Finite Group Databasebeta

← #13167371

Subgroups — #13167371

Order 1728.

400 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 63 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#11724341C212yesmaximal
2#11724335C212yesmaximal
3#11723649C212yesmaximal
4C3×C8×#157C211yesmaximal, characteristic
5#10593295—31nomaximal
6#89040C2214yes—
7C3×C4×#157C2211yescharacteristic
8C3×#3787C2211yescharacteristic
9#88943—24no—
10#88940—24no—
11C3×#3746—22no—
12C3×C8×C3⋊D3C2211yescharacteristic
13#88812—22no—
14#64228—32no—
15#64222—32no—
16#63802—32no—
17D3×Dic12—62no—
18C8×#157D311yescharacteristic
19C2×C3×#157D412yes—
20C3×C4×C3⋊D3C2311yescharacteristic
21C3×#634—24no—
22#6296—24no—
23#6294—24no—
24C3×#622D412yes—
25C33×C8D411yescharacteristic
26C3×C3⋊(C3⋊C8)D411yescharacteristic
27#4699—91nomaximal
28#3797—34no—
29D3×Dic6—64no—
30C4×#157C2×D311yescharacteristic
31#3787C2×D311yescharacteristic
32#3776—64no—
33#3773—64no—
34#3746—22no—
35C8×C3⋊D3C2×D311yescharacteristic
36C3×Dic12—62no—
37C3×Dic12—62no—
38C3×Dic12—122no—
39C3×D3×C8—62no—
40#3719—64no—
41#3718—62no—
42C2×C3×C3⋊D3C2×D411yescharacteristic
43C3×#157Q1614yescomplemented
44C33×C4C2×D411yescharacteristic
45C3×C3⋊Dic3—44no—
46C3×C3⋊Dic3C2×D411yescharacteristic
47D3×Q16—182no—
48C2×Dic12—92no—
49C2×Dic12—92no—
50#869—92no—
51C3×C4×C8—91no—
52C2×#157D1212yes—
53C4×C3⋊D3C22×D311yescharacteristic
54#634—24no—
55C3×C4×D3—62no—
56C3×Dic6—64no—
57C3×Dic6—64no—
58C3×Dic6—124no—
59#627—64no—
60#627—124no—
61#624—64no—
62D3×Dic3—124no—
63#622D1212yes—
64C32×C8—22no—
65C32×C8—42no—
66C32×C8D1211yescharacteristic
67C3⋊(C3⋊C8)D1211yescharacteristic
68C3×C3⋊C8—62no—
69#494—271noSylow
70C2×C33#12711yescharacteristic
71C3×C3⋊D3C2×Q1612yes—
72D3×Q8—184no—
73C2×Dic6—94no—
74C3×Q16—182no—
75C2×C3×C8—91no—
76C2×C3×C8—91no—
77C3×C42—91no—
78#216—184no—
79C4⋊Dic3—184no—
80Dic12—62no—
81Dic12—62no—
82Dic12—122no—
83Dic12—182no—
84Dic12—184no—
85D3×C8—62no—
86C2×C3⋊D3C2×D1211yescharacteristic
87C2×C3×D3—62no—
88#157Dic1214yescomplemented
89C32×C4—22no—
90C32×C4—42no—
91C32×C4C2×D1211yescharacteristic
92C3⋊Dic3—44no—
93C3⋊Dic3C2×D1211yescharacteristic
94C3×Dic3—62no—
95C3×Dic3—124no—
96C3×Dic3—124no—
97C3×Dic3—244no—
98C2×Q16—272no—
99C2×Q16—272no—
100C4⋊Q8—272no—
101C4×C8—91no—
102C33#49411yescharacteristic, complemented, Sylow
103C3×Q8—184no—
104C2×C3×C4—92no—
105C2×C3×C4—91no—
106C2×Dic3—184no—
107C4×D3—62no—
108C4×D3—364no—
109Dic6—64no—
110Dic6—64no—
111Dic6—124no—
112Dic6—184no—
113Dic6—364no—
114C3×C8—22no—
115C3×C8—42no—
116C3×C8—92no—
117C3×C8—91no—
118C3×C8GO+(4,𝔽2)11yescharacteristic
119C3⋊C8—62no—
120C2×C32—22no—
121C2×C32—42no—
122C2×C32#87411yescharacteristic
123C3⋊D3C2×Dic1212yes—
124C3×D3—64no—
125C2×Q8—274no—
126Q16—182no—
127Q16—542no—
128Q16—544no—
129C2×C8—91no—
130C2×C8—91no—
131C4⋊C4—544no—
132C42—91no—
133C22×C3—91no—
134C2×D3—62no—
135C3×C4—22no—
136C3×C4—42no—
137C3×C4—94no—
138C3×C4—91no—
139C3×C4—364no—
140C3×C4C2×GO+(4,𝔽2)11yescharacteristic
141Dic3—62no—
142Dic3—124no—
143Dic3—124no—
144Dic3—244no—
145C32—22no—
146C32—42no—
147C32#469911yescharacteristic, complemented
148Q8—184no—
149Q8—544no—
150C2×C4—92no—
151C2×C4—91no—
152C2×C4—544no—
153C8—92no—
154C8—91no—
155C8#157⋊D311yescharacteristic
156C2×C3—22no—
157C2×C3—42no—
158C2×C3—92no—
159C2×C3#6423111yescharacteristic
160D3—64no—
161C22—91no—
162C4—94no—
163C4—91no—
164C4—364no—
165C4C2×#157⋊D311yescharacteristic
166C3—22no—
167C3—42no—
168C3#1059329511yescharacteristic, complemented
169C2—92no—
170C2#1172434411yescharacteristic
subgroupG/H
centerC2#11724344
commutator, cosocleC3×C4×C3⋊D3C23
FrattiniC4C2×#157⋊D3
FittingC33×C8D4
socleC2×C33#127
prime radicalC2×C3×C3⋊D3C2×D4
hypercenterC8#157⋊D3
hypocenterC33#494
LCSG ▷ C3×C4×C3⋊D3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
43. C3×#157Q16semidirect
88. #157Dic12semidirect
102. C33#494semidirect
147. C32#4699semidirect
168. C3#10593295semidirect

Maximal subgroups

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

subgroupindex
#117243412
#117243352
#117236492
C3×C8×#1572
#105932953
#46999