Finite Group Databasebeta

← C3×#2003 #72718

Subgroups — C3×#2003 #72718

Order 384.

382 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 92 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×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C3×#536C211yesmaximal, characteristic, complemented
3C2×C3×#124C211yesmaximal, characteristic, complemented
4C2×C3×#104C211yesmaximal, characteristic, complemented
5C3×#418C211yesmaximal, characteristic, complemented
6C2×C3×#101C211yesmaximal, characteristic, complemented
7C3×#339C211yesmaximal, characteristic, complemented
8#2003C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#41C2211yescharacteristic
10C22×C3×Q8C2211yescharacteristic
11C22×C3×D4C2211yescharacteristic
12C3×AGL(1,ℤ/8)—22no—
13C3×AGL(1,ℤ/8)—22no—
14C2×C3×SD16—21no—
15C2×C3×D8—21no—
16C2×C3×M16C2211yescharacteristic
17C2×C3×M16C2211yescharacteristic
18C3×C4⋊Q8—21no—
19C3×#124—22no—
20C3×#124—21no—
21C3×#124—22no—
22C3×#123—22no—
23C3×C22⋊Q8—22no—
24C3×#117C2211yescharacteristic
25C2×C3×C22⋊C4—22no—
26C2×C3×C42C2211yescharacteristic
27C3×#104—22no—
28C3×#102—22no—
29C3×#101—22no—
30C2×AGL(1,ℤ/8)C2×C311yescharacteristic
31#536C2×C311yescharacteristic, complemented
32C2×#124C2×C311yescharacteristic, complemented
33C2×#104C2×C311yescharacteristic, complemented
34#418C2×C311yescharacteristic, complemented
35C2×#101C2×C311yescharacteristic, complemented
36#339C2×C311yescharacteristic, complemented
37C24×C3—21no—
38C3×#41—21no—
39C3×#41—21no—
40C3×#41—21no—
41C3×#41—21no—
42C2×C3×Q8—21no—
43C2×C3×Q8—21no—
44C2×C3×Q8—21no—
45C2×C3×Q8—21no—
46C2×C3×Q8—21no—
47C2×C3×Q8—21no—
48C2×C3×Q8D411yescharacteristic
49C2×C3×D4—21no—
50C2×C3×D4—21no—
51C2×C3×D4—21no—
52C2×C3×D4—21no—
53C2×C3×D4—21no—
54C2×C3×D4—21no—
55C2×C3×D4D411yescharacteristic
56C2×C3×D4D412yes—
57C22×C3×C4—21no—
58C22×C3×C4—21no—
59C22×C3×C4D412yes—
60C22×C3×C4C2311yescharacteristic
61C3×SD16—24no—
62C3×D8—24no—
63C3×M16—21no—
64C3×M16—21no—
65C3×M16—21no—
66C3×M16D412yes—
67C2×C3×C8—21no—
68C2×C3×C8—21no—
69C3×C4⋊C4—24no—
70C3×C4⋊C4D412yes—
71C3×C22⋊C4—28no—
72C3×C22⋊C4—24no—
73C3×C22⋊C4—21no—
74C3×C42—21no—
75C3×C42—21no—
76C3×C42D412yes—
77C2×#41C22×C311yescharacteristic
78C22×Q8C22×C311yescharacteristic
79C22×D4C22×C311yescharacteristic
80AGL(1,ℤ/8)—22no—
81AGL(1,ℤ/8)—22no—
82C2×SD16—21no—
83C2×D8—21no—
84C2×M16C22×C311yescharacteristic
85C2×M16C22×C311yescharacteristic
86C4⋊Q8—21no—
87#124—22no—
88#124—21no—
89#124—22no—
90#123—22no—
91C22⋊Q8—22no—
92#117C22×C311yescharacteristic
93C2×C22⋊C4—22no—
94C2×C42C22×C311yescharacteristic
95#104—22no—
96#102—22no—
97#101—22no—
98C23×C3—22no—
99C23×C3—21no—
100C23×C3—44no—
101C23×C3—41no—
102C23×C3—41no—
103C23×C3C2×D411yescharacteristic
104C3×Q8—21no—
105C3×Q8—21no—
106C3×Q8—41no—
107C3×Q8—41no—
108C3×Q8—41no—
109C3×Q8—41no—
110C3×D4—21no—
111C3×D4—22no—
112C3×D4—22no—
113C3×D4—21no—
114C3×D4—41no—
115C3×D4—41no—
116C3×D4—41no—
