Finite Group Databasebeta

← C3×#29657 #10648264

Subgroups — C3×#29657 #10648264

Order 768.

1148 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 242 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×#543C211yesmaximal, characteristic
2C3×#2483C213yesmaximal
3C3×#2062C214yesmaximal, complemented
4C3×#2030C214yesmaximal, complemented
5C3×#1819C213yesmaximal
6#29657C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#543—26no—
8C3×#543C2214yes—
9C2×C3×C22⋊Q8—23no—
10C2×C3×C22⋊Q8—23no—
11C22×C3×C22⋊C4C2211yescharacteristic
12C22×C3×C22⋊C4C2213yes—
13C3×#397—26no—
14C3×#385—23no—
15C3×#385C2216yes—
16C3×#381—23no—
17C3×#381C2216yes—
18C3×#379C2213yes—
19C3×#342C22112yes—
20C2×#543C2×C311yescharacteristic
21#2483C2×C313yes—
22#2062C2×C314yescomplemented
23#2030C2×C314yescomplemented
24#1819C2×C313yes—
25C25×C3C2311yescharacteristic
26C22×C3×Q8—21no—
27C22×C3×Q8—21no—
28C23×C3×C4D413yes—
29C23×C3×C4D413yes—
30C23×C3×C4C2311yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C3×C22⋊Q8—212no—
33C3×C22⋊Q8—212no—
34C3×C22⋊Q8—212no—
35C3×C22⋊Q8—212no—
36C2×C3×C4⋊C4—23no—
37C2×C3×C4⋊C4—23no—
38C2×C3×C4⋊C4—26no—
39C2×C3×C22⋊C4—26no—
40C2×C3×C22⋊C4—26no—
41C2×C3×C22⋊C4—23no—
42C2×C3×C22⋊C4—23no—
43C2×C3×C22⋊C4—23no—
44C2×C3×C22⋊C4—23no—
45C2×C3×C22⋊C4D416yes—
46C2×C3×C22⋊C4C2316yes—
47C2×C3×C22⋊C4C2316yes—
48C3×#95—26no—
49C3×#95—26no—
50C3×#95—26no—
51#543—26no—
52#543C22×C314yes—
53C2×C22⋊Q8—23no—
54C2×C22⋊Q8—23no—
55C22×C22⋊C4C22×C311yescharacteristic
56C22×C22⋊C4C22×C313yes—
57#397—26no—
58#385—23no—
59#385C22×C316yes—
60#381—23no—
61#381C22×C316yes—
62#379C22×C313yes—
63#342C22×C3112yes—
64C24×C3—24no—
65C24×C3—42no—
66C24×C3—81no—
67C24×C3C2×D413yes—
68C24×C3C2×D413yes—
69C24×C3C2411yescharacteristic
70C2×C3×Q8—24no—
71C2×C3×Q8—24no—
72C2×C3×Q8—82no—
73C2×C3×Q8—82no—
74C22×C3×C4—23no—
75C22×C3×C4—23no—
76C22×C3×C4—23no—
77C22×C3×C4—23no—
78C22×C3×C4—23no—
79C22×C3×C4—26no—
80C22×C3×C4—23no—
81C22×C3×C4—24no—
82C22×C3×C4—46no—
83C22×C3×C4—81no—
84C22×C3×C4—83no—
85C22×C3×C4C2×D416yes—
86C22×C3×C4C2×D416yes—
87C3×C4⋊C4—212no—
88C3×C4⋊C4—212no—
89C3×C4⋊C4—86no—
90C3×C22⋊C4—212no—
91C3×C22⋊C4—212no—
92C3×C22⋊C4—212no—
93C3×C22⋊C4—46no—
94C3×C22⋊C4—43no—
95C3×C22⋊C4—41no—
96C3×C22⋊C4—83no—
97C25C23×C311yescharacteristic
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4C3×D413yes—
101C23×C4C3×D413yes—
102C23×C4C23×C311yescharacteristic
103C23×C4C23×C311yescharacteristic
104C22⋊Q8—212no—
105C22⋊Q8—212no—
106C22⋊Q8—212no—
107C22⋊Q8—212no—
108C2×C4⋊C4—23no—
109C2×C4⋊C4—23no—
110C2×C4⋊C4—26no—
111C2×C22⋊C4—26no—
112C2×C22⋊C4—26no—
113C2×C22⋊C4—23no—
114C2×C22⋊C4—23no—
115C2×C22⋊C4—23no—
116C2×C22⋊C4—23no—
117C2×C22⋊C4C3×D416yes—
118C2×C22⋊C4C23×C316yes—
119C2×C22⋊C4C23×C316yes—
120#95—26no—
121#95—26no—
122#95—26no—
123C23×C3—26no—
124C23×C3—26no—
