Finite Group Databasebeta

← C3×C8×GO+(4,𝔽2) #13166650

Subgroups — C3×C8×GO+(4,𝔽2) #13166650

Order 1728.

464 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 98 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×C4×GO+(4,𝔽2)C211yesmaximal, characteristic
2C3×D32×C8C212yesmaximal, complemented
3C3×C8×#157C211yesmaximal, characteristic, complemented
4C3×#63732C211yesmaximal, characteristic, complemented
5C3×D32⋊C8C212yesmaximal, complemented
6C8×GO+(4,𝔽2)C311yesmaximal, characteristic, direct product factor
7C2×C3×GO+(4,𝔽2)C411yescharacteristic
8C3×C4×D32C2212yes—
9C3×C4×#157C2211yescharacteristic
10C3×#3787C2211yescharacteristic
11C3×#3773C411yescharacteristic
12C3×D32⋊C4C412yes—
13C3×C8×C3⋊D3C2211yescharacteristic
14C32×D3×C8—22no—
15C3×#3710C2212yes—
16C3×D3×C3⋊C8—22no—
17C4×GO+(4,𝔽2)C2×C311yescharacteristic
18C2×C3×D3×C8—62no—
19D32×C8C2×C312yescomplemented
20C8×#157C2×C311yescharacteristic, complemented
21#63732C2×C311yescharacteristic, complemented
22D32⋊C8C2×C312yescomplemented
23C2×C3×D32C2×C412yes—
24C2×C3×#157C2×C411yescharacteristic
25C2×C3×#157C2×C411yescharacteristic
26C3×GO+(4,𝔽2)C814yesdirect product factor
27C3×C4×C3⋊D3C2311yescharacteristic
28C32×C4×D3—22no—
29C3×#624C2×C412yes—
30C3×D3×Dic3—22no—
31C3×#622—21no—
32C33×C8D411yescharacteristic, complemented
33C3×C3⋊(C3⋊C8)D411yescharacteristic, complemented
34C32×C3⋊C8—22no—
35C3×C8×D4—91nomaximal
36C2×GO+(4,𝔽2)C3×C411yescharacteristic
37C2×C3×C4×D3—62no—
38C4×D32C22×C312yes—
39C4×#157C22×C311yescharacteristic
40#3787C22×C311yescharacteristic
41#3773C3×C411yescharacteristic
42D32⋊C4C3×C412yes—
43C2×C32×C8—62no—
44C8×C3⋊D3C22×C311yescharacteristic
45C2×C3×C3⋊C8—62no—
46C3×D3×C8—22no—
47C3×D3×C8—22no—
48C3×D3×C8—42no—
49C3×D3×C8—62no—
50C3×D3×C8—62no—
51C3×D3×C8—62no—
52#3710C22×C312yes—
53D3×C3⋊C8—22no—
54C2×C3×C3⋊D3C22×C411yescharacteristic
55C2×C32×D3—22no—
56C3×D32—22no—
57C3×D32C2×C814yescomplemented
58C3×#157—21no—
59C3×#157C2×C812yescomplemented
60C33×C4C2×D411yescharacteristic
61C3×C3⋊Dic3#4111yescharacteristic
62C32×Dic3—22no—
63C22×C3×C8—92no—
64C3×C4×D4—91no—
65C2×D3×C8—62no—
66C3×C4⋊C8—91no—
67C3×C22⋊C8—92no—
68C3×C4×C8—91no—
69C22×C3×D3—62no—
70C2×D32C2×C3×C412yes—
71C2×#157C2×C3×C411yescharacteristic
72C2×#157C2×C3×C411yescharacteristic
73GO+(4,𝔽2)C3×C814yesdirect product factor
74C2×C32×C4—62no—
75C4×C3⋊D3C23×C311yescharacteristic
76C2×C3×Dic3—62no—
77C3×C4×D3—22no—
78C3×C4×D3—22no—
79C3×C4×D3—42no—
80C3×C4×D3—62no—
81C3×C4×D3—62no—
82C3×C4×D3—62no—
83#624C2×C3×C412yes—
84D3×Dic3—22no—
85#622—21no—
86C32×C8—22no—
87C32×C8—42no—
88C32×C8—62no—
89C32×C8C3×D411yescharacteristic, complemented
90C3⋊(C3⋊C8)C3×D411yescharacteristic, complemented
91C3×C3⋊C8—22no—
92C3×C3⋊C8—22no—
93C3×C3⋊C8—42no—
94C3×C3⋊C8—62no—
95C8×D4—91noSylow
96C2×C33C4×D411yescharacteristic
97C3×C3⋊D3C22×C811yescharacteristic
98C3×C3⋊D3#13111yescharacteristic
99C32×D3—24no—
100C2×C3×D4—91no—
101C22×C3×C4—92no—
102C2×C4×D3—62no—
103C2×C3×C8—62no—
104C2×C3×C8—62no—
105C2×C3×C8—92no—
106C2×C3×C8—91no—
107C2×C3×C8—91no—
108C2×C3×C8—122no—
109C2×C3×C8—182no—
110C3×C4⋊C4—91no—
111C3×C22⋊C4—92no—
112C3×C42—91no—
113C2×C3⋊C8—62no—
114D3×C8—22no—
115D3×C8—62no—
116D3×C8—62no—
117D3×C8—62no—
118C22×C32—62no—
119C2×C3⋊D3C22×C3×C411yescharacteristic
120C2×C3×D3—22no—
121C2×C3×D3—22no—
122C2×C3×D3—42no—
123C2×C3×D3—64no—
124C2×C3×D3—62no—
125C2×C3×D3—64no—
126D32—22no—
127D32C2×C3×C814yescomplemented
128#157—21no—
129#157C2×C3×C812yescomplemented
130C32×C4—22no—
131C32×C4—42no—
132C32×C4—62no—
133C32×C4C2×C3×D411yescharacteristic
134C3⋊Dic3C3×#4111yescharacteristic
135C3×Dic3—22no—
136C3×Dic3—22no—
137C3×Dic3—42no—
138C3×Dic3—62no—
139C22×C8—92no—
140C4×D4—91no—
141C4⋊C8—91no—
142C22⋊C8—92no—
143C4×C8—91no—
144C33C8×D411yescharacteristic, complemented, Sylow
145C23×C3—92no—
146C22×D3—62no—
147C3×D4—94no—
148C2×C3×C4—62no—
149C2×C3×C4—62no—
150C2×C3×C4—92no—
151C2×C3×C4—91no—
152C2×C3×C4—91no—
153C2×C3×C4—91no—
154C2×C3×C4—122no—
155C2×C3×C4—182no—
156C2×Dic3—62no—
157C4×D3—22no—
158C4×D3—62no—
159C4×D3—62no—
160C4×D3—62no—
161C3×C8—22no—
162C3×C8—42no—
163C3×C8—62no—
164C3×C8—62no—
165C3×C8—91no—
166C3×C8—122no—
167C3×C8—181no—
168C3×C8GO+(4,𝔽2)11yescharacteristic, direct product factor
169C3⋊C8—22no—
170C3⋊C8—62no—
171C2×C32—22no—
172C2×C32—42no—
173C2×C32—64no—
174C2×C32C3×C4×D411yescharacteristic
175C3⋊D3C22×C3×C811yescharacteristic
176C3⋊D3C3×#13111yescharacteristic
177C3×D3—24no—
178C3×D3—24no—
179C3×D3—44no—
180C3×D3—62no—
181C3×D3—62no—
182C2×D4—91no—
183C22×C4—92no—
184C2×C8—62no—
185C2×C8—92no—
186C2×C8—91no—
187C2×C8—91no—
188C2×C8—182no—
189C4⋊C4—91no—
190C22⋊C4—92no—
191C42—91no—
192C22×C3—62no—
193C22×C3—62no—
194C22×C3—94no—
195C22×C3—91no—
196C22×C3—122no—
197C22×C3—182no—
198C2×D3—22no—
199C2×D3—64no—
200C2×D3—62no—
201C2×D3—64no—
202C3×C4—22no—
203C3×C4—42no—
204C3×C4—62no—
205C3×C4—62no—
206C3×C4—92no—
207C3×C4—91no—
208C3×C4—122no—
209C3×C4—181no—
210C3×C4C2×GO+(4,𝔽2)11yescharacteristic
211Dic3—22no—
212Dic3—62no—
213C32—22no—
214C32—42no—
215C32C3×C8×D411yescharacteristic, complemented
216C23—92no—
217D4—94no—
218C2×C4—62no—
219C2×C4—92no—
220C2×C4—91no—
221C2×C4—91no—
222C2×C4—91no—
223C2×C4—182no—
224C8—62no—
225C8—91no—
226C8—181no—
227C8C3×GO+(4,𝔽2)11yescharacteristic, direct product factor
228C2×C3—22no—
229C2×C3—42no—
230C2×C3—64no—
231C2×C3—64no—
232C2×C3—91no—
233C2×C3—91no—
234C2×C3—124no—
235C2×C3C4×GO+(4,𝔽2)11yescharacteristic
236D3—24no—
237D3—62no—
238D3—62no—
239C22—62no—
240C22—94no—
241C22—91no—
242C22—182no—
243C4—62no—
244C4—92no—
245C4—91no—
246C4—181no—
247C4C2×C3×GO+(4,𝔽2)11yescharacteristic
248C3—22no—
249C3—42no—
250C3C8×GO+(4,𝔽2)11yescharacteristic, direct product factor
251C2—64no—
252C2—91no—
253C2—91no—
254C2C3×C4×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
center, hypercenterC3×C8GO+(4,𝔽2)
commutatorC3⋊D3C22×C3×C8
FrattiniC4C2×C3×GO+(4,𝔽2)
FittingC33×C8D4
socleC2×C33C4×D4
prime radicalC2×C3×GO+(4,𝔽2)C4
cosocleC4×C3⋊D3C23×C3
hypocenterC32C3×C8×D4
LCSG ▷ C3⋊D3 ▷ C32
UCSC1 ◁ C3×C8

