Finite Group Databasebeta

← C3×#10592761 #13163100

Subgroups — C3×#10592761 #13163100

Order 1728.

1180 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1180 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, complemented
2C33×#124C211yesmaximal, characteristic, complemented
3C3×#64141C212yesmaximal, complemented
4C3×#63663C211yesmaximal, characteristic, complemented
5C3×#63632C212yesmaximal, complemented
6#10592761C311yesmaximal, characteristic, direct product factor
7C3×#5062—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C33×D4C2211yescharacteristic
10C3×#3832—22no—
11C3×Q8×C3⋊D3—22no—
12C3×#3828—22no—
13C3×#3828C2212yes—
14C33×C22⋊C4—22no—
15C33×C42C2211yescharacteristic
16C3×#3754—22no—
17C3×#3753—22no—
18C3×#3748C2212yes—
19#64364C2×C311yescharacteristic, complemented
20C3×#1007—34no—
21C32×#124—24no—
22C32×#124D314yescomplemented
23C32×#124C2×C311yescharacteristic, complemented
24#64141C2×C312yescomplemented
25C3×#932—38no—
26#63663C2×C311yescharacteristic, complemented
27#63632C2×C312yescomplemented
28C3×#836—34no—
29C3×#805—38no—
30C23×C33—21no—
31C33×Q8—22no—
32C33×D4—22no—
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×#454—91no—
45#5062—34no—
46C2×C32×Q8—24no—
47C2×C32×Q8C2×D314yes—
48C2×C32×Q8C22×C311yescharacteristic
49C2×C32×D4—24no—
50C2×C32×D4C2×D314yes—
51C2×C32×D4C22×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×C22⋊C4—216no—
61C32×C22⋊C4—28no—
62C32×C22⋊C4—22no—
63C32×C42—24no—
64C32×C42C2×D314yes—
65C32×C42C22×C311yescharacteristic
66#3754—22no—
67#3753—22no—
68#3748C22×C312yes—
69C3×#214—68no—
70C3×#213—68no—
71C3×#208—38no—
72C22×C33—42no—
73C22×C33—41no—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C33×C4—22no—
77C33×C4—41no—
78C33×C4C2×D412yes—
79C3×C3⋊Dic3—22no—
80C3×2−1+4—91no—
81#1007—34no—
82C3×AGL(1,ℤ/8)—92no—
83C3×#124—24no—
84C3×#124C3×D314yescomplemented
85C3×#124C3⋊D311yescharacteristic, complemented
86#932—38no—
87C3×#104—92no—
88C3×#101—91no—
89#836—34no—
90#805—38no—
91C23×C32—28no—
92C23×C32—24no—
93C23×C32—21no—
94C32×Q8—22no—
95C32×Q8—28no—
96C32×Q8—48no—
97C32×D4—22no—
98C32×D4—28no—
99C32×D4—48no—
100C2×C32×C4—28no—
101C2×C32×C4—24no—
102C2×C32×C4—21no—
103C2×C32×C4—28no—
104C2×C32×C4—24no—
105C2×C32×C4#6718yes—
106C2×C32×C4C3×D412yes—
107C2×C32×C4C22×D314yes—
108C2×C32×C4C23×C311yescharacteristic
109#638—22no—
110C4⋊(C3⋊D3)—22no—
111C4⋊(C3⋊D3)C3×D412yes—
112C4×C3⋊D3—24no—
113C4×C3⋊D3—22no—
114#634—22no—
115#634C3×D412yes—
116C3×#67—68no—
117C3×D12—38no—
118C3×D12—68no—
119C3×C4×D3—616no—
120C3×C4×D3—68no—
121C3×Dic6—38no—
122C3×Dic6—68no—
123C3⋊(C3⋊C8)—22no—
124C3×C3⋊C8—68no—
125#454—91noSylow
126C2×C33—21no—
127C2×C33—81no—
128C2×C33#12011yescharacteristic
129C3×C3⋊D3—42no—
130C3×#41—92no—
131C3×#41—182no—
132C3×#41—182no—
133C2×C3×Q8—24no—
134C2×C3×Q8—182no—
135C2×C3×Q8C2×C3×D314yes—
136C2×C3×Q8C2×C3⋊D311yescharacteristic
137C2×C3×D4—24no—
138C2×C3×D4C2×C3×D314yes—
139C2×C3×D4C2×C3⋊D311yescharacteristic
140#239—68no—
141D3×Q8—68no—
142#235—38no—
143#235—68no—
144C3×SD16—182no—
145C3×D8—182no—
146C3×M16—92no—
147C3×C22⋊C4—216no—
148C3×C22⋊C4—28no—
149C3×C22⋊C4—22no—
150C3×C42—24no—
151C3×C42C2×C3×D314yes—
152C3×C42C2×C3⋊D311yescharacteristic
153#214—68no—
154#213—68no—
155#208—38no—
156C22×C32—24no—
157C22×C32—48no—
158C22×C32—416no—
159C22×C32—48no—
160C22×C32—42no—
161C22×C32—44no—
162C22×C32—41no—
163C22×C32C2×#6714yes—
164C22×C32C2×C3×D411yescharacteristic
165C2×C3⋊D3—22no—
166C2×C3×D3—68no—
167C32×C4—22no—
