Finite Group Databasebeta

← C3×Q8×GO+(4,𝔽2) #13181042

Subgroups — C3×Q8×GO+(4,𝔽2) #13181042

Order 1728.

948 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 194 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×D32×Q8C212yesmaximal, complemented
2C3×Q8×#157C211yesmaximal, characteristic, complemented
3C3×C4×GO+(4,𝔽2)C213yesmaximal
4C3×#64222C213yesmaximal, complemented
5C3×D32⋊Q8C216yesmaximal, complemented
6Q8×GO+(4,𝔽2)C311yesmaximal, characteristic, direct product factor
7C2×C3×GO+(4,𝔽2)C2211yescharacteristic
8C3×Q8×C3⋊D3C2211yescharacteristic
9C32×D3×Q8—22no—
10C3×C4×D32C2216yes—
11C3×#3797C2216yes—
12C3×D3×Dic6—26no—
13C3×C4⋊#157C2213yes—
14C3×C4×#157C2213yes—
15C3×#3773C2216yes—
16C3×#3773C2213yes—
17C3×D32⋊C4C2216yes—
18C2×C3×D3×Q8—62no—
19D32×Q8C2×C312yescomplemented
20Q8×#157C2×C311yescharacteristic, complemented
21C4×GO+(4,𝔽2)C2×C313yes—
22#64222C2×C313yescomplemented
23D32⋊Q8C2×C316yescomplemented
24C2×C3×D32C2312yes—
25C2×C3×#157C2313yes—
26C2×C3×#157C2311yescharacteristic
27C3×GO+(4,𝔽2)Q814yesdirect product factor
28C33×Q8D411yescharacteristic, complemented
29C3×C4×C3⋊D3C2313yes—
30C3×#634D413yescomplemented
31C32×C4×D3—26no—
32C32×Dic6—26no—
33C3×#627—26no—
34C3×#624C2316yes—
35C3×D3×Dic3—26no—
36C3×D4×Q8—91nomaximal
37C2×GO+(4,𝔽2)C22×C311yescharacteristic
38C2×C32×Q8—62no—
39Q8×C3⋊D3C22×C311yescharacteristic
40C3×D3×Q8—22no—
41C3×D3×Q8—22no—
42C3×D3×Q8—42no—
43C3×D3×Q8—66no—
44C3×D3×Q8—66no—
45C3×D3×Q8—62no—
46C2×C3×C4×D3—66no—
47C2×C3×Dic6—66no—
48C4×D32C22×C316yes—
49#3797C22×C316yes—
50D3×Dic6—26no—
51C4⋊#157C22×C313yes—
52C4×#157C22×C313yes—
53#3773C22×C316yes—
54#3773C22×C313yes—
55D32⋊C4C22×C316yes—
56C2×C3×C3⋊D3C2411yescharacteristic
57C2×C32×D3—22no—
58C3×D32—22no—
59C3×D32C2×Q814yescomplemented
60C3×#157—23no—
61C3×#157C2×Q812yescomplemented
62C33×C4C2×D413yes—
63C3×C3⋊Dic3C2×D413yes—
64C32×Dic3—26no—
65C22×C3×Q8—92no—
66C2×D3×Q8—62no—
67C3×C4⋊Q8—93no—
68C3×C22⋊Q8—96no—
69C3×C4×Q8—91no—
70C3×C4×D4—93no—
71C22×C3×D3—62no—
72C2×D32C23×C312yes—
73C2×#157C23×C313yes—
74C2×#157C23×C311yescharacteristic
75GO+(4,𝔽2)C3×Q814yesdirect product factor
76C32×Q8—22no—
77C32×Q8—42no—
78C32×Q8—66no—
79C32×Q8C3×D411yescharacteristic, complemented
80C2×C32×C4—66no—
81C4×C3⋊D3C23×C313yes—
82#634C3×D413yescomplemented
83C2×C3×Dic3—66no—
84C3×C4×D3—26no—
85C3×C4×D3—26no—
86C3×C4×D3—46no—
87C3×C4×D3—66no—
88C3×C4×D3—66no—
89C3×C4×D3—66no—
90C3×Dic6—26no—
91C3×Dic6—26no—
92C3×Dic6—46no—
93C3×Dic6—66no—
94C3×Dic6—612no—
95#627—26no—
96#624C23×C316yes—
97D3×Dic3—26no—
98D4×Q8—91noSylow
99C2×C33C22×D411yescharacteristic
100C3×C3⋊D3C22×Q811yescharacteristic
101C3×C3⋊D32−1+411yescharacteristic
102C32×D3—24no—
103C2×C3×Q8—62no—
104C2×C3×Q8—62no—
105C2×C3×Q8—96no—
106C2×C3×Q8—91no—
107C2×C3×Q8—122no—
108C2×C3×Q8—186no—
109C2×C3×D4—91no—
110C22×C3×C4—96no—
111D3×Q8—22no—
112D3×Q8—66no—
113D3×Q8—62no—
114D3×Q8—66no—
115C2×C4×D3—66no—
116C2×Dic6—66no—
117C3×C4⋊C4—93no—
118C3×C4⋊C4—96no—
119C3×C4⋊C4—93no—
120C3×C22⋊C4—96no—
121C3×C42—93no—
122C22×C32—62no—
123C2×C3⋊D3C24×C311yescharacteristic
124C2×C3×D3—22no—
125C2×C3×D3—22no—
126C2×C3×D3—42no—
127C2×C3×D3—64no—
128C2×C3×D3—62no—
129C2×C3×D3—64no—
130D32—22no—
131D32C2×C3×Q814yescomplemented
132#157—23no—
133#157C2×C3×Q812yescomplemented
134C32×C4—26no—
