Finite Group Databasebeta

← #10590518

Subgroups — #10590518

Order 576.

246 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 66 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
1C3×#897C211yesmaximal, characteristic
2#63856C211yesmaximal, characteristic
3C2×#3718C211yesmaximal, characteristic, complemented
4C2×D3×C3⋊C8C211yesmaximal, characteristic
5#63634C211yesmaximal, characteristic, complemented
6C3×#810C211yesmaximal, characteristic, complemented
7#63566C211yesmaximal, characteristic, complemented
8#5051—31nomaximal
9#4813—31nomaximal
10C2×C3×C4×D3C2211yescharacteristic
11C2×C3×Dic6C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C32×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×(C2×D3)⋊C4—21no—
17C3×C4⋊Dic3—21no—
18C2×C3×C3⋊C8C2211yescharacteristic
19#3724—21no—
20(C2×D3)⋊Dic3—21no—
21#3718—22no—
22D3×C3⋊C8—22no—
23C3×C22⋊Q8—31no—
24#946—31no—
25C2×#216—31no—
26C2×#216—31no—
27#924—31no—
28C22×C3⋊C8—31no—
29C2×D3×C8—31no—
30#897D311yescharacteristic
31C3×#103—31no—
32#835—31no—
33#816—31no—
34#810D311yescharacteristic, complemented
35#807—31no—
36#807—31no—
37#807—61no—
38C22×C3×D3D411yescharacteristic
39C2×C32×C4—21no—
40C2×C32×C4C2311yescharacteristic
41C2×C3⋊Dic3—21no—
42C2×C3×Dic3—21no—
43C2×C3×Dic3—21no—
44C2×C3×Dic3D411yescharacteristic
45C3×C4×D3—21no—
46C3×C4×D3—21no—
47C3×Dic6—21no—
48C3×Dic6—21no—
49C3×Dic6—21no—
50C3×Dic6—21no—
51C3⋊(C3⋊C8)D412yes—
52C3×C3⋊C8—21no—
53#467—91noSylow
54C2×C3×Q8—31no—
55C2×C3×Q8—31no—
56C22×C3×C4—31no—
57C2×C4×D3C2×D311yescharacteristic
58C2×Dic6C2×D311yescharacteristic
59C2×Dic6C2×D311yescharacteristic
60C2×C3×C8—31no—
61C3×C4⋊C4—21no—
62C3×C4⋊C4—61no—
63C3×C4⋊C4C2×D311yescharacteristic
64C3×C4⋊C4C2×D311yescharacteristic
65C3×C22⋊C4—61no—
66#217—61no—
67#216—62no—
68#216—62no—
69(C2×D3)⋊C4—21no—
70(C2×D3)⋊C4—61no—
71C4⋊Dic3—21no—
72C4⋊Dic3—31no—
73C4⋊Dic3—31no—
74C4⋊Dic3—61no—
75Dic3⋊C4—61no—
76Dic3⋊C4—61no—
77C2×C3⋊C8—31no—
78C2×C3⋊C8—31no—
79C2×C3⋊C8—61no—
80C2×C3⋊C8—62no—
81C2×C3⋊C8C2×D311yescharacteristic
82D3×C8—62no—
83C22×C32C2×D411yescharacteristic
84C2×C3×D3—21no—
85C2×C3×D3—21no—
86C2×C3×D3Q1612yescomplemented
87C32×C4—41no—
88C32×C4C2×D411yescharacteristic
89C32×C4#4111yescharacteristic
90C3⋊Dic3—41no—
91C3×Dic3—21no—
92C3×Dic3—41no—
93C3×Dic3—41no—
94C2×Q16—91no—
95C22×C8—91no—
96C22⋊Q8—31no—
97C22⋊Q8—91no—
98#107—91no—
99#103—31no—
100#103—91no—
101C23×C3—31no—
102C22×D3#6711yescharacteristic
103C3×Q8—61no—
104C3×Q8—61no—
105C3×Q8—61no—
106C3×Q8—61no—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—31no—
111C2×C3×C4—41no—
112C2×C3×C4—61no—
113C2×C3×C4—61no—
114C2×C3×C4—61no—
115C2×C3×C4—61no—
116C2×C3×C4C22×D311yescharacteristic
