Finite Group Databasebeta

← C3×#13415 #10632022

Subgroups — C3×#13415 #10632022

Order 768.

436 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 114 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×#494C211yesmaximal, characteristic
2C2×C3×#362C212yesmaximal
3C3×#2141C212yesmaximal
4C3×#1945C211yesmaximal, characteristic
5C3×#1376C211yesmaximal, characteristic
6#13415C311yesmaximal, characteristic, direct product factor, Sylow
7C22×C3×Q16—21no—
8C22×C3×Q16C2212yes—
9C2×C3×C4⋊Q8—21no—
10C2×C3×M32C2212yes—
11C3×#494—24no—
12C2×C3×#108C2211yescharacteristic
13C3×#427C2211yescharacteristic
14C3×#424—21no—
15C2×C3×C4×C8C2211yescharacteristic
16C3×#362—24no—
17C3×#358—24no—
18C3×#339—21no—
19C2×#494C2×C311yescharacteristic
20C2×#362C2×C312yes—
21#2141C2×C312yes—
22#1945C2×C311yescharacteristic
23#1376C2×C311yescharacteristic
24C22×C3×Q8—22no—
25C2×C3×Q16—24no—
26C2×C3×Q16—22no—
27C2×C3×Q16—22no—
28C2×C3×Q16—22no—
29C2×C3×Q16—22no—
30C2×C3×Q16—22no—
31C2×C3×Q16—42no—
32C2×C3×Q16—42no—
33C2×C3×Q16D414yes—
34C2×C3×M16—21no—
35C22×C3×C8D411yescharacteristic
36C22×C3×C8C2311yescharacteristic
37C3×C4⋊Q8—24no—
38C3×C4⋊Q8—42no—
39C3×#117—21no—
40C2×C3×C4⋊C4—42no—
41C2×C3×C42D411yescharacteristic
42C3×M32—22no—
43C3×M32—22no—
44C2×C3×C16—22no—
45C3×#108—22no—
46C3×#107D412yes—
47C3×#106—21no—
48C3×C4⋊C8—42no—
49C3×C4×C8—22no—
50C22×Q16—21no—
51C22×Q16C22×C312yes—
52C2×C4⋊Q8—21no—
53C2×M32C22×C312yes—
54#494—24no—
55C2×#108C22×C311yescharacteristic
56#427C22×C311yescharacteristic
57#424—21no—
58C2×C4×C8C22×C311yescharacteristic
59#362—24no—
60#358—24no—
61#339—21no—
62C2×C3×Q8—24no—
63C2×C3×Q8—42no—
64C2×C3×Q8—42no—
65C2×C3×Q8—42no—
66C2×C3×Q8—42no—
67C2×C3×Q8—42no—
68C22×C3×C4—21no—
69C22×C3×C4—42no—
70C22×C3×C4C2×D411yescharacteristic
71C3×Q16—24no—
72C3×Q16—24no—
73C3×Q16—48no—
74C3×Q16—42no—
75C3×Q16—42no—
76C3×M16—41no—
77C3×M16—41no—
78C2×C3×C8—21no—
79C2×C3×C8—21no—
80C2×C3×C8—41no—
81C2×C3×C8D812yes—
82C2×C3×C8C2×D411yescharacteristic
83C2×C3×C8C2×D412yes—
84C2×C3×C8#4112yes—
85C2×C3×C8#4111yescharacteristic
86C3×C4⋊C4—22no—
87C3×C4⋊C4—48no—
88C3×C22⋊C4—41no—
89C3×C42—21no—
90C3×C42—41no—
91C3×C42SD1612yes—
92C3×C16—42no—
93C22×Q8—22no—
94C2×Q16—24no—
95C2×Q16—22no—
96C2×Q16—22no—
97C2×Q16—22no—
98C2×Q16—22no—
99C2×Q16—22no—
100C2×Q16—42no—
101C2×Q16—42no—
102C2×Q16C3×D414yes—
103C2×M16—21no—
104C22×C8C3×D411yescharacteristic
105C22×C8C23×C311yescharacteristic
106C4⋊Q8—24no—
107C4⋊Q8—42no—
108#117—21no—
109C2×C4⋊C4—42no—
110C2×C42C3×D411yescharacteristic
111M32—22no—
112M32—22no—
113C2×C16—22no—
114#108—22no—
115#107C3×D412yes—
116#106—21no—
117C4⋊C8—42no—
118C4×C8—22no—
119C23×C3AGL(1,ℤ/8)11yescharacteristic
120C3×Q8—44no—
