Finite Group Databasebeta

← C3×#10592794 #13163133

Subgroups — C3×#10592794 #13163133

Order 1728.

1124 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 174 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×#64364C211yesmaximal, characteristic
2C33×C4⋊Q8C211yesmaximal, characteristic, complemented
3C3×#64151C212yesmaximal, complemented
4C3×#63663C211yesmaximal, characteristic, complemented
5C3×#63632C212yesmaximal, complemented
6#10592794C311yesmaximal, characteristic, direct product factor
7C3×#5095—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C33×Q8C2211yescharacteristic
10C3×#3832—22no—
11C3×Q8×C3⋊D3—22no—
12C3×#3828—22no—
13C3×#3828C2212yes—
14C33×C4⋊C4—22no—
15C33×C42C2211yescharacteristic
16C3×#3756—22no—
17C3×#3755—22no—
18C3×#3748C2212yes—
19#64364C2×C311yescharacteristic
20C3×#1007—34no—
21C32×C4⋊Q8—24no—
22C32×C4⋊Q8D314yescomplemented
23C32×C4⋊Q8C2×C311yescharacteristic, complemented
24#64151C2×C312yescomplemented
25C3×#942—38no—
26#63663C2×C311yescharacteristic, complemented
27#63632C2×C312yescomplemented
28C3×#836—34no—
29C3×#805—38no—
30C33×Q8—22no—
31C33×Q8—22no—
32C2×C33×C4—21no—
33C2×C33×C4—21no—
34C2×C33×C4D412yes—
35C2×C33×C4C2311yescharacteristic
36C3×#638—22no—
37C3×C4⋊(C3⋊D3)—22no—
38C3×C4⋊(C3⋊D3)D412yes—
39C3×C4×C3⋊D3—24no—
40C3×C4×C3⋊D3—22no—
41C3×#634—22no—
42C3×#634D412yes—
43C3×C3⋊(C3⋊C8)—22no—
44C3×#456—91no—
45#5095—34no—
46C2×C32×Q8—24no—
47C2×C32×Q8—24no—
48C2×C32×Q8C2×D314yes—
49C2×C32×Q8C2×D314yes—
50C2×C32×Q8C22×C311yescharacteristic
51C2×C32×Q8C22×C311yescharacteristic
52#3832—22no—
53Q8×C3⋊D3—22no—
54#3828—22no—
55#3828C22×C312yes—
56C3×#239—68no—
57C3×D3×Q8—68no—
58C3×#235—38no—
59C3×#235—68no—
60C32×C4⋊C4—216no—
61C32×C4⋊C4—28no—
62C32×C4⋊C4—22no—
63C32×C42—24no—
64C32×C42C2×D314yes—
65C32×C42C22×C311yescharacteristic
66#3756—22no—
67#3755—22no—
68#3748C22×C312yes—
69C3×#216—68no—
70C3×#215—68no—
71C3×#208—38no—
72C22×C33C2×D411yescharacteristic
73C2×C3×C3⋊D3—22no—
74C33×C4—22no—
75C33×C4—22no—
76C33×C4—41no—
77C33×C4C2×D412yes—
78C3×C3⋊Dic3—22no—
79C3×2−1+4—91no—
80#1007—34no—
81C3×#137—92no—
82C3×C4⋊Q8—24no—
83C3×C4⋊Q8C3×D314yescomplemented
84C3×C4⋊Q8C3⋊D311yescharacteristic, complemented
85#942—38no—
86C3×#104—92no—
87C3×#101—91no—
88#836—34no—
89#805—38no—
90C32×Q8—22no—
91C32×Q8—22no—
92C32×Q8—28no—
93C32×Q8—28no—
94C32×Q8—48no—
95C32×Q8—48no—
96C2×C32×C4—28no—
97C2×C32×C4—24no—
98C2×C32×C4—21no—
99C2×C32×C4—28no—
100C2×C32×C4—24no—
101C2×C32×C4—21no—
102C2×C32×C4—28no—
103C2×C32×C4—24no—
104C2×C32×C4#6718yes—
105C2×C32×C4C3×D412yes—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C23×C311yescharacteristic
108#638—22no—
109C4⋊(C3⋊D3)—22no—
110C4⋊(C3⋊D3)C3×D412yes—
111C4×C3⋊D3—24no—
112C4×C3⋊D3—22no—
113#634—22no—
114#634C3×D412yes—
115C3×#67—68no—
116C3×D12—38no—
117C3×D12—68no—
118C3×C4×D3—616no—
119C3×C4×D3—68no—
120C3×Dic6—38no—
121C3×Dic6—68no—
122C3⋊(C3⋊C8)—22no—
123C3×C3⋊C8—68no—
124#456—91noSylow
125C2×C33—21no—
126C2×C33#12011yescharacteristic
127C3×C3⋊D3—42no—
128C3×#41—92no—
129C3×#41—182no—
130C3×#41—182no—
131C2×C3×Q8—24no—
132C2×C3×Q8—24no—
133C2×C3×Q8—182no—
134C2×C3×Q8C2×C3×D314yes—
135C2×C3×Q8C2×C3×D314yes—
136C2×C3×Q8C2×C3⋊D311yescharacteristic
137C2×C3×Q8C2×C3⋊D311yescharacteristic
138#239—68no—
139D3×Q8—68no—
140#235—38no—
141#235—68no—
142C3×Q16—182no—
143C3×SD16—182no—
144C3×M16—92no—
145C3×C4⋊C4—216no—
146C3×C4⋊C4—28no—
147C3×C4⋊C4—22no—
148C3×C42—24no—
149C3×C42C2×C3×D314yes—
150C3×C42C2×C3⋊D311yescharacteristic
