Finite Group Databasebeta

← C3×#2023 #72738

Subgroups — C3×#2023 #72738

Order 384.

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

normal subgroups only

This group has 326 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
1C2×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#102C212yesmaximal
3C3×#390C211yesmaximal, characteristic
4C3×#340C212yesmaximal, complemented
5C3×#336C211yesmaximal, characteristic, complemented
6#2023C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×Q8—21no—
8C22×C3×D4C2211yescharacteristic
9C22×C3×C8C2212yes—
10C3×C4⋊Q8—24no—
11C2×C3×C4⋊C4—21no—
12C2×C3×C4⋊C4C2211yescharacteristic
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C22⋊C4—21no—
15C2×C3×C22⋊C4—21no—
16C2×C3×C42C2211yescharacteristic
17C3×#102—24no—
18C2×C4⋊Q8C2×C311yescharacteristic, complemented
19C2×#102C2×C312yes—
20#390C2×C311yescharacteristic
21#340C2×C312yescomplemented
22#336C2×C311yescharacteristic, complemented
23C24×C3—21no—
24C2×C3×Q8—24no—
25C2×C3×Q8—42no—
26C2×C3×Q8—42no—
27C2×C3×D4—22no—
28C2×C3×D4—22no—
29C2×C3×D4—21no—
30C2×C3×D4Q812yescomplemented
31C22×C3×C4—22no—
32C22×C3×C4—21no—
33C22×C3×C4D411yescharacteristic
34C22×C3×C4D411yescharacteristic
35C22×C3×C4C2311yescharacteristic
36C2×C3×C8—22no—
37C2×C3×C8—22no—
38C2×C3×C8—22no—
39C3×C4⋊C4—24no—
40C3×C4⋊C4—22no—
41C3×C4⋊C4—22no—
42C3×C4⋊C4D414yes—
43C3×C22⋊C4—42no—
44C3×C22⋊C4—42no—
45C3×C42—22no—
46C22×Q8—21no—
47C22×D4C22×C311yescharacteristic
48C22×C8C22×C312yes—
49C4⋊Q8—24no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4C22×C311yescharacteristic
52C2×C4⋊C4C22×C312yes—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C42C22×C311yescharacteristic
56#102—24no—
57C23×C3—22no—
58C23×C3—41no—
59C23×C3—41no—
60C23×C3—41no—
61C23×C3—41no—
62C23×C3—41no—
63C23×C3—41no—
64C23×C3C2×D411yescharacteristic
65C3×Q8—48no—
66C3×D4—24no—
67C3×D4—42no—
68C2×C3×C4—21no—
69C2×C3×C4—21no—
70C2×C3×C4—21no—
71C2×C3×C4—21no—
72C2×C3×C4—24no—
73C2×C3×C4—42no—
74C2×C3×C4—42no—
75C2×C3×C4—41no—
76C2×C3×C4—41no—
77C2×C3×C4—41no—
78C2×C3×C4D812yes—
79C2×C3×C4SD1612yes—
80C2×C3×C4C2×D412yes—
81C2×C3×C4C2×Q811yescharacteristic
82C2×C3×C4#4112yes—
83C2×C3×C4#4111yescharacteristic
84C3×C8—42no—
85C24—21no—
86C2×Q8—24no—
87C2×Q8—42no—
88C2×Q8—42no—
89C2×D4—22no—
90C2×D4—22no—
91C2×D4—21no—
92C2×D4C3×Q812yescomplemented
93C22×C4—22no—
94C22×C4—21no—
95C22×C4C3×D411yescharacteristic
96C22×C4C3×D411yescharacteristic
97C22×C4C23×C311yescharacteristic
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—22no—
101C4⋊C4—24no—
102C4⋊C4—22no—
103C4⋊C4—22no—
104C4⋊C4C3×D414yes—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C42—22no—
108C22×C3—42no—
109C22×C3—42no—
110C22×C3—42no—
111C22×C3—81no—
112C22×C3—81no—
113C22×C3—81no—
114C22×C3—81no—
115C22×C3#12011yescharacteristic
116C22×C3C22⋊Q811yescharacteristic
117C22×C3#12411yescharacteristic
118C22×C3C2×D811yescharacteristic
119C22×C3C2×SD1611yescharacteristic
120C22×C3AGL(1,ℤ/8)11yescharacteristic
121C22×C3#13711yescharacteristic
122C3×C4—22no—
123C3×C4—22no—
124C3×C4—44no—
125C3×C4—81no—
126C3×C4C22⋊Q812yes—
127C3×C4#12412yes—
128C23—22no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23C2×C3×D411yescharacteristic
136Q8—48no—
137D4—24no—
138D4—42no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—24no—
144C2×C4—42no—
145C2×C4—42no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4C3×D812yes—
150C2×C4C3×SD1612yes—
151C2×C4C2×C3×D412yes—
152C2×C4C2×C3×Q811yescharacteristic
153C2×C4C3×#4112yes—
154C2×C4C3×#4111yescharacteristic
155C8—42no—
156C2×C3—82no—
157C2×C3#39311yescharacteristic
158C2×C3#44711yescharacteristic
159C2×C3#44811yescharacteristic
160C2×C3#47411yescharacteristic
161C2×C3#47611yescharacteristic
162C2×C3#48611yescharacteristic
163C2×C3#48911yescharacteristic
164C22—42no—
165C22—42no—
166C22—42no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22C3×#12011yescharacteristic
172C22C3×C22⋊Q811yescharacteristic
173C22C3×#12411yescharacteristic
174C22C2×C3×D811yescharacteristic
175C22C2×C3×SD1611yescharacteristic
176C22C3×AGL(1,ℤ/8)11yescharacteristic
177C22C3×#13711yescharacteristic
178C4—22no—
179C4—22no—
180C4—44no—
181C4—81no—
182C4C3×C22⋊Q812yes—
183C4C3×#12412yes—
184C3#202311yescharacteristic, direct product factor, Sylow
185C2—82no—
186C2C3×#39311yescharacteristic
187C2C3×#44711yescharacteristic
188C2C3×#44811yescharacteristic
189C2C3×#47411yescharacteristic
190C2C3×#47611yescharacteristic
191C2C3×#48611yescharacteristic
192C2C3×#48911yescharacteristic
subgroupG/H
center, socleC23×C3C2×D4
commutator, Frattini, cosocleC22×C4C23×C3
prime radicalC22×C3×D4C22
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. C2×C3×C4⋊Q8C2semidirect
4. C3×#340C2semidirect
5. C3×#336C2semidirect
6. #2023C3direct (N × H)
18. C2×C4⋊Q8C2×C3semidirect
21. #340C2×C3semidirect
22. #336C2×C3semidirect
30. C2×C3×D4Q8semidirect
92. C2×D4C3×Q8semidirect
184. C3#2023direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊Q82
C2×C3×#1022
C3×#3902
C3×#3402
C3×#3362
#20233