168C32×C4—28no—
169C32×C4—28no—
170C32×C4—48no—
171C32×C4—48no—
172C32×C4—44no—
173C32×C4—41no—
174C32×C4D3×D418yes—
175C32×C4C2×C3×D412yes—
176C3⋊Dic3—22no—
177C3×Dic3—68no—
1782−1+4—91no—
179AGL(1,ℤ/8)—92no—
180#124C3×C3⋊D311yescharacteristic, complemented
181#104—92no—
182#101—91no—
183C33#45411yescharacteristic, complemented, Sylow
184C23×C3—28no—
185C23×C3—24no—
186C23×C3—21no—
187C3×Q8—28no—
188C3×Q8—22no—
189C3×Q8—48no—
190C3×Q8—92no—
191C3×Q8—182no—
192C3×D4—28no—
193C3×D4—22no—
194C3×D4—48no—
195C3×D4—92no—
196C3×D4—182no—
197C3×D4—182no—
198C2×C3×C4—28no—
199C2×C3×C4—24no—
200C2×C3×C4—21no—
201C2×C3×C4—28no—
202C2×C3×C4—24no—
203C2×C3×C4—184no—
204C2×C3×C4—182no—
205C2×C3×C4C3×#6718yes—
206C2×C3×C4#63812yes—
207C2×C3×C4C22×C3×D314yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209#67—68no—
210D12—38no—
211D12—68no—
212C4×D3—616no—
213C4×D3—68no—
214Dic6—38no—
215Dic6—68no—
216C3×C8—182no—
217C3⋊C8—68no—
218C2×C32—28no—
219C2×C32—24no—
220C2×C32—21no—
221C2×C32—24no—
222C2×C32—88no—
223C2×C32—84no—
224C2×C32—81no—
225C2×C32#93714yes—
226C2×C32C3×#12011yescharacteristic
227C3⋊D3—42no—
228C3×D3—128no—
229#41—92no—
230#41—182no—
231#41—182no—
232C2×Q8—182no—
233C2×Q8C2×C3×C3⋊D311yescharacteristic
234C2×D4C2×C3×C3⋊D311yescharacteristic
235SD16—182no—
236D8—182no—
237M16—92no—
238C22⋊C4—22no—
239C42C2×C3×C3⋊D311yescharacteristic
240C22×C3—24no—
241C22×C3—48no—
242C22×C3—416no—
243C22×C3—48no—
244C22×C3—42no—
245C22×C3—44no—
246C22×C3—41no—
247C22×C3—182no—
248C22×C3C2×C3×#6714yes—
249C22×C3C2×#63811yescharacteristic
250C2×D3—68no—
251C3×C4—28no—
252C3×C4—22no—
253C3×C4—28no—
254C3×C4—48no—
255C3×C4—48no—
256C3×C4—44no—
257C3×C4—41no—
258C3×C4—182no—
259C3×C4C3×D3×D418yes—
260C3×C4D4×C3⋊D312yes—
261Dic3—68no—
262C32—24no—
263C32#506214yescomplemented
264C32C3×#45411yescharacteristic, complemented
265C23—21no—
266Q8—22no—
267Q8—92no—
268Q8—182no—
269D4—22no—
270D4—92no—
271D4—182no—
272D4—182no—
273C2×C4—21no—
274C2×C4—184no—
275C2×C4—182no—
276C2×C4C3×#63812yes—
277C2×C4C22×C3×C3⋊D311yescharacteristic
278C8—182no—
279C2×C3—28no—
280C2×C3—24no—
281C2×C3—21no—
282C2×C3—24no—
283C2×C3—88no—
284C2×C3—84no—
285C2×C3—81no—
286C2×C3—362no—
287C2×C3C3×#93714yes—
288C2×C3#6414611yescharacteristic
289D3—128no—
290C22—42no—
291C22—41no—
292C22—182no—
293C22C2×C3×#63811yescharacteristic
294C4—22no—
295C4—41no—
296C4—182no—
297C4C3×D4×C3⋊D312yes—
298C3—24no—
299C3C3×#506214yescomplemented
300C3#1059276111yescharacteristic, direct product factor
301C2—21no—
302C2—81no—
303C2—362no—
304C2C3×#6414611yescharacteristic
subgroupG/H
centerC2×C3#64146
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#124C2
socleC2×C33#120
hypercenterC3×#124C3⋊D3
hypocenterC32C3×#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#124

Complements

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

NH ≅ G/Ntype
1. C3×#64364C2semidirect
2. C33×#124C2semidirect
3. C3×#64141C2semidirect
4. C3×#63663C2semidirect
5. C3×#63632C2semidirect
6. #10592761C3direct (N × H)
19. #64364C2×C3semidirect
22. C32×#124D3semidirect
23. C32×#124C2×C3semidirect
24. #64141C2×C3semidirect
26. #63663C2×C3semidirect
27. #63632C2×C3semidirect
84. C3×#124C3×D3semidirect
85. C3×#124C3⋊D3semidirect
180. #124C3×C3⋊D3semidirect
183. C33#454semidirect
263. C32#5062semidirect
264. C32C3×#454semidirect
299. C3C3×#5062semidirect
300. C3#10592761direct (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×#1242
C3×#641412
C3×#636632
C3×#636322
#105927613
C3×#50623