Finite Group Databasebeta

← C3×#2077 #72792

Subgroups — C3×#2077 #72792

Order 384.

256 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 96 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×#106C211yesmaximal, characteristic, complemented
2C2×C3×C22⋊C8C211yesmaximal, characteristic
3C3×#380C211yesmaximal, characteristic
4C3×#380C211yesmaximal, characteristic
5C3×#340C211yesmaximal, characteristic, complemented
6C3×#340C211yesmaximal, characteristic, complemented
7C3×#340C211yesmaximal, characteristic, complemented
8#2077C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×C4C2211yescharacteristic
10C22×C3×C8C2211yescharacteristic
11C22×C3×C8C2211yescharacteristic
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C22⋊C4—21no—
17C2×C3×C22⋊C4—21no—
18C3×#106—22no—
19C3×C22⋊C8—22no—
20C3×#95—21no—
21C3×#95—21no—
22C2×#106C2×C311yescharacteristic, complemented
23C2×C22⋊C8C2×C311yescharacteristic
24#380C2×C311yescharacteristic
25#380C2×C311yescharacteristic
26#340C2×C311yescharacteristic, complemented
27#340C2×C311yescharacteristic, complemented
28#340C2×C311yescharacteristic, complemented
29C24×C3D411yescharacteristic
30C22×C3×C4—21no—
31C22×C3×C4—21no—
32C22×C3×C4—21no—
33C22×C3×C4—21no—
34C22×C3×C4—21no—
35C22×C3×C4—21no—
36C22×C3×C4—21no—
37C22×C3×C4—21no—
38C22×C3×C4—21no—
39C22×C3×C4—21no—
40C22×C3×C4D411yescharacteristic
41C22×C3×C4C2311yescharacteristic
42C2×C3×C8—21no—
43C2×C3×C8—21no—
44C2×C3×C8—21no—
45C2×C3×C8—21no—
46C2×C3×C8—21no—
47C2×C3×C8D412yes—
48C3×C4⋊C4—21no—
49C3×C4⋊C4—21no—
50C3×C4⋊C4—21no—
51C3×C4⋊C4—21no—
52C3×C4⋊C4—21no—
53C3×C4⋊C4—21no—
54C3×C4⋊C4—21no—
55C3×C4⋊C4—21no—
56C3×C22⋊C4—42no—
57C3×C22⋊C4—42no—
58C23×C4C22×C311yescharacteristic
59C22×C8C22×C311yescharacteristic
60C22×C8C22×C311yescharacteristic
61C2×C4⋊C4C22×C311yescharacteristic
62C2×C4⋊C4C22×C311yescharacteristic
63C2×C4⋊C4C22×C311yescharacteristic
64C2×C4⋊C4C22×C311yescharacteristic
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67#106—22no—
68C22⋊C8—22no—
69#95—21no—
70#95—21no—
71C23×C3—21no—
72C23×C3—21no—
73C23×C3—21no—
74C23×C3—21no—
75C23×C3—21no—
76C23×C3—21no—
77C23×C3SD1612yes—
78C23×C3C2×D411yescharacteristic
79C2×C3×C4—21no—
80C2×C3×C4—21no—
81C2×C3×C4—21no—
82C2×C3×C4—41no—
83C2×C3×C4—41no—
84C2×C3×C4—41no—
85C2×C3×C4—41no—
86C2×C3×C4—41no—
87C2×C3×C4—41no—
88C2×C3×C4—41no—
89C2×C3×C4—41no—
90C2×C3×C4—41no—
91C2×C3×C4—41no—
92C2×C3×C4—41no—
93C2×C3×C4—41no—
94C2×C3×C4—41no—
95C2×C3×C4—41no—
96C2×C3×C4—41no—
97C2×C3×C4—41no—
98C2×C3×C4C2×D411yescharacteristic
99C2×C3×C4#4111yescharacteristic
100C2×C3×C4#4111yescharacteristic
101C2×C3×C4#4111yescharacteristic
102C2×C3×C4#4111yescharacteristic
103C2×C3×C4#4111yescharacteristic
104C3×C8—22no—
105C3×C8—41no—
106C24C3×D411yescharacteristic
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—21no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4C3×D411yescharacteristic
118C22×C4C23×C311yescharacteristic
119C2×C8—21no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8C3×D412yes—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C22⋊C4—42no—
134C22⋊C4—42no—
135C22×C3—22no—
136C22×C3—22no—
137C22×C3—22no—
138C22×C3—41no—
139C22×C3—41no—
140C22×C3—41no—
141C22×C3—41no—
142C22×C3#12111yescharacteristic
143C22×C3#12311yescharacteristic
144C22×C3#12311yescharacteristic
145C22×C3C2×SD1611yescharacteristic
146C22×C3#13511yescharacteristic
147C22×C3AGL(1,ℤ/8)11yescharacteristic
148C22×C3#13711yescharacteristic
149C3×C4—41no—
150C3×C4—81no—
151C3×C4—81no—
152C3×C4—81no—
153C3×C4—81no—
154C3×C4#12311yescharacteristic
155C3×C4#12411yescharacteristic
156C3×C4#12611yescharacteristic
157C3×C4#12611yescharacteristic
158C23—21no—
159C23—21no—
160C23—21no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23C3×SD1612yes—
165C23C2×C3×D411yescharacteristic
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4C2×C3×D411yescharacteristic
186C2×C4C3×#4111yescharacteristic
187C2×C4C3×#4111yescharacteristic
188C2×C4C3×#4111yescharacteristic
189C2×C4C3×#4111yescharacteristic
190C2×C4C3×#4111yescharacteristic
191C8—22no—
192C8—41no—
193C2×C3—42no—
194C2×C3#39711yescharacteristic
195C2×C3#46511yescharacteristic
196C2×C3#46811yescharacteristic
197C2×C3#48111yescharacteristic
198C2×C3#48211yescharacteristic
199C2×C3#48311yescharacteristic
200C2×C3#48511yescharacteristic
201C22—22no—
202C22—22no—
203C22—22no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22C3×#12111yescharacteristic
209C22C3×#12311yescharacteristic
210C22C3×#12311yescharacteristic
211C22C2×C3×SD1611yescharacteristic
212C22C3×#13511yescharacteristic
213C22C3×AGL(1,ℤ/8)11yescharacteristic
214C22C3×#13711yescharacteristic
215C4—41no—
216C4—81no—
217C4—81no—
218C4—81no—
219C4—81no—
220C4C3×#12311yescharacteristic
221C4C3×#12411yescharacteristic
222C4C3×#12611yescharacteristic
223C4C3×#12611yescharacteristic
224C3#207711yescharacteristic, direct product factor, Sylow
225C2—42no—
226C2C3×#39711yescharacteristic
227C2C3×#46511yescharacteristic
228C2C3×#46811yescharacteristic
229C2C3×#48111yescharacteristic
230C2C3×#48211yescharacteristic
231C2C3×#48311yescharacteristic
232C2C3×#48511yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC24×C3D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#106C2semidirect
5. C3×#340C2semidirect
6. C3×#340C2semidirect
7. C3×#340C2semidirect
8. #2077C3direct (N × H)
22. C2×#106C2×C3semidirect
26. #340C2×C3semidirect
27. #340C2×C3semidirect
28. #340C2×C3semidirect
224. C3#2077direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#1062
C2×C3×C22⋊C82
C3×#3802
C3×#3402
#20773