Finite Group Databasebeta

← #13173066

Subgroups — #13173066

Order 1728.

708 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C32×Q8×A4C211yesmaximal, characteristic
2#11724463C211yesmaximal, characteristic
3C3⋊(C3⋊(A4⋊C8))C211yesmaximal, characteristic
4#10593650—312nomaximal
5#10592927—31nomaximal
6C32×C4×A4—21no—
7C32×C4×A4C2211yescharacteristic
8C3⋊(C3⋊SL(2,ℤ/4))—21no—
9#88863—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C3×Q8×A4D3112yes—
12#64259—312no—
13C2×#3756—31no—
14C2×#3756—31no—
15#64133—31no—
16C3⋊(A4⋊C8)—312no—
17#63662—31no—
18#63662—31no—
19#63657—31no—
20C2×C32×A4D411yescharacteristic
21C33×Q8—41no—
22#6308—41no—
23C3⋊(C3⋊(C3⋊C8))—41no—
24#5416—99no—
25#5228—94no—
26C2×C32×Q8—31no—
27C2×C32×Q8—31no—
28C2×C32×Q8—61no—
29C22×C32×C4—21no—
30C22×C32×C4C2×D311yescharacteristic
31C2×#634—31no—
32C3×C4×A4—212no—
33C3×C4×A4C2×D3112yes—
34C3⋊SL(2,ℤ/4)—612no—
35#3757—61no—
36#3756—62no—
37#3756—61no—
38#3756—61no—
39#3756—1212no—
40C4⋊(C3⋊Dic3)—31no—
41#3750—61no—
42C2×C3⋊(C3⋊C8)—31no—
43C2×C3⋊(C3⋊C8)—31no—
44C32×A4Q1611yescharacteristic, complemented
45C33×C4—41no—
46C33×C4—81no—
47C3⋊(C3⋊Dic3)—81no—
48C22×C3×Q8C3⋊D314yes—
49Q8×A4C3⋊D319yes—
50A4⋊Q8—99no—
51C2×#216—94no—
52C2×#216—94no—
53#924—94no—
54A4⋊C8—99no—
55#835—94no—
56#835—94no—
57#830—94no—
58C23×C32#6711yescharacteristic
59C2×C3×A4#67112yes—
60C32×Q8—32no—
61C32×Q8—31no—
62C32×Q8—412no—
63C32×Q8—61no—
64C32×Q8S411yescharacteristic
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×C32×C4—61no—
68C2×C32×C4—61no—
69C2×C3⋊Dic3—61no—
70C2×C3⋊Dic3—61no—
71#634—61no—
72#634—61no—
73#634—1212no—
74C3⋊(C3⋊C8)—61no—
75C3⋊(C3⋊C8)—61no—
76C3⋊(C3⋊C8)—1212no—
77#451—271noSylow
78C2×C33—41no—
79C2×C3×Q8—34no—
80C2×C3×Q8—34no—
81C2×C3×Q8—64no—
82C22×C3×C4—24no—
83C22×C3×C4C2×C3⋊D314yes—
84C2×Dic6—94no—
85C4×A4—29no—
86C4×A4C2×C3⋊D319yes—
87SL(2,ℤ/4)—189no—
88#217—184no—
89#216—188no—
90#216—184no—
91#216—184no—
92#216—369no—
93C4⋊Dic3—94no—
94Dic3⋊C4—184no—
95C2×C3⋊C8—94no—
96C2×C3⋊C8—94no—
97C22×C32—31no—
98C22×C32—31no—
99C22×C32#21611yescharacteristic, complemented
100C3×A4#216112yescomplemented
101C32×C4—21no—
102C32×C4—31no—
103C32×C4—412no—
104C32×C4—61no—
105C32×C4—812no—
106C32×C4C2×S411yescharacteristic
107C3⋊Dic3—121no—
108C3⋊Dic3—121no—
109C3⋊Dic3—2412no—
110C22×Q8C3⋊(C3⋊D3)11yescharacteristic
111C2×Q16—271no—
112C2×Q16—271no—
113C22⋊Q8—271no—
114#103—271no—
115#103—271no—
116C22⋊C8—271no—
117C33—41noSylow
118C23×C3#63814yes—
119C2×A4#63819yes—
120C3×Q8—38no—
121C3×Q8—34no—
122C3×Q8—49no—
123C3×Q8—64no—
124C3×Q8C3⋊S414yes—
125C2×C3×C4—34no—
126C2×C3×C4—34no—
127C2×C3×C4—64no—
128C2×C3×C4—64no—
129C2×Dic3—184no—
130C2×Dic3—184no—
131Dic6—184no—
132Dic6—184no—
133Dic6—369no—
134C3⋊C8—184no—
135C3⋊C8—184no—
136C3⋊C8—369no—
137C2×C32—31no—
138C2×C32—31no—
139C2×C32—412no—
140C2×C32GL(2,ℤ/4)11yescharacteristic
141C2×Q8—31no—
142C2×Q8—31no—
143C2×Q8—61no—
144C2×Q8—271no—
145C22×C4—21no—
146C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
147Q16—542no—
148Q16—541no—
149Q16—541no—
150C2×C8—271no—
151C2×C8—271no—
152C4⋊C4—271no—
153C4⋊C4—541no—
154C22⋊C4—541no—
155C22×C3—34no—
156C22×C3—34no—
157C22×C3#375614yescomplemented
158A4#375619yescomplemented
159C3×C4—24no—
160C3×C4—34no—
161C3×C4—49no—
162C3×C4—64no—
163C3×C4—89no—
164C3×C4C2×C3⋊S414yes—
165Dic3—364no—
166Dic3—364no—
167Dic3—729no—
168C32—412no—
169C32#541611yescharacteristic, complemented
170C23#631211yescharacteristic
171Q8—32no—
172Q8—31no—
173Q8—61no—
174Q8—541no—
175Q8—541no—
176Q8C3⋊(C3⋊S4)11yescharacteristic
177C2×C4—31no—
178C2×C4—31no—
179C2×C4—61no—
180C2×C4—61no—
181C2×C4—541no—
182C2×C4—541no—
183C8—541no—
184C8—541no—
185C2×C3—34no—
186C2×C3—34no—
187C2×C3—49no—
188C2×C3#6426914yes—
189C22—31no—
190C22—31no—
191C22#8886311yescharacteristic, complemented
192C4—21no—
193C4—31no—
194C4—61no—
195C4—1081no—
196C4—1081no—
197C4C2×C3⋊(C3⋊S4)11yescharacteristic
198C3—49no—
199C3#1059365014yescomplemented
200C2—31no—
201C2—31no—
202C2#1172447311yescharacteristic
subgroupG/H
centerC2#11724473
commutator, cosocleC32×C4×A4C22
FrattiniC4C2×C3⋊(C3⋊S4)
FittingC22×C32×Q8D3
socleC23×C32#67
prime radicalC2×C32×A4D4
hypercenterQ8C3⋊(C3⋊S4)
hypocenterC32×A4Q16
LCSG ▷ C32×C4×A4 ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
44. C32×A4Q16semidirect
99. C22×C32#216semidirect
100. C3×A4#216semidirect
157. C22×C3#3756semidirect
158. A4#3756semidirect
169. C32#5416semidirect
191. C22#88863semidirect
199. C3#10593650semidirect

Maximal subgroups

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

subgroupindex
C32×Q8×A42
#117244632
C3⋊(C3⋊(A4⋊C8))2
#105936503
#105929273
#888634