Finite Group Databasebeta

← #10591321

Subgroups — #10591321

Order 576.

594 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C22×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×C3×Dic3⋊C4C211yesmaximal, characteristic, complemented
3C2×#3722C212yesmaximal, complemented
4C2×#3722C212yesmaximal, complemented
5C3×C3⋊#95C211yesmaximal, characteristic, complemented
6#4991—31nomaximal
7#4672—31nomaximal
8C23×C3⋊D3C2212yes—
9C22×C32×C4C2211yescharacteristic
10C2×C4⋊(C3⋊D3)—24no—
11C2×C4⋊(C3⋊D3)—22no—
12C22×C3×Dic3C2211yescharacteristic
13C22×C3×Dic3C2212yes—
14C22×C3×Dic3C2211yescharacteristic
15C3×Dic3⋊C4—22no—
16#3722—24no—
17#3722—24no—
18C22×D12—31no—
19C22×D12—31no—
20C22×D12—61no—
21C2×C3×C4⋊C4—31no—
22C2×(C2×D3)⋊C4—32no—
23C2×(C2×D3)⋊C4—32no—
24C2×(C2×D3)⋊C4—32no—
25C2×(C2×D3)⋊C4—32no—
26C2×Dic3⋊C4D311yescharacteristic, complemented
27C3×#95—31no—
28C3⋊#95D311yescharacteristic, complemented
29C23×C32C2311yescharacteristic
30C22×C3⋊D3—22no—
31C22×C3⋊D3—22no—
32C22×C3⋊D3—22no—
33C22×C3⋊D3—22no—
34C22×C3⋊D3—22no—
35C22×C3⋊D3—22no—
36C22×C3⋊D3D414yes—
37C2×C32×C4—22no—
38C2×C32×C4D412yes—
39C4⋊(C3⋊D3)—28no—
40C2×C3×Dic3—21no—
41C2×C3×Dic3—21no—
42C2×C3×Dic3—22no—
43C2×C3×Dic3—22no—
44C2×C3×Dic3—22no—
45C2×C3×Dic3—21no—
46C2×C3×Dic3—22no—
47C2×C3×Dic3D412yescomplemented
48#394—91noSylow
49C23×D3—32no—
50C23×D3—32no—
51C23×D3—62no—
52C22×C3×C4—21no—
53C22×C3×C4—31no—
54C22×C3×C4—32no—
55C22×C3×C4—31no—
56C22×C3×C4C2×D311yescharacteristic
57C22×C3×C4C2×D311yescharacteristic
58C22×Dic3C2×D312yes—
59C22×Dic3C2×D311yescharacteristic
60C22×Dic3C2×D311yescharacteristic
61C2×D12—64no—
62C2×D12—64no—
63C2×D12—68no—
64C2×D12—64no—
65C2×D12—62no—
66C2×D12—62no—
67C3×C4⋊C4—62no—
68(C2×D3)⋊C4—64no—
69(C2×D3)⋊C4—64no—
70(C2×D3)⋊C4—64no—
71(C2×D3)⋊C4—64no—
72Dic3⋊C4—22no—
73C22×C32C2×D411yescharacteristic
74C22×C32C2×D411yescharacteristic
75C22×C32C2×D412yes—
76C22×C32#4111yescharacteristic
77C22×C32#4112yes—
78C2×C3⋊D3—24no—
79C2×C3⋊D3—24no—
80C2×C3⋊D3—24no—
81C2×C3⋊D3—42no—
82C2×C3⋊D3—42no—
83C2×C3⋊D3—42no—
84C2×C3⋊D3—42no—
85C32×C4—22no—
86C3×Dic3—22no—
87C3×Dic3—41no—
88C3×Dic3—42no—
89C22×D4—91no—
90C2×C4⋊C4—31no—
91C2×C22⋊C4—92no—
92C2×C22⋊C4—92no—
93#95—31no—
94C23×C3—21no—
95C23×C3C22×D311yescharacteristic
96C23×C3C22×D311yescharacteristic
97C22×D3—34no—
98C22×D3—34no—
99C22×D3—64no—
100C22×D3—64no—
101C22×D3—64no—
102C22×D3—62no—
103C22×D3—62no—
104C22×D3—62no—
105C22×D3—62no—
106C22×D3—62no—
107C22×D3—62no—
108C22×D3—62no—
109C22×D3—62no—
110C22×D3—62no—
111C22×D3—62no—
112C22×D3—62no—
113C22×D3—62no—
114C22×D3—122no—
115C22×D3—122no—
116C22×D3—122no—
117C22×D3—122no—
118C2×C3×C4—24no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—32no—
123C2×C3×C4—61no—
124C2×C3×C4—61no—
125C2×C3×C4—62no—
126C2×C3×C4—61no—
127C2×C3×C4—62no—
128C2×C3×C4—62no—
129C2×C3×C4—62no—
130C2×C3×C4D1212yes—
131C2×C3×C4#6712yes—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—21no—
139C2×Dic3D1212yescomplemented
140D12—616no—
