Finite Group Databasebeta

← #10590517

Subgroups — #10590517

Order 576.

226 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 64 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×D3×C4⋊C4C211yesmaximal, characteristic
2#63856C211yesmaximal, characteristic
3#63634C211yesmaximal, characteristic, complemented
4C3×#810C211yesmaximal, characteristic, complemented
5#63575C211yesmaximal, characteristic, complemented
6#63565C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8#4970—31nomaximal
9#4811—31nomaximal
10C2×C3×C4×D3—21no—
11C2×C3×C4×D3C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C32×C4⋊C4C2211yescharacteristic
14C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊C8)C2211yescharacteristic
16C3×C4⋊Dic3C2211yescharacteristic
17C3×Dic3⋊C4—21no—
18C2×C3×C3⋊C8C2211yescharacteristic
19#3724—21no—
20(C2×D3)⋊Dic3—21no—
21C2×C3×C4⋊C4—31no—
22#946—31no—
23#924—31no—
24D3×C4⋊C4D311yescharacteristic
25C3×#103—31no—
26#835—31no—
27#830—31no—
28(C2×D3)⋊C8—31no—
29#818—31no—
30#810—31no—
31#810D311yescharacteristic, complemented
32#807—31no—
33#807—31no—
34#807—31no—
35#807—61no—
36C22×C3×D3D411yescharacteristic
37C2×C32×C4—21no—
38C2×C32×C4C2311yescharacteristic
39C2×C3⋊Dic3—21no—
40C2×C3×Dic3—21no—
41C2×C3×Dic3—21no—
42C2×C3×Dic3D411yescharacteristic
43C3×C4×D3—21no—
44C3×C4×D3—21no—
45C3×C4×D3—42no—
46C3×Dic6—21no—
47C3×Dic6—21no—
48C3⋊(C3⋊C8)—21no—
49C3×C3⋊C8—21no—
50#484—91noSylow
51C2×C3×Q8—31no—
52C22×C3×C4—31no—
53C22×C3×C4—61no—
54C2×C4×D3—21no—
55C2×C4×D3C2×D311yescharacteristic
56C2×Dic6C2×D311yescharacteristic
57C2×C3×C8—31no—
58C3×C4⋊C4—21no—
59C3×C4⋊C4—31no—
60C3×C4⋊C4—61no—
61C3×C4⋊C4C2×D311yescharacteristic
62C3×C4⋊C4C2×D311yescharacteristic
63#217—61no—
64(C2×D3)⋊C4—61no—
65C4⋊Dic3—31no—
66C4⋊Dic3—31no—
67C4⋊Dic3—61no—
68C4⋊Dic3C2×D311yescharacteristic
69Dic3⋊C4—21no—
70Dic3⋊C4—61no—
71Dic3⋊C4—61no—
72C2×C3⋊C8—31no—
73C2×C3⋊C8—31no—
74C2×C3⋊C8—61no—
75C2×C3⋊C8C2×D311yescharacteristic
76C22×C32C2×D411yescharacteristic
77C2×C3×D3—21no—
78C2×C3×D3—21no—
79C2×C3×D3Q1612yes—
80C32×C4—41no—
81C32×C4#4111yescharacteristic
82C32×C4#4111yescharacteristic
83C3⋊Dic3—41no—
84C3×Dic3—21no—
85C3×Dic3—41no—
86C3×Dic3—41no—
87C22⋊Q8—91no—
88C2×C4⋊C4—31no—
89#107—91no—
90#107—91no—
91#103—31no—
92#103—91no—
93C22⋊C8—91no—
94C23×C3—31no—
95C22×D3#6711yescharacteristic
96C3×Q8—61no—
97C3×Q8—61no—
98C2×C3×C4—21no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—31no—
102C2×C3×C4—41no—
103C2×C3×C4—61no—
104C2×C3×C4—61no—
105C2×C3×C4—61no—
106C2×C3×C4—61no—
107C2×C3×C4—122no—
