Finite Group Databasebeta

← #10590699

Subgroups — #10590699

Order 576.

212 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 70 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
1#63855C211yesmaximal, characteristic
2#63642C211yesmaximal, characteristic, complemented
3C3×(C2×D3)⋊C8C211yesmaximal, characteristic
4C3×Dic3⋊C8C211yesmaximal, characteristic, complemented
5Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
7#63553C211yesmaximal, characteristic, complemented
8#4840—31nomaximal
9#4720—31nomaximal
10C2×C3×C4×D3C2211yescharacteristic
11C2×C32×C8C2211yescharacteristic
12C4×C3⋊Dic3C2211yescharacteristic
13C2×C3⋊(C3⋊C8)C2211yescharacteristic
14C3×C4×Dic3C2211yescharacteristic
15C2×C3×C3⋊C8C2211yescharacteristic
16C2×C3×C3⋊C8C2211yescharacteristic
17Dic3⋊Dic3C411yescharacteristic
18Dic3⋊Dic3C411yescharacteristic
19(C2×D3)⋊Dic3C411yescharacteristic
20(C2×D3)⋊Dic3C411yescharacteristic
21#926—31no—
22#872—31no—
23C3×C4⋊C8—31no—
24C3×C22⋊C8—31no—
25#830—31no—
26(C2×D3)⋊C8—31no—
27(C2×D3)⋊C8D311yescharacteristic
28#815—31no—
29#815—31no—
30#815—61no—
31Dic3⋊C8—31no—
32Dic3⋊C8D311yescharacteristic, complemented
33C8×Dic3—31no—
34C8×Dic3—31no—
35C4⋊(C3⋊C8)—31no—
36C22×C3×D3C2×C411yescharacteristic
37C2×C32×C4C2311yescharacteristic
38C2×C3⋊Dic3C2×C411yescharacteristic
39C2×C3⋊Dic3C2×C411yescharacteristic
40C2×C3×Dic3C2×C411yescharacteristic
41C2×C3×Dic3C2×C411yescharacteristic
42C2×C3×Dic3C2×C411yescharacteristic
43C3×C4×D3—21no—
44C3×C4×D3—21no—
45C32×C8—21no—
46C3⋊(C3⋊C8)—21no—
47C3×C3⋊C8—21no—
48C3×C3⋊C8—21no—
49#433—91noSylow
50C22×C3×C4—31no—
51C2×C4×D3C2×D311yescharacteristic
52C2×C3×C8—21no—
53C2×C3×C8—31no—
54C2×C3×C8—31no—
55C2×C3×C8C2×D311yescharacteristic
56C2×C3×C8C2×D311yescharacteristic
57C3×C42—31no—
58#217—31no—
59#217—31no—
60(C2×D3)⋊C4—31no—
61(C2×D3)⋊C4—31no—
62C4⋊Dic3—31no—
63C4⋊Dic3—31no—
64Dic3⋊C4—31no—
65Dic3⋊C4—31no—
66C4×Dic3—31no—
67C4×Dic3—31no—
68C4×Dic3—61no—
69C4×Dic3C2×D311yescharacteristic
70C2×C3⋊C8—31no—
71C2×C3⋊C8—31no—
72C2×C3⋊C8—61no—
73C2×C3⋊C8C2×D311yescharacteristic
74C2×C3⋊C8C2×D311yescharacteristic
75C22×C32C22×C411yescharacteristic
76C2×C3×D3—21no—
77C2×C3×D3—21no—
78C2×C3×D3—21no—
79C32×C4#4111yescharacteristic
80C32×C4#4111yescharacteristic
81C3⋊Dic3—21no—
82C3⋊Dic3—21no—
83C3×Dic3—21no—
84C3×Dic3—21no—
85C3×Dic3—21no—
86#117—91no—
87C4⋊C8—31no—
88C4⋊C8—91no—
89C22⋊C8—31no—
90C22⋊C8—91no—
91#97—91no—
92C4×C8—91no—
93C23×C3—31no—
94C22×D3C4×D311yescharacteristic
95C2×C3×C4—21no—
96C2×C3×C4—31no—
97C2×C3×C4—31no—
98C2×C3×C4—31no—
99C2×C3×C4—61no—
100C2×C3×C4—61no—
101C2×C3×C4C22×D311yescharacteristic
102C2×C3×C4C22×D311yescharacteristic
103C2×Dic3—31no—
104C2×Dic3—31no—
105C2×Dic3—31no—
