Finite Group Databasebeta

← C3×#13309 #10631916

Subgroups — C3×#13309 #10631916

Order 768.

568 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 114 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×#3414C211yesmaximal, characteristic, complemented
2C3×#3050C212yesmaximal, complemented
3C3×#2145C212yesmaximal, complemented
4C3×#2112C211yesmaximal, characteristic, complemented
5C3×#1384C211yesmaximal, characteristic, complemented
6#13309C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#135C2212yes—
8C23×C3×C8C2211yescharacteristic
9C3×#525—22no—
10C3×#510—22no—
11C3×#509—22no—
12C3×#508—22no—
13C2×C3×M32C2211yescharacteristic
14C2×C3×M32C2211yescharacteristic
15C3×#467—22no—
16C3×#466—22no—
17C3×#465—24no—
18C3×#427C2212yes—
19C3×#416—22no—
20C3×#360—24no—
21C3×C22⋊C16—22no—
22#3414C2×C311yescharacteristic, complemented
23#3050C2×C312yescomplemented
24#2145C2×C312yescomplemented
25#2112C2×C311yescharacteristic, complemented
26#1384C2×C311yescharacteristic, complemented
27C2×C3×#41—22no—
28C23×C3×C4D411yescharacteristic
29C3×#135—22no—
30C3×#135—22no—
31C3×#135—22no—
32C3×#135—22no—
33C2×C3×Q16D412yescomplemented
34C2×C3×SD16—22no—
35C2×C3×D8D412yescomplemented
36C22×C3×C8—22no—
37C22×C3×C8—22no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C22×C3×C8D411yescharacteristic
41C22×C3×C8C2311yescharacteristic
42C3×#123—42no—
43C3×C22⋊Q8—24no—
44C3×#121—24no—
45C3×#120—42no—
46C3×C4×D4—44no—
47C3×#117—22no—
48C3×Q32—22no—
49C3×SD32—24no—
50C3×D16—22no—
51C3×M32—22no—
52C3×M32—22no—
53C2×C3×C16—21no—
54C2×C3×C16D412yescomplemented
55C3×#107—22no—
56C3×#106D414yescomplemented
57C3×#103—24no—
58C3×#102—24no—
59C2×#135C22×C312yes—
60C23×C8C22×C311yescharacteristic
61#525—22no—
62#510—22no—
63#509—22no—
64#508—22no—
65C2×M32C22×C311yescharacteristic
66C2×M32C22×C311yescharacteristic
67#467—22no—
68#466—22no—
69#465—24no—
70#427C22×C312yes—
71#416—22no—
72#360—24no—
73C22⋊C16—22no—
74C24×C3D811yescharacteristic
75C3×#41—42no—
76C3×#41—42no—
77C3×#41—42no—
78C3×#41—42no—
79C2×C3×Q8—22no—
80C2×C3×D4—22no—
81C2×C3×D4—44no—
82C2×C3×D4—42no—
83C22×C3×C4—22no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—22no—
87C22×C3×C4—42no—
88C22×C3×C4D811yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C3×Q16—22no—
91C3×Q16—22no—
92C3×SD16—42no—
93C3×SD16—42no—
94C3×D8—22no—
95C3×D8—22no—
96C2×C3×C8—22no—
97C2×C3×C8—21no—
98C2×C3×C8—41no—
99C2×C3×C8—42no—
100C2×C3×C8—41no—
101C2×C3×C8C2×D412yes—
102C2×C3×C8C2×D411yescharacteristic
103C2×C3×C8C2×D412yes—
104C2×C3×C8#4111yescharacteristic
105C3×C4⋊C4—24no—
106C3×C4⋊C4—44no—
107C3×C22⋊C4—44no—
108C3×C22⋊C4—44no—
109C3×C22⋊C4—42no—
110C3×C42—42no—
111C3×C16—22no—
112C3×C16—41no—
113C2×#41—22no—
114C23×C4C3×D411yescharacteristic
115#135—22no—
116#135—22no—
117#135—22no—
118#135—22no—
119C2×Q16C3×D412yescomplemented
120C2×SD16—22no—
121C2×D8C3×D412yescomplemented
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8C3×D411yescharacteristic
127C22×C8C23×C311yescharacteristic
128#123—42no—
129C22⋊Q8—24no—
130#121—24no—
131#120—42no—
132C4×D4—44no—
133#117—22no—
134Q32—22no—
135SD32—24no—
136D16—22no—
137M32—22no—
138M32—22no—
139C2×C16—21no—
140C2×C16C3×D412yescomplemented
141#107—22no—
142#106C3×D414yescomplemented
143#103—24no—
144#102—24no—
145C23×C3—22no—
