Finite Group Databasebeta

← #330537

Subgroups — #330537

Order 512.

532 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 154 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#33294C211yesmaximal, characteristic
2#29999C211yesmaximal, characteristic
3#29799C211yesmaximal, characteristic
4#21343C211yesmaximal, characteristic
5#10594C212yesmaximal
6#10590C212yesmaximal
7#10479C212yesmaximal
8#10478C212yesmaximal
9#9897C211yesmaximal, characteristic
10#9692C211yesmaximal, characteristic
11#8808C211yesmaximal, characteristic
12C2×#462C2211yescharacteristic
13C2×#462C2211yescharacteristic
14#2868C2211yescharacteristic
15#2859C2211yescharacteristic
16#2295C2211yescharacteristic
17#2087—22no—
18#2068—22no—
19#2043—22no—
20#2016—22no—
21#1992—22no—
22#1990C2212yes—
23#1990C2212yes—
24#1984C2212yes—
25#1981C2211yescharacteristic
26#1981C2211yescharacteristic
27#1947C2212yes—
28#1939C2212yes—
29#1917—22no—
30#1842C2211yescharacteristic
31#1836—21no—
32#1836—21no—
33#1831C2211yescharacteristic
34#1803C2212yes—
35#1768—21no—
36C4×C4⋊C8C2211yescharacteristic
37#1746—21no—
38C4×#97C2211yescharacteristic
39C2×#335C2212yes—
40C2×#334C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×Q16—21no—
46C22×Q16—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#125—21no—
51C2×C4×Q8C2311yescharacteristic
52#462—42no—
53#462—42no—
54C2×#107—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2311yescharacteristic
60C2×C4⋊C8C2311yescharacteristic
61C2×#103—21no—
62C2×#103—21no—
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—22no—
66C2×#97D411yescharacteristic
67C2×#97D412yes—
68C2×#97D411yescharacteristic
69C2×C4×C8—21no—
70C2×C4×C8C2312yes—
71#391D412yes—
72#389—22no—
73#389—22no—
74#389D412yes—
75#387—22no—
76#384—21no—
77#384—21no—
78#384—22no—
79#384C2312yes—
80#382—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4C2311yescharacteristic
83#376—21no—
84C43C2311yescharacteristic
85#340—22no—
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#336—22no—
95#336—21no—
96#335—42no—
97#334—42no—
98#332—42no—
99#332—42no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8C2×D411yescharacteristic
103C2×Q16—42no—
104C2×Q16—42no—
105C2×Q16—82no—
106C2×Q16—82no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8C2×D412yes—
112C22×C8C2×D412yes—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—82no—
118C4⋊Q8—82no—
119#125—44no—
120C4×Q8—22no—
121C4×Q8—22no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D412yes—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42#4112yes—
142C2×C42C2411yescharacteristic
143#107—82no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—44no—
151C4⋊C8—82no—
152#103—42no—
153#103—42no—
154#103—82no—
155#103—82no—
156#103—82no—
157#97—22no—
158#97—42no—
159#97Q1614yescomplemented
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—42no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—21no—
167#95—42no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—41no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4#12112yes—
187C22×C4#12112yes—
188C22×C4C22×D411yescharacteristic
189C22×C4C22×D411yescharacteristic
190C22×C4C2×#4111yescharacteristic
191Q16—84no—
192Q16—84no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8C2×Q1614yes—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—44no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—84no—
224C4⋊C4—84no—
225C4⋊C4—84no—
226C42—22no—
227C42—22no—
228C42—22no—
229C42—42no—
230C42—42no—
231C42—42no—
232C42C2×Q1612yes—
233C42#13712yes—
234C42H(5,𝔽2)12yes—
235C422−1+412yes—
236C23C2×#12111yescharacteristic
237Q8—41no—
238Q8—41no—
239Q8—42no—
240Q8—81no—
241Q8—81no—
242Q8—81no—
243Q8—81no—
244Q8—82no—
245Q8—82no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4#46214yes—
267C2×C4#53511yescharacteristic
268C2×C4#53611yescharacteristic
269C2×C4#53712yes—
270C2×C4#54212yes—
271C2×C4C22×Q1611yescharacteristic
272C2×C4C2×#13711yescharacteristic
273C2×C4#57512yes—
274C2×C4#57612yes—
275C2×C4#57712yes—
276C8—24no—
277C8—42no—
278C8—82no—
279C8—81no—
280C8—81no—
281C8—82no—
282C22#262011yescharacteristic
283C22C2×#46211yescharacteristic
284C22#304011yescharacteristic
285C22#306212yes—
286C22#306912yes—
287C4—24no—
288C4—22no—
289C4—42no—
290C4—81no—
291C4—81no—
292C4—82no—
293C4—161no—
294C4—161no—
295C4—162no—
296C4#322011yescharacteristic
297C4#322211yescharacteristic
298C4#324112yes—
299C4#324912yes—
300C4#325212yes—
301C2#1924012yes—
302C2#1938312yes—
303C2#2983011yescharacteristic
304C2#2985511yescharacteristic
305C2#2986211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
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
159. #97Q16semidirect

Maximal subgroups

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

subgroupindex
#332942
#299992
#297992
#213432
#105942
#105902
#104792
#104782
#98972
#96922
#88082