121C3×Q8—44no—
122C3×Q8—82no—
123C3×Q8—82no—
124C2×C3×C4—24no—
125C2×C3×C4—22no—
126C2×C3×C4—44no—
127C2×C3×C4—42no—
128C2×C3×C4—82no—
129C2×C3×C4—82no—
130C2×C3×C4#12011yescharacteristic
131C2×C3×C4#12111yescharacteristic
132C2×C3×C4#12411yescharacteristic
133C2×C3×C4C2×D811yescharacteristic
134C2×C3×C4C2×SD1611yescharacteristic
135C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
136C3×C8—22no—
137C3×C8—81no—
138C3×C8#12112yes—
139C3×C8#12312yes—
140C2×Q8—24no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—42no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4C2×C3×D411yescharacteristic
149Q16—24no—
150Q16—24no—
151Q16—48no—
152Q16—42no—
153Q16—42no—
154M16—41no—
155M16—41no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—41no—
159C2×C8C3×D812yes—
160C2×C8C2×C3×D411yescharacteristic
161C2×C8C2×C3×D412yes—
162C2×C8C3×#4112yes—
163C2×C8C3×#4111yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—48no—
166C22⋊C4—41no—
167C42—21no—
168C42—41no—
169C42C3×SD1612yes—
170C16—42no—
171C22×C3—22no—
172C22×C3#45011yescharacteristic
173C22×C3#45911yescharacteristic
174C22×C3#49011yescharacteristic
175C3×C4—24no—
176C3×C4—82no—
177C3×C4—84no—
178C3×C4#39411yescharacteristic
179C3×C4#44711yescharacteristic
180C3×C4#46011yescharacteristic
181C3×C4#48611yescharacteristic
182C23C3×AGL(1,ℤ/8)11yescharacteristic
183Q8—44no—
184Q8—44no—
185Q8—82no—
186Q8—82no—
187C2×C4—24no—
188C2×C4—22no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—82no—
192C2×C4—82no—
193C2×C4C3×#12011yescharacteristic
194C2×C4C3×#12111yescharacteristic
195C2×C4C3×#12411yescharacteristic
196C2×C4C2×C3×D811yescharacteristic
197C2×C4C2×C3×SD1611yescharacteristic
198C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
199C8—22no—
200C8—81no—
201C8C3×#12112yes—
202C8C3×#12312yes—
203C2×C3—21no—
204C2×C3—21no—
205C2×C3#204111yescharacteristic
206C2×C3#221412yes—
207C22—22no—
208C22C3×#45011yescharacteristic
209C22C3×#45911yescharacteristic
210C22C3×#49011yescharacteristic
211C4—24no—
212C4—84no—
213C4—82no—
214C4C3×#39411yescharacteristic
215C4C3×#44711yescharacteristic
216C4C3×#46011yescharacteristic
217C4C3×#48611yescharacteristic
218C3#1341511yescharacteristic, direct product factor, Sylow
219C2—21no—
220C2—21no—
221C2C3×#204111yescharacteristic
222C2C3×#221412yes—
subgroupG/H
center, socleC22×C3#459
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC23×C3AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×C4×C8 ◁ G

Complements

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

NH ≅ G/Ntype
6. #13415C3direct (N × H)
218. C3#13415direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4942
C2×C3×#3622
C3×#21412
C3×#19452
C3×#13762
#134153