Finite Group Databasebeta

← #13166259

Subgroups — #13166259

Order 1728.

372 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 60 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#11724245C212yesmaximal
2C3⋊D3×Dic12C211yesmaximal, characteristic
3#11723092C211yesmaximal, characteristic
4C3×#63768C211yesmaximal, characteristic
5#11722708C212yesmaximal
6#10593786—31nomaximal
7C3⋊D3×Dic6C2212yes—
8C4×C3⋊#157C2211yescharacteristic
9#88988C2211yescharacteristic
10C3×C4⋊#157C2212yes—
11Dic3⋊#157—22no—
12Dic3×#157—22no—
13#88853C411yescharacteristic, complemented
14C3×C8×C3⋊D3C2211yescharacteristic
15C32×Dic12C411yescharacteristic, complemented
16#88799C412yescomplemented
17Q8×#157—32no—
18Q16×C3⋊D3—31no—
19D3×Dic12—62no—
20#63784—32no—
21#63768D311yescharacteristic
22#63767—31no—
23C2×C3⋊#157D412yes—
24C2×C3×#157—22no—
25#6308C2×C412yes—
26C3×C4×C3⋊D3C2311yescharacteristic
27C32×Dic6C2×C412yes—
28#6293—22no—
29Dic3×C3⋊D3—22no—
30C3⋊#622—21no—
31C33×C8C2×C411yescharacteristic
32C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
33#4853—91nomaximal
34Q8×C3⋊D3—32no—
35D3×Dic6—64no—
36C4⋊#157—62no—
37C4⋊#157C2×D312yes—
38C4×#157—31no—
39C4×#157—62no—
40#3787—31no—
41C32×Q16—31no—
42#3756—32no—
43#3746—31no—
44#3746—62no—
45#3746—122no—
46C8×C3⋊D3C2×D311yescharacteristic
47C3×Dic12—22no—
48C3×D3×C8—62no—
49#3719—64no—
50C2×C3×C3⋊D3C2×D411yescharacteristic
51C3⋊#157—22no—
52C3×#157—42no—
53C33×C4C22×C411yescharacteristic
54C3⋊(C3⋊Dic3)—22no—
55C3×C3⋊Dic3#4111yescharacteristic
56C32×Dic3—22no—
57D3×Q16—182no—
58C2×Dic12—91no—
59#887—92no—
60C3×#106—91no—
61#815—91no—
62#810—92no—
63C2×#157—22no—
64C2×#157—32no—
65C32×Q8—32no—
66C4×C3⋊D3—62no—
67C4×C3⋊D3C22×D311yescharacteristic
68#634—32no—
69#634—64no—
70#634—62no—
71#634—124no—
72C3×C4×D3—62no—
73C3×Dic6—24no—
74#627—124no—
75D3×Dic3—124no—
76#622—61no—
77C32×C8—22no—
78C32×C8—42no—
79C32×C8C4×D311yescharacteristic
80C3⋊(C3⋊C8)C4×D311yescharacteristic
81C3×C3⋊C8—62no—
82#441—271noSylow
83C2×C33C4×D411yescharacteristic
84C3×C3⋊D3#13712yes—
85D3×Q8—184no—
86C2×Dic6—92no—
87C3×Q16—62no—
88C2×C3×C8—91no—
89C3×C4⋊C4—92no—
90#216—184no—
91Dic3⋊C4—182no—
92C4×Dic3—91no—
93C4×Dic3—182no—
94C2×C3⋊C8—91no—
95Dic12—92no—
96Dic12—91no—
97Dic12—182no—
98Dic12—362no—
99Dic12#15711yescharacteristic, complemented
100D3×C8—62no—
101C2×C3⋊D3D3×D411yescharacteristic
102C2×C3×D3—62no—
103#157—42no—
104#157—62no—
105C32×C4—22no—
106C32×C4—42no—
107C32×C4—62no—
108C32×C4C2×C4×D311yescharacteristic
109C3⋊Dic3—62no—
110C3⋊Dic3—124no—
111C3⋊Dic3—244no—
112C3⋊Dic3#23911yescharacteristic
113C3×Dic3—44no—
114C3×Dic3—62no—
115C2×Q16—271no—
116C4×Q8—272no—
117#106—91no—
118#103—272no—
119#97—271no—
120C33#44111yescharacteristic, complemented, Sylow
121C3×Q8—64no—
122C2×C3×C4—91no—
123C2×C3×C4—182no—
124C2×Dic3—92no—
125C2×Dic3—182no—
126C4×D3—62no—
127C4×D3—364no—
128Dic6—92no—
129Dic6—184no—
130Dic6—182no—
131Dic6—364no—
132Dic6—364no—
133Dic6C2×#15712yes—
134C3×C8—22no—
135C3×C8—42no—
136C3×C8—91no—
137C3×C8C2×#15711yescharacteristic
138C3⋊C8—62no—
139C3⋊C8—181no—
140C2×C32—22no—
141C2×C32—42no—
142C2×C32#89311yescharacteristic
143C3⋊D3#91512yes—
144C3×D3—64no—
145C2×Q8—272no—
146Q16—31no—
147Q16—272no—
148Q16—271no—
149C2×C8—91no—
150C2×C8—271no—
151C4⋊C4—92no—
152C4⋊C4—542no—
153C42—271no—
154C42—542no—
155C22×C3—91no—
156C2×D3—62no—
157C3×C4—22no—
158C3×C4—42no—
159C3×C4—91no—
160C3×C4—124no—
161C3×C4—362no—
162C3×C4C22×#15711yescharacteristic
163Dic3—22no—
164Dic3—62no—
165Dic3—182no—
166Dic3—182no—
167Dic3—364no—
168Dic3—724no—
169C32—22no—
170C32—42no—
171C32#485311yescharacteristic, complemented
172Q8—32no—
173Q8—272no—
174Q8—542no—
175C2×C4—91no—
176C2×C4—182no—
177C2×C4—272no—
178C2×C4—542no—
179C8—91no—
180C8—541no—
181C8D3×#15711yescharacteristic
182C2×C3—22no—
183C2×C3—42no—
184C2×C3—92no—
185C2×C3D4×#15711yescharacteristic
186D3—64no—
187C22—91no—
188C4—62no—
189C4—91no—
190C4—362no—
191C4—542no—
192C4—542no—
193C4C2×D3×#15711yescharacteristic
194C3—22no—
195C3—42no—
196C3#1059378611yescharacteristic, complemented
197C2—92no—
198C2#1172425911yescharacteristic
subgroupG/H
centerC2#11724259
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC33×C8C2×C4
socleC2×C33C4×D4
prime radicalC2×C3×C3⋊D3C2×D4
cosocleC3×C4×C3⋊D3C23
hypercenterC8D3×#157
hypocenterC33#441
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
13. #88853C4semidirect
15. C32×Dic12C4semidirect
16. #88799C4semidirect
99. Dic12#157semidirect
120. C33#441semidirect
171. C32#4853semidirect
196. C3#10593786semidirect

Maximal subgroups

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

subgroupindex
#117242452
C3⋊D3×Dic122
#117230922
C3×#637682
#117227082
#105937863
#48539