146C23×C3—21no—
147C23×C3—42no—
148C23×C3—42no—
149C23×C3C2×D811yescharacteristic
150C3×Q8—42no—
151C3×Q8—42no—
152C3×D4—42no—
153C3×D4—42no—
154C3×D4—88no—
155C3×D4—82no—
156C3×D4—82no—
157C2×C3×C4—22no—
158C2×C3×C4—21no—
159C2×C3×C4—41no—
160C2×C3×C4—42no—
161C2×C3×C4—41no—
162C2×C3×C4—44no—
163C2×C3×C4—42no—
164C2×C3×C4—82no—
165C2×C3×C4—82no—
166C2×C3×C4#12012yes—
167C2×C3×C4#12111yescharacteristic
168C2×C3×C4#13511yescharacteristic
169C2×C3×C4AGL(1,ℤ/8)12yes—
170C3×C8—41no—
171C3×C8#12011yescharacteristic
172C3×C8#12112yes—
173C3×C8#12711yescharacteristic
174C24C3×D811yescharacteristic
175#41—42no—
176#41—42no—
177#41—42no—
178#41—42no—
179C2×Q8—22no—
180C2×D4—22no—
181C2×D4—44no—
182C2×D4—42no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4C3×D811yescharacteristic
189C22×C4C2×C3×D411yescharacteristic
190Q16—22no—
191Q16—22no—
192SD16—42no—
193SD16—42no—
194D8—22no—
195D8—22no—
196C2×C8—22no—
197C2×C8—21no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—41no—
201C2×C8C2×C3×D412yes—
202C2×C8C2×C3×D411yescharacteristic
203C2×C8C2×C3×D412yes—
204C2×C8C3×#4111yescharacteristic
205C4⋊C4—24no—
206C4⋊C4—44no—
207C22⋊C4—44no—
208C22⋊C4—44no—
209C22⋊C4—42no—
210C42—42no—
211C16—22no—
212C16—41no—
213C22×C3—24no—
214C22×C3—22no—
215C22×C3—41no—
216C22×C3—42no—
217C22×C3—41no—
218C22×C3—41no—
219C22×C3—84no—
220C22×C3—82no—
221C22×C3#44712yes—
222C22×C3#46611yescharacteristic
223C3×C4—41no—
224C3×C4—84no—
225C3×C4—82no—
226C3×C4#39211yescharacteristic
227C3×C4#44912yes—
228C3×C4#46911yescharacteristic
229C23—22no—
230C23—21no—
231C23—42no—
232C23—42no—
233C23C2×C3×D811yescharacteristic
234Q8—42no—
235Q8—42no—
236D4—42no—
237D4—42no—
238D4—88no—
239D4—82no—
240D4—82no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4C3×#12012yes—
251C2×C4C3×#12111yescharacteristic
252C2×C4C3×#13511yescharacteristic
253C2×C4C3×AGL(1,ℤ/8)12yes—
254C8—41no—
255C8C3×#12011yescharacteristic
256C8C3×#12112yes—
257C8C3×#12711yescharacteristic
258C2×C3—22no—
259C2×C3—21no—
260C2×C3—42no—
261C2×C3—162no—
262C2×C3#199911yescharacteristic
263C22—24no—
264C22—22no—
265C22—41no—
266C22—42no—
267C22—41no—
268C22—41no—
269C22—84no—
270C22—82no—
271C22C3×#44712yes—
272C22C3×#46611yescharacteristic
273C4—41no—
274C4—84no—
275C4—82no—
276C4C3×#39211yescharacteristic
277C4C3×#44912yes—
278C4C3×#46911yescharacteristic
279C3#1330911yescharacteristic, direct product factor, Sylow
280C2—22no—
281C2—21no—
282C2—42no—
283C2—162no—
284C2C3×#199911yescharacteristic
subgroupG/H
centerC3×C4#469
commutator, Frattini, cosocleC22×C8C23×C3
socleC2×C3#1999
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C23×C3×C8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3414C2semidirect
2. C3×#3050C2semidirect
3. C3×#2145C2semidirect
4. C3×#2112C2semidirect
5. C3×#1384C2semidirect
6. #13309C3direct (N × H)
22. #3414C2×C3semidirect
23. #3050C2×C3semidirect
24. #2145C2×C3semidirect
25. #2112C2×C3semidirect
26. #1384C2×C3semidirect
33. C2×C3×Q16D4semidirect
35. C2×C3×D8D4semidirect
54. C2×C3×C16D4semidirect
56. C3×#106D4semidirect
119. C2×Q16C3×D4semidirect
121. C2×D8C3×D4semidirect
140. C2×C16C3×D4semidirect
142. #106C3×D4semidirect
279. C3#13309direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34142
C3×#30502
C3×#21452
C3×#21122
C3×#13842
#133093