Finite Group Databasebeta

← C2×C3×Q8×#157 #13181499

Subgroups — C2×C3×Q8×#157 #13181499

Order 1728.

984 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 312 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
1C2×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2C3×Q8×#157C218yesmaximal, direct product factor
3C2×C3×C4⋊#157C213yesmaximal
4C2×C3×C4×#157C213yesmaximal
5C2×Q8×#157C311yesmaximal, characteristic, direct product factor
6C22×C3×#157C2213yes—
7C22×C3×#157C2211yescharacteristic
8C2×C33×Q8C411yescharacteristic, complemented
9C3×Q8×C3⋊D3C414yescomplemented
10C3×Q8×C3⋊D3C2214yes—
11C2×C3×C4×C3⋊D3C2213yes—
12C2×C3×#634C413yescomplemented
13C3×C4⋊#157C22112yes—
14C3×C4×#157C22112yes—
15C2×Q8×C3⋊D3C2×C311yescharacteristic
16C2×C3×D3×Q8—62no—
17Q8×#157C2×C318yesdirect product factor
18C2×C4⋊#157C2×C313yes—
19C2×C4×#157C2×C313yes—
20C22×C3×C3⋊D3C2311yescharacteristic
21C2×C3×#157—26no—
22C2×C3×#157Q814yesdirect product factor
23C2×C3×#157C2316yes—
24C2×C3×#157C2312yes—
25C33×Q8C2×C414yescomplemented
26C2×C33×C4C2×C413yes—
27C2×C3×C3⋊Dic3C2×C413yes—
28C3×C4×C3⋊D3C2×C416yes—
29C3×C4×C3⋊D3C2316yes—
30C3×#634C2×C4112yescomplemented
31C2×C3×C4×Q8—91nomaximal
32C22×#157C22×C313yes—
33C22×#157C22×C311yescharacteristic
34C2×C32×Q8—22no—
35C2×C32×Q8—42no—
36C2×C32×Q8C3×C411yescharacteristic, complemented
37Q8×C3⋊D3C3×C414yescomplemented
38Q8×C3⋊D3C22×C314yes—
39C2×C4×C3⋊D3C22×C313yes—
40C2×#634C3×C413yescomplemented
41C3×D3×Q8—68no—
42C3×D3×Q8—68no—
43C2×C3×C4×D3—66no—
44C2×C3×Dic6—66no—
45C4⋊#157C22×C3112yes—
46C4×#157C22×C3112yes—
47C22×C33C22×C411yescharacteristic
48C2×C3×C3⋊D3C22×C411yescharacteristic
49C2×C3×C3⋊D3C2×Q812yes—
50C2×C3×C3⋊D3#4112yes—
51C2×C3×C3⋊D3C2411yescharacteristic
52C3×#157—26no—
53C3×#157C2×Q814yesdirect product factor
54C33×C4C22×C416yes—
55C3×C3⋊Dic3C22×C416yes—
56C22×C3×Q8—91no—
57C2×D3×Q8—62no—
58C3×C4×Q8—98no—
59C2×C3×C4⋊C4—93no—
60C2×C3×C42—93no—
61C22×C3⋊D3C23×C311yescharacteristic
62C22×C3×D3—62no—
63C2×#157—26no—
64C2×#157C3×Q814yesdirect product factor
65C2×#157C23×C316yes—
66C2×#157C23×C312yes—
67C32×Q8—28no—
68C32×Q8—48no—
69C32×Q8C2×C3×C414yescomplemented
70C2×C32×C4—26no—
71C2×C32×C4—46no—
72C2×C32×C4C2×C3×C413yes—
73C2×C3⋊Dic3C2×C3×C413yes—
74C4×C3⋊D3C2×C3×C416yes—
75C4×C3⋊D3C23×C316yes—
76#634C2×C3×C4112yescomplemented
77C2×C3×Dic3—66no—
78C3×C4×D3—612no—
79C3×C4×D3—612no—
80C3×Dic6—624no—
81C2×C4×Q8—91noSylow
82C2×C33C4×Q812yescomplemented
83C2×C33C23×C411yescharacteristic
84C3×C3⋊D3C4×Q812yescomplemented
85C3×C3⋊D3C22×Q811yescharacteristic
86C3×C3⋊D3C2×#4111yescharacteristic
87C2×C3×Q8—22no—
88C2×C3×Q8—42no—
89C2×C3×Q8—94no—
90C2×C3×Q8—94no—
91C2×C3×Q8—93no—
92C2×C3×Q8#15711yescharacteristic, direct product factor
93C22×C3×C4—91no—
94C22×C3×C4—93no—
95C22×C3×C4—93no—
96D3×Q8—68no—
97D3×Q8—68no—
98C2×C4×D3—66no—
99C2×Dic6—66no—
100C3×C4⋊C4—912no—
101C3×C42—912no—
102C22×C32—22no—
103C22×C32—42no—
104C22×C32C22×C3×C411yescharacteristic
105C2×C3⋊D3C22×C3×C411yescharacteristic
106C2×C3⋊D3C2×C3×Q812yes—
107C2×C3⋊D3C3×#4112yes—
108C2×C3⋊D3C24×C311yescharacteristic
109C2×C3×D3—62no—
110C2×C3×D3—64no—
111C2×C3×D3—64no—
112C2×C3×D3—62no—
113#157—26no—
114#157C2×C3×Q814yesdirect product factor
115C32×C4—212no—
116C32×C4—412no—
117C32×C4C22×C3×C416yes—
118C3⋊Dic3C22×C3×C416yes—
119C3×Dic3—612no—
120C22×Q8—91no—
121C4×Q8—98no—
122C2×C4⋊C4—93no—
123C2×C42—93no—
124C33C2×C4×Q811yescharacteristic, complemented, Sylow
125C23×C3—91no—
126C22×D3—62no—
127C3×Q8—28no—
128C3×Q8—48no—
129C3×Q8—912no—
130C3×Q8C2×#15714yesdirect product factor
131C2×C3×C4—26no—
132C2×C3×C4—46no—
133C2×C3×C4—94no—
134C2×C3×C4—92no—
135C2×C3×C4—93no—
136C2×C3×C4—96no—
137C2×C3×C4—96no—
138C2×C3×C4—96no—
139C2×C3×C4—186no—
140C2×C3×C4C2×#15713yes—
141C2×Dic3—66no—
142C4×D3—612no—
143C4×D3—612no—
144Dic6—624no—
145C2×C32—24no—
146C2×C32—22no—
147C2×C32—44no—
148C2×C32—42no—
149C2×C32C3×C4×Q812yescomplemented
150C2×C32C23×C3×C411yescharacteristic
151C3⋊D3C3×C4×Q812yescomplemented
152C3⋊D3C22×C3×Q811yescharacteristic
153C3⋊D3C2×C3×#4111yescharacteristic
154C3×D3—64no—
155C3×D3—62no—
156C3×D3—62no—
157C2×Q8—94no—
158C2×Q8—94no—
159C2×Q8—93no—
160C2×Q8C3×#15711yescharacteristic, direct product factor
161C22×C4—91no—
162C22×C4—93no—
163C22×C4—93no—
164C4⋊C4—912no—
165C42—912no—
166C22×C3—22no—
167C22×C3—42no—
168C22×C3—92no—
169C22×C3—92no—
170C22×C3—91no—
171C22×C3—91no—
172C22×C3C22×#15711yescharacteristic
173C2×D3—62no—
174C2×D3—62no—
175C2×D3—64no—
176C2×D3—64no—
177C3×C4—212no—
178C3×C4—412no—
179C3×C4—94no—
180C3×C4—96no—
181C3×C4—186no—
182C3×C4C22×#15716yes—
183Dic3—612no—
184C32—22no—
185C32—42no—
186C32C2×C3×C4×Q811yescharacteristic, complemented
187C23—91no—
188Q8—912no—
189Q8C2×C3×#15714yesdirect product factor
190C2×C4—92no—
191C2×C4—94no—
192C2×C4—93no—
193C2×C4—96no—
194C2×C4—96no—
195C2×C4—96no—
196C2×C4—186no—
197C2×C4C2×C3×#15713yes—
198C2×C3—22no—
199C2×C3—24no—
200C2×C3—42no—
201C2×C3—44no—
202C2×C3—91no—
203C2×C3—91no—
204C2×C3—92no—
205C2×C3Q8×#15712yesdirect product factor
206C2×C3C23×#15711yescharacteristic
207D3—64no—
208D3—62no—
209D3—62no—
210C22—91no—
211C22—91no—
212C22—92no—
213C22—92no—
214C22C22×C3×#15711yescharacteristic
215C4—94no—
216C4—96no—
217C4—186no—
218C4C22×C3×#15716yes—
219C3—22no—
220C3—42no—
221C3C2×Q8×#15711yescharacteristic, direct product factor
222C2—91no—
223C2—91no—
224C2—92no—
225C2C3×Q8×#15712yesdirect product factor
226C2C23×C3×#15711yescharacteristic
subgroupG/H
centerC22×C3C22×#157
commutatorC2×C32C23×C3×C4
FrattiniC2C23×C3×#157
FittingC2×C33×Q8C4
socleC22×C33C22×C4
prime radicalC22×C3×C3⋊D3C23
cosocleC2×C3⋊D3C24×C3
hypercenterC2×C3×Q8#157
hypocenterC32C2×C3×C4×Q8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×Q8

