Finite Group Databasebeta

← #345337

Subgroups — #345337

Order 512.

535 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 144 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#26279C212yesmaximal
2#24682C212yesmaximal
3#10593C212yesmaximal
4#10483C212yesmaximal
5#10057C212yesmaximal
6#9835C211yesmaximal, characteristic
7#9662C211yesmaximal, characteristic
8#9596C211yesmaximal, characteristic
9#8895C212yesmaximal
10C2×#494C2211yescharacteristic
11C2×#477C2212yes—
12C2×C4×Q16C2211yescharacteristic
13#2567C2212yes—
14#2355C2212yes—
15#2101—22no—
16#2085—22no—
17#2072—22no—
18#2056—22no—
19#1990C2212yes—
20#1990C2212yes—
21#1987C2212yes—
22#1984C2212yes—
23#1947—22no—
24#1947C2212yes—
25#1939C2212yes—
26#1928—22no—
27#1919—22no—
28#1918—22no—
29#1834C2211yescharacteristic
30#1803C2212yes—
31C4×#106C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#333C2212yes—
35C2×Q8⋊C8C2212yes—
36C2×#322C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39C22×Q16—41no—
40C22×Q16D411yescharacteristic
41C2×C4⋊Q8C2312yes—
42C2×C4×Q8C2312yes—
43#494—22no—
44#494—41no—
45#477—42no—
46#477—42no—
47C4×Q16—41no—
48C4×Q16—41no—
49C2×#107—22no—
50C2×#107—22no—
51C2×#107D411yescharacteristic
52C2×#106D411yescharacteristic
53C2×#106D411yescharacteristic
54C2×C4⋊C8—22no—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×#103—22no—
58C2×#103—22no—
59C2×#103D412yes—
60C2×C4×C8C2311yescharacteristic
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63#395—42no—
64#391—22no—
65#391—22no—
66#389—22no—
67#389—22no—
68#387—42no—
69#384—22no—
70#384C2312yes—
71#382—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4C2312yes—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—41no—
84#340D412yes—
85#336—22no—
86#336—22no—
87#336—22no—
88#335—41no—
89#335—41no—
90#333—42no—
91Q8⋊C8—24no—
92#322—41no—
93C22×Q8—22no—
94C22×Q8C2×D412yes—
95C2×Q16—42no—
96C2×Q16—41no—
97C2×Q16—44no—
98C2×Q16—84no—
99C22×C8—21no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4×Q8—22no—
111C4×Q8—22no—
112C4×Q8—22no—
113C4×Q8—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4C2×D412yes—
123C2×C4⋊C4C2×D412yes—
124C2×C4⋊C4C2×D412yes—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—42no—
128C2×C42C2411yescharacteristic
129#107—41no—
130#107—44no—
131#107—44no—
132#106—41no—
133#106Q1614yescomplemented
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—44no—
139#103—42no—
140#103—44no—
141#103—44no—
142C4×C8—21no—
143C4×C8—21no—
144C4×C8—21no—
145C4×C8—21no—
146C4×C8—21no—
147C4×C8#4112yes—
148#95—22no—
149#95—42no—
150#95—42no—
151#95—42no—
152#95—42no—
153#95—42no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—82no—
163C2×Q8—82no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4C22×D411yescharacteristic
171C22×C4C22×D411yescharacteristic
172C22×C4H(5,𝔽2)11yescharacteristic
173Q16—44no—
174Q16—88no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8C2×Q1612yes—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—82no—
209C4⋊C4—162no—
210C4⋊C4—162no—
211C4⋊C4C2×Q1614yes—
212C42—42no—
213C42—42no—
214C42—42no—
215C42—42no—
216C42—162no—
217C42C2×#4111yescharacteristic
218C42H(5,𝔽2)11yescharacteristic
219C42H(5,𝔽2)11yescharacteristic
220C422−1+411yescharacteristic
221C23D4211yescharacteristic
222Q8—42no—
223Q8—42no—
224Q8—42no—
225Q8—42no—
226Q8—84no—
227Q8—162no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4—82no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4#53411yescharacteristic
248C2×C4#53711yescharacteristic
249C2×C4#53811yescharacteristic
250C2×C4#54711yescharacteristic
251C2×C4#55011yescharacteristic
252C2×C4#56211yescharacteristic
253C2×C4C22×Q1611yescharacteristic
254C2×C4#57512yes—
255C2×C4#57511yescharacteristic
256C2×C4#57812yes—
257C2×C4#57811yescharacteristic
258C2×C4#57811yescharacteristic
259C8—22no—
260C8—42no—
261C8—41no—
262C8—41no—
263C8—41no—
264C8—82no—
265C8—82no—
266C22#268211yescharacteristic
267C22#326912yes—
268C22#327711yescharacteristic
269C22D4×Q1611yescharacteristic
270C22#328812yes—
271C4—21no—
272C4—21no—
273C4—44no—
274C4—82no—
275C4—82no—
276C4—82no—
277C4—162no—
278C4—162no—
279C4#319711yescharacteristic
280C4#320211yescharacteristic
281C4#324911yescharacteristic
282C4#326111yescharacteristic
283C4#334912yes—
284C4#339511yescharacteristic
285C4#340211yescharacteristic
286C2#1981612yes—
287C2#2023212yes—
288C2#3120511yescharacteristic
289C2#3121311yescharacteristic
290C2#3123111yescharacteristic
subgroupG/H
center, socle, prime radicalC23D42
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
133. #106Q16semidirect

Maximal subgroups

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

subgroupindex
#262792
#246822
#105932
#104832
#100572
#98352
#96622
#95962
#88952