Finite Group Databasebeta

← #10589415⋊C3 #13135652

Subgroups — #10589415⋊C3 #13135652

Order 1728.

307 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 96 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
1C22×C8×H(3,𝔽3)C211yesmaximal, characteristic
2C2×(C3⋊(C3⋊C16))⋊C3C212yesmaximal, complemented
3#10589415C311yesmaximal, characteristic, complemented
4C3×#4549—31nomaximal
5C22×C4×H(3,𝔽3)C411yescharacteristic
6C2×C8×H(3,𝔽3)—22no—
7C2×C8×H(3,𝔽3)C411yescharacteristic
8C2×C8×H(3,𝔽3)C2211yescharacteristic
9(C3⋊(C3⋊C16))⋊C3—22no—
10C22×C32×C8—21no—
11C22×C32×C8D311yescharacteristic
12C22×C32×C8C2×C311yescharacteristic
13C2×C3⋊(C3⋊C16)C2×C312yescomplemented
14C2×C3×C3⋊C16—32no—
15C23×H(3,𝔽3)C811yescharacteristic
16C2×C4×H(3,𝔽3)—21no—
17C2×C4×H(3,𝔽3)—21no—
18C2×C4×H(3,𝔽3)C811yescharacteristic
19C2×C4×H(3,𝔽3)C2×C411yescharacteristic
20C8×H(3,𝔽3)—21no—
21C8×H(3,𝔽3)D412yes—
22C3×C22⋊C16—91no—
23#4549—31no—
24#4549—91no—
25C22×C32×C4—21no—
26C22×C32×C4Dic311yescharacteristic
27C22×C32×C4C3×C411yescharacteristic
28C2×C32×C8—24no—
29C2×C32×C8—21no—
30C2×C32×C8—21no—
31C2×C32×C8—22no—
32C2×C32×C8—22no—
33C2×C32×C8Dic311yescharacteristic
34C2×C32×C8C3×C411yescharacteristic
35C2×C32×C8C2×D311yescharacteristic
36C2×C32×C8C22×C311yescharacteristic
37C3⋊(C3⋊C16)—22no—
38C3×C3⋊C16—62no—
39C22×H(3,𝔽3)—21no—
40C22×H(3,𝔽3)—21no—
41C22×H(3,𝔽3)C1612yescomplemented
42C22×H(3,𝔽3)C2×C811yescharacteristic
43C4×H(3,𝔽3)—21no—
44C4×H(3,𝔽3)C22⋊C411yescharacteristic
45C4×H(3,𝔽3)M1611yescharacteristic
46C22×C3×C8—31no—
47C22×C3×C8—31no—
48C22×C3×C8—61no—
49C22×C3×C8C3×D311yescharacteristic
50C2×C3×C16—92no—
51C2×C3⋊C16—32no—
52C2×C3⋊C16—92no—
53C23×C32—21no—
54C23×C32C3⋊C811yescharacteristic
55C23×C32C3×C811yescharacteristic
56C2×C32×C4—22no—
57C2×C32×C4—22no—
58C2×C32×C4—21no—
59C2×C32×C4—21no—
60C2×C32×C4—21no—
61C2×C32×C4—21no—
62C2×C32×C4—21no—
63C2×C32×C4—21no—
64C2×C32×C4C3⋊C811yescharacteristic
65C2×C32×C4C3×C811yescharacteristic
66C2×C32×C4C2×Dic311yescharacteristic
67C2×C32×C4C2×C3×C411yescharacteristic
68C32×C8—22no—
69C32×C8—21no—
70C32×C8—21no—
71C32×C8—22no—
72C32×C8#6712yes—
73C32×C8C3×D412yes—
74C22⋊C16—91noSylow
75C2×H(3,𝔽3)—22no—
76C2×H(3,𝔽3)C22⋊C811yescharacteristic
77C2×H(3,𝔽3)C2×C1611yescharacteristic
78C2×H(3,𝔽3)M3211yescharacteristic
79C22×C3×C4—31no—
80C22×C3×C4—31no—
81C22×C3×C4—61no—
82C22×C3×C4C3×Dic311yescharacteristic
83C2×C3×C8—22no—
84C2×C3×C8—31no—
85C2×C3×C8—31no—
86C2×C3×C8—31no—
87C2×C3×C8—31no—
88C2×C3×C8—62no—
89C2×C3×C8—62no—
90C2×C3×C8—64no—
91C2×C3×C8—61no—
92C2×C3×C8—61no—
93C2×C3×C8C3×Dic311yescharacteristic
94C2×C3×C8C2×C3×D311yescharacteristic
95C3×C16—182no—
96C3⋊C16—62no—
97C3⋊C16—182no—
98C22×C32—22no—
99C22×C32—22no—
100C22×C32—21no—
101C22×C32—21no—
102C22×C32—21no—
103C22×C32—21no—
104C22×C32—22no—
105C22×C32—21no—
106C22×C32C3⋊C1612yescomplemented
107C22×C32C3×C1612yescomplemented
108C22×C32C2×C3⋊C811yescharacteristic
109C22×C32C2×C3×C811yescharacteristic
110C32×C4—22no—
111C32×C4—21no—
112C32×C4—21no—
113C32×C4—21no—
114C32×C4—21no—
115C32×C4#20811yescharacteristic
116C32×C4#21711yescharacteristic
117C32×C4C3×C22⋊C411yescharacteristic
118C32×C4C3×M1611yescharacteristic
119C22×C8(C3⋊D3)⋊C311yescharacteristic
120C2×C16—92no—
121H(3,𝔽3)C22⋊C1611yescharacteristic, complemented, Sylow