Complements

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

NH ≅ G/Ntype
2. C3×D32×C8C2semidirect
3. C3×C8×#157C2semidirect
4. C3×#63732C2semidirect
5. C3×D32⋊C8C2semidirect
6. C8×GO+(4,𝔽2)C3direct (N × H)
19. D32×C8C2×C3semidirect
20. C8×#157C2×C3semidirect
21. #63732C2×C3semidirect
22. D32⋊C8C2×C3semidirect
26. C3×GO+(4,𝔽2)C8direct (N × H)
32. C33×C8D4semidirect
33. C3×C3⋊(C3⋊C8)D4semidirect
57. C3×D32C2×C8semidirect
59. C3×#157C2×C8semidirect
73. GO+(4,𝔽2)C3×C8direct (N × H)
89. C32×C8C3×D4semidirect
90. C3⋊(C3⋊C8)C3×D4semidirect
127. D32C2×C3×C8semidirect
129. #157C2×C3×C8semidirect
144. C33C8×D4semidirect
168. C3×C8GO+(4,𝔽2)direct (N × H)
215. C32C3×C8×D4semidirect
227. C8C3×GO+(4,𝔽2)direct (N × H)
250. C3C8×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×C4×GO+(4,𝔽2)2
C3×D32×C82
C3×C8×#1572
C3×#637322
C3×D32⋊C82
C8×GO+(4,𝔽2)3
C3×C8×D49