Finite Group Databasebeta

← C3×#2012 #72727

Subgroups — C3×#2012 #72727

Order 384.

426 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C22×C3×SD16C211yesmaximal, characteristic, complemented
2C22×C3×D8C211yesmaximal, characteristic
3C2×C3×#124C211yesmaximal, characteristic, complemented
4C2×C3×#103C211yesmaximal, characteristic, complemented
5C2×C3×#102C211yesmaximal, characteristic, complemented
6C3×#390C211yesmaximal, characteristic, complemented
7C3×#336C211yesmaximal, characteristic, complemented
8#2012C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×Q8C2211yescharacteristic
10C22×C3×D4C2211yescharacteristic
11C22×C3×D4C2211yescharacteristic
12C2×C3×SD16—22no—
13C2×C3×SD16—22no—
14C2×C3×SD16—22no—
15C2×C3×D8—22no—
16C2×C3×D8—22no—
17C2×C3×D8—22no—
18C22×C3×C8C2211yescharacteristic
19C22×C3×C8C2211yescharacteristic
20C3×#124—22no—
21C3×#124—22no—
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4—22no—
24C2×C3×C22⋊C4—22no—
25C2×C3×C42C2211yescharacteristic
26C3×#103—22no—
27C3×#102—22no—
28C22×SD16C2×C311yescharacteristic, complemented
29C22×D8C2×C311yescharacteristic
30C2×#124C2×C311yescharacteristic, complemented
31C2×#103C2×C311yescharacteristic, complemented
32C2×#102C2×C311yescharacteristic, complemented
33#390C2×C311yescharacteristic, complemented
34#336C2×C311yescharacteristic, complemented
35C24×C3—21no—
36C24×C3—21no—
37C2×C3×Q8—21no—
38C2×C3×Q8—21no—
39C2×C3×Q8—21no—
40C2×C3×Q8—21no—
41C2×C3×Q8—21no—
42C2×C3×Q8D412yes—
43C2×C3×D4—21no—
44C2×C3×D4—21no—
45C2×C3×D4—21no—
46C2×C3×D4—21no—
47C2×C3×D4—21no—
48C2×C3×D4—21no—
49C2×C3×D4—21no—
50C2×C3×D4—21no—
51C2×C3×D4—21no—
52C2×C3×D4—21no—
53C2×C3×D4D412yes—
54C2×C3×D4D412yes—
55C22×C3×C4—21no—
56C22×C3×C4—21no—
57C22×C3×C4D412yes—
58C22×C3×C4C2311yescharacteristic
59C3×SD16—28no—
60C3×D8—28no—
61C2×C3×C8—21no—
62C2×C3×C8—21no—
63C2×C3×C8—21no—
64C2×C3×C8—21no—
65C2×C3×C8D412yes—
66C2×C3×C8D412yes—
67C3×C4⋊C4—21no—
68C3×C4⋊C4—21no—
69C3×C22⋊C4—28no—
70C3×C22⋊C4—44no—
71C3×C42—21no—
72C3×C42—21no—
73C22×Q8C22×C311yescharacteristic
74C22×D4C22×C311yescharacteristic
75C22×D4C22×C311yescharacteristic
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×D8—22no—
80C2×D8—22no—
81C2×D8—22no—
82C22×C8C22×C311yescharacteristic
83C22×C8C22×C311yescharacteristic
84#124—22no—
85#124—22no—
86C2×C4⋊C4C22×C311yescharacteristic
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C42C22×C311yescharacteristic
90#103—22no—
91#102—22no—
92C23×C3—22no—
93C23×C3—22no—
94C23×C3—41no—
95C23×C3—41no—
96C23×C3—41no—
97C23×C3—41no—
98C23×C3—42no—
99C23×C3—42no—
100C23×C3—42no—
101C23×C3—42no—
102C23×C3C2×D411yescharacteristic
103C3×Q8—22no—
104C3×Q8—22no—
105C3×Q8—41no—
106C3×Q8—41no—
107C3×D4—22no—
108C3×D4—22no—
109C3×D4—22no—
110C3×D4—22no—
111C3×D4—41no—
112C3×D4—41no—
113C3×D4—41no—
114C3×D4—41no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—41no—
120C2×C3×C4—41no—
121C2×C3×C4—41no—
122C2×C3×C4—42no—
