Finite Group Databasebeta

← #345331

Subgroups — #345331

Order 512.

595 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 595 conjugacy classes of subgroups — too many to draw. Only its 148 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#26278C212yesmaximal
2#24732C212yesmaximal
3#10600C212yesmaximal
4#10478C212yesmaximal
5#10067C212yesmaximal
6#9834C211yesmaximal, characteristic
7#9662C211yesmaximal, characteristic
8#9596C211yesmaximal, characteristic
9#8896C212yesmaximal
10C2×#500C2211yescharacteristic
11C2×#462—22no—
12C2×#462C2212yes—
13C2×C4×Q16C2211yescharacteristic
14#2546C2212yes—
15Q8×C4⋊C4C2212yes—
16#2100—22no—
17#2084—22no—
18#2068—22no—
19#2043—22no—
20#1990C2212yes—
21#1988C2212yes—
22#1984C2212yes—
23#1956C2212yes—
24#1948C2212yes—
25#1939C2212yes—
26#1919—22no—
27#1919—22no—
28#1878—22no—
29#1834C2211yescharacteristic
30#1803C2212yes—
31C4×#106C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#335C2211yescharacteristic
34C2×#332C2212yes—
35C2×Q8⋊C8C2212yes—
36C2×#322C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39C22×Q16—22no—
40C22×Q16D411yescharacteristic
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8C2312yes—
43C2×C4×Q8—22no—
44C2×C4×Q8C2312yes—
45#500—22no—
46#500—41no—
47#462—42no—
48#462—42no—
49#462—48no—
50C4×Q16—41no—
51C4×Q16—41no—
52C2×#107—41no—
53C2×#107D411yescharacteristic
54C2×#106—22no—
55C2×#106D411yescharacteristic
56C2×#106D411yescharacteristic
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#103D412yes—
65C2×C4×C8C2311yescharacteristic
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#398—42no—
69#391—22no—
70#389—22no—
71#389—22no—
72#389—22no—
73#389—42no—
74#384—22no—
75#384C2312yes—
76C4×C4⋊C4—22no—
77C4×C4⋊C4C2312yes—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—22no—
83#340—22no—
84#340—41no—
85#340D412yes—
86#336—22no—
87#336—22no—
88#336—22no—
89#335—41no—
90#335—41no—
91#332—42no—
92Q8⋊C8—24no—
93#322—41no—
94C22×Q8—22no—
95C22×Q8C2×D412yes—
96C2×Q16—24no—
97C2×Q16—41no—
98C2×Q16—44no—
99C2×Q16—44no—
100C2×Q16—44no—
101C22×C8—21no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C22×C8C2×D411yescharacteristic
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—44no—
113C4⋊Q8—82no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—44no—
119C4×Q8—82no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D412yes—
132C2×C4⋊C4C2×D412yes—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—42no—
136C2×C42C2411yescharacteristic
137#107—41no—
138#107—44no—
139#106—41no—
140#106—44no—
141#106Q1614yescomplemented
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—44no—
147#103—42no—
148#103—44no—
149#103—44no—
150#103—44no—
151#103—82no—
152C4×C8—21no—
153C4×C8—21no—
154C4×C8—21no—
155C4×C8—21no—
156C4×C8—21no—
157C4×C8#4112yes—
158#95—22no—
159#95—42no—
160#95—42no—
161#95—42no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8AGL(1,ℤ/8)14yes—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4C22×D411yescharacteristic
181C22×C4C22×D411yescharacteristic
182C22×C4H(5,𝔽2)11yescharacteristic
183Q16—44no—
184Q16—88no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8C2×Q1612yes—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—84no—
219C4⋊C4—84no—
220C4⋊C4—82no—
221C4⋊C4—84no—
222C4⋊C4—162no—
223C4⋊C4C2×Q1614yes—
224C42—42no—
225C42—42no—
226C42—42no—
227C42—42no—
228C42—84no—
229C42C2×#4111yescharacteristic
230C42H(5,𝔽2)11yescharacteristic
231C42H(5,𝔽2)11yescharacteristic
232C422−1+411yescharacteristic
233C23D4211yescharacteristic
234Q8—24no—
235Q8—24no—
236Q8—42no—
237Q8—42no—
238Q8—82no—
239Q8—82no—
240Q8—82no—
241Q8—82no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4#53411yescharacteristic
262C2×C4#53711yescharacteristic
263C2×C4#53811yescharacteristic
264C2×C4#54711yescharacteristic
265C2×C4#55011yescharacteristic
266C2×C4#56211yescharacteristic
267C2×C4C22×Q1611yescharacteristic
268C2×C4C2×AGL(1,ℤ/8)12yes—
269C2×C4#57511yescharacteristic
270C2×C4#57712yes—
271C2×C4#57811yescharacteristic
272C2×C4#57811yescharacteristic
273C8—22no—
274C8—42no—
275C8—41no—
276C8—41no—
277C8—41no—
278C8—82no—
279C8—82no—
280C22#268211yescharacteristic
281C22#326512yes—
282C22#327711yescharacteristic
283C22D4×Q1611yescharacteristic
284C22#328912yes—
285C4—21no—
286C4—21no—
287C4—44no—
288C4—44no—
289C4—82no—
290C4—82no—
291C4—162no—
292C4—162no—
293C4#319211yescharacteristic
294C4#320411yescharacteristic
295C4#324911yescharacteristic
296C4#326111yescharacteristic
297C4#334512yes—
298C4#339511yescharacteristic
299C4#340211yescharacteristic
300C2#1981212yes—
301C2#2020012yes—
302C2#3120411yescharacteristic
303C2#3121311yescharacteristic
304C2#3123311yescharacteristic
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
141. #106Q16semidirect

Maximal subgroups

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

subgroupindex
#262782
#247322
#106002
#104782
#100672
#98342
#96622
#95962
#88962