Finite Group Databasebeta

← #13173383

Subgroups — #13173383

Order 1728.

372 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 71 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
1Q8×C3⋊#157C212yesmaximal
2C3×Q16×C3⋊D3C211yesmaximal, characteristic
3#11723109C212yesmaximal
4#11723093C211yesmaximal, characteristic
5#11723092C211yesmaximal, characteristic
6#10593786—31nomaximal
7C3×Q8×C3⋊D3C2212yes—
8C4⋊(C3⋊#157)—22no—
9C4⋊(C3⋊#157)C2212yes—
10C4×C3⋊#157—22no—
11C4×C3⋊#157C2211yescharacteristic
12#88988C2211yescharacteristic
13C33×Q16C411yescharacteristic, complemented
14C3×#3756C412yescomplemented
15C3×#3746C411yescharacteristic, complemented
16C3×C8×C3⋊D3C2211yescharacteristic
17Q8×#157—32no—
18Q16×C3⋊D3D311yescharacteristic
19C3×D3×Q16—62no—
20#63784—32no—
21#63768—31no—
22#63767—31no—
23C2×C3⋊#157—22no—
24C2×C3⋊#157D412yes—
25C33×Q8C2×C412yes—
26C3×C4×C3⋊D3—22no—
27C3×C4×C3⋊D3C2311yescharacteristic
28C3×#634—22no—
29C3×#634C2×C412yes—
30C3⋊#622—21no—
31C33×C8C2×C411yescharacteristic
32C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
33#5184—91nomaximal
34Q8×C3⋊D3C2×D312yes—
35C3×D3×Q8—64no—
36C4⋊#157—32no—
37C4⋊#157—62no—
38C4×#157—31no—
39C4×#157—62no—
40#3787—31no—
41C32×Q16—22no—
42C32×Q16—42no—
43C32×Q16Dic311yescharacteristic, complemented
44#3756Dic312yescomplemented
45#3746Dic311yescharacteristic, complemented
46C8×C3⋊D3C2×D311yescharacteristic
47C3×#216—64no—
48C3×Dic12—62no—
49C3×D3×C8—62no—
50C2×C3×C3⋊D3C2×D411yescharacteristic
51C3⋊#157—22no—
52C3⋊#157—42no—
53C33×C4—22no—
54C33×C4C22×C411yescharacteristic
55C3×C3⋊Dic3—22no—
56C3×C3⋊Dic3#4111yescharacteristic
57C2×C3×Q16—91no—
58Q8×Dic3—92no—
59D3×Q16—62no—
60#835—92no—
61#817—91no—
62#815—91no—
63C2×#157—32no—
64C2×#157—62no—
65C32×Q8—24no—
66C32×Q8—44no—
67C32×Q8C2×Dic312yes—
68C4×C3⋊D3—22no—
69C4×C3⋊D3C22×D311yescharacteristic
70#634—22no—
71#634C2×Dic312yes—
72C3×C4×D3—62no—
73C3×C4×D3—124no—
74C3×Dic6—64no—
75C3×Dic6—124no—
76#622—61no—
77C32×C8—22no—
78C32×C8—42no—
79C32×C8C2×Dic311yescharacteristic
80C3⋊(C3⋊C8)C2×Dic311yescharacteristic
81C3×C3⋊C8—62no—
82#441—271noSylow
83C2×C33C4×D411yescharacteristic
84C3×C3⋊D3#13712yes—
85C2×C3×Q8—92no—
86D3×Q8—64no—
87C3×Q16—22no—
88C3×Q16—42no—
89C3×Q16—92no—
90C3×Q16—91no—
91C3×Q16#15711yescharacteristic, complemented
92C2×C3×C8—91no—
93#216—64no—
94C4⋊Dic3—92no—
95C4⋊Dic3—182no—
96C4×Dic3—91no—
97C4×Dic3—182no—
98C2×C3⋊C8—91no—
99Dic12—62no—
100D3×C8—62no—
101C2×C3⋊D3D3×D411yescharacteristic
102C2×C3×D3—62no—
103#157—62no—
104#157—122no—
105C32×C4—22no—
106C32×C4—22no—
107C32×C4—42no—
108C32×C4—44no—
109C32×C4—84no—
110C32×C4C22×Dic311yescharacteristic
111C3⋊Dic3—22no—
112C3⋊Dic3#23711yescharacteristic
113C3×Dic3—62no—
114C3×Dic3—124no—
115C2×Q16—91no—
116C4×Q8—272no—
117#106—271no—
118#103—272no—
119#97—271no—
120C33#44111yescharacteristic, complemented, Sylow
121C3×Q8—24no—
122C3×Q8—44no—
123C3×Q8—92no—
124C3×Q8—182no—
125C3×Q8C2×#15712yes—
126C2×C3×C4—91no—
127C2×C3×C4—182no—
128C2×Dic3—92no—
129C2×Dic3—182no—
130C4×D3—62no—
131C4×D3—124no—
132Dic6—64no—
133Dic6—124no—
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×C32D4×Dic311yescharacteristic
143C3⋊D3#91812yes—
144C3×D3—64no—
145C2×Q8—92no—
146Q16—92no—
147Q16—91no—
148Q16C3⋊#15711yescharacteristic, complemented
149C2×C8—91no—
150C2×C8—271no—
151C4⋊C4—272no—
152C4⋊C4—542no—
153C42—271no—
154C42—542no—
155C22×C3—91no—
156C2×D3—62no—
157C3×C4—22no—
158C3×C4—22no—
159C3×C4—44no—
160C3×C4—42no—
161C3×C4—84no—
162C3×C4—91no—
163C3×C4—182no—
164C3×C4C22×#15711yescharacteristic
165Dic3—62no—
166Dic3—124no—
167Dic3—182no—
168Dic3—362no—
169C32—22no—
170C32—42no—
171C32#518411yescharacteristic, complemented
172Q8—92no—
173Q8—182no—
174Q8C2×C3⋊#15712yes—
175C2×C4—91no—
176C2×C4—182no—
177C2×C4—272no—
178C2×C4—542no—
179C8—91no—
180C8—541no—
181C8C2×C3⋊#15711yescharacteristic
182C2×C3—22no—
183C2×C3—42no—
184C2×C3—92no—
185C2×C3D4×#15711yescharacteristic
186D3—64no—
187C22—91no—
188C4—22no—
189C4—91no—
190C4—182no—
191C4—542no—
192C4—1082no—
193C4C22×C3⋊#15711yescharacteristic
194C3—22no—
195C3—42no—
196C3#1059378611yescharacteristic, complemented
197C2—92no—
198C2D4×C3⋊#15711yescharacteristic
subgroupG/H
centerC2D4×C3⋊#157
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC33×Q16C4
socleC2×C33C4×D4
prime radicalC2×C3×C3⋊D3C2×D4
cosocleC3×C4×C3⋊D3C23
hypercenterQ16C3⋊#157
hypocenterC33#441
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q16

Complements

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

NH ≅ G/Ntype
13. C33×Q16C4semidirect
14. C3×#3756C4semidirect
15. C3×#3746C4semidirect
43. C32×Q16Dic3semidirect
44. #3756Dic3semidirect
45. #3746Dic3semidirect
91. C3×Q16#157semidirect
120. C33#441semidirect
148. Q16C3⋊#157semidirect
171. C32#5184semidirect
196. C3#10593786semidirect

Maximal subgroups

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

subgroupindex
Q8×C3⋊#1572
C3×Q16×C3⋊D32
#117231092
#117230932
#117230922
#105937863
#51849