Finite Group Databasebeta

← C3×#10592785 #13163123

Subgroups — C3×#10592785 #13163123

Order 1728.

1016 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 198 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
1C33×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#3756C212yesmaximal, complemented
3C2×C3×#3755C212yesmaximal
4C3×#64084C211yesmaximal, characteristic
5C3×#63630C211yesmaximal, characteristic, complemented
6#10592785C311yesmaximal, characteristic, direct product factor
7C3×#5086—34nomaximal
8C2×C33×Q8C2212yes—
9C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C3×#634C2211yescharacteristic
11C33×C4⋊C4—22no—
12C33×C42C2211yescharacteristic
13C3×#3756—24no—
14C3×#3755—24no—
15C3×#3752—22no—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C32×C4⋊Q8—24no—
18C32×C4⋊Q8D314yescomplemented
19C32×C4⋊Q8C2×C311yescharacteristic, complemented
20C2×#3756C2×C312yescomplemented
21C2×#3755C2×C312yes—
22#64084C2×C311yescharacteristic
23C2×C3×#216—38no—
24C2×C3×#215—38no—
25C3×#875—34no—
26#63630C2×C311yescharacteristic, complemented
27C3×#803—34no—
28C22×C3×C3⋊D3—21no—
29C33×Q8—22no—
30C33×Q8—22no—
31C2×C33×C4—22no—
32C2×C33×C4D412yes—
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—21no—
35C3×C4⋊(C3⋊D3)—22no—
36C3×#634—22no—
37C3×C3⋊(C3⋊C8)D414yes—
38C3×#497—91no—
39#5086—34no—
40C2×C32×Q8—28no—
41C2×C32×Q8C2×D318yes—
42C2×C32×Q8C22×C312yes—
43C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
44C2×#634C22×C311yescharacteristic
45C2×C3×D12—34no—
46C2×C3×Dic6—34no—
47C32×C4⋊C4—216no—
48C32×C4⋊C4—28no—
49C32×C4⋊C4—22no—
50C32×C42—24no—
51C32×C42C2×D314yes—
52C32×C42C22×C311yescharacteristic
53#3756—24no—
54#3755—24no—
55#3752—22no—
56C2×C3⋊(C3⋊C8)C22×C312yes—
57C3×#216—616no—
58C3×#215—616no—
59C3×(C2×D3)⋊C4—68no—
60C2×C3×C3⋊C8—38no—
61C22×C33C2×D411yescharacteristic
62C2×C3×C3⋊D3—41no—
63C2×C3×C3⋊D3—42no—
64C33×C4—22no—
65C33×C4—42no—
66C33×C4C2×D412yes—
67C3×C3⋊Dic3—41no—
68C2×C3×Q16—92no—
69C2×C3×SD16—92no—
70C3×C4⋊Q8—24no—
71C3×C4⋊Q8C3×D314yescomplemented
72C3×C4⋊Q8C3⋊D311yescharacteristic, complemented
73C3×#124—91no—
74C2×#216—38no—
75C2×#215—38no—
76#875—34no—
77C3×#97—91no—
78#803—34no—
79C22×C3⋊D3—21no—
80C22×C3×D3—64no—
81C32×Q8—22no—
82C32×Q8—22no—
83C32×Q8—28no—
84C32×Q8—28no—
85C32×Q8—48no—
86C32×Q8—48no—
87C2×C32×C4—216no—
88C2×C32×C4—28no—
89C2×C32×C4—22no—
90C2×C32×C4—28no—
91C2×C32×C4—24no—
92C2×C32×C4#6718yes—
93C2×C32×C4C3×D412yes—
94C2×C32×C4C22×D314yes—
95C2×C32×C4C23×C311yescharacteristic
96C2×C3⋊Dic3—21no—
97C4⋊(C3⋊D3)—22no—
98#634—22no—
99C2×C3×Dic3—64no—
100C3×D12—68no—
101C3×Dic6—68no—
102C3⋊(C3⋊C8)C3×D414yes—
103C3×C3⋊C8—316no—
104#497—91noSylow
105C2×C33#12711yescharacteristic
106C2×C33#13712yes—
107C3×C3⋊D3—81no—
108C2×C3×Q8—28no—
109C2×C3×Q8—91no—
110C2×C3×Q8C2×C3×D318yes—
111C2×C3×Q8C2×C3⋊D312yes—
112C2×C3×D4—91no—
113C2×D12—34no—
114C2×Dic6—34no—
115C3×Q16—184no—
116C3×SD16—184no—
117C2×C3×C8—92no—
118C3×C4⋊C4—216no—
119C3×C4⋊C4—28no—
120C3×C4⋊C4—22no—
121C3×C22⋊C4—182no—
122C3×C42—24no—
123C3×C42C2×C3×D314yes—
124C3×C42C2×C3⋊D311yescharacteristic
125#216—616no—
126#215—616no—