117C3×D4—41no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4—22no—
124C2×C3×C4—42no—
125C2×C3×C4—41no—
126C2×C3×C4—41no—
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2×D411yescharacteristic
132C2×C3×C4#4111yescharacteristic
133C3×C8—22no—
134C3×C8—41no—
135C24—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139#41—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8C3×D411yescharacteristic
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C3×D411yescharacteristic
154C2×D4C3×D412yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4C3×D412yes—
158C22×C4C23×C311yescharacteristic
159SD16—24no—
160D8—24no—
161M16—21no—
162M16—21no—
163M16—21no—
164M16C3×D412yes—
165C2×C8—21no—
166C2×C8—21no—
167C4⋊C4—24no—
168C4⋊C4C3×D412yes—
169C22⋊C4—28no—
170C22⋊C4—24no—
171C22⋊C4—21no—
172C42—21no—
173C42—21no—
174C42C3×D412yes—
175C22×C3—22no—
176C22×C3—44no—
177C22×C3—41no—
178C22×C3—42no—
179C22×C3—42no—
180C22×C3—82no—
181C22×C3—82no—
182C22×C3#12011yescharacteristic
183C22×C3#12011yescharacteristic
184C22×C3#12111yescharacteristic
185C3×C4—42no—
186C3×C4—42no—
187C3×C4—41no—
188C3×C4—42no—
189C3×C4#12011yescharacteristic
190C3×C4#12111yescharacteristic
191C3×C4#12111yescharacteristic
192C3×C4#12711yescharacteristic
193C23—21no—
194C23—22no—
195C23—44no—
196C23—41no—
197C23—41no—
198C23C2×C3×D411yescharacteristic
199Q8—21no—
200Q8—21no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205D4—21no—
206D4—22no—
207D4—22no—
208D4—21no—
209D4—41no—
210D4—41no—
211D4—41no—
212D4—41no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
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—41no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—81no—
233C2×C3—82no—
234C2×C3#39211yescharacteristic
235C2×C3#45412yes—
236C22—22no—
237C22—44no—
238C22—41no—
239C22—42no—
240C22—42no—
241C22—82no—
242C22—82no—
243C22C3×#12011yescharacteristic
244C22C3×#12011yescharacteristic
245C22C3×#12111yescharacteristic
246C4—42no—
247C4—42no—
248C4—41no—
249C4—42no—
250C4C3×#12011yescharacteristic
251C4C3×#12111yescharacteristic
252C4C3×#12111yescharacteristic
253C4C3×#12711yescharacteristic
254C3#200311yescharacteristic, direct product factor, Sylow
255C2—21no—
256C2—21no—
257C2—82no—
258C2—81no—
259C2C3×#39211yescharacteristic
260C2C3×#45412yes—
subgroupG/H
center, socleC22×C3#120
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC2×C3×AGL(1,ℤ/8)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#536C2semidirect
3. C2×C3×#124C2semidirect
4. C2×C3×#104C2semidirect
5. C3×#418C2semidirect
6. C2×C3×#101C2semidirect
7. C3×#339C2semidirect
8. #2003C3direct (N × H)
31. #536C2×C3semidirect
32. C2×#124C2×C3semidirect
33. C2×#104C2×C3semidirect
34. #418C2×C3semidirect
35. C2×#101C2×C3semidirect
36. #339C2×C3semidirect
254. C3#2003direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×AGL(1,ℤ/8)2
C3×#5362
C2×C3×#1242
C2×C3×#1042
C3×#4182
C2×C3×#1012
C3×#3392
#20033