Finite Group Databasebeta

← C2×SL(2,𝔽3)×#157 #13180669

Subgroups — C2×SL(2,𝔽3)×#157 #13180669

Order 1728.

366 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 52 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×C3⋊D3×SL(2,𝔽3)C211yesmaximal, characteristic
2SL(2,𝔽3)×#157C212yesmaximal, direct product factor
3C2×Q8×#157C311yesmaximal, characteristic, complemented
4C22×C3×#157—41nomaximal
5C2×C32×SL(2,𝔽3)C411yescharacteristic, complemented
6C3⋊D3×SL(2,𝔽3)C411yescharacteristic, complemented
7C3⋊D3×SL(2,𝔽3)C2211yescharacteristic
8C2×Q8×C3⋊D3C2×C311yescharacteristic
9Q8×#157—32no—
10Q8×#157C2×C312yescomplemented
11C2×C4⋊#157—31no—
12C2×C4×#157—31no—
13C2×D3×SL(2,𝔽3)—62no—
14C22×C3×C3⋊D3—41no—
15C2×C3×#157—44no—
16C2×C3×#157—42no—
17C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
18C2×C4×SL(2,𝔽3)—91nomaximal
19C22×#157—31no—
20C22×#157A411yescharacteristic
21C2×C32×Q8C3×C411yescharacteristic, complemented
22Q8×C3⋊D3—31no—
23Q8×C3⋊D3—31no—
24Q8×C3⋊D3C3×C411yescharacteristic, complemented
25Q8×C3⋊D3C22×C311yescharacteristic
26C2×C4×C3⋊D3—31no—
27C2×#634—31no—
28C2×C3×SL(2,𝔽3)—22no—
29C2×C3×SL(2,𝔽3)—42no—
30C4⋊#157—32no—
31C4⋊#157—32no—
32C4×#157—32no—
33C4×#157—32no—
34D3×SL(2,𝔽3)—62no—
35D3×SL(2,𝔽3)—62no—
36C22×C33—41no—
37C2×C3×C3⋊D3—41no—
38C2×C3×C3⋊D3—42no—
39C2×C3×C3⋊D3—42no—
40C2×C3×C3⋊D3—41no—
41C3×#157—44no—
42C2×D3×Q8—62no—
43C22×SL(2,𝔽3)—91no—
44C4×SL(2,𝔽3)—92no—
45C22×C3⋊D3C2×A411yescharacteristic
46C22×C3×D3—242no—
47C2×#157—32no—
48C2×#157—62no—
49C2×#157SL(2,𝔽3)14yesdirect product factor
50C2×#157C2×A412yes—
51C32×Q8—31no—
52C32×Q8C2×C3×C411yescharacteristic, complemented
53C2×C32×C4—31no—
54C2×C3⋊Dic3—31no—
55C4×C3⋊D3—31no—
56C4×C3⋊D3—31no—
57C4×C3⋊D3—31no—
58C4×C3⋊D3—31no—
59#634—32no—
60#634—31no—
61#634—31no—
62C3×SL(2,𝔽3)—22no—
63C3×SL(2,𝔽3)—42no—
64C2×C4×Q8—91noSylow
65C2×C33—42no—
66C2×C33—41no—
67C3×C3⋊D3—42no—
68C3×C3⋊D3—41no—
69C3×C3⋊D3—41no—
70C2×C3×Q8—22no—
71C22×C3×C4—361no—
72D3×Q8—62no—
73D3×Q8—62no—
74D3×Q8—182no—
75D3×Q8—182no—
76C2×C4×D3—182no—
77C2×Dic6—182no—
78C2×SL(2,𝔽3)—42no—
79C2×SL(2,𝔽3)—91no—
80C2×SL(2,𝔽3)—91no—
81C2×SL(2,𝔽3)#15711yescharacteristic, direct product factor
82C22×C32—82no—
83C22×C32—162no—
84C22×C32C4×A411yescharacteristic
85C2×C3⋊D3C4×A411yescharacteristic
86C2×C3⋊D3C2×SL(2,𝔽3)12yes—
87C2×C3⋊D3#41⋊C312yes—
88C2×C3⋊D3C22×A411yescharacteristic
89C2×C3×D3—244no—
90C2×C3×D3—244no—
91C2×C3×D3—242no—
92C2×C3×D3—242no—
93#157—62no—
94#157C2×SL(2,𝔽3)14yesdirect product factor
95C32×C4—31no—
96C32×C4—31no—
97C3⋊Dic3—31no—
98C3⋊Dic3—31no—
99C22×Q8—91no—
100C4×Q8—92no—
101C4×Q8—272no—
102C2×C4⋊C4—271no—
103C2×C42—271no—
104C33—41noSylow
105C23×C3—361no—
106C22×D3—62no—
107C3×Q8—22no—
108C3×Q8—62no—
109C2×C3×C4—62no—
110C2×C3×C4—362no—
111C2×C3×C4—364no—
112C2×Dic3—182no—
