Finite Group Databasebeta

← #319042

Subgroups — #319042

Order 512.

842 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33043C211yesmaximal, characteristic
2#27972C211yesmaximal, characteristic
3#27950C211yesmaximal, characteristic
4#25856C211yesmaximal, characteristic
5#25813C211yesmaximal, characteristic
6C2×#1990C211yesmaximal, characteristic
7#10679C212yesmaximal, complemented
8#10313C212yesmaximal, complemented
9#9823C212yesmaximal, complemented
10#9821C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C2×#500C2212yes—
14C2×#487C2212yes—
15C2×#487C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2857C2211yescharacteristic
18#2857C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#2083—22no—
21#2072—22no—
22#2057—22no—
23#2043—22no—
24#1990—22no—
25#1990C2211yescharacteristic
26#1990C2212yes—
27#1990C2212yes—
28#1990C2212yes—
29#1990C2212yes—
30#1956C2212yes—
31#1956C2211yescharacteristic
32#1956C2212yes—
33#1933—22no—
34#1892—22no—
35#1872—22no—
36#1852—22no—
37#1852—22no—
38#1845C2212yes—
39#1834C2211yescharacteristic
40#1834C2212yes—
41#1787—22no—
42C4×C22⋊C8C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C2×#340—22no—
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#487—24no—
68#487—42no—
69#487—42no—
70C2×#107—24no—
71C2×#107—24no—
72C2×#103—22no—
73C2×#103—24no—
74C2×#103—22no—
75C2×#103—24no—
76C2×#103—24no—
77C2×C22⋊C8D412yes—
78C2×C22⋊C8D412yes—
79C2×C4×C8—42no—
80C2×C4×C8D412yes—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2312yes—
83C2×C4×C8C2312yes—
84C2×C4×C8C2311yescharacteristic
85#391—21no—
86#391—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384C2311yescharacteristic
95#384C2311yescharacteristic
96#384C2312yes—
97#380—21no—
98#380—21no—
99#380—21no—
100#380—21no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—22no—
105#340—22no—
106#340—24no—
107#340—24no—
108#336D412yes—
109#336D412yes—
110C82—42no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—44no—
115C22×Q8—41no—
116C22×Q8—44no—
117C22×Q8—82no—
118C23×C4—41no—
119C23×C4—41no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C22×C8—24no—
124C22×C8—24no—
125C22×C8—42no—
126C22×C8C2×D412yes—
127C22×C8C2×D414yes—
128C22×C8C2×D414yes—
129C22×C8C2×D412yes—
130C4⋊Q8—24no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—44no—
139C4⋊Q8—44no—
140C4⋊Q8—42no—
141C4⋊Q8#4114yes—
142C22⋊Q8—44no—
143C22⋊Q8—44no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
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—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2411yescharacteristic
180#107—48no—
181#107—84no—
182#103—48no—
183#103—44no—
184#103—44no—
185#103—84no—
186#103—84no—
187C22⋊C8—24no—
188C22⋊C8Q1618yes—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—42no—
196#95—41no—
197#95—41no—
198#95—41no—
199#95—41no—
200C24#12711yescharacteristic
201C2×Q8—24no—
202C2×Q8—44no—
203C2×Q8—42no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—42no—
207C2×Q8—44no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—44no—
212C2×Q8—42no—
213C2×Q8—816no—
214C2×Q8—82no—
215C2×Q8—82no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—42no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4#12711yescharacteristic
236C22×C4#12711yescharacteristic
237C22×C4#12711yescharacteristic
238C22×C4C2×Q1614yes—
239C22×C4#13514yes—
240C22×C4#13514yes—
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C2×C8—22no—
245C2×C8—24no—
246C2×C8—22no—
247C2×C8—24no—
248C2×C8—22no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—24no—
254C2×C8—48no—
255C2×C8C2×Q1618yes—
256C2×C8#13714yes—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C22⋊C4—82no—
283C22⋊C4—82no—
284C42—21no—
285C42—21no—
286C42—22no—
287C42—22no—
288C42C2×#4112yes—
289C42H(5,𝔽2)12yes—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#49412yes—
295C23#49512yes—
296C23#49512yes—
297C23C2×#12711yescharacteristic
298Q8—48no—
299Q8—88no—
300Q8—84no—
301Q8—82no—
302Q8—82no—
303Q8—84no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—28no—
313C2×C4—28no—
314C2×C4—44no—
315C2×C4—88no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
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—81no—
332C2×C4—81no—
333C2×C4—164no—
334C2×C4#49412yes—
335C2×C4#49512yes—
336C2×C4#49512yes—
337C2×C4#52511yescharacteristic
338C2×C4#53411yescharacteristic
339C2×C4#54612yes—
340C2×C4#54612yes—
341C2×C4C22×Q1612yes—
342C2×C4C2×#13512yes—
343C2×C4C2×#13512yes—
344C2×C4C2×#13712yes—
345C2×C4#57612yes—
346C2×C4#57712yes—
347C8—24no—
348C8—28no—
349C8—42no—
350C8—44no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22#171014yes—
358C22#253911yescharacteristic
359C22C2×#49411yescharacteristic
360C22C2×#49511yescharacteristic
361C22C2×#49511yescharacteristic
362C22#315211yescharacteristic
363C22#315411yescharacteristic
364C22#315511yescharacteristic
365C4—22no—
366C4—22no—
367C4—22no—
368C4—84no—
369C4—161no—
370C4—161no—
371C4—161no—
372C4—161no—
373C4—162no—
374C4#310812yes—
375C4#319012yes—
376C4#329714yes—
377C4#329914yes—
378C2—24no—
379C2C2×#171011yescharacteristic
380C2#1962611yescharacteristic
381C2#1963311yescharacteristic
382C2#1965811yescharacteristic
383C2#2830611yescharacteristic
384C2#2833111yescharacteristic
385C2#2833311yescharacteristic
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. #10679C2semidirect
8. #10313C2semidirect
9. #9823C2semidirect
10. #9821C2semidirect

Maximal subgroups

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

subgroupindex
#330432
#279722
#279502
#258562
#258132
C2×#19902
#106792
#103132
#98232
#98212
C8×C22⋊C82