Finite Group Databasebeta

← #10593952

Subgroups — #10593952

Order 576.

420 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 120 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#64079C211yesmaximal, characteristic, complemented
2C3×#875C211yesmaximal, characteristic
3C3×#870C211yesmaximal, characteristic, complemented
4#63899C212yesmaximal, complemented
5#63893C212yesmaximal, complemented
6#63884C212yesmaximal
7#63861C212yesmaximal, complemented
8#63858C212yesmaximal, complemented
9#63839C212yesmaximal, complemented
10#5699—31nomaximal
11#5672—31nomaximal
12C2×C3×D12C2211yescharacteristic
13C2×C3×Dic6C2211yescharacteristic
14C2×#627C2212yes—
15C2×D3×Dic3C2212yes—
16C32×C42C2211yescharacteristic
17C4⋊(C3⋊Dic3)C2211yescharacteristic
18C4⋊(C3⋊Dic3)C2211yescharacteristic
19#3750C2212yes—
20#3750C2212yes—
21C3×(C2×D3)⋊C4C2212yes—
22C3×(C2×D3)⋊C4C2212yes—
23C3×C4⋊Dic3C2211yescharacteristic
24C3×C4⋊Dic3C2211yescharacteristic
25C3×Dic3⋊C4C2212yes—
26C3×Dic3⋊C4C2212yes—
27#3724C2212yes—
28Dic3⋊Dic3C2212yes—
29(C2×D3)⋊Dic3C2212yes—
30(C2×D3)⋊Dic3C2212yes—
31(C2×D3)⋊Dic3C2212yes—
32Dic32C2212yes—
33C3×#125—31no—
34C3×#124—31no—
35#944—32no—
36#935—32no—
37#898—32no—
38#898—32no—
39#897—32no—
40#896—32no—
41#894—32no—
42#888—32no—
43#879—32no—
44#879—32no—
45#878—32no—
46#878—32no—
47#875D311yescharacteristic
48#870—31no—
49#870—31no—
50#870—61no—
51#870D311yescharacteristic, complemented
52C22×C3×D3C2312yes—
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55C2×C32×C4C2311yescharacteristic
56C2×C3⋊Dic3C2312yes—
57C2×C3⋊Dic3C2312yes—
58C2×C3×Dic3C2312yes—
59C2×C3×Dic3C2312yes—
60C2×C3×Dic3C2312yes—
61C3×D12—41no—
62C3×Dic6—41no—
63#627—42no—
64D3×Dic3—42no—
65#563—91noSylow
66C2×C3×Q8—31no—
67C2×C3×D4—31no—
68C22×Dic3—32no—
69C2×D12C2×D311yescharacteristic
70C2×C4×D3—32no—
71C2×Dic6—32no—
72C2×Dic6—32no—
73C2×Dic6C2×D311yescharacteristic
74C3×C4⋊C4—32no—
75C3×C4⋊C4—31no—
76C3×C4⋊C4—32no—
77C3×C4⋊C4—31no—
78C3×C22⋊C4—32no—
79C3×C22⋊C4—32no—
80C3×C42—21no—
81C3×C42C2×D311yescharacteristic
82C3×C42C2×D311yescharacteristic
83#217—32no—
84#217—32no—
85#217—32no—
86(C2×D3)⋊C4—32no—
87(C2×D3)⋊C4—32no—
88(C2×D3)⋊C4—32no—
89(C2×D3)⋊C4C2×D312yes—
90(C2×D3)⋊C4C2×D312yes—
91C4⋊Dic3—31no—
92C4⋊Dic3—31no—
93C4⋊Dic3—31no—
94C4⋊Dic3—31no—
95C4⋊Dic3—32no—
96C4⋊Dic3—61no—
97C4⋊Dic3—61no—
98C4⋊Dic3C2×D311yescharacteristic
99C4⋊Dic3C2×D311yescharacteristic
100Dic3⋊C4—32no—
101Dic3⋊C4—32no—
102Dic3⋊C4—32no—
103Dic3⋊C4—32no—
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4—32no—
107Dic3⋊C4—62no—
108Dic3⋊C4—62no—
109Dic3⋊C4C2×D312yes—
110Dic3⋊C4C2×D312yes—
111C4×Dic3—32no—
112C4×Dic3—32no—
113C22×C32C2411yescharacteristic
114C2×C3×D3—22no—
115C2×C3×D3—22no—
116C2×C3×D3—22no—
117C32×C4—21no—
118C32×C4—21no—
119C32×C4—21no—
120C3⋊Dic3—22no—
121C3⋊Dic3—22no—
122C3×Dic3—22no—
123C3×Dic3—22no—
124C3×Dic3—22no—
125C4⋊Q8—92no—
126#126—92no—
127#126—92no—
128#125—31no—
129#125—91no—
130#124—31no—
131#123—92no—
132C22⋊Q8—92no—
133C22⋊Q8—92no—
134C23×C3—32no—
135C22×D3C22×D312yes—
136C3×Q8—121no—
