Finite Group Databasebeta

← C3×D8×#157 #13173248

Subgroups — C3×D8×#157 #13173248

Order 1728.

464 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 88 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
1C3×D4×#157C212yesmaximal, complemented
2C3×D8×C3⋊D3C211yesmaximal, characteristic
3C3×#63782C212yesmaximal, complemented
4C3×#63769C211yesmaximal, characteristic, complemented
5C3×C8×#157C211yesmaximal, characteristic, complemented
6D8×#157C311yesmaximal, characteristic, direct product factor
7C22×C3×#157—22no—
8C3×D4×C3⋊D3C2212yes—
9C3×C22⋊#157—22no—
10C3×C4⋊#157C2212yes—
11C3×C4×#157C2211yescharacteristic
12C3×#3787C2211yescharacteristic
13C33×D8C411yescharacteristic, complemented
14C3×#3753C412yescomplemented
15C3×#3745C411yescharacteristic, complemented
16C3×C8×C3⋊D3C2211yescharacteristic
17D4×#157C2×C312yescomplemented
18D8×C3⋊D3C2×C311yescharacteristic
19C3×D3×D8—62no—
20#63782C2×C312yescomplemented
21#63769C2×C311yescharacteristic, complemented
22C8×#157C2×C311yescharacteristic, complemented
23C22×C3×C3⋊D3—22no—
24C2×C3×#157—22no—
25C2×C3×#157—44no—
26C2×C3×#157D411yescharacteristic
27C2×C3×#157D411yescharacteristic
28C33×D4C2×C412yescomplemented
29C3×#638—22no—
30C3×C4⋊(C3⋊D3)C2×C412yescomplemented
31C3×C4×C3⋊D3C2311yescharacteristic
32C3×#622—21no—
33C33×C8C2×C411yescharacteristic, complemented
34C3×C3⋊(C3⋊C8)C2×C411yescharacteristic, complemented
35C3×C4×D8—91nomaximal
36C22×#157—22no—
37D4×C3⋊D3C22×C312yes—
38C3×D3×D4—64no—
39C22⋊#157—22no—
40C4⋊#157C22×C312yes—
41C4×#157C22×C311yescharacteristic
42#3787C22×C311yescharacteristic
43C32×D8—22no—
44C32×D8—42no—
45C32×D8C3×C411yescharacteristic, complemented
46#3753C3×C412yescomplemented
47#3745C3×C411yescharacteristic, complemented
48C8×C3⋊D3C22×C311yescharacteristic
49C3×#213—64no—
50C3×D24—62no—
51C3×D3×C8—62no—
52C22×C33—22no—
53C2×C3×C3⋊D3—22no—
54C2×C3×C3⋊D3—42no—
55C2×C3×C3⋊D3—42no—
56C2×C3×C3⋊D3C2×D411yescharacteristic
57C3×#157—21no—
58C3×#157—42no—
59C3×#157D812yesdirect product factor
60C33×C4C22×C411yescharacteristic
61C3×C3⋊Dic3#4111yescharacteristic
62C2×C3×D8—91no—
63C3×C4×D4—92no—
64D3×D8—62no—
65C3×#107—91no—
66C3×#102—92no—
67C3×C4×C8—91no—
68C22×C3⋊D3—22no—
69C22×C3×D3—124no—
70C2×#157—22no—
71C2×#157—44no—
72C2×#157C3×D411yescharacteristic
73C2×#157C3×D411yescharacteristic
74C32×D4—24no—
75C32×D4—44no—
76C32×D4C2×C3×C412yescomplemented
77#638—22no—
78C4⋊(C3⋊D3)C2×C3×C412yescomplemented
79C4×C3⋊D3C23×C311yescharacteristic
80C3×#67—124no—
81C3×D12—64no—
82C3×C4×D3—62no—
83#622—21no—
84C32×C8—22no—
85C32×C8—42no—
86C32×C8C2×C3×C411yescharacteristic, complemented
87C3⋊(C3⋊C8)C2×C3×C411yescharacteristic, complemented
88C3×C3⋊C8—62no—
89C4×D8—91noSylow
90C2×C33—42no—
91C2×C33C4×D411yescharacteristic
92C3×C3⋊D3—42no—
93C3×C3⋊D3C2×D811yescharacteristic
94C3×C3⋊D3#13511yescharacteristic
95C2×C3×D4—92no—
96C22×C3×C4—182no—
97D3×D4—64no—
98C3×D8—22no—
99C3×D8—42no—
100C3×D8—92no—
101C3×D8—91no—
102C3×D8#15711yescharacteristic, direct product factor
103C2×C3×C8—91no—
104C2×C3×C8—91no—
105C3×C4⋊C4—92no—
106C3×C22⋊C4—182no—
107C3×C42—91no—
108#213—64no—
109D24—62no—
110D3×C8—62no—
111C22×C32—22no—
112C22×C32—44no—
113C22×C32—84no—
114C2×C3⋊D3—22no—
115C2×C3⋊D3—42no—
116C2×C3⋊D3—42no—
117C2×C3⋊D3C2×C3×D411yescharacteristic
118C2×C3×D3—62no—
119C2×C3×D3—124no—
120C2×C3×D3—244no—
121C2×C3×D3—244no—
122#157—21no—
123#157—42no—
124#157C3×D812yesdirect product factor
125C32×C4—22no—
126C32×C4—42no—
127C32×C4C22×C3×C411yescharacteristic
128C3⋊Dic3C3×#4111yescharacteristic
129C3×Dic3—62no—
130C2×D8—91no—
131C4×D4—92no—
132#107—91no—
133#102—92no—
134C4×C8—91no—
135C33C4×D811yescharacteristic, complemented, Sylow
136C23×C3—182no—
137C22×D3—124no—
138C3×D4—24no—
139C3×D4—44no—
140C3×D4—92no—
141C3×D4—182no—
142C3×D4C2×#15712yescomplemented
143C2×C3×C4—91no—
144C2×C3×C4—91no—
145C2×C3×C4—91no—
146C2×C3×C4—182no—
147C2×C3×C4—364no—
148#67—124no—
149D12—64no—
150C4×D3—62no—
151C3×C8—22no—
152C3×C8—42no—
153C3×C8—91no—
154C3×C8—181no—
155C3×C8C2×#15711yescharacteristic, complemented
156C3⋊C8—62no—
157C2×C32—22no—
158C2×C32—42no—
159C2×C32—42no—
160C2×C32—84no—
161C2×C32—164no—
162C2×C32C3×C4×D411yescharacteristic
163C3⋊D3—42no—
164C3⋊D3C2×C3×D811yescharacteristic
165C3⋊D3C3×#13511yescharacteristic
166C3×D3—62no—
167C3×D3—62no—
168C3×D3—244no—
169C2×D4—92no—
170C22×C4—182no—
171D8—92no—
172D8—91no—
173D8C3×#15711yescharacteristic, direct product factor
174C2×C8—91no—
175C2×C8—91no—
176C4⋊C4—92no—
177C22⋊C4—182no—
178C42—91no—
179C22×C3—22no—
180C22×C3—44no—
181C22×C3—84no—
182C22×C3—91no—
183C22×C3—182no—
184C22×C3—362no—
185C22×C3—362no—
186C2×D3—62no—
187C2×D3—124no—
188C2×D3—244no—
189C2×D3—244no—
190C3×C4—22no—
191C3×C4—42no—
192C3×C4—92no—
193C3×C4—91no—
194C3×C4—181no—
195C3×C4—362no—
196C3×C4C22×#15711yescharacteristic
197Dic3—62no—
198C32—22no—
199C32—42no—
200C32C3×C4×D811yescharacteristic, complemented
201C23—182no—
202D4—92no—
203D4—182no—
204D4C2×C3×#15712yescomplemented
205C2×C4—91no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4—182no—
209C2×C4—364no—
210C8—91no—
211C8—181no—
212C8C2×C3×#15711yescharacteristic, complemented
213C2×C3—22no—
214C2×C3—42no—
215C2×C3—42no—
216C2×C3—84no—
217C2×C3—91no—
218C2×C3—91no—
219C2×C3—164no—
220C2×C3—362no—
221C2×C3D4×#15711yescharacteristic
222D3—62no—
223D3—62no—
224D3—244no—
225C22—22no—
226C22—91no—
227C22—182no—
228C22—362no—
229C22—362no—
230C4—92no—
231C4—91no—
232C4—181no—
233C4—362no—
234C4C22×C3×#15711yescharacteristic
235C3—22no—
236C3—42no—
237C3D8×#15711yescharacteristic, direct product factor
238C2—42no—
239C2—91no—
240C2—91no—
241C2—362no—
242C2C3×D4×#15711yescharacteristic
subgroupG/H
centerC2×C3D4×#157
commutatorC32×C4C22×C3×C4
FrattiniC4C22×C3×#157
FittingC33×D8C4
socleC2×C33C4×D4
prime radicalC3×D8×C3⋊D3C2
cosocleC4×C3⋊D3C23×C3
hypercenterC3×D8#157
hypocenterC32C3×C4×D8
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D8

