Finite Group Databasebeta

← C3×#13315 #10631922

Subgroups — C3×#13315 #10631922

Order 768.

516 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 118 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×Q32C211yesmaximal, characteristic, complemented
2C2×C3×#473C212yesmaximal
3C2×C3×#362C212yesmaximal, complemented
4C3×#2113C211yesmaximal, characteristic
5C3×#1381C211yesmaximal, characteristic, complemented
6#13315C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#137—22no—
8C22×C3×Q16C2212yes—
9C2×C3×#131C2211yescharacteristic
10C2×C3×Q32—24no—
11C2×C3×Q32—22no—
12C2×C3×M32C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14C3×#473—24no—
15C3×#473—24no—
16C2×C3×#108C2212yes—
17C2×C3×#101—22no—
18C3×#362—24no—
19C3×C22⋊C16—22no—
20C22×Q32C2×C311yescharacteristic, complemented
21C2×#473C2×C312yes—
22C2×#362C2×C312yescomplemented
23#2113C2×C311yescharacteristic
24#1381C2×C311yescharacteristic, complemented
25C2×C3×#41D411yescharacteristic
26C22×C3×Q8—22no—
27C3×#137—28no—
28C3×#137—44no—
29C2×C3×Q16—22no—
30C2×C3×Q16—22no—
31C2×C3×Q16—22no—
32C2×C3×Q16—22no—
33C2×C3×Q16—22no—
34C2×C3×Q16—42no—
35C2×C3×Q16D414yes—
36C2×C3×SD16—42no—
37C3×#131—22no—
38C3×#131—22no—
39C2×C3×M16—21no—
40C2×C3×M16—22no—
41C2×C3×M16D411yescharacteristic
42C22×C3×C8—21no—
43C22×C3×C8C2311yescharacteristic
44C3×Q32—28no—
45C3×M32—22no—
46C2×C3×C16—22no—
47C2×C3×C16—21no—
48C2×C3×C16D412yes—
49C3×#108—22no—
50C3×#108D414yes—
51C3×#101—28no—
52C2×#137—22no—
53C22×Q16C22×C312yes—
54C2×#131C22×C311yescharacteristic
55C2×Q32—24no—
56C2×Q32—22no—
57C2×M32C22×C311yescharacteristic
58C22×C16C22×C311yescharacteristic
59#473—24no—
60#473—24no—
61C2×#108C22×C312yes—
62C2×#101—22no—
63#362—24no—
64C22⋊C16—22no—
65C3×#41—22no—
66C3×#41—22no—
67C2×C3×Q8—24no—
68C2×C3×Q8—42no—
69C2×C3×Q8—42no—
70C2×C3×Q8—42no—
71C2×C3×Q8—42no—
72C2×C3×Q8—42no—
73C2×C3×Q8D811yescharacteristic
74C2×C3×D4—21no—
75C2×C3×D4D811yescharacteristic
76C22×C3×C4—21no—
77C22×C3×C4—42no—
78C22×C3×C4C2×D411yescharacteristic
79C3×Q16—24no—
80C3×Q16—24no—
81C3×Q16—48no—
82C3×Q16—42no—
83C3×Q16—42no—
84C3×SD16—48no—
85C3×M16—22no—
86C3×M16—24no—
87C3×M16—42no—
88C3×M16—42no—
89C2×C3×C8—21no—
90C2×C3×C8—42no—
91C2×C3×C8—41no—
92C2×C3×C8—41no—
93C2×C3×C8C2×D412yes—
94C2×C3×C8C2×D411yescharacteristic
95C2×C3×C8C2×D412yes—
96C2×C3×C8#4111yescharacteristic
97C3×C16—22no—
98C3×C16—41no—
99C2×#41C3×D411yescharacteristic
100C22×Q8—22no—
101#137—28no—
102#137—44no—
103C2×Q16—22no—
104C2×Q16—22no—
105C2×Q16—22no—
106C2×Q16—22no—
107C2×Q16—22no—
108C2×Q16—42no—
109C2×Q16C3×D414yes—
110C2×SD16—42no—
111#131—22no—
112#131—22no—
113C2×M16—21no—
114C2×M16—22no—
115C2×M16C3×D411yescharacteristic
116C22×C8—21no—
117C22×C8C23×C311yescharacteristic
118Q32—28no—
