×#387—22no—
36C2×#385—22no—
37C2×#384C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C2×#380—22no—
40C2×#380C2212yes—
41#2100C2218yescomplemented
42#2074C2218yescomplemented
43#2043C2218yescomplemented
44#2026C2218yescomplemented
45#1992C2214yescomplemented
46#1960C2214yescomplemented
47#1933C22116yescomplemented
48#1894C22116yes—
49#1872C2218yes—
50#1872C2218yes—
51#1836C2214yescomplemented
52#1823C2218yes—
53#1787C2218yes—
54#1749C2212yes—
55#1749C2212yes—
56C4×C22⋊C8C2212yes—
57C4×C22⋊C8C2212yes—
58#1728C2212yes—
59C23×Q8C2312yes—
60C24×C4D412yes—
61C24×C4C2311yescharacteristic
62C22×M16C2314yes—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—44no—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×#125—22no—
68C2×#125—44no—
69C2×#125C2312yes—
70C2×#125C2312yes—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—216no—
74C2×C22⋊Q8C2314yes—
75C2×C22⋊Q8C2314yes—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—22no—
86C22×C42—44no—
87C22×C42D412yes—
88C22×C42D412yes—
89C22×C42D412yes—
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#491—48no—
97C2×#103C23116yes—
98C2×#103C23116yes—
99#407—28no—
100C2×C22⋊C8C2314yes—
101C2×C22⋊C8C2314yes—
102C4×M16—44no—
103C2×#97C2314yes—
104C2×C4×C8C2314yes—
105#391—24no—
106#391C2314yes—
107#391C2314yes—
108#389—22no—
109#389—22no—
110#389—24no—
111#389—24no—
112#389—24no—
113#389—24no—
114#389C2314yes—
115#389C2314yes—
116#387—24no—
117#387—24no—
118#387—24no—
119#387—24no—
120#385—24no—
121#385—24no—
122#385—24no—
123#385—24no—
124#384—22no—
125#384C2312yes—
126#384C2312yes—
127#382—22no—
128#382C2312yes—
129#382C2312yes—
130#380—22no—
131#380—22no—
132#380—24no—
133#380—24no—
134#380—24no—
135#380—24no—
136#380C2314yes—
137#380C2314yes—
138C2×#95—24no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—22no—
142#340C23116yes—
143#340C23116yes—
144#336C2314yes—
145#336C2314yes—
146C25C2×D411yescharacteristic
147C22×Q8—22no—
148C22×Q8—28no—
149C22×Q8—44no—
150C22×Q8—44no—
151C22×Q8C2414yes—
152C23×C4—28no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—44no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4#4118yes—
171C23×C4C2411yescharacteristic
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C2×M16—28no—
175C2×M16—44no—
176C2×M16—44no—
177C22×C8C2418yes—
178C4⋊Q8—28no—
179C4⋊Q8—44no—
180C4⋊Q8—48no—
181#125—28no—
182#125—44no—
183#125—48no—
184C22⋊Q8—216no—
185C22⋊Q8—216no—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—48no—
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—48no—
217C2×C22⋊C4—48no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—24no—
221C2×C42—44no—
222C2×C42—44no—
223C2×C42—44no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2411yescharacteristic
234C2×C42C2411yescharacteristic
235C2×C42C2411yescharacteristic
236C2×C42C2411yescharacteristic
237#103—416no—
238#103—416no—
239C22⋊C8—28no—
240C22⋊C8—28no—
241#97—28no—
242C4×C8—44no—
243#95—28no—
244#95—28no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251C24—24no—
252C24—24no—
253C24—41no—
254C24—41no—
255C24#12414yes—
256C24C22×D411yescharacteristic
257C24C22×D411yescharacteristic
258C24C22×D411yescharacteristic
259C2×Q8—28no—
260C2×Q8—28no—
261C2×Q8—28no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—816no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—416no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—432no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—48no—
312C22×C4#12414yes—
313C22×C4#12414yes—
314C22×C4#12414yes—
315C22×C4#13718yes—
316C22×C4C22×D412yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D412yes—
321C22×C4C22×D412yes—
322C22×C4C22×D412yes—
323C22×C4C2×#4114yes—
324C22×C4C2×#4114yes—
325C22×C4C2×#4114yes—
326C22×C4C2511yescharacteristic
327M16—416no—
328C2×C8—28no—
329C2×C8—28no—
330C2×C8—28no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—44no—
335C4⋊C4—48no—
336C4⋊C4—48no—
337C4⋊C4—44no—
338C4⋊C4—48no—
339C4⋊C4—48no—
340C4⋊C4—44no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C4⋊C4—48no—
344C4⋊C4—48no—
345C4⋊C4—48no—
346C4⋊C4—48no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—48no—
350C22⋊C4—48no—
351C42—24no—
352C42—24no—
353C42—24no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42H(5,𝔽2)14yes—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—22no—
366C23—22no—
367C23—28no—
368C23—28no—
369C23—21no—
370C23—21no—
371C23—22no—
372C23—22no—
373C23—44no—
374C23—48no—
375C23—48no—
376C23C2×#12412yes—
377C23C2×#12412yes—
378C23C2×#12412yes—
379C23C2×#13714yes—
380C23#57514yes—
381C23C23×D411yescharacteristic
382Q8—416no—
383Q8—88no—
384C2×C4—216no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—48no—
409C2×C4—48no—
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—48no—
422C2×C4—88no—
423C2×C4—88no—
424C2×C4—88no—
425C2×C4—88no—
426C2×C4C2×#12412yes—
427C2×C4C2×#12412yes—
428C2×C4C2×#12412yes—
429C2×C4#53414yes—
430C2×C4C2×#13714yes—
431C2×C4C2×#13714yes—
432C2×C4C22×#4112yes—
433C2×C4C2×H(5,𝔽2)12yes—
434C2×C4#58512yes—
435C8—48no—
436C22—24no—
437C22—24no—
438C22—24no—
439C22—24no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—416no—
448C22—42no—
449C22—42no—
450C22#313914yes—
451C22#314214yes—
452C22C22×#12411yescharacteristic
453C22C2×#53411yescharacteristic
454C22#344911yescharacteristic
455C22C22×#13712yes—
456C22C2×#57512yes—
457C4—24no—
458C4—24no—
459C4—24no—
460C4—22no—
461C4—22no—
462C4—22no—
463C4—42no—
464C4—84no—
465C4—84no—
466C4—84no—
467C4—84no—
468C4#350514yes—
469C2—24no—
470C2—24no—
471C2—42no—
472C2#3501911yescharacteristic
473C2C2×#313911yescharacteristic
474C2C2×#314211yescharacteristic
475C2#6075112yes—
476C2#6095012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
3. #33059C2semidirect
4. #30202C2semidirect
5. #29329C2semidirect
6. #29106C2semidirect
7. #27983C2semidirect
8. #27966C2semidirect
9. #25912C2semidirect
10. #25861C2semidirect
17. C2×#491C22semidirect
41. #2100C22semidirect
42. #2074C22semidirect
43. #2043C22semidirect
44. #2026C22semidirect
45. #1992C22semidirect
46. #1960C22semidirect
47. #1933C22semidirect
51. #1836C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#28512
#330592
#302022
#293292
#291062
#279832
#279662
#259122
#258612
#217652
C4×#4072