113C4×D3—182no—
114C4×D3—182no—
115C4×D3—182no—
116C4×D3—182no—
117Dic6—184no—
118Dic6—182no—
119Dic6—182no—
120SL(2,𝔽3)—42no—
121SL(2,𝔽3)C2×#15711yescharacteristic, direct product factor
122C2×C32—82no—
123C2×C32—84no—
124C2×C32—162no—
125C2×C32—164no—
126C2×C32C4×SL(2,𝔽3)12yescomplemented
127C2×C32C2×C4×A411yescharacteristic
128C3⋊D3C4×SL(2,𝔽3)12yescomplemented
129C3⋊D3C22×SL(2,𝔽3)11yescharacteristic
130C3⋊D3C2×#41⋊C311yescharacteristic
131C3×D3—244no—
132C3×D3—242no—
133C3×D3—242no—
134C2×Q8—91no—
135C2×Q8—91no—
136C2×Q8—271no—
137C2×Q8—271no—
138C2×Q8—271no—
139C2×Q8C3×#15711yescharacteristic, complemented
140C22×C4—91no—
141C22×C4—271no—
142C22×C4—271no—
143C4⋊C4—272no—
144C4⋊C4—272no—
145C42—272no—
146C42—272no—
147C22×C3—22no—
148C22×C3—41no—
149C22×C3—162no—
150C22×C3—362no—
151C22×C3—361no—
152C22×C3—362no—
153C22×C3—361no—
154C2×D3—62no—
155C2×D3—62no—
156C2×D3—64no—
157C2×D3—64no—
158C3×C4—62no—
159C3×C4—62no—
160C3×C4—364no—
161Dic3—182no—
162Dic3—182no—
163C32—82no—
164C32—162no—
165C32C2×C4×SL(2,𝔽3)11yescharacteristic, complemented
166C23—91no—
167Q8—31no—
168Q8—271no—
169Q8—271no—
170Q8—272no—
171Q8C2×C3×#15711yescharacteristic, complemented
172C2×C4—31no—
173C2×C4—92no—
174C2×C4—94no—
175C2×C4—271no—
176C2×C4—271no—
177C2×C4—271no—
178C2×C4—271no—
179C2×C4—271no—
180C2×C4—272no—
181C2×C4—542no—
182C2×C3—24no—
183C2×C3—22no—
184C2×C3—42no—
185C2×C3—41no—
186C2×C3—164no—
187C2×C3—162no—
188C2×C3—361no—
189C2×C3—361no—
190C2×C3—362no—
191D3—64no—
192D3—62no—
193D3—62no—
194C22—92no—
195C22—92no—
196C22—91no—
197C22—91no—
198C22A4×#15711yescharacteristic
199C4—31no—
200C4—31no—
201C4—94no—
202C4—271no—
203C4—271no—
204C4—542no—
205C3—22no—
206C3—41no—
207C3—162no—
208C2—91no—
209C2—91no—
210C2—92no—
211C2SL(2,𝔽3)×#15712yesdirect product factor
212C2C2×A4×#15711yescharacteristic
subgroupG/H
center, hypercenterC22A4×#157
commutator, hypocenterC32×Q8C2×C3×C4
FrattiniC2C2×A4×#157
FittingC2×C32×Q8C3×C4
socleC22×C32C4×A4
prime radicalC2×C3⋊D3×SL(2,𝔽3)C2
cosocleQ8×C3⋊D3C22×C3
LCSG ▷ C32×Q8
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)×#157C2direct (N × H)
3. C2×Q8×#157C3semidirect
5. C2×C32×SL(2,𝔽3)C4semidirect
6. C3⋊D3×SL(2,𝔽3)C4semidirect
10. Q8×#157C2×C3semidirect
17. C32×SL(2,𝔽3)C2×C4semidirect
21. C2×C32×Q8C3×C4semidirect
24. Q8×C3⋊D3C3×C4semidirect
49. C2×#157SL(2,𝔽3)direct (N × H)
52. C32×Q8C2×C3×C4semidirect
81. C2×SL(2,𝔽3)#157direct (N × H)
94. #157C2×SL(2,𝔽3)direct (N × H)
121. SL(2,𝔽3)C2×#157direct (N × H)
126. C2×C32C4×SL(2,𝔽3)semidirect
128. C3⋊D3C4×SL(2,𝔽3)semidirect
139. C2×Q8C3×#157semidirect
165. C32C2×C4×SL(2,𝔽3)semidirect
171. Q8C2×C3×#157semidirect
211. C2SL(2,𝔽3)×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×SL(2,𝔽3)2
SL(2,𝔽3)×#1572
C2×Q8×#1573
C22×C3×#1574
C2×C4×SL(2,𝔽3)9