106C2×Dic3—31no—
107C2×Dic3—61no—
108C2×Dic3—61no—
109C2×Dic3C4×D311yescharacteristic
110C2×Dic3C4×D311yescharacteristic
111C2×Dic3C4×D311yescharacteristic
112C4×D3—21no—
113C4×D3—21no—
114C3×C8—22no—
115C3×C8—21no—
116C3×C8—21no—
117C3×C8—61no—
118C3×C8—61no—
119C3⋊C8—21no—
120C3⋊C8—21no—
121C3⋊C8—62no—
122C3⋊C8—61no—
123C3⋊C8—61no—
124C2×C32#11711yescharacteristic
125C2×C32#13111yescharacteristic
126C2×C32#13111yescharacteristic
127C3×D3—41no—
128C22×C4—31no—
129C2×C8—31no—
130C2×C8—31no—
131C2×C8—91no—
132C2×C8D3211yescharacteristic
133C4⋊C4—91no—
134C4⋊C4—91no—
135C22⋊C4—91no—
136C22⋊C4—91no—
137C42—31no—
138C42—91no—
139C22×C3—21no—
140C22×C3—61no—
141C22×C3—61no—
142C22×C3—61no—
143C22×C3C2×C4×D311yescharacteristic
144C22×C3C2×C4×D311yescharacteristic
145C2×D3—21no—
146C2×D3—21no—
147C2×D3—21no—
148C3×C4—21no—
149C3×C4—21no—
150C3×C4—61no—
151C3×C4—61no—
152C3×C4—61no—
153C3×C4#23711yescharacteristic
154C3×C4#23711yescharacteristic
155C3×C4#23711yescharacteristic
156C3×C4#23911yescharacteristic
157Dic3—21no—
158Dic3—21no—
159Dic3—21no—
160Dic3—61no—
161Dic3—61no—
162Dic3—61no—
163Dic3—61no—
164Dic3—121no—
165Dic3—121no—
166C32#43311yescharacteristic, complemented, Sylow
167C23—31no—
168C2×C4—31no—
169C2×C4—31no—
170C2×C4—31no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—91no—
174C2×C4—91no—
175C2×C4C2×D3211yescharacteristic
176C8—21no—
177C8—61no—
178C8—61no—
179C8—181no—
180C2×C3—21no—
181C2×C3—21no—
182C2×C3—21no—
183C2×C3—121no—
184C2×C3#87711yescharacteristic
185C2×C3#89211yescharacteristic
186C2×C3#90111yescharacteristic
187C2×C3#90111yescharacteristic
188C2×C3#90711yescharacteristic
189C2×C3#90711yescharacteristic
190D3—41no—
191C22—61no—
192C22—61no—
193C22—61no—
194C22C4×D3211yescharacteristic
195C4—61no—
196C4—61no—
197C4—61no—
198C4—181no—
199C4—181no—
200C4#379611yescharacteristic
201C4#380511yescharacteristic
202C3—21no—
203C3#472011yescharacteristic, complemented
204C3#484011yescharacteristic, complemented
205C2—121no—
206C2#6380411yescharacteristic
207C2#6380411yescharacteristic
208C2#6387811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D32
commutator, socleC22×C32C22×C4
FittingC2×C32×C8C22
prime radicalC22×C3×D3C2×C4
cosocleC2×C32×C4C23
hypercenterC2×C8D32
hypocenterC32#433
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. #63642C2semidirect
4. C3×Dic3⋊C8C2semidirect
5. Dic3⋊(C3⋊C8)C2semidirect
7. #63553C2semidirect
32. Dic3⋊C8D3semidirect
166. C32#433semidirect
203. C3#4720semidirect
204. C3#4840semidirect

Maximal subgroups

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

subgroupindex
#638552
#636422
C3×(C2×D3)⋊C82
C3×Dic3⋊C82
Dic3⋊(C3⋊C8)2
(C2×D3)⋊(C3⋊C8)2
#635532
#48403
#47203