137C3×D4—121no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—32no—
142C2×C3×C4—32no—
143C2×C3×C4—32no—
144C2×C3×C4C22×D311yescharacteristic
145C2×C3×C4C22×D311yescharacteristic
146C2×C3×C4C22×D311yescharacteristic
147C2×C3×C4C22×D311yescharacteristic
148C2×C3×C4C22×D311yescharacteristic
149C2×C3×C4C22×D311yescharacteristic
150C2×Dic3—32no—
151C2×Dic3—32no—
152C2×Dic3—32no—
153C2×Dic3—32no—
154C2×Dic3—62no—
155C2×Dic3—62no—
156C2×Dic3—122no—
157C2×Dic3C22×D312yes—
158C2×Dic3C22×D312yes—
159C2×Dic3C22×D312yes—
160D12—41no—
161C4×D3—122no—
162Dic6—41no—
163Dic6—122no—
164Dic6—122no—
165C2×C32H(5,𝔽2)11yescharacteristic
166C2×C322−1+411yescharacteristic
167C2×C322−1+411yescharacteristic
168C3×D3—42no—
169C2×Q8—31no—
170C2×Q8—92no—
171C2×D4—31no—
172C22×C4—92no—
173C4⋊C4—31no—
174C4⋊C4—32no—
175C4⋊C4—32no—
176C4⋊C4—31no—
177C4⋊C4—92no—
178C4⋊C4—92no—
179C4⋊C4—91no—
180C4⋊C4—92no—
181C4⋊C4—92no—
182C4⋊C4—91no—
183C22⋊C4—32no—
184C22⋊C4—32no—
185C22⋊C4—92no—
186C22⋊C4—92no—
187C22⋊C4—92no—
188C42—92no—
189C42D3211yescharacteristic
190C22×C3—21no—
191C22×C3—62no—
192C22×C3—62no—
193C22×C3—62no—
194C22×C3C23×D311yescharacteristic
195C22×C3C23×D311yescharacteristic
196C2×D3—22no—
197C2×D3—22no—
198C2×D3—22no—
199C3×C4—22no—
200C3×C4—21no—
201C3×C4—21no—
202C3×C4—21no—
203C3×C4—21no—
204C3×C4—21no—
205C3×C4—21no—
206C3×C4—41no—
207C3×C4—41no—
208C3×C4—62no—
209C3×C4—62no—
210C3×C4—62no—
211Dic3—22no—
212Dic3—22no—
213Dic3—22no—
214Dic3—62no—
215Dic3—62no—
216Dic3—62no—
217Dic3—62no—
218Dic3—122no—
219Dic3—122no—
220C32#56311yescharacteristic, complemented, Sylow
221C23—32no—
222Q8—121no—
223Q8—362no—
224D4—121no—
225C2×C4—32no—
226C2×C4—32no—
227C2×C4—32no—
228C2×C4—92no—
229C2×C4—92no—
230C2×C4—362no—
231C2×C4C2×D3211yescharacteristic
232C2×C4C2×D3211yescharacteristic
233C2×C4C2×D3211yescharacteristic
234C2×C3—21no—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3—122no—
238C2×C3#100411yescharacteristic
239C2×C3#100711yescharacteristic
240C2×C3#100911yescharacteristic
241C2×C3#101011yescharacteristic
242C2×C3#101011yescharacteristic
243C2×C3#101011yescharacteristic
244D3—42no—
245C22—62no—
246C22—62no—
247C22—62no—
248C22C22×D3211yescharacteristic
249C4—21no—
250C4—21no—
251C4—21no—
252C4—62no—
253C4—62no—
254C4—62no—
255C4—182no—
256C4—182no—
257C3—21no—
258C3#567211yescharacteristic, complemented
259C3#569911yescharacteristic, complemented
260C2—122no—
261C2#6429711yescharacteristic
262C2#6429811yescharacteristic
263C2#6430711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
prime radicalC2×C3×D12C22
hypercenterC42D32
hypocenterC32#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #64079C2semidirect
3. C3×#870C2semidirect
4. #63899C2semidirect
5. #63893C2semidirect
7. #63861C2semidirect
8. #63858C2semidirect
9. #63839C2semidirect
51. #870D3semidirect
220. C32#563semidirect
258. C3#5672semidirect
259. C3#5699semidirect

Maximal subgroups

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

subgroupindex
#640792
C3×#8752
C3×#8702
#638992
#638932
#638842
#638612
#638582
#638392
#56993
#56723