Finite Group Databasebeta

← #345161

Subgroups — #345161

Order 512.

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

normal subgroups only

This group has 557 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#26282C212yesmaximal
2#24817C212yesmaximal
3#10589C212yesmaximal
4#10483C212yesmaximal
5#10057C212yesmaximal
6#9835C211yesmaximal, characteristic
7#9658C211yesmaximal, characteristic
8#9598C211yesmaximal, characteristic
9#8892C212yesmaximal
10C2×#494C2211yescharacteristic
11C2×#477C2212yes—
12C2×C4×Q16C2211yescharacteristic
13#2567C2212yes—
14Q8×C4⋊C4C2212yes—
15#2100—22no—
16#2083—22no—
17#2083—22no—
18#2068—22no—
19#1990C2212yes—
20#1990C2212yes—
21#1987C2212yes—
22#1984C2212yes—
23#1947C2212yes—
24#1939C2212yes—
25#1921—22no—
26#1921—22no—
27#1878—22no—
28#1834C2211yescharacteristic
29#1803—22no—
30#1803C2212yes—
31C4×#107C2211yescharacteristic
32C2×#335C2211yescharacteristic
33C2×#334C2211yescharacteristic
34C2×#333C2212yes—
35C2×Q8⋊C8C2212yes—
36C2×#322C2211yescharacteristic
37#1296C2212yes—
38#1279C2212yes—
39C22×Q16—41no—
40C22×Q16D411yescharacteristic
41C2×C4⋊Q8C2312yes—
42C2×C4×Q8—22no—
43C2×C4×Q8C2312yes—
44#494—22no—
45#494—41no—
46#477—42no—
47#477—42no—
48C4×Q16—41no—
49C4×Q16—41no—
50C2×#107—22no—
51C2×#107—22no—
52C2×#107D412yes—
53C2×#107D411yescharacteristic
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#389—22no—
66#389—22no—
67#389—22no—
68#389—42no—
69#384—22no—
70#384—22no—
71#384C2312yes—
72C4×C4⋊C4—22no—
73C4×C4⋊C4C2312yes—
74#340—22no—
75#340—22no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#340—41no—
83#340D411yescharacteristic
84#340D411yescharacteristic
85#336—22no—
86#336—22no—
87#336—22no—
88#335—41no—
89#334—41no—
90#333—42no—
91Q8⋊C8—24no—
92#322—41no—
93C22×Q8—22no—
94C22×Q8C2×D412yes—
95C2×Q16—41no—
96C2×Q16—41no—
97C2×Q16—44no—
98C2×Q16—84no—
99C2×Q16Q1614yescomplemented
100C22×C8—21no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8C2×D411yescharacteristic
104C22×C8C2×D411yescharacteristic
105C22×C8C2×D411yescharacteristic
106C22×C8C2×D411yescharacteristic
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4×Q8—22no—
112C4×Q8—22no—
113C4×Q8—22no—
114C4×Q8—22no—
115C4×Q8—44no—
116C4×Q8—82no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×D412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C4⋊C4C2×D412yes—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—42no—
133C2×C42C2411yescharacteristic
134#107—24no—
135#107—41no—
136#107—44no—
137#107—44no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—82no—
143#103—42no—
144#103—44no—
145#103—82no—
146C4×C8—21no—
147C4×C8—21no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8#4112yes—
152#95—22no—
153#95—42no—
154#95—42no—
155#95—42no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—22no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×Q8—82no—
164C2×Q8—82no—
165C2×Q8C2×Q1614yes—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4C22×D411yescharacteristic
173C22×C4C22×D411yescharacteristic
174C22×C4H(5,𝔽2)11yescharacteristic
175Q16—44no—
176Q16—88no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—22no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8C2×Q1612yes—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
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—42no—
209C4⋊C4—42no—
210C4⋊C4—82no—
211C4⋊C4—84no—
212C4⋊C4—162no—
213C4⋊C4—162no—
214C4⋊C4—162no—
215C4⋊C4#13714yes—
216C42—42no—
217C42—42no—
218C42—42no—
219C42—42no—
220C42—84no—
221C42C2×#4111yescharacteristic
222C42H(5,𝔽2)11yescharacteristic
223C42H(5,𝔽2)11yescharacteristic
224C422−1+411yescharacteristic
225C23D4211yescharacteristic
226Q8—24no—
227Q8—24no—
228Q8—42no—
229Q8—42no—
230Q8—84no—
231Q8—162no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—82no—
251C2×C4#53411yescharacteristic
252C2×C4#53711yescharacteristic
253C2×C4#53811yescharacteristic
254C2×C4#54711yescharacteristic
255C2×C4#55011yescharacteristic
256C2×C4#56211yescharacteristic
257C2×C4C22×Q1611yescharacteristic
258C2×C4C2×#13712yes—
259C2×C4#57511yescharacteristic
260C2×C4#57611yescharacteristic
261C2×C4#57811yescharacteristic
262C2×C4#57812yes—
263C8—22no—
264C8—42no—
265C8—41no—
266C8—41no—
267C8—41no—
268C8—82no—
269C8—82no—
270C22#268211yescharacteristic
271C22#327012yes—
272C22#327311yescharacteristic
273C22D4×Q1611yescharacteristic
274C22#328812yes—
275C4—21no—
276C4—21no—
277C4—44no—
278C4—44no—
279C4—82no—
280C4—82no—
281C4—162no—
282C4—162no—
283C4#318911yescharacteristic
284C4#320311yescharacteristic
285C4#325511yescharacteristic
286C4#326111yescharacteristic
287C4#333811yescharacteristic
288C4#335211yescharacteristic
289C4#339812yes—
290C2#1981712yes—
291C2#2021012yes—
292C2#3119011yescharacteristic
293C2#3122511yescharacteristic
294C2#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
99. C2×Q16Q16semidirect

Maximal subgroups

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

subgroupindex
#262822
#248172
#105892
#104832
#100572
#98352
#96582
#95982
#88922