Finite Group Databasebeta

← #345351

Subgroups — #345351

Order 512.

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

normal subgroups only

This group has 553 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#26281C212yesmaximal
2#24632C212yesmaximal
3#10592C212yesmaximal
4#10478C212yesmaximal
5#10067C212yesmaximal
6#9834C211yesmaximal, characteristic
7#9662C211yesmaximal, characteristic
8#9596C211yesmaximal, characteristic
9#8895C212yesmaximal
10C2×#500C2211yescharacteristic
11C2×#462C2212yes—
12C2×C4×Q16C2211yescharacteristic
13#2546C2212yes—
14#2355C2212yes—
15#2101—22no—
16#2085—22no—
17#2056—22no—
18#2016—22no—
19#1990C2212yes—
20#1988C2212yes—
21#1984C2212yes—
22#1981—22no—
23#1956C2212yes—
24#1947C2212yes—
25#1939C2212yes—
26#1928—22no—
27#1921—22no—
28#1920—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×Q8C2312yes—
44#500—22no—
45#500—41no—
46#462—42no—
47#462—42no—
48C4×Q16—41no—
49C4×Q16—41no—
50C2×#107—22no—
51C2×#107—41no—
52C2×#107D411yescharacteristic
53C2×#106D411yescharacteristic
54C2×#106D411yescharacteristic
55C2×C4⋊C8—22no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×#103—22no—
59C2×#103—22no—
60C2×#103—22no—
61C2×#103—22no—
62C2×#103D412yes—
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65C2×C4×C8C2311yescharacteristic
66#398—42no—
67#391—22no—
68#391—22no—
69#389—22no—
70#389—22no—
71#387—42no—
72#384C2312yes—
73#382—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4C2312yes—
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#332—42no—
91Q8⋊C8—24no—
92#322—41no—
93C22×Q8—22no—
94C22×Q8C2×D412yes—
95C2×Q16—42no—
96C2×Q16—41no—
97C2×Q16—44no—
98C2×Q16—82no—
99C2×Q16—82no—
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—44no—
112C4⋊Q8—82no—
113C4×Q8—22no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×Q8—22no—
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⋊C4C2×D412yes—
126C2×C4⋊C4C2×D412yes—
127C2×C4⋊C4C2×D412yes—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—42no—
131C2×C42C2411yescharacteristic
132#107—41no—
133#107—44no—
134#107—44no—
135#106—41no—
136#106Q1614yescomplemented
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—82no—
142#103—42no—
143#103—44no—
144#103—44no—
145#103—82no—
146#103—82no—
147C4×C8—21no—
148C4×C8—21no—
149C4×C8—21no—
150C4×C8—21no—
151C4×C8—21no—
152C4×C8#4112yes—
153#95—22no—
154#95—42no—
155#95—42no—
156#95—42no—
157#95—42no—
158#95—42no—
159C2×Q8—22no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4C22×D411yescharacteristic
178C22×C4C22×D411yescharacteristic
179C22×C4H(5,𝔽2)11yescharacteristic
180Q16—44no—
181Q16—88no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8C2×Q1612yes—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—82no—
216C4⋊C4—84no—
217C4⋊C4—84no—
218C4⋊C4C2×Q1614yes—
219C42—42no—
220C42—42no—
221C42—42no—
222C42—42no—
223C42—162no—
224C42C2×#4111yescharacteristic
225C42H(5,𝔽2)11yescharacteristic
226C42H(5,𝔽2)11yescharacteristic
227C422−1+411yescharacteristic
228C23D4211yescharacteristic
229Q8—42no—
230Q8—42no—
231Q8—42no—
232Q8—42no—
233Q8—82no—
234Q8—82no—
235Q8—82no—
236Q8—82no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—82no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4#53411yescharacteristic
257C2×C4#53711yescharacteristic
258C2×C4#53811yescharacteristic
259C2×C4#54711yescharacteristic
260C2×C4#55011yescharacteristic
261C2×C4#56211yescharacteristic
262C2×C4C22×Q1611yescharacteristic
263C2×C4#57512yes—
264C2×C4#57511yescharacteristic
265C2×C4#57612yes—
266C2×C4#57811yescharacteristic
267C2×C4#57811yescharacteristic
268C8—22no—
269C8—42no—
270C8—41no—
271C8—41no—
272C8—41no—
273C8—82no—
274C8—82no—
275C22#268211yescharacteristic
276C22#326812yes—
277C22#327711yescharacteristic
278C22D4×Q1611yescharacteristic
279C22#328912yes—
280C4—21no—
281C4—21no—
282C4—44no—
283C4—82no—
284C4—82no—
285C4—82no—
286C4—162no—
287C4—162no—
288C4#319711yescharacteristic
289C4#319911yescharacteristic
290C4#324911yescharacteristic
291C4#326111yescharacteristic
292C4#334712yes—
293C4#339511yescharacteristic
294C4#340211yescharacteristic
295C2#1981512yes—
296C2#2026912yes—
297C2#3120711yescharacteristic
298C2#3121311yescharacteristic
299C2#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
136. #106Q16semidirect

Maximal subgroups

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

subgroupindex
#262812
#246322
#105922
#104782
#100672
#98342
#96622
#95962
#88952