Complements

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

NH ≅ G/Ntype
1. C3×D4×#157C2semidirect
3. C3×#63782C2semidirect
4. C3×#63769C2semidirect
5. C3×C8×#157C2semidirect
6. D8×#157C3direct (N × H)
13. C33×D8C4semidirect
14. C3×#3753C4semidirect
15. C3×#3745C4semidirect
17. D4×#157C2×C3semidirect
20. #63782C2×C3semidirect
21. #63769C2×C3semidirect
22. C8×#157C2×C3semidirect
28. C33×D4C2×C4semidirect
30. C3×C4⋊(C3⋊D3)C2×C4semidirect
33. C33×C8C2×C4semidirect
34. C3×C3⋊(C3⋊C8)C2×C4semidirect
45. C32×D8C3×C4semidirect
46. #3753C3×C4semidirect
47. #3745C3×C4semidirect
59. C3×#157D8direct (N × H)
76. C32×D4C2×C3×C4semidirect
78. C4⋊(C3⋊D3)C2×C3×C4semidirect
86. C32×C8C2×C3×C4semidirect
87. C3⋊(C3⋊C8)C2×C3×C4semidirect
102. C3×D8#157direct (N × H)
124. #157C3×D8direct (N × H)
135. C33C4×D8semidirect
142. C3×D4C2×#157semidirect
155. C3×C8C2×#157semidirect
173. D8C3×#157direct (N × H)
200. C32C3×C4×D8semidirect
204. D4C2×C3×#157semidirect
212. C8C2×C3×#157semidirect
237. C3D8×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D4×#1572
C3×D8×C3⋊D32
C3×#637822
C3×#637692
C3×C8×#1572
D8×#1573
C3×C4×D89