Finite Group Databasebeta

← #10591326

Subgroups — #10591326

Order 576.

406 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C2×#3752C211yesmaximal, characteristic
2C2×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#3722C212yesmaximal
4C3×C3⋊#95C211yesmaximal, characteristic, complemented
5#63579C212yesmaximal, complemented
6#4992—31nomaximal
7#4674—31nomaximal
8C23×C3⋊D3C2211yescharacteristic
9C22×C32×C4C2211yescharacteristic
10C22×C3⋊Dic3C2211yescharacteristic
11C22×C3×Dic3C2212yes—
12C22×C3×Dic3C2212yes—
13#3752—22no—
14C3×C4⋊Dic3—22no—
15#3722—24no—
16C2×C3×C4⋊C4—31no—
17C2×(C2×D3)⋊C4—31no—
18C2×(C2×D3)⋊C4—31no—
19C2×(C2×D3)⋊C4—32no—
20C2×(C2×D3)⋊C4—32no—
21C2×(C2×D3)⋊C4—61no—
22C2×C4⋊Dic3D311yescharacteristic, complemented
23C3×#95—31no—
24C3⋊#95—32no—
25C3⋊#95—32no—
26C3⋊#95D311yescharacteristic, complemented
27C23×C32C2311yescharacteristic
28C22×C3⋊D3—21no—
29C22×C3⋊D3—21no—
30C22×C3⋊D3—22no—
31C22×C3⋊D3—22no—
32C22×C3⋊D3D412yes—
33C2×C32×C4—22no—
34C2×C32×C4D412yes—
35C2×C3⋊Dic3—21no—
36C2×C3⋊Dic3—22no—
37C2×C3×Dic3—22no—
38C2×C3×Dic3—22no—
39C2×C3×Dic3—22no—
40C2×C3×Dic3—22no—
41C2×C3×Dic3—22no—
42C2×C3×Dic3—22no—
43#397—91noSylow
44C23×D3—31no—
45C23×D3—31no—
46C23×D3—61no—
47C22×C3×C4—21no—
48C22×C3×C4—32no—
49C22×C3×C4—32no—
50C22×C3×C4C2×D311yescharacteristic
51C22×C3×C4C2×D311yescharacteristic
52C22×Dic3—31no—
53C22×Dic3—31no—
54C22×Dic3—61no—
55C22×Dic3C2×D312yes—
56C22×Dic3C2×D312yes—
57C3×C4⋊C4—62no—
58(C2×D3)⋊C4—64no—
59(C2×D3)⋊C4—64no—
60(C2×D3)⋊C4—64no—
61(C2×D3)⋊C4—62no—
62(C2×D3)⋊C4—62no—
63C4⋊Dic3—22no—
64C22×C32C2×D411yescharacteristic
65C22×C32C2×D411yescharacteristic
66C22×C32#4112yes—
67C22×C32#4112yes—
68C22×C32#4111yescharacteristic
69C2×C3⋊D3—22no—
70C2×C3⋊D3—24no—
71C2×C3⋊D3—41no—
72C2×C3⋊D3—41no—
73C2×C3⋊D3—42no—
74C32×C4—22no—
75C3⋊Dic3—41no—
76C3×Dic3—42no—
77C3×Dic3—42no—
78C2×C4⋊C4—31no—
79C2×C22⋊C4—92no—
80C2×C22⋊C4—91no—
81#95—31no—
82#95—92no—
83C23×C3—21no—
84C23×C3C22×D311yescharacteristic
85C23×C3C22×D311yescharacteristic
86C22×D3—32no—
87C22×D3—32no—
88C22×D3—62no—
89C22×D3—62no—
90C22×D3—62no—
91C22×D3—61no—
92C22×D3—62no—
93C22×D3—62no—
94C22×D3—61no—
95C22×D3—62no—
96C22×D3—61no—
97C22×D3—62no—
98C22×D3—61no—
99C22×D3—122no—
100C22×D3—122no—
101C2×C3×C4—22no—
102C2×C3×C4—22no—
103C2×C3×C4—22no—
104C2×C3×C4—42no—
105C2×C3×C4—62no—
106C2×C3×C4—62no—
107C2×C3×C4—62no—
108C2×C3×C4—62no—
109C2×C3×C4—62no—
110C2×C3×C4—62no—
111C2×C3×C4D1212yes—
112C2×C3×C4#6712yes—
113C2×Dic3—22no—
114C2×Dic3—22no—
115C2×Dic3—22no—
116C2×Dic3—22no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—62no—
120C2×Dic3—61no—
121C2×Dic3—62no—
122C2×Dic3—62no—
123C2×Dic3—61no—
124C2×Dic3—122no—
125C2×C32#12111yescharacteristic