122C23×C3—31no—
123C23×C3—31no—
124C23×C3—61no—
125C23×C3C3×C3⋊C811yescharacteristic
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—31no—
129C2×C3×C4—31no—
130C2×C3×C4—31no—
131C2×C3×C4—31no—
132C2×C3×C4—61no—
133C2×C3×C4—61no—
134C2×C3×C4—61no—
135C2×C3×C4—61no—
136C2×C3×C4—62no—
137C2×C3×C4—62no—
138C2×C3×C4—61no—
139C2×C3×C4—61no—
140C2×C3×C4C3×C3⋊C811yescharacteristic
141C2×C3×C4C2×C3×Dic311yescharacteristic
142C3×C8—21no—
143C3×C8—32no—
144C3×C8—32no—
145C3×C8—61no—
146C3×C8—61no—
147C3×C8—62no—
148C3×C8—62no—
149C3×C8C3×#6712yes—
150C2×C32—24no—
151C2×C32—22no—
152C2×C32—22no—
153C2×C32—21no—
154C2×C32—21no—
155C2×C32—21no—
156C2×C32C2×C3⋊C1611yescharacteristic
157C2×C32#81211yescharacteristic
158C2×C32#83011yescharacteristic
159C2×C32C3×C22⋊C811yescharacteristic
160C2×C32C2×C3×C1611yescharacteristic
161C2×C32C3×M3211yescharacteristic
162C22×C4(C3⋊Dic3)⋊C311yescharacteristic
163C2×C8—22no—
164C2×C8(C3⋊Dic3)⋊C311yescharacteristic
165C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
166C16—182no—
167C22×C3—21no—
168C22×C3—21no—
169C22×C3—32no—
170C22×C3—32no—
171C22×C3—31no—
172C22×C3—31no—
173C22×C3—61no—
174C22×C3—61no—
175C22×C3—61no—
176C22×C3—61no—
177C22×C3—62no—
178C22×C3—62no—
179C22×C3—62no—
180C22×C3—61no—
181C22×C3C3×C3⋊C1612yes—
182C22×C3C2×C3×C3⋊C811yescharacteristic
183C3×C4—21no—
184C3×C4—31no—
185C3×C4—31no—
186C3×C4—31no—
187C3×C4—31no—
188C3×C4—61no—
189C3×C4—61no—
190C3×C4—62no—
191C3×C4—61no—
192C3×C4—61no—
193C3×C4C3×#20811yescharacteristic
194C3×C4C3×#21711yescharacteristic
195C32—21no—
196C32#454911yescharacteristic, complemented
197C32C3×C22⋊C1611yescharacteristic, complemented
198C23(C3⋊(C3⋊C8))⋊C311yescharacteristic
199C2×C4—21no—
200C2×C4—21no—
201C2×C4(C3⋊(C3⋊C8))⋊C311yescharacteristic
202C2×C4C2×(C3⋊Dic3)⋊C311yescharacteristic
203C8—21no—
204C8#638⋊C312yes—
205C2×C3—22no—
206C2×C3—31no—
207C2×C3—31no—
208C2×C3—31no—
209C2×C3—31no—
210C2×C3—31no—
211C2×C3—31no—
212C2×C3—62no—
213C2×C3—62no—
214C2×C3—64no—
215C2×C3—61no—
216C2×C3—61no—
217C2×C3—61no—
218C2×C3C2×C3×C3⋊C1611yescharacteristic
219C2×C3C3×#81211yescharacteristic
220C2×C3C3×#83011yescharacteristic
221C22—21no—
222C22—21no—
223C22(C3⋊(C3⋊C16))⋊C312yescomplemented
224C22C2×(C3⋊(C3⋊C8))⋊C311yescharacteristic
225C4—21no—
226C4#3748⋊C311yescharacteristic
227C4#3757⋊C311yescharacteristic
228C3—31no—
229C3—31no—
230C3—61no—
231C3C3×#454911yescharacteristic
232C2—22no—
233C2C2×(C3⋊(C3⋊C16))⋊C311yescharacteristic
234C2#63639⋊C311yescharacteristic
235C2#63657⋊C311yescharacteristic
subgroupG/H
centerC2×C8C2×(C3⋊D3)⋊C3
commutatorC2×C32C2×C3×C16
FrattiniC2×C3×C8C2×C3×D3
FittingC22×C8×H(3,𝔽3)C2
socleC22×C3C2×C3×C3⋊C8
prime radicalC23×H(3,𝔽3)C8
cosocleC2×C32×C8C22×C3
hypercenterC22×C8(C3⋊D3)⋊C3
hypocenterC32C3×C22⋊C16
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C8 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C2×(C3⋊(C3⋊C16))⋊C3C2semidirect
3. #10589415C3semidirect
13. C2×C3⋊(C3⋊C16)C2×C3semidirect
41. C22×H(3,𝔽3)C16semidirect
106. C22×C32C3⋊C16semidirect
107. C22×C32C3×C16semidirect
121. H(3,𝔽3)C22⋊C16semidirect
196. C32#4549semidirect
197. C32C3×C22⋊C16semidirect
223. C22(C3⋊(C3⋊C16))⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C8×H(3,𝔽3)2
C2×(C3⋊(C3⋊C16))⋊C32
#105894153
C3×#45493