125C23×C3—26no—
126C23×C3—43no—
127C23×C3—41no—
128C23×C3—41no—
129C23×C3—43no—
130C23×C3—43no—
131C23×C3—88no—
132C23×C3#12013yes—
133C23×C3#12011yescharacteristic
134C23×C3C22×D413yes—
135C23×C32−1+414yes—
136C3×Q8—84no—
137C3×Q8—84no—
138C2×C3×C4—26no—
139C2×C3×C4—26no—
140C2×C3×C4—46no—
141C2×C3×C4—46no—
142C2×C3×C4—46no—
143C2×C3×C4—46no—
144C2×C3×C4—46no—
145C2×C3×C4—46no—
146C2×C3×C4—46no—
147C2×C3×C4—41no—
148C2×C3×C4—41no—
149C2×C3×C4—46no—
150C2×C3×C4—46no—
151C2×C3×C4—83no—
152C2×C3×C4—83no—
153C24—24no—
154C24—42no—
155C24—81no—
156C24C2×C3×D413yes—
157C24C2×C3×D413yes—
158C24C24×C311yescharacteristic
159C2×Q8—24no—
160C2×Q8—24no—
161C2×Q8—82no—
162C2×Q8—82no—
163C22×C4—23no—
164C22×C4—23no—
165C22×C4—23no—
166C22×C4—23no—
167C22×C4—23no—
168C22×C4—26no—
169C22×C4—23no—
170C22×C4—24no—
171C22×C4—46no—
172C22×C4—81no—
173C22×C4—83no—
174C22×C4C2×C3×D416yes—
175C22×C4C2×C3×D416yes—
176C4⋊C4—212no—
177C4⋊C4—212no—
178C4⋊C4—86no—
179C22⋊C4—212no—
180C22⋊C4—212no—
181C22⋊C4—212no—
182C22⋊C4—46no—
183C22⋊C4—43no—
184C22⋊C4—41no—
185C22⋊C4—83no—
186C22×C3—21no—
187C22×C3—23no—
188C22×C3—21no—
189C22×C3—23no—
190C22×C3—23no—
191C22×C3—23no—
192C22×C3—23no—
193C22×C3—44no—
194C22×C3—412no—
195C22×C3—86no—
196C22×C3#45812yes—
197C22×C3C2×#12011yescharacteristic
198C22×C3#53616yes—
199C3×C4—46no—
200C3×C4—46no—
201C3×C4—86no—
202C3×C4—81no—
203C3×C4—81no—
204C23—26no—
205C23—26no—
206C23—26no—
207C23—43no—
208C23—41no—
209C23—41no—
210C23—43no—
211C23—43no—
212C23—88no—
213C23C3×#12013yes—
214C23C3×#12011yescharacteristic
215C23C22×C3×D413yes—
216C23C3×2−1+414yes—
217Q8—84no—
218Q8—84no—
219C2×C4—26no—
220C2×C4—26no—
221C2×C4—46no—
222C2×C4—46no—
223C2×C4—46no—
224C2×C4—46no—
225C2×C4—46no—
226C2×C4—46no—
227C2×C4—46no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—46no—
231C2×C4—46no—
232C2×C4—83no—
233C2×C4—83no—
234C2×C3—26no—
235C2×C3—22no—
236C2×C3—43no—
237C2×C3#261411yescharacteristic
238C2×C3C2×#45811yescharacteristic
239C2×C3#302611yescharacteristic
240C22—23no—
241C22—21no—
242C22—23no—
243C22—21no—
244C22—23no—
245C22—23no—
246C22—23no—
247C22—44no—
248C22—412no—
249C22—86no—
250C22C3×#45812yes—
251C22C2×C3×#12011yescharacteristic
252C22C3×#53616yes—
253C4—46no—
254C4—46no—
255C4—86no—
256C4—81no—
257C4—81no—
258C3#2965711yescharacteristic, direct product factor, Sylow
259C2—26no—
260C2—22no—
261C2—43no—
262C2C3×#261411yescharacteristic
263C2C2×C3×#45811yescharacteristic
264C2C3×#302611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C25×C3 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2062C2semidirect
4. C3×#2030C2semidirect
6. #29657C3direct (N × H)
22. #2062C2×C3semidirect
23. #2030C2×C3semidirect
258. C3#29657direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5432
C3×#24832
C3×#20622
C3×#20302
C3×#18192
#296573