126C2×C32#12312yes—
127C2×C32#12311yescharacteristic
128C2×C32#12411yescharacteristic
129C2×C32#12612yes—
130C3⋊D3—42no—
131C24—91no—
132C22×C4—32no—
133C22×C4—32no—
134C22×C4—91no—
135C22×C4D3211yescharacteristic
136C4⋊C4—62no—
137C22⋊C4—184no—
138C22⋊C4—182no—
139C22×C3—21no—
140C22×C3—22no—
141C22×C3—21no—
142C22×C3—22no—
143C22×C3—21no—
144C22×C3C2×D1211yescharacteristic
145C22×C3#23512yes—
146C22×C3#23512yes—
147C22×C3D3×D411yescharacteristic
148C22×C3D3×D411yescharacteristic
149C22×C3#23711yescharacteristic
150C22×C3#23712yes—
151C22×C3#23911yescharacteristic
152C22×C3#23912yes—
153C22×C3C2×#6711yescharacteristic
154C2×D3—64no—
155C2×D3—64no—
156C2×D3—64no—
157C2×D3—62no—
158C2×D3—62no—
159C2×D3—124no—
160C2×D3—124no—
161C2×D3—122no—
162C2×D3—122no—
163C2×D3—121no—
164C2×D3—121no—
165C2×D3—122no—
166C2×D3—122no—
167C2×D3—121no—
168C2×D3—121no—
169C3×C4—22no—
170C3×C4—22no—
171C3×C4—42no—
172C3×C4—122no—
173C3×C4—122no—
174Dic3—42no—
175Dic3—42no—
176Dic3—122no—
177Dic3—121no—
178Dic3—121no—
179C32#39711yescharacteristic, complemented, Sylow
180C23—92no—
181C23—181no—
182C23—182no—
183C23—182no—
184C23—181no—
185C23C2×D3211yescharacteristic
186C2×C4—22no—
187C2×C4—62no—
188C2×C4—62no—
189C2×C4—62no—
190C2×C4—62no—
191C2×C4—62no—
192C2×C4—62no—
193C2×C4—182no—
194C2×C4—181no—
195C2×C4#62612yes—
196C2×C3—22no—
197C2×C3—22no—
198C2×C3—21no—
199C2×C3—21no—
200C2×C3—21no—
201C2×C3#87511yescharacteristic
202C2×C3#88312yes—
203C2×C3#88611yescharacteristic
204C2×C3#89412yes—
205C2×C3#89511yescharacteristic
206C2×C3#89812yes—
207C2×C3#89812yes—
208C2×C3#92911yescharacteristic
209C2×C3#93811yescharacteristic
210C2×C3#94711yescharacteristic
211D3—124no—
212D3—122no—
213D3—122no—
214C22—184no—
215C22—182no—
216C22—362no—
217C22—361no—
218C22—361no—
219C22#379611yescharacteristic
220C22#379912yes—
221C22#380211yescharacteristic
222C22#380412yes—
223C22C2×#62611yescharacteristic
224C4—22no—
225C4—122no—
226C4—122no—
227C4—361no—
228C3—21no—
229C3#467411yescharacteristic, complemented
230C3#499211yescharacteristic, complemented
231C2—362no—
232C2#6384812yes—
233C2#6385412yes—
234C2#6386411yescharacteristic
235C2#6391611yescharacteristic
236C2#6396611yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC22×C32×C4C22
prime radicalC23×C3⋊D3C22
hypercenterC22×C4D32
hypocenterC32#397
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×C4⋊Dic3C2semidirect
4. C3×C3⋊#95C2semidirect
5. #63579C2semidirect
22. C2×C4⋊Dic3D3semidirect
26. C3⋊#95D3semidirect
179. C32#397semidirect
229. C3#4674semidirect
230. C3#4992semidirect

Maximal subgroups

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

subgroupindex
C2×#37522
C2×C3×C4⋊Dic32
C2×#37222
C3×C3⋊#952
#635792
#49923
#46743