117C2×C3×C4C22×D311yescharacteristic
118C2×Dic3—21no—
119C2×Dic3—21no—
120C2×Dic3—61no—
121C2×Dic3—61no—
122C2×Dic3—121no—
123C2×Dic3#6711yescharacteristic
124C4×D3—21no—
125C4×D3—21no—
126Dic6—21no—
127Dic6—21no—
128Dic6—21no—
129Dic6—21no—
130C3×C8—61no—
131C3⋊C8—21no—
132C3⋊C8—32no—
133C3⋊C8—32no—
134C3⋊C8—62no—
135C2×C32#12111yescharacteristic
136C2×C32C2×Q1611yescharacteristic
137C2×C32#13511yescharacteristic
138C3×D3—22no—
139C2×Q8—31no—
140C2×Q8—31no—
141C22×C4—31no—
142Q16—182no—
143C2×C8—31no—
144C2×C8—91no—
145C2×C8—182no—
146C4⋊C4—61no—
147C4⋊C4—91no—
148C4⋊C4—181no—
149C4⋊C4D3211yescharacteristic
150C22⋊C4—61no—
151C22⋊C4—181no—
152C22×C3—21no—
153C22×C3—32no—
154C22×C3—61no—
155C22×C3—61no—
156C22×C3D3×D411yescharacteristic
157C22×C3C2×#6711yescharacteristic
158C2×D3—21no—
159C2×D3—21no—
160C2×D3#21612yescomplemented
161C3×C4—21no—
162C3×C4—21no—
163C3×C4—41no—
164C3×C4—41no—
165C3×C4—61no—
166C3×C4—81no—
167C3×C4—121no—
168C3×C4—121no—
169C3×C4#23511yescharacteristic
170C3×C4D3×D411yescharacteristic
171C3×C4D3×D411yescharacteristic
172C3×C4#23711yescharacteristic
173Dic3—21no—
174Dic3—41no—
175Dic3—41no—
176Dic3—121no—
177Dic3—121no—
178Dic3—241no—
179C32#46711yescharacteristic, complemented, Sylow
180C23—31no—
181Q8—61no—
182Q8—61no—
183Q8—61no—
184Q8—61no—
185C2×C4—21no—
186C2×C4—31no—
187C2×C4—61no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—181no—
192C2×C4C2×D3211yescharacteristic
193C8—61no—
194C8—92no—
195C2×C3—21no—
196C2×C3—21no—
197C2×C3—21no—
198C2×C3—62no—
199C2×C3#88411yescharacteristic
200C2×C3#91611yescharacteristic
201C2×C3D3×Q1611yescharacteristic
202C2×C3#93911yescharacteristic
203C2×C3C2×#21611yescharacteristic
204C2×C3#95011yescharacteristic
205D3—22no—
206C22—32no—
207C22—61no—
208C22—61no—
209C22D3×#6711yescharacteristic
210C4—41no—
211C4—61no—
212C4—121no—
213C4—121no—
214C4—361no—
215C4#379511yescharacteristic
216C4#380211yescharacteristic
217C3—21no—
218C3#481311yescharacteristic, complemented
219C3#505111yescharacteristic, complemented
220C2—62no—
221C2#6388511yescharacteristic
222C2#6393511yescharacteristic
223C2D3×#21611yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radicalC22×C3×D3D4
hypercenterC4⋊C4D32
hypocenterC32#467
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#3718C2semidirect
5. #63634C2semidirect
6. C3×#810C2semidirect
7. #63566C2semidirect
34. #810D3semidirect
86. C2×C3×D3Q16semidirect
160. C2×D3#216semidirect
179. C32#467semidirect
218. C3#4813semidirect
219. C3#5051semidirect

Maximal subgroups

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

subgroupindex
C3×#8972
#638562
C2×#37182
C2×D3×C3⋊C82
#636342
C3×#8102
#635662
#50513
#48133