151#216—68no—
152#215—68no—
153#208—38no—
154C22×C32—24no—
155C22×C32C2×#6714yes—
156C22×C32C2×C3×D411yescharacteristic
157C2×C3⋊D3—22no—
158C2×C3×D3—68no—
159C32×C4—216no—
160C32×C4—22no—
161C32×C4—28no—
162C32×C4—22no—
163C32×C4—28no—
164C32×C4—28no—
165C32×C4—48no—
166C32×C4—48no—
167C32×C4—44no—
168C32×C4—41no—
169C32×C4D3×D418yes—
170C32×C4C2×C3×D412yes—
171C3⋊Dic3—22no—
172C3×Dic3—68no—
1732−1+4—91no—
174#137—92no—
175C4⋊Q8C3×C3⋊D311yescharacteristic, complemented
176#104—92no—
177#101—91no—
178C33#45611yescharacteristic, complemented, Sylow
179C3×Q8—28no—
180C3×Q8—22no—
181C3×Q8—28no—
182C3×Q8—22no—
183C3×Q8—48no—
184C3×Q8—48no—
185C3×Q8—92no—
186C3×Q8—182no—
187C3×D4—92no—
188C3×D4—182no—
189C3×D4—182no—
190C2×C3×C4—28no—
191C2×C3×C4—24no—
192C2×C3×C4—21no—
193C2×C3×C4—28no—
194C2×C3×C4—24no—
195C2×C3×C4—21no—
196C2×C3×C4—28no—
197C2×C3×C4—24no—
198C2×C3×C4—184no—
199C2×C3×C4—182no—
200C2×C3×C4C3×#6718yes—
201C2×C3×C4#63812yes—
202C2×C3×C4C22×C3×D314yes—
203C2×C3×C4C22×C3⋊D311yescharacteristic
204#67—68no—
205D12—38no—
206D12—68no—
207C4×D3—616no—
208C4×D3—68no—
209Dic6—38no—
210Dic6—68no—
211C3×C8—182no—
212C3⋊C8—68no—
213C2×C32—28no—
214C2×C32—21no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32#93714yes—
218C2×C32C3×#12011yescharacteristic
219C3⋊D3—42no—
220C3×D3—128no—
221#41—92no—
222#41—182no—
223#41—182no—
224C2×Q8—182no—
225C2×Q8C2×C3×C3⋊D311yescharacteristic
226C2×Q8C2×C3×C3⋊D311yescharacteristic
227Q16—182no—
228SD16—182no—
229M16—92no—
230C4⋊C4—22no—
231C42C2×C3×C3⋊D311yescharacteristic
232C22×C3—24no—
233C22×C3—182no—
234C22×C3C2×C3×#6714yes—
235C22×C3C2×#63811yescharacteristic
236C2×D3—68no—
237C3×C4—216no—
238C3×C4—22no—
239C3×C4—28no—
240C3×C4—28no—
241C3×C4—22no—
242C3×C4—28no—
243C3×C4—48no—
244C3×C4—48no—
245C3×C4—44no—
246C3×C4—41no—
247C3×C4—182no—
248C3×C4C3×D3×D418yes—
249C3×C4D4×C3⋊D312yes—
250Dic3—68no—
251C32—24no—
252C32#509514yescomplemented
253C32C3×#45611yescharacteristic, complemented
254Q8—22no—
255Q8—22no—
256Q8—92no—
257Q8—182no—
258D4—92no—
259D4—182no—
260D4—182no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—184no—
264C2×C4—182no—
265C2×C4C3×#63812yes—
266C2×C4C22×C3×C3⋊D311yescharacteristic
267C8—182no—
268C2×C3—28no—
269C2×C3—21no—
270C2×C3—24no—
271C2×C3—24no—
272C2×C3—362no—
273C2×C3C3×#93714yes—
274C2×C3#6414611yescharacteristic
275D3—128no—
276C22—182no—
277C22C2×C3×#63811yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—182no—
282C4C3×D4×C3⋊D312yes—
283C3—24no—
284C3C3×#509514yescomplemented
285C3#1059279411yescharacteristic, direct product factor
286C2—21no—
287C2—362no—
288C2C3×#6414611yescharacteristic
subgroupG/H
centerC2×C3#64146
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C4⋊Q8C2
socleC2×C33#120
prime radicalC3×#64364C2
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterC32C3×#456
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C33×C4⋊Q8C2semidirect
3. C3×#64151C2semidirect
4. C3×#63663C2semidirect
5. C3×#63632C2semidirect
6. #10592794C3direct (N × H)
22. C32×C4⋊Q8D3semidirect
23. C32×C4⋊Q8C2×C3semidirect
24. #64151C2×C3semidirect
26. #63663C2×C3semidirect
27. #63632C2×C3semidirect
83. C3×C4⋊Q8C3×D3semidirect
84. C3×C4⋊Q8C3⋊D3semidirect
175. C4⋊Q8C3×C3⋊D3semidirect
178. C33#456semidirect
252. C32#5095semidirect
253. C32C3×#456semidirect
284. C3C3×#5095semidirect
285. C3#10592794direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643642
C33×C4⋊Q82
C3×#641512
C3×#636632
C3×#636322
#105927943
C3×#50953