127(C2×D3)⋊C4—68no—
128C2×C3⋊C8—38no—
129C22×C32—24no—
130C22×C32C2×#6714yes—
131C22×C32C2×C3×D411yescharacteristic
132C2×C3⋊D3—41no—
133C2×C3⋊D3—42no—
134C2×C3×D3—124no—
135C2×C3×D3—128no—
136C32×C4—216no—
137C32×C4—28no—
138C32×C4—22no—
139C32×C4—28no—
140C32×C4—416no—
141C32×C4—48no—
142C32×C4—42no—
143C32×C4D3×D418yes—
144C32×C4C2×C3×D412yes—
145C3⋊Dic3—41no—
146C3×Dic3—124no—
147C2×Q16—92no—
148C2×SD16—92no—
149C4⋊Q8C3×C3⋊D311yescharacteristic, complemented
150#124—91no—
151#97—91no—
152C33#49711yescharacteristic, complemented, Sylow
153C23×C3—181no—
154C22×D3—64no—
155C3×Q8—28no—
156C3×Q8—28no—
157C3×Q8—22no—
158C3×Q8—22no—
159C3×Q8—48no—
160C3×Q8—48no—
161C3×Q8—182no—
162C3×D4—182no—
163C2×C3×C4—216no—
164C2×C3×C4—28no—
165C2×C3×C4—22no—
166C2×C3×C4—28no—
167C2×C3×C4—24no—
168C2×C3×C4—181no—
169C2×C3×C4C3×#6718yes—
170C2×C3×C4#63812yes—
171C2×C3×C4C22×C3×D314yes—
172C2×C3×C4C22×C3⋊D311yescharacteristic
173C2×Dic3—64no—
174D12—68no—
175Dic6—68no—
176C3×C8—94no—
177C3⋊C8—316no—
178C2×C32—24no—
179C2×C32—28no—
180C2×C32#94014yes—
181C2×C32#94218yes—
182C2×C32C3×#12711yescharacteristic
183C2×C32C3×#13712yes—
184C3⋊D3—81no—
185C3×D3—244no—
186C2×Q8—91no—
187C2×Q8C2×C3×C3⋊D312yes—
188C2×D4—91no—
189Q16—184no—
190SD16—184no—
191C2×C8—92no—
192C4⋊C4—22no—
193C22⋊C4—182no—
194C42C2×C3×C3⋊D311yescharacteristic
195C22×C3—24no—
196C22×C3—361no—
197C22×C3—362no—
198C22×C3C2×C3×#6714yes—
199C22×C3C2×#63811yescharacteristic
200C2×D3—124no—
201C2×D3—128no—
202C3×C4—216no—
203C3×C4—28no—
204C3×C4—22no—
205C3×C4—28no—
206C3×C4—416no—
207C3×C4—48no—
208C3×C4—42no—
209C3×C4—361no—
210C3×C4C3×D3×D418yes—
211C3×C4D4×C3⋊D312yes—
212Dic3—124no—
213C32—24no—
214C32#508614yescomplemented
215C32C3×#49711yescharacteristic, complemented
216C23—181no—
217Q8—22no—
218Q8—22no—
219Q8—182no—
220D4—182no—
221C2×C4—22no—
222C2×C4—181no—
223C2×C4C3×#63812yes—
224C2×C4C22×C3×C3⋊D311yescharacteristic
225C8—94no—
226C2×C3—24no—
227C2×C3—28no—
228C2×C3—721no—
229C2×C3C3×#94014yes—
230C2×C3C3×#94218yes—
231C2×C3#6414911yescharacteristic
232C2×C3#6415112yes—
233D3—244no—
234C22—361no—
235C22—362no—
236C22C2×C3×#63811yescharacteristic
237C4—22no—
238C4—42no—
239C4—361no—
240C4C3×D4×C3⋊D312yes—
241C3—24no—
242C3C3×#508614yescomplemented
243C3#1059278511yescharacteristic, direct product factor
244C2—721no—
245C2C3×#6414911yescharacteristic
246C2C3×#6415112yes—
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4⋊Q8C2
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterC32C3×#497
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C33×C4⋊Q8C2semidirect
2. C2×C3×#3756C2semidirect
5. C3×#63630C2semidirect
6. #10592785C3direct (N × H)
18. C32×C4⋊Q8D3semidirect
19. C32×C4⋊Q8C2×C3semidirect
20. C2×#3756C2×C3semidirect
26. #63630C2×C3semidirect
71. C3×C4⋊Q8C3×D3semidirect
72. C3×C4⋊Q8C3⋊D3semidirect
149. C4⋊Q8C3×C3⋊D3semidirect
152. C33#497semidirect
214. C32#5086semidirect
215. C32C3×#497semidirect
242. C3C3×#5086semidirect
243. C3#10592785direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4⋊Q82
C2×C3×#37562
C2×C3×#37552
C3×#640842
C3×#636302
#105927853
C3×#50863