Finite Group Databasebeta

← #316939

Subgroups — #316939

Order 512.

611 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28159C212yesmaximal
2#22247C211yesmaximal, characteristic, complemented
3#22214C211yesmaximal, characteristic
4#10623C211yesmaximal, characteristic, complemented
5#10621C211yesmaximal, characteristic
6#10265C211yesmaximal, characteristic, complemented
7#10264C211yesmaximal, characteristic
8#9226C212yesmaximal, complemented
9#9225C212yesmaximal
10#8788C211yesmaximal, characteristic, complemented
11#8591C212yesmaximal, complemented
12C2×#500C2211yescharacteristic
13C2×#462C2212yes—
14C2×#461C2212yes—
15C2×#446C2211yescharacteristic
16C2×#436C2211yescharacteristic
17C4×C4⋊Q8C2212yes—
18C4×#124C2211yescharacteristic
19#2074—22no—
20#2015—22no—
21#1990C2212yes—
22#1984C2211yescharacteristic
23#1954C2211yescharacteristic
24#1924—22no—
25#1918—22no—
26#1918—22no—
27#1880—22no—
28#1878—22no—
29#1851—22no—
30#1842C2211yescharacteristic
31#1834C2211yescharacteristic
32#1831C2212yes—
33#1803C2212yes—
34#1802C2212yes—
35#1744C2211yescharacteristic
36C2×#332C2212yes—
37C2×#332C2212yes—
38#1296C2212yes—
39#1296C2212yes—
40#1277C2212yes—
41#1277C2211yescharacteristic
42#1277C2211yescharacteristic
43C22×Q16—22no—
44C22×SD16—22no—
45C22×M16—21no—
46C2×C4⋊Q8C2312yes—
47C2×#124D411yescharacteristic
48C2×C4×Q8—22no—
49C2×C4×Q8C2311yescharacteristic
50C2×C4×D4C2311yescharacteristic
51#500—24no—
52#462—24no—
53#462—42no—
54#461—24no—
55#461—42no—
56#446—42no—
57#436—42no—
58C2×#107—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#102—22no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—22no—
71C2×#97—21no—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#391—22no—
75#391D411yescharacteristic
76#390D411yescharacteristic
77#389—22no—
78#389—22no—
79#388—22no—
80#384C2312yes—
81#382D411yescharacteristic
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C22⋊C4—22no—
85C43C2311yescharacteristic
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#336—22no—
92#336—22no—
93#336—22no—
94#336D412yes—
95#336D412yes—
96#332—42no—
97#332—42no—
98C22×Q8—22no—
99C22×Q8C2×D411yescharacteristic
100C22×D4C2×D411yescharacteristic
101C23×C4—21no—
102C2×Q16—44no—
103C2×Q16—44no—
104C2×Q16—82no—
105C2×SD16—44no—
106C2×SD16—44no—
107C2×SD16—82no—
108C2×M16—42no—
109C2×M16—82no—
110C22×C8—21no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—21no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D412yes—
116C22×C8C2×D411yescharacteristic
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8#4114yes—
121#124—42no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—82no—
125C4×Q8—82no—
126C4×D4—22no—
127C4×D4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C22⋊C4—22no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—22no—
140C2×C42—21no—
141C2×C42—42no—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D412yes—
145C2×C42#4112yes—
146C2×C42C2411yescharacteristic
147#107—28no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—42no—
155C4⋊C8—82no—
156#103—44no—
157#103—44no—
158#103—82no—
159#102—82no—
160C22⋊C8—42no—
161C22⋊C8—82no—
162#97—42no—
163C4×C8—22no—
164C4×C8—22no—
165#95—22no—
166#95—22no—
167#95—42no—
168#95—42no—
169C24—21no—
170C2×Q8—24no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—21no—
183C2×D4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4#12112yes—
194C22×C4#12112yes—
195C22×C4C22×D411yescharacteristic
196C22×C4C22×D411yescharacteristic
197C22×C4C2×#4111yescharacteristic
198Q16—88no—
199SD16—88no—
200M16—84no—
201C2×C8—22no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—24no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—82no—
230C4⋊C4—84no—
231C22⋊C4—44no—
232C22⋊C4—42no—
233C22⋊C4—82no—
234C42—22no—
235C42—21no—
236C42—21no—
237C42—22no—
238C42—22no—
239C42—42no—
240C42—41no—
241C42—41no—
242C42—84no—
243C42C2×#4112yes—
244C42H(5,𝔽2)12yes—
245C23—42no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—42no—
250C23C2×#12111yescharacteristic
251Q8—44no—
252Q8—42no—
253Q8—44no—
254Q8—42no—
255Q8—82no—
256Q8—82no—
257D4—42no—
258D4—42no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—44no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4#52511yescharacteristic
283C2×C4#53511yescharacteristic
284C2×C4#54612yes—
285C2×C4#55312yes—
286C2×C4#57512yes—
287C2×C4#57611yescharacteristic
288C2×C4#57611yescharacteristic
289C2×C4#57712yes—
290C2×C4#57812yes—
291C8—42no—
292C8—41no—
293C8—41no—
294C8—81no—
295C8—81no—
296C8—82no—
297C8—82no—
298C22—81no—
299C22—81no—
300C22—82no—
301C22—81no—
302C22—82no—
303C22#253511yescharacteristic
304C22#304612yes—
305C22#305411yescharacteristic
306C22#305811yescharacteristic
307C22#306912yes—
308C4—22no—
309C4—42no—
310C4—42no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—84no—
315C4—162no—
316C4#312112yes—
317C4#322211yescharacteristic
318C4#322311yescharacteristic
319C4#332212yes—
320C4#332212yes—
321C2—161no—
322C2#1945412yes—
323C2#1954612yes—
324C2#2814611yescharacteristic
325C2#2817311yescharacteristic
326C2#2817911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22247C2semidirect
4. #10623C2semidirect
6. #10265C2semidirect
8. #9226C2semidirect
10. #8788C2semidirect
11. #8591C2semidirect

Maximal subgroups

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

subgroupindex
#281592
#222472
#222142
#106232
#106212
#102652
#102642
#92262
#92252
#87882
#85912