119M32—22no—
120C2×C16—22no—
121C2×C16—21no—
122C2×C16C3×D412yes—
123#108—22no—
124#108C3×D414yes—
125#101—28no—
126C23×C3—21no—
127C23×C3#13511yescharacteristic
128C3×Q8—22no—
129C3×Q8—44no—
130C3×Q8—44no—
131C3×Q8—82no—
132C3×Q8—82no—
133C3×D4—22no—
134C3×D4—42no—
135C2×C3×C4—21no—
136C2×C3×C4—41no—
137C2×C3×C4—41no—
138C2×C3×C4—44no—
139C2×C3×C4—82no—
140C2×C3×C4—82no—
141C2×C3×C4#12012yes—
142C2×C3×C4#12111yescharacteristic
143C2×C3×C4C2×D811yescharacteristic
144C2×C3×C4AGL(1,ℤ/8)12yes—
145C3×C8—41no—
146C3×C8—44no—
147C3×C8#12011yescharacteristic
148C3×C8#12112yes—
149C3×C8#12711yescharacteristic
150#41—22no—
151#41—22no—
152C2×Q8—24no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8C3×D811yescharacteristic
159C2×D4—21no—
160C2×D4C3×D811yescharacteristic
161C22×C4—21no—
162C22×C4—42no—
163C22×C4C2×C3×D411yescharacteristic
164Q16—24no—
165Q16—24no—
166Q16—48no—
167Q16—42no—
168Q16—42no—
169SD16—48no—
170M16—22no—
171M16—24no—
172M16—42no—
173M16—42no—
174C2×C8—21no—
175C2×C8—42no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8C2×C3×D412yes—
179C2×C8C2×C3×D411yescharacteristic
180C2×C8C2×C3×D412yes—
181C2×C8C3×#4111yescharacteristic
182C16—22no—
183C16—41no—
184C22×C3—22no—
185C22×C3—42no—
186C22×C3—41no—
187C22×C3#44912yes—
188C22×C3#46611yescharacteristic
189C3×C4—41no—
190C3×C4—84no—
191C3×C4#39211yescharacteristic
192C3×C4#44712yes—
193C3×C4#46911yescharacteristic
194C23—21no—
195C23C3×#13511yescharacteristic
196Q8—22no—
197Q8—44no—
198Q8—44no—
199Q8—82no—
200Q8—82no—
201D4—22no—
202D4—42no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—44no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4C3×#12012yes—
210C2×C4C3×#12111yescharacteristic
211C2×C4C2×C3×D811yescharacteristic
212C2×C4C3×AGL(1,ℤ/8)12yes—
213C8—41no—
214C8—44no—
215C8C3×#12011yescharacteristic
216C8C3×#12112yes—
217C8C3×#12711yescharacteristic
218C2×C3—22no—
219C2×C3—81no—
220C2×C3#199911yescharacteristic
221C2×C3#222212yes—
222C22—22no—
223C22—41no—
224C22—42no—
225C22C3×#44912yes—
226C22C3×#46611yescharacteristic
227C4—41no—
228C4—84no—
229C4C3×#39211yescharacteristic
230C4C3×#44712yes—
231C4C3×#46911yescharacteristic
232C3#1331511yescharacteristic, direct product factor, Sylow
233C2—22no—
234C2—81no—
235C2C3×#199911yescharacteristic
236C2C3×#222212yes—
subgroupG/H
center, socleC22×C3#466
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC2×C3×#41D4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q32C2semidirect
3. C2×C3×#362C2semidirect
5. C3×#1381C2semidirect
6. #13315C3direct (N × H)
20. C22×Q32C2×C3semidirect
22. C2×#362C2×C3semidirect
24. #1381C2×C3semidirect
232. C3#13315direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q322
C2×C3×#4732
C2×C3×#3622
C3×#21132
C3×#13812
#133153