135C32×C4—46no—
136C32×C4—66no—
137C32×C4C2×C3×D413yes—
138C3⋊Dic3C2×C3×D413yes—
139C3×Dic3—26no—
140C3×Dic3—26no—
141C3×Dic3—46no—
142C3×Dic3—66no—
143C22×Q8—92no—
144C4⋊Q8—93no—
145C22⋊Q8—96no—
146C4×Q8—91no—
147C4×D4—93no—
148C33D4×Q811yescharacteristic, complemented, Sylow
149C23×C3—92no—
150C22×D3—62no—
151C3×Q8—22no—
152C3×Q8—42no—
153C3×Q8—66no—
154C3×Q8—66no—
155C3×Q8—93no—
156C3×Q8—126no—
157C3×Q8—186no—
158C3×Q8GO+(4,𝔽2)11yescharacteristic, direct product factor
159C3×D4—94no—
160C2×C3×C4—66no—
161C2×C3×C4—66no—
162C2×C3×C4—96no—
163C2×C3×C4—91no—
164C2×C3×C4—93no—
165C2×C3×C4—93no—
166C2×C3×C4—126no—
167C2×C3×C4—186no—
168C2×Dic3—66no—
169C4×D3—26no—
170C4×D3—66no—
171C4×D3—66no—
172C4×D3—66no—
173Dic6—26no—
174Dic6—66no—
175Dic6—612no—
176C2×C32—22no—
177C2×C32—42no—
178C2×C32—64no—
179C2×C32C22×C3×D411yescharacteristic
180C3⋊D3C22×C3×Q811yescharacteristic
181C3⋊D3C3×2−1+411yescharacteristic
182C3×D3—24no—
183C3×D3—24no—
184C3×D3—44no—
185C3×D3—62no—
186C3×D3—62no—
187C2×Q8—62no—
188C2×Q8—96no—
189C2×Q8—91no—
190C2×Q8—186no—
191C2×D4—91no—
192C22×C4—96no—
193C4⋊C4—93no—
194C4⋊C4—93no—
195C4⋊C4—96no—
196C22⋊C4—96no—
197C42—93no—
198C22×C3—62no—
199C22×C3—62no—
200C22×C3—94no—
201C22×C3—91no—
202C22×C3—122no—
203C22×C3—182no—
204C2×D3—22no—
205C2×D3—64no—
206C2×D3—62no—
207C2×D3—64no—
208C3×C4—26no—
209C3×C4—46no—
210C3×C4—66no—
211C3×C4—66no—
212C3×C4—92no—
213C3×C4—93no—
214C3×C4—126no—
215C3×C4—183no—
216C3×C4C2×GO+(4,𝔽2)13yes—
217Dic3—26no—
218Dic3—66no—
219C32—22no—
220C32—42no—
221C32C3×D4×Q811yescharacteristic, complemented
222C23—92no—
223Q8—66no—
224Q8—93no—
225Q8—186no—
226Q8C3×GO+(4,𝔽2)11yescharacteristic, direct product factor
227D4—94no—
228C2×C4—66no—
229C2×C4—96no—
230C2×C4—91no—
231C2×C4—93no—
232C2×C4—93no—
233C2×C4—186no—
234C2×C3—22no—
235C2×C3—42no—
236C2×C3—64no—
237C2×C3—64no—
238C2×C3—91no—
239C2×C3—91no—
240C2×C3—124no—
241C2×C3C22×GO+(4,𝔽2)11yescharacteristic
242D3—24no—
243D3—62no—
244D3—62no—
245C22—62no—
246C22—94no—
247C22—91no—
248C22—182no—
249C4—66no—
250C4—92no—
251C4—93no—
252C4—183no—
253C4C2×C3×GO+(4,𝔽2)13yes—
254C3—22no—
255C3—42no—
256C3Q8×GO+(4,𝔽2)11yescharacteristic, direct product factor
257C2—64no—
258C2—91no—
259C2—91no—
260C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC33×Q8D4
socleC2×C33C22×D4
prime radicalC2×C3×GO+(4,𝔽2)C22
hypercenterC3×Q8GO+(4,𝔽2)
hypocenterC32C3×D4×Q8
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
1. C3×D32×Q8C2semidirect
2. C3×Q8×#157C2semidirect
4. C3×#64222C2semidirect
5. C3×D32⋊Q8C2semidirect
6. Q8×GO+(4,𝔽2)C3direct (N × H)
19. D32×Q8C2×C3semidirect
20. Q8×#157C2×C3semidirect
22. #64222C2×C3semidirect
23. D32⋊Q8C2×C3semidirect
27. C3×GO+(4,𝔽2)Q8direct (N × H)
28. C33×Q8D4semidirect
30. C3×#634D4semidirect
59. C3×D32C2×Q8semidirect
61. C3×#157C2×Q8semidirect
75. GO+(4,𝔽2)C3×Q8direct (N × H)
79. C32×Q8C3×D4semidirect
82. #634C3×D4semidirect
131. D32C2×C3×Q8semidirect
133. #157C2×C3×Q8semidirect
148. C33D4×Q8semidirect
158. C3×Q8GO+(4,𝔽2)direct (N × H)
221. C32C3×D4×Q8semidirect
226. Q8C3×GO+(4,𝔽2)direct (N × H)
256. C3Q8×GO+(4,𝔽2)direct (N × H)

Maximal subgroups

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

subgroupindex
C3×D32×Q82
C3×Q8×#1572
C3×C4×GO+(4,𝔽2)2
C3×#642222
C3×D32⋊Q82
Q8×GO+(4,𝔽2)3
C3×D4×Q89