Finite Group Databasebeta

← #319050

Subgroups — #319050

Order 512.

906 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33043C211yesmaximal, characteristic
2#27948C211yesmaximal, characteristic
3#27910C211yesmaximal, characteristic
4#25813C211yesmaximal, characteristic
5#25811C211yesmaximal, characteristic
6C2×#1956C211yesmaximal, characteristic
7#10680C212yesmaximal, complemented
8#10314C212yesmaximal, complemented
9#9823C212yesmaximal, complemented
10#9819C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#500C2212yes—
14C2×#498C2212yes—
15C2×#487C2211yescharacteristic
16C2×#484—22no—
17C22×#103—22no—
18C22×C4×C8C2211yescharacteristic
19#2857C2211yescharacteristic
20#2857C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2095—22no—
23#2084—22no—
24#2058—22no—
25#2057—22no—
26#1990C2211yescharacteristic
27#1990C2212yes—
28#1990C2212yes—
29#1990C2212yes—
30#1956—22no—
31#1956C2212yes—
32#1956C2211yescharacteristic
33#1956C2212yes—
34#1956C2212yes—
35#1933—22no—
36#1892—22no—
37#1853—22no—
38#1853—22no—
39#1852—22no—
40#1845C2212yes—
41#1844C2212yes—
42#1834C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45C2×C82C2212yes—
46#1273C2212yes—
47C23×Q8—21no—
48C23×C8D412yes—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2312yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C42C2311yescharacteristic
65#500—24no—
66#500—42no—
67#498—24no—
68#498—42no—
69#487—42no—
70#484—48no—
71C2×#107—24no—
72C2×#107—24no—
73C2×#106—24no—
74C2×#106—24no—
75C2×#103—24no—
76C2×#103—22no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—22no—
80C2×#103—48no—
81C2×C22⋊C8D412yes—
82C2×C22⋊C8D412yes—
83C2×C4×C8—42no—
84C2×C4×C8D412yes—
85C2×C4×C8C2311yescharacteristic
86C2×C4×C8C2312yes—
87C2×C4×C8C2312yes—
88C2×C4×C8C2311yescharacteristic
89#391—21no—
90#391—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98#384C2311yescharacteristic
99#384C2311yescharacteristic
100#384C2312yes—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—22no—
109#340—22no—
110#336D412yes—
111#336D412yes—
112C82—42no—
113C22×Q8—24no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—42no—
118C22×Q8—44no—
119C22×Q8—41no—
120C22×Q8—44no—
121C23×C4—41no—
122C23×C4—41no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C22×C8—24no—
127C22×C8—24no—
128C22×C8—42no—
129C22×C8C2×D412yes—
130C22×C8C2×D414yes—
131C22×C8C2×D414yes—
132C22×C8C2×D412yes—
133C4⋊Q8—24no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—44no—
142C4⋊Q8—44no—
143C4⋊Q8—42no—
144C4⋊Q8#4114yes—
145C22⋊Q8—44no—
146C22⋊Q8—44no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4—82no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2411yescharacteristic
182#107—48no—
183#107—48no—
184#106—48no—
185#106—84no—
186#103—48no—
187#103—44no—
188#103—48no—
189#103—44no—
190#103—48no—
191C22⋊C8SD1618yes—
192C22⋊C8Q1618yes—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—42no—
200#95—41no—
201#95—41no—
202#95—41no—
203#95—41no—
204C24#12711yescharacteristic
205C2×Q8—24no—
206C2×Q8—44no—
207C2×Q8—416no—
208C2×Q8—42no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—44no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—82no—
219C2×Q8—88no—
220C2×Q8—82no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—42no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4—82no—
240C22×C4#12711yescharacteristic
241C22×C4#12711yescharacteristic
242C22×C4#12711yescharacteristic
243C22×C4C2×SD1614yes—
244C22×C4C2×Q1614yes—
245C22×C4#13514yes—
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C2×C8—22no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—22no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—48no—
259C2×C8C2×SD1618yes—
260C2×C8C2×Q1618yes—
261C2×C8AGL(1,ℤ/8)14yes—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—44no—
265C4⋊C4—44no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—82no—
283C4⋊C4—82no—
284C4⋊C4—82no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C22⋊C4—82no—
288C22⋊C4—82no—
289C42—21no—
290C42—21no—
291C42—22no—
292C42—22no—
293C42C2×#4112yes—
294C42H(5,𝔽2)12yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#49212yes—
300C23#49412yes—
301C23#49512yes—
302C23C2×#12711yescharacteristic
303Q8—48no—
304Q8—88no—
305Q8—84no—
306Q8—82no—
307Q8—82no—
308Q8—84no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—28no—
318C2×C4—28no—
319C2×C4—44no—
320C2×C4—88no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—164no—
339C2×C4#49212yes—
340C2×C4#49412yes—
341C2×C4#49512yes—
342C2×C4#52511yescharacteristic
343C2×C4#53411yescharacteristic
344C2×C4#54612yes—
345C2×C4#54612yes—
346C2×C4C22×SD1612yes—
347C2×C4C22×Q1612yes—
348C2×C4C2×#13512yes—
349C2×C4C2×AGL(1,ℤ/8)12yes—
350C2×C4#57712yes—
351C2×C4#57812yes—
352C8—24no—
353C8—28no—
354C8—28no—
355C8—42no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22#170714yes—
363C22#253911yescharacteristic
364C22C2×#49211yescharacteristic
365C22C2×#49411yescharacteristic
366C22C2×#49511yescharacteristic
367C22#315111yescharacteristic
368C22#315411yescharacteristic
369C22#315611yescharacteristic
370C4—22no—
371C4—22no—
372C4—22no—
373C4—84no—
374C4—161no—
375C4—161no—
376C4—162no—
377C4—161no—
378C4—161no—
379C4#310512yes—
380C4#318812yes—
381C4#329814yes—
382C4#329914yes—
383C2—24no—
384C2C2×#170711yescharacteristic
385C2#1962111yescharacteristic
386C2#1963211yescharacteristic
387C2#1965711yescharacteristic
388C2#2830311yescharacteristic
389C2#2833311yescharacteristic
390C2#2833611yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10680C2semidirect
8. #10314C2semidirect
9. #9823C2semidirect
10. #9819C2semidirect

Maximal subgroups

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

subgroupindex
#330432
#279482
#279102
#258132
#258112
C2×#19562
#106802
#103142
#98232
#98192
C8×C22⋊C82