Finite Group Databasebeta

← #10590950

Subgroups — #10590950

Order 576.

321 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 321 conjugacy classes of subgroups — too many to draw. Only its 68 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×#3797C211yesmaximal, characteristic
2C2×C3×Dic12C211yesmaximal, characteristic, complemented
3#63871C211yesmaximal, characteristic
4C2×#3718C211yesmaximal, characteristic, complemented
5#63647C211yesmaximal, characteristic
6C3×#816C211yesmaximal, characteristic, complemented
7#63565C211yesmaximal, characteristic, complemented
8#5186—31nomaximal
9#4809—31nomaximal
10C2×C4×C3⋊D3C2211yescharacteristic
11C2×C3×Dic6C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C2×C3×Dic6C2211yescharacteristic
14C2×#627—21no—
15C2×#624—21no—
16#3797—22no—
17#3797—22no—
18C2×C32×C8C2211yescharacteristic
19C2×C3⋊(C3⋊C8)C2211yescharacteristic
20C3×C4⋊Dic3C2211yescharacteristic
21C3×Dic12—22no—
22Dic3⋊Dic3—21no—
23#3722—21no—
24#3718—22no—
25C2×D3×Q8—31no—
26C2×D3×Q8—31no—
27C2×C3×Q16—31no—
28#946—31no—
29C2×#216—31no—
30C2×#216—31no—
31C2×Dic12D311yescharacteristic, complemented
32#897—31no—
33C3×#103—31no—
34#835—31no—
35(C2×D3)⋊C8—31no—
36(C2×D3)⋊C8—31no—
37(C2×D3)⋊C8—61no—
38#816D311yescharacteristic, complemented
39#810—31no—
40C22×C3⋊D3D411yescharacteristic
41C2×C32×C4C2311yescharacteristic
42C2×C3⋊Dic3D411yescharacteristic
43C4×C3⋊D3—21no—
44C4×C3⋊D3—21no—
45C2×C3×Dic3—21no—
46C2×C3×Dic3—21no—
47C2×C3×Dic3—21no—
48C2×C3×Dic3—21no—
49C3×Dic6—21no—
50C3×Dic6—21no—
51C3×Dic6—21no—
52C3×Dic6—21no—
53C3×Dic6D412yes—
54C3×Dic6D412yes—
55#627—24no—
56#624—24no—
57C32×C8—21no—
58C3⋊(C3⋊C8)—21no—
59#451—91noSylow
60C2×C3×Q8—31no—
61C2×C3×Q8—31no—
62C2×C3×Q8—31no—
63D3×Q8—62no—
64D3×Q8—62no—
65D3×Q8—62no—
66D3×Q8—62no—
67C2×C4×D3—31no—
68C2×C4×D3—31no—
69C2×C4×D3—61no—
70C2×C4×D3—61no—
71C2×C4×D3—61no—
72C2×Dic6—61no—
73C2×Dic6—61no—
74C2×Dic6C2×D311yescharacteristic
75C2×Dic6C2×D311yescharacteristic
76C2×Dic6C2×D311yescharacteristic
77C3×Q16—62no—
78C2×C3×C8—21no—
79C2×C3×C8C2×D311yescharacteristic
80C2×C3×C8C2×D311yescharacteristic
81C3×C4⋊C4—31no—
82#216—62no—
83#216—62no—
84(C2×D3)⋊C4—61no—
85(C2×D3)⋊C4—61no—
86C4⋊Dic3—61no—
87C4⋊Dic3C2×D311yescharacteristic
88Dic3⋊C4—61no—
89C2×C3⋊C8—31no—
90C2×C3⋊C8—31no—
91C2×C3⋊C8—61no—
92Dic12—22no—
93C22×C32C2×D411yescharacteristic
94C2×C3⋊D3—21no—
95C2×C3⋊D3—21no—
96C2×C3⋊D3Q1612yescomplemented
97C32×C4C2×D411yescharacteristic
98C32×C4C2×D411yescharacteristic
99C3⋊Dic3—21no—
100C3×Dic3—22no—
101C3×Dic3—22no—
102C3×Dic3—41no—
103C3×Dic3—41no—
104C22×Q8—91no—
105C2×Q16—31no—
106C2×Q16—91no—
107C22⋊Q8—91no—
108#103—31no—
109#103—91no—
110C22⋊C8—91no—
111C22×D3—31no—
112C22×D3—31no—
113C22×D3—61no—
114C3×Q8—32no—
115C3×Q8—32no—
116C3×Q8—61no—
117C3×Q8—61no—
118C3×Q8—61no—
119C3×Q8—61no—
120C2×C3×C4—21no—
121C2×C3×C4—61no—
122C2×C3×C4—61no—
123C2×C3×C4—61no—
124C2×C3×C4—61no—
125C2×C3×C4C22×D311yescharacteristic
126C2×C3×C4C22×D311yescharacteristic
127C2×Dic3—21no—