Complements

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

NH ≅ G/Ntype
2. C3×Q8×#157C2direct (N × H)
5. C2×Q8×#157C3direct (N × H)
8. C2×C33×Q8C4semidirect
9. C3×Q8×C3⋊D3C4semidirect
12. C2×C3×#634C4semidirect
17. Q8×#157C2×C3direct (N × H)
22. C2×C3×#157Q8direct (N × H)
25. C33×Q8C2×C4semidirect
30. C3×#634C2×C4semidirect
36. C2×C32×Q8C3×C4semidirect
37. Q8×C3⋊D3C3×C4semidirect
40. C2×#634C3×C4semidirect
53. C3×#157C2×Q8direct (N × H)
64. C2×#157C3×Q8direct (N × H)
69. C32×Q8C2×C3×C4semidirect
76. #634C2×C3×C4semidirect
82. C2×C33C4×Q8semidirect
84. C3×C3⋊D3C4×Q8semidirect
92. C2×C3×Q8#157direct (N × H)
114. #157C2×C3×Q8direct (N × H)
124. C33C2×C4×Q8semidirect
130. C3×Q8C2×#157direct (N × H)
149. C2×C32C3×C4×Q8semidirect
151. C3⋊D3C3×C4×Q8semidirect
160. C2×Q8C3×#157direct (N × H)
186. C32C2×C3×C4×Q8semidirect
189. Q8C2×C3×#157direct (N × H)
205. C2×C3Q8×#157direct (N × H)
221. C3C2×Q8×#157direct (N × H)
225. C2C3×Q8×#157direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
C3×Q8×#1572
C2×C3×C4⋊#1572
C2×C3×C4×#1572
C2×Q8×#1573
C2×C3×C4×Q89