Finite Group Databasebeta

← C3×#10259 #10628866

Subgroups — C3×#10259 #10628866

Order 768.

474 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3249C211yesmaximal, characteristic
2C3×#3078C212yesmaximal
3C3×#1984C211yesmaximal, characteristic
4C3×#1493C212yesmaximal
5C3×#1292C211yesmaximal, characteristic
6#10259C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#533C2212yes—
8C2×C3×C4×Q8C2211yescharacteristic
9C3×#479—24no—
10C3×#477—22no—
11C3×#477—22no—
12C3×#475—22no—
13C3×#467—22no—
14C3×#462—22no—
15C3×#428—22no—
16C3×#423C2212yes—
17C2×C3×C4⋊C8C2211yescharacteristic
18C3×#419—22no—
19C2×C3×C4×C8C2211yescharacteristic
20C3×#336—22no—
21C3×Q8⋊C8—24no—
22#3249C2×C311yescharacteristic
23#3078C2×C312yes—
24#1984C2×C311yescharacteristic
25#1493C2×C312yes—
26#1292C2×C311yescharacteristic
27C22×C3×Q8D411yescharacteristic
28C2×C3×Q16—22no—
29C2×C3×M16—22no—
30C22×C3×C8—21no—
31C22×C3×C8D412yes—
32C3×C4⋊Q8D412yes—
33C3×C4⋊Q8D412yes—
34C3×#125—22no—
35C3×C22⋊Q8—22no—
36C3×C22⋊Q8—22no—
37C3×C4×Q8—22no—
38C3×C4×Q8—22no—
39C3×C4×Q8—22no—
40C3×C4×Q8—22no—
41C3×#117—22no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C42—21no—
45C2×C3×C42C2311yescharacteristic
46C3×#107—24no—
47C3×#107—22no—
48C3×#106—24no—
49C3×C4⋊C8—22no—
50C3×C4⋊C8—22no—
51C3×C4⋊C8D414yes—
52C3×#103—24no—
53C3×#103—24no—
54C3×#103—24no—
55C3×C4×C8—22no—
56#533C22×C312yes—
57C2×C4×Q8C22×C311yescharacteristic
58#479—24no—
59#477—22no—
60#477—22no—
61#475—22no—
62#467—22no—
63#462—22no—
64#428—22no—
65#423C22×C312yes—
66C2×C4⋊C8C22×C311yescharacteristic
67#419—22no—
68C2×C4×C8C22×C311yescharacteristic
69#336—22no—
70Q8⋊C8—24no—
71C2×C3×Q8—21no—
72C2×C3×Q8—22no—
73C2×C3×Q8—22no—
74C2×C3×Q8—22no—
75C2×C3×Q8—22no—
76C2×C3×Q8Q1612yescomplemented
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4C2×D411yescharacteristic
80C22×C3×C4C2×D411yescharacteristic
81C22×C3×C4#4111yescharacteristic
82C3×Q16—82no—
83C3×M16—42no—
84C3×M16—42no—
85C2×C3×C8—22no—
86C2×C3×C8—22no—
87C2×C3×C8—24no—
88C2×C3×C8—41no—
89C2×C3×C8—41no—
90C2×C3×C8—42no—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—42no—
97C3×C4⋊C4—42no—
98C3×C4⋊C4—42no—
99C3×C22⋊C4—42no—
100C3×C42—41no—
101C3×C42—42no—
102C3×C42—41no—
103C3×C42C2×D412yes—
104C3×C42C2×D412yes—
105C22×Q8C3×D411yescharacteristic
106C2×Q16—22no—
107C2×M16—22no—
108C22×C8—21no—
109C22×C8C3×D412yes—
110C4⋊Q8C3×D412yes—
111C4⋊Q8C3×D412yes—
112#125—22no—
113C22⋊Q8—22no—
114C22⋊Q8—22no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—22no—
119#117—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4C3×D411yescharacteristic
122C2×C42—21no—
123C2×C42C23×C311yescharacteristic
124#107—24no—
125#107—22no—
126#106—24no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8C3×D414yes—
130#103—24no—
131#103—24no—
132#103—24no—
133C4×C8—22no—
134C23×C3#12111yescharacteristic
135C3×Q8—24no—
136C3×Q8—44no—
137C3×Q8—42no—
138C3×Q8—42no—
139C3×Q8—42no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—22no—
143C2×C3×C4—41no—
144C2×C3×C4—41no—
145C2×C3×C4—41no—
146C2×C3×C4—42no—
147C2×C3×C4—41no—
148C2×C3×C4—42no—
149C2×C3×C4—42no—
150C2×C3×C4#12012yes—
151C2×C3×C4#12011yescharacteristic
152C2×C3×C4#12112yes—
153C2×C3×C4#12711yescharacteristic
154C2×C3×C4C2×Q1611yescharacteristic
155C2×C3×C4#13511yescharacteristic
156C2×C3×C4#13712yes—
157C3×C8—42no—
158C3×C8—44no—
159C3×C8—42no—
160C2×Q8—21no—
161C2×Q8—22no—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8C3×Q1612yescomplemented
166C22×C4—21no—
167C22×C4—21no—
168C22×C4C2×C3×D411yescharacteristic
169C22×C4C2×C3×D411yescharacteristic
170C22×C4C3×#4111yescharacteristic
171Q16—82no—
172M16—42no—
173M16—42no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—24no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—42no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—42no—
188C22⋊C4—42no—
189C42—41no—
190C42—42no—
191C42—41no—
192C42C2×C3×D412yes—
193C42C2×C3×D412yes—
194C22×C3—21no—
195C22×C3—21no—
196C22×C3#39211yescharacteristic
197C22×C3#46711yescharacteristic
198C22×C3#47011yescharacteristic
199C3×C4—21no—
200C3×C4—22no—
201C3×C4—42no—
202C3×C4—82no—
203C3×C4—81no—
204C3×C4—82no—
205C3×C4#45112yes—
206C3×C4#45212yes—
207C3×C4#45511yescharacteristic
208C3×C4#45611yescharacteristic
209C23C3×#12111yescharacteristic
210Q8—24no—
211Q8—44no—
212Q8—42no—
213Q8—42no—
214Q8—42no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4C3×#12012yes—
226C2×C4C3×#12011yescharacteristic
227C2×C4C3×#12112yes—
228C2×C4C3×#12711yescharacteristic
229C2×C4C2×C3×Q1611yescharacteristic
230C2×C4C3×#13511yescharacteristic
231C2×C4C3×#13712yes—
232C8—42no—
233C8—44no—
234C8—42no—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3#200111yescharacteristic
238C2×C3#200711yescharacteristic
239C2×C3#201711yescharacteristic
240C22—21no—
241C22—21no—
242C22C3×#39211yescharacteristic
243C22C3×#46711yescharacteristic
244C22C3×#47011yescharacteristic
245C4—21no—
246C4—22no—
247C4—42no—
248C4—82no—
249C4—81no—
250C4—82no—
251C4C3×#45112yes—
252C4C3×#45212yes—
253C4C3×#45511yescharacteristic
254C4C3×#45611yescharacteristic
255C3#1025911yescharacteristic, direct product factor, Sylow
256C2—21no—
257C2—21no—
258C2C3×#200111yescharacteristic
259C2C3×#200711yescharacteristic
260C2C3×#201711yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC23×C3#121
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. #10259C3direct (N × H)
76. C2×C3×Q8Q16semidirect
165. C2×Q8C3×Q16semidirect
255. C3#10259direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32492
C3×#30782
C3×#19842
C3×#14932
C3×#12922
#102593