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4#4111yescharacteristic
129C3×C8—22no—
130C3×C8—22no—
131C24—21no—
132C24—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8C3×D412yes—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4C3×D412yes—
150C2×D4C3×D412yes—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4C3×D412yes—
154C22×C4C23×C311yescharacteristic
155SD16—28no—
156D8—28no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8C3×D412yes—
162C2×C8C3×D412yes—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C22⋊C4—28no—
166C22⋊C4—44no—
167C42—21no—
168C42—21no—
169C22×C3—42no—
170C22×C3—42no—
171C22×C3—44no—
172C22×C3—44no—
173C22×C3—82no—
174C22×C3—82no—
175C22×C3—81no—
176C22×C3—81no—
177C22×C3—81no—
178C22×C3—81no—
179C22×C3#12011yescharacteristic
180C22×C3#12111yescharacteristic
181C22×C3#12711yescharacteristic
182C22×C3#13512yes—
183C22×C3AGL(1,ℤ/8)12yes—
184C3×C4—42no—
185C3×C4—42no—
186C3×C4—81no—
187C3×C4#12011yescharacteristic
188C3×C4#12011yescharacteristic
189C3×C4#12111yescharacteristic
190C3×C4#12111yescharacteristic
191C23—22no—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23C2×C3×D411yescharacteristic
202Q8—22no—
203Q8—22no—
204Q8—41no—
205Q8—41no—
206D4—22no—
207D4—22no—
208D4—22no—
209D4—22no—
210D4—41no—
211D4—41no—
212D4—41no—
213D4—41no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4C2×C3×D411yescharacteristic
223C2×C4C2×C3×D411yescharacteristic
224C2×C4C2×C3×D411yescharacteristic
225C2×C4C2×C3×D411yescharacteristic
226C2×C4C2×C3×D411yescharacteristic
227C2×C4C3×#4111yescharacteristic
228C8—22no—
229C8—22no—
230C2×C3—82no—
231C2×C3—82no—
232C2×C3#39211yescharacteristic
233C2×C3#44912yes—
234C2×C3#46312yes—
235C2×C3#49511yescharacteristic
236C2×C3#49611yescharacteristic
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—82no—
242C22—82no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22C3×#12011yescharacteristic
248C22C3×#12111yescharacteristic
249C22C3×#12711yescharacteristic
250C22C3×#13512yes—
251C22C3×AGL(1,ℤ/8)12yes—
252C4—42no—
253C4—42no—
254C4—81no—
255C4C3×#12011yescharacteristic
256C4C3×#12011yescharacteristic
257C4C3×#12111yescharacteristic
258C4C3×#12111yescharacteristic
259C3#201211yescharacteristic, direct product factor, Sylow
260C2—82no—
261C2—82no—
262C2C3×#39211yescharacteristic
263C2C3×#44912yes—
264C2C3×#46312yes—
265C2C3×#49511yescharacteristic
266C2C3×#49611yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC22×C3×D8C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×SD16C2semidirect
3. C2×C3×#124C2semidirect
4. C2×C3×#103C2semidirect
5. C2×C3×#102C2semidirect
6. C3×#390C2semidirect
7. C3×#336C2semidirect
8. #2012C3direct (N × H)
28. C22×SD16C2×C3semidirect
30. C2×#124C2×C3semidirect
31. C2×#103C2×C3semidirect
32. C2×#102C2×C3semidirect
33. #390C2×C3semidirect
34. #336C2×C3semidirect
259. C3#2012direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×SD162
C22×C3×D82
C2×C3×#1242
C2×C3×#1032
C2×C3×#1022
C3×#3902
C3×#3362
#20123