Finite Group Databasebeta

← Q16×C3⋊#157 #13173382

Subgroups — Q16×C3⋊#157 #13173382

Order 1728.

376 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 73 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#11723094C211yesmaximal, characteristic
5C8×C3⋊#157C211yesmaximal, characteristic
6Q16×#157—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#63769—31no—
22C8×#157—31no—
23C2×C3⋊#157—22no—
24C2×C3⋊#157D411yescharacteristic
25C2×C3⋊#157D411yescharacteristic
26C33×Q8C2×C412yes—
27C3×C4×C3⋊D3—22no—
28C3×C4×C3⋊D3C2311yescharacteristic
29C3×#634—22no—
30C3×#634C2×C412yes—
31C3⋊#622—21no—
32C33×C8C2×C411yescharacteristic
33C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
34Dic3×Q16—91nomaximal
35Q8×C3⋊D3C2×D312yes—
36C3×D3×Q8—64no—
37C4⋊#157—32no—
38C4⋊#157—62no—
39C4×#157—31no—
40C4×#157—62no—
41#3787—31no—
42C32×Q16—22no—
43C32×Q16—42no—
44C32×Q16Dic311yescharacteristic, complemented
45#3756Dic312yescomplemented
46#3746Dic311yescharacteristic, complemented
47C8×C3⋊D3C2×D311yescharacteristic
48C3×#216—64no—
49C3×Dic12—62no—
50C3×D3×C8—62no—
51C2×C3×C3⋊D3C2×D411yescharacteristic
52C3⋊#157—21no—
53C3⋊#157—42no—
54C3⋊#157Q1612yesdirect product factor
55C33×C4—22no—
56C33×C4C22×C411yescharacteristic
57C3×C3⋊Dic3—22no—
58C3×C3⋊Dic3#4111yescharacteristic
59C2×C3×Q16—91no—
60Q8×Dic3—92no—
61D3×Q16—62no—
62#835—92no—
63#818—91no—
64C8×Dic3—91no—
65C2×#157—31no—
66C2×#157—31no—
67C2×#157—62no—
68C32×Q8—24no—
69C32×Q8—44no—
70C32×Q8C2×Dic312yes—
71C4×C3⋊D3—22no—
72C4×C3⋊D3C22×D311yescharacteristic
73#634—22no—
74#634C2×Dic312yes—
75C3×C4×D3—62no—
76C3×C4×D3—124no—
77C3×Dic6—64no—
78C3×Dic6—124no—
79#622—61no—
80C32×C8—22no—
81C32×C8—42no—
82C32×C8C2×Dic311yescharacteristic
83C3⋊(C3⋊C8)C2×Dic311yescharacteristic
84C3×C3⋊C8—62no—
85C4×Q16—271noSylow
86C2×C33C4×D411yescharacteristic
87C3×C3⋊D3C2×Q1611yescharacteristic
88C3×C3⋊D3#13511yescharacteristic
89C2×C3×Q8—92no—
90D3×Q8—64no—
91C3×Q16—22no—
92C3×Q16—42no—
93C3×Q16—92no—
94C3×Q16—91no—
95C3×Q16#15711yescharacteristic, complemented
96C2×C3×C8—91no—
97#216—64no—
98C4⋊Dic3—92no—
99C4⋊Dic3—182no—
100C4×Dic3—91no—
101C4×Dic3—182no—
102C2×C3⋊C8—91no—
103Dic12—62no—
104D3×C8—62no—
105C2×C3⋊D3D3×D411yescharacteristic
106C2×C3×D3—62no—
107#157—32no—
108#157—61no—
109#157—122no—
110C32×C4—22no—
111C32×C4—22no—
112C32×C4—42no—
113C32×C4—44no—
114C32×C4—84no—
115C32×C4C22×Dic311yescharacteristic
116C3⋊Dic3—22no—
117C3⋊Dic3#23711yescharacteristic
118C3×Dic3—62no—
119C3×Dic3—124no—
120C2×Q16—91no—
121C4×Q8—272no—
122#107—271no—
123#103—272no—
124C4×C8—271no—
125C33C4×Q1611yescharacteristic, complemented, Sylow
126C3×Q8—24no—
127C3×Q8—44no—
128C3×Q8—92no—
129C3×Q8—182no—
130C3×Q8C2×#15712yes—
131C2×C3×C4—91no—
132C2×C3×C4—182no—
133C2×Dic3—91no—
134C2×Dic3—91no—
135C2×Dic3—182no—
136C4×D3—62no—
137C4×D3—124no—
138Dic6—64no—
139Dic6—124no—
140C3×C8—22no—
141C3×C8—42no—
142C3×C8—91no—
143C3×C8C2×#15711yescharacteristic
144C3⋊C8—62no—
145C3⋊C8—181no—
146C2×C32—22no—
147C2×C32—42no—
148C2×C32D4×Dic311yescharacteristic
149C3⋊D3D3×Q1611yescharacteristic
150C3⋊D3#91911yescharacteristic
151C3×D3—62no—
152C3×D3—62no—
153C2×Q8—92no—
154Q16—92no—
155Q16—91no—
156Q16C3⋊#15711yescharacteristic, direct product factor
157C2×C8—91no—
158C2×C8—271no—
159C4⋊C4—272no—
160C4⋊C4—542no—
161C42—271no—
162C42—542no—
163C22×C3—91no—
164C2×D3—62no—
165C3×C4—22no—
166C3×C4—22no—
167C3×C4—44no—
168C3×C4—42no—
169C3×C4—84no—
170C3×C4—91no—
171C3×C4—182no—
172C3×C4C22×#15711yescharacteristic
173Dic3—62no—
174Dic3—92no—
175Dic3—124no—
176Dic3—181no—
177Dic3—362no—
178C32—22no—
179C32—42no—
180C32Dic3×Q1611yescharacteristic, complemented
181Q8—92no—
182Q8—182no—
183Q8C2×C3⋊#15712yes—
184C2×C4—91no—
185C2×C4—182no—
186C2×C4—271no—
187C2×C4—271no—
188C2×C4—542no—
189C8—91no—
190C8—541no—
191C8C2×C3⋊#15711yescharacteristic
192C2×C3—22no—
193C2×C3—42no—
194C2×C3—91no—
195C2×C3—91no—
196C2×C3D4×#15711yescharacteristic
197D3—62no—
198D3—62no—
199C22—91no—
200C4—22no—
201C4—91no—
202C4—182no—
203C4—272no—
204C4—541no—
205C4—1082no—
206C4C22×C3⋊#15711yescharacteristic
207C3—22no—
208C3—42no—
209C3Q16×#15711yescharacteristic, complemented
210C2—91no—
211C2—91no—
212C2D4×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
hypocenterC33C4×Q16
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
44. C32×Q16Dic3semidirect
45. #3756Dic3semidirect
46. #3746Dic3semidirect
54. C3⋊#157Q16direct (N × H)
95. C3×Q16#157semidirect
125. C33C4×Q16semidirect
156. Q16C3⋊#157direct (N × H)
180. C32Dic3×Q16semidirect
209. C3Q16×#157semidirect

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
#117230942
C8×C3⋊#1572
Q16×#1573
Dic3×Q169