141D12—68no—
142D12—68no—
143C2×C32#12011yescharacteristic
144C2×C32#12112yes—
145C2×C32#12111yescharacteristic
146C2×C32#12312yes—
147C2×C32#12411yescharacteristic
148C3⋊D3—44no—
149C24—92no—
150C2×D4—184no—
151C2×D4—182no—
152C22×C4—31no—
153C22×C4—32no—
154C22×C4—31no—
155C22×C4D3211yescharacteristic
156C4⋊C4—62no—
157C22⋊C4—184no—
158C22⋊C4—184no—
159C22×C3—22no—
160C22×C3—21no—
161C22×C3—21no—
162C22×C3—21no—
163C22×C3—22no—
164C22×C3C2×D1211yescharacteristic
165C22×C3C2×D1211yescharacteristic
166C22×C3#23512yes—
167C22×C3#23511yescharacteristic
168C22×C3D3×D411yescharacteristic
169C22×C3D3×D412yes—
170C22×C3D3×D412yes—
171C22×C3#23911yescharacteristic
172C22×C3#23912yes—
173C22×C3C2×#6711yescharacteristic
174C2×D3—68no—
175C2×D3—64no—
176C2×D3—64no—
177C2×D3—64no—
178C2×D3—64no—
179C2×D3—64no—
180C2×D3—64no—
181C2×D3—124no—
182C2×D3—124no—
183C2×D3—124no—
184C2×D3—124no—
185C2×D3—124no—
186C2×D3—124no—
187C2×D3—122no—
188C2×D3—122no—
189C2×D3—122no—
190C2×D3—122no—
191C2×D3—122no—
192C2×D3—122no—
193C2×D3—122no—
194C2×D3—122no—
195C3×C4—24no—
196C3×C4—22no—
197C3×C4—22no—
198C3×C4—62no—
199C3×C4—121no—
200C3×C4—122no—
201Dic3—22no—
202Dic3—42no—
203Dic3—41no—
204C32#39411yescharacteristic, complemented, Sylow
205C23—94no—
206C23—182no—
207C23—182no—
208C23—182no—
209C23—182no—
210C23—182no—
211C23—182no—
212C23C2×D3211yescharacteristic
213D4—188no—
214C2×C4—22no—
215C2×C4—32no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—61no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4#62612yes—
224C2×C3—21no—
225C2×C3—22no—
226C2×C3—21no—
227C2×C3—22no—
228C2×C3—21no—
229C2×C3#87511yescharacteristic
230C2×C3#88211yescharacteristic
231C2×C3#88412yes—
232C2×C3#89412yes—
233C2×C3#89412yes—
234C2×C3#89511yescharacteristic
235C2×C3#89512yes—
236C2×C3#93011yescharacteristic
237C2×C3#93711yescharacteristic
238C2×C3#94711yescharacteristic
239D3—128no—
240D3—124no—
241D3—124no—
242C22—184no—
243C22—184no—
244C22—184no—
245C22—362no—
246C22—362no—
247C22—362no—
248C22—362no—
249C22#379912yes—
250C22#379911yescharacteristic
251C22D3×D1211yescharacteristic
252C22C2×#62611yescharacteristic
253C22#381112yes—
254C4—22no—
255C4—62no—
256C4—122no—
257C4—121no—
258C3—21no—
259C3#467211yescharacteristic, complemented
260C3#499111yescharacteristic, complemented
261C2—364no—
262C2#6386411yescharacteristic
263C2#6389712yes—
264C2#6390911yescharacteristic
265C2#6391012yes—
266C2#6398111yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC22×C32×C4C22
prime radicalC22×C4⋊(C3⋊D3)C2
hypercenterC22×C4D32
hypocenterC32#394
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic3⋊C4C2semidirect
3. C2×#3722C2semidirect
4. C2×#3722C2semidirect
5. C3×C3⋊#95C2semidirect
26. C2×Dic3⋊C4D3semidirect
28. C3⋊#95D3semidirect
47. C2×C3×Dic3D4semidirect
139. C2×Dic3D12semidirect
204. C32#394semidirect
259. C3#4672semidirect
260. C3#4991semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊(C3⋊D3)2
C2×C3×Dic3⋊C42
C2×#37222
C3×C3⋊#952
#49913
#46723