128C2×Dic3—21no—
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3—31no—
132C2×Dic3—31no—
133C2×Dic3—61no—
134C4×D3—64no—
135C4×D3—64no—
136C4×D3—61no—
137C4×D3—61no—
138C4×D3—61no—
139C4×D3—61no—
140C4×D3—121no—
141C4×D3—121no—
142Dic6—21no—
143Dic6—21no—
144Dic6—21no—
145Dic6—21no—
146Dic6—64no—
147Dic6—64no—
148Dic6D1212yes—
149Dic6#6712yes—
150C3×C8—21no—
151C3×C8—21no—
152C3×C8—41no—
153C3⋊C8—61no—
154C3⋊C8—61no—
155C3⋊C8—121no—
156C2×C32#12011yescharacteristic
157C2×C32C2×Q1611yescharacteristic
158C2×C32#13711yescharacteristic
159C3⋊D3—22no—
160C2×Q8—31no—
161C2×Q8—31no—
162C2×Q8—31no—
163C2×Q8—181no—
164C2×Q8—182no—
165C2×Q8—182no—
166C22×C4—91no—
167C22×C4—181no—
168Q16—62no—
169Q16—182no—
170C2×C8—91no—
171C2×C8D3211yescharacteristic
172C4⋊C4—31no—
173C4⋊C4—181no—
174C22⋊C4—181no—
175C22×C3—21no—
176C22×C3D3×D411yescharacteristic
177C22×C3D3×D411yescharacteristic
178C2×D3—32no—
179C2×D3—32no—
180C2×D3—62no—
181C2×D3—61no—
182C2×D3—61no—
183C2×D3—61no—
184C2×D3—61no—
185C2×D3—121no—
186C2×D3—121no—
187C3×C4—21no—
188C3×C4—21no—
189C3×C4—62no—
190C3×C4—62no—
191C3×C4—121no—
192C3×C4—121no—
193C3×C4C2×D1211yescharacteristic
194C3×C4D3×D411yescharacteristic
195C3×C4D3×D411yescharacteristic
196C3×C4C2×#6711yescharacteristic
197Dic3—22no—
198Dic3—22no—
199Dic3—41no—
200Dic3—41no—
201Dic3—61no—
202Dic3—61no—
203Dic3—121no—
204C32#45111yescharacteristic, complemented, Sylow
205C23—91no—
206Q8—32no—
207Q8—32no—
208Q8—61no—
209Q8—61no—
210Q8—61no—
211Q8—61no—
212Q8—184no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—91no—
218C2×C4—184no—
219C2×C4—181no—
220C2×C4—181no—
221C2×C4C2×D3211yescharacteristic
222C8—21no—
223C8—181no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—21no—
227C2×C3#88211yescharacteristic
228C2×C3#91511yescharacteristic
229C2×C3D3×Q1611yescharacteristic
230C2×C3D3×Q1611yescharacteristic
231C2×C3#91811yescharacteristic
232C2×C3#93711yescharacteristic
233D3—62no—
234D3—62no—
235D3—122no—
236C22—92no—
237C22—181no—
238C22—181no—
239C22#380211yescharacteristic
240C4—62no—
241C4—62no—
242C4—121no—
243C4—121no—
244C4—181no—
245C4D3×D1211yescharacteristic
246C4D3×#6711yescharacteristic
247C3—21no—
248C3#480911yescharacteristic, complemented
249C3#518611yescharacteristic, complemented
250C2—182no—
251C2#6380211yescharacteristic
252C2#6381111yescharacteristic
253C2#6392211yescharacteristic
subgroupG/H
centerC22#3802
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×C8C22
socleC22×C32C2×D4
prime radicalC22×C3⋊D3D4
hypercenterC2×C8D32
hypocenterC32#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic12C2semidirect
4. C2×#3718C2semidirect
6. C3×#816C2semidirect
7. #63565C2semidirect
31. C2×Dic12D3semidirect
38. #816D3semidirect
96. C2×C3⋊D3Q16semidirect
204. C32#451semidirect
248. C3#4809semidirect
249. C3#5186semidirect

Maximal subgroups

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

subgroupindex
C2×#37972
C2×C3×Dic122
#638712
C2×#37182
#636472
C3×#8162
#635652
#51863
#48093