108C2×C3×C4C22×D311yescharacteristic
109C2×C3×C4C22×D311yescharacteristic
110C2×Dic3—21no—
111C2×Dic3—21no—
112C2×Dic3—61no—
113C2×Dic3—61no—
114C2×Dic3—121no—
115C2×Dic3#6711yescharacteristic
116C4×D3—21no—
117C4×D3—21no—
118C4×D3—42no—
119Dic6—21no—
120Dic6—21no—
121C3×C8—61no—
122C3⋊C8—21no—
123C3⋊C8—62no—
124C3⋊C8—61no—
125C3⋊C8—61no—
126C2×C32#12311yescharacteristic
127C2×C32C2×Q1611yescharacteristic
128C2×C32AGL(1,ℤ/8)11yescharacteristic
129C3×D3—22no—
130C2×Q8—31no—
131C22×C4—31no—
132C22×C4—61no—
133C2×C8—31no—
134C2×C8—91no—
135C4⋊C4—31no—
136C4⋊C4—61no—
137C4⋊C4—91no—
138C4⋊C4—181no—
139C4⋊C4D3211yescharacteristic
140C22⋊C4—181no—
141C22×C3—21no—
142C22×C3—32no—
143C22×C3—61no—
144C22×C3—61no—
145C22×C3D3×D411yescharacteristic
146C22×C3C2×#6711yescharacteristic
147C2×D3—21no—
148C2×D3—21no—
149C2×D3#21612yes—
150C3×C4—21no—
151C3×C4—21no—
152C3×C4—41no—
153C3×C4—41no—
154C3×C4—61no—
155C3×C4—81no—
156C3×C4—121no—
157C3×C4—121no—
158C3×C4#23511yescharacteristic
159C3×C4#23511yescharacteristic
160C3×C4#23511yescharacteristic
161C3×C4#23711yescharacteristic
162Dic3—21no—
163Dic3—41no—
164Dic3—41no—
165Dic3—121no—
166Dic3—121no—
167Dic3—241no—
168C32#48411yescharacteristic, complemented, Sylow
169C23—31no—
170Q8—61no—
171Q8—61no—
172C2×C4—21no—
173C2×C4—31no—
174C2×C4—61no—
175C2×C4—61no—
176C2×C4—61no—
177C2×C4—61no—
178C2×C4—122no—
179C2×C4—181no—
180C2×C4C2×D3211yescharacteristic
181C8—61no—
182C8—181no—
183C2×C3—21no—
184C2×C3—21no—
185C2×C3—21no—
186C2×C3—62no—
187C2×C3#88311yescharacteristic
188C2×C3#91411yescharacteristic
189C2×C3D3×Q1611yescharacteristic
190C2×C3#92911yescharacteristic
191C2×C3#93211yescharacteristic
192C2×C3C2×#21611yescharacteristic
193D3—22no—
194C22—32no—
195C22—61no—
196C22—61no—
197C22D3×#6711yescharacteristic
198C4—41no—
199C4—61no—
200C4—121no—
201C4—121no—
202C4—361no—
203C4#379511yescharacteristic
204C4#379811yescharacteristic
205C3—21no—
206C3#481111yescharacteristic, complemented
207C3#497011yescharacteristic, complemented
208C2—62no—
209C2#6390511yescharacteristic
210C2#6393411yescharacteristic
211C2D3×#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#484
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. #63634C2semidirect
4. C3×#810C2semidirect
5. #63575C2semidirect
6. #63565C2semidirect
31. #810D3semidirect
168. C32#484semidirect
206. C3#4811semidirect
207. C3#4970semidirect

Maximal subgroups

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

subgroupindex
C3×D3×C4⋊C42
#638562
#636342
C3×#8102
#635752
#635652
(C2×D3)⋊(C3⋊C8)2
#49703
#48113