Finite Group Databasebeta

← C3×#10217 #10628824

Subgroups — C3×#10217 #10628824

Order 768.

430 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C3×#3255C211yesmaximal, characteristic
2C3×#3237C212yesmaximal
3C3×#1995C211yesmaximal, characteristic
4C3×#1506C212yesmaximal
5C3×#1307C211yesmaximal, characteristic
6#10217C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#533C2211yescharacteristic
8C3×#533C2212yes—
9C3×#501—22no—
10C3×#491—22no—
11C3×#485—24no—
12C3×#477—22no—
13C3×#475—22no—
14C3×#475—22no—
15C3×#470—22no—
16C3×#462—22no—
17C3×#432C2212yes—
18C3×#423C2211yescharacteristic
19C3×#376C2211yescharacteristic
20C3×#339—22no—
21C3×#328—24no—
22#3255C2×C311yescharacteristic
23#3237C2×C312yes—
24#1995C2×C311yescharacteristic
25#1506C2×C312yes—
26#1307C2×C311yescharacteristic
27C2×C3×Q16—22no—
28C2×C3×M16D412yes—
29C3×C4⋊Q8D412yes—
30C3×C4⋊Q8D412yes—
31C3×C4⋊Q8D412yes—
32C3×#125—22no—
33C3×#125—21no—
34C3×C22⋊Q8—22no—
35C3×C22⋊Q8—22no—
36C3×C22⋊Q8—22no—
37C3×C4×Q8—22no—
38C3×C4×Q8—22no—
39C3×C4×Q8—22no—
40C3×#117—21no—
41C3×#117—22no—
42C2×C3×C42—21no—
43C2×C3×C42C2311yescharacteristic
44C3×#107—24no—
45C3×#106—22no—
46C3×#106—24no—
47C3×C4⋊C8—22no—
48C3×C4⋊C8—22no—
49C3×#103—24no—
50C3×#103—24no—
51C3×#103—24no—
52C3×C22⋊C8D414yes—
53C3×#97—22no—
54C3×#95—22no—
55#533C22×C311yescharacteristic
56#533C22×C312yes—
57#501—22no—
58#491—22no—
59#485—24no—
60#477—22no—
61#475—22no—
62#475—22no—
63#470—22no—
64#462—22no—
65#432C22×C312yes—
66#423C22×C311yescharacteristic
67#376C22×C311yescharacteristic
68#339—22no—
69#328—24no—
70C2×C3×Q8—22no—
71C2×C3×Q8—22no—
72C2×C3×Q8—22no—
73C22×C3×C4—41no—
74C22×C3×C4C2×D411yescharacteristic
75C22×C3×C4C2×D412yes—
76C3×Q16—82no—
77C3×M16—42no—
78C2×C3×C8—24no—
79C2×C3×C8—22no—
80C3×C4⋊C4—22no—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—22no—
83C3×C4⋊C4—22no—
84C3×C4⋊C4—22no—
85C3×C4⋊C4—22no—
86C3×C4⋊C4—42no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4—42no—
89C3×C22⋊C4—41no—
90C3×C22⋊C4—42no—
91C3×C42—22no—
92C3×C42—41no—
93C3×C42—41no—
94C3×C42—42no—
95C3×C42C2×D411yescharacteristic
96C3×C42C2×D412yes—
97C3×C42#4111yescharacteristic
98C2×Q16—22no—
99C2×M16C3×D412yes—
100C4⋊Q8C3×D412yes—
101C4⋊Q8C3×D412yes—
102C4⋊Q8C3×D412yes—
103#125—22no—
104#125—21no—
105C22⋊Q8—22no—
106C22⋊Q8—22no—
107C22⋊Q8—22no—
108C4×Q8—22no—
109C4×Q8—22no—
110C4×Q8—22no—
111#117—21no—
112#117—22no—
113C2×C42—21no—
114C2×C42C23×C311yescharacteristic
115#107—24no—
116#106—22no—
117#106—24no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120#103—24no—
121#103—24no—
122#103—24no—
123C22⋊C8C3×D414yes—
124#97—22no—
125#95—22no—
126C23×C3#12711yescharacteristic
127C3×Q8—44no—
128C3×Q8—42no—
129C3×Q8—42no—
130C3×Q8—82no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—42no—
137C2×C3×C4—42no—
138C2×C3×C4—82no—
139C2×C3×C4#12012yes—
140C2×C3×C4#12011yescharacteristic
141C2×C3×C4#12112yes—
142C2×C3×C4#12111yescharacteristic
143C3×C8—44no—
144C3×C8—42no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—22no—
148C22×C4—41no—
149C22×C4C2×C3×D411yescharacteristic
150C22×C4C2×C3×D412yes—
151Q16—82no—
152M16—42no—
153C2×C8—24no—
154C2×C8—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C22⋊C4—41no—
165C22⋊C4—42no—
166C42—22no—
167C42—41no—
168C42—41no—
169C42—42no—
170C42C2×C3×D411yescharacteristic
171C42C2×C3×D412yes—
172C42C3×#4111yescharacteristic
173C22×C3—21no—
174C22×C3—22no—
175C22×C3#39211yescharacteristic
176C3×C4—21no—
177C3×C4—22no—
178C3×C4—22no—
179C3×C4—82no—
180C3×C4—82no—
181C3×C4—82no—
182C3×C4—82no—
183C3×C4#45511yescharacteristic
184C3×C4#45611yescharacteristic
185C23C3×#12711yescharacteristic
186Q8—44no—
187Q8—42no—
188Q8—42no—
189Q8—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—82no—
198C2×C4C3×#12012yes—
199C2×C4C3×#12011yescharacteristic
200C2×C4C3×#12112yes—
201C2×C4C3×#12111yescharacteristic
202C8—44no—
203C8—42no—
204C2×C3—41no—
205C2×C3#200711yescharacteristic
206C2×C3#200912yes—
207C22—21no—
208C22—22no—
209C22C3×#39211yescharacteristic
210C4—21no—
211C4—22no—
212C4—22no—
213C4—82no—
214C4—82no—
215C4—82no—
216C4—82no—
217C4C3×#45511yescharacteristic
218C4C3×#45611yescharacteristic
219C3#1021711yescharacteristic, direct product factor, Sylow
220C2—41no—
221C2C3×#200711yescharacteristic
222C2C3×#200912yes—
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC23×C3#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
6. #10217C3direct (N × H)
219. C3#10217direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32552
C3×#32372
C3×#19952
C3×#15062
C3×#13072
#102173