Finite Group Databasebeta

← C3×#10592760 #13163099

Subgroups — C3×#10592760 #13163099

Order 1728.

1216 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1216 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×#64361C211yesmaximal, characteristic
2C33×#124C211yesmaximal, characteristic, complemented
3C3×#64151C212yesmaximal, complemented
4C3×#63660C211yesmaximal, characteristic, complemented
5C3×#63632C212yesmaximal, complemented
6#10592760C311yesmaximal, characteristic, direct product factor
7C3×#5061—34nomaximal
8C2×C33×Q8C2211yescharacteristic, complemented
9C2×C33×D4C2211yescharacteristic
10C2×C3×#638—22no—
11C3×#3830—22no—
12C3×D4×C3⋊D3—22no—
13C3×#3828C2212yes—
14C33×C22⋊C4—22no—
15C33×C42C2211yescharacteristic, complemented
16C3×#3756—22no—
17C3×#3755—22no—
18C3×#3748C2212yescomplemented
19#64361C2×C311yescharacteristic
20C3×#1004—34no—
21C32×#124—24no—
22C32×#124D314yescomplemented
23C32×#124C2×C311yescharacteristic, complemented
24#64151C2×C312yescomplemented
25C3×#942—38no—
26#63660C2×C311yescharacteristic, complemented
27#63632C2×C312yescomplemented
28C3×#833—34no—
29C3×#805—38no—
30C23×C33D412yes—
31C22×C3×C3⋊D3—22no—
32C33×Q8—22no—
33C33×D4—22no—
34C2×C33×C4—21no—
35C2×C33×C4—21no—
36C2×C33×C4C2311yescharacteristic
37C3×#638—24no—
38C3×#638—22no—
39C2×C3×C3⋊Dic3—22no—
40C3×C4⋊(C3⋊D3)D412yes—
41C3×C4×C3⋊D3—22no—
42C3×#634D412yes—
43C3×C3⋊(C3⋊C8)—22no—
44C3×#455—91no—
45#5061—34no—
46C2×C32×Q8—24no—
47C2×C32×Q8C2×D314yescomplemented
48C2×C32×Q8C22×C311yescharacteristic, complemented
49C2×C32×D4—24no—
50C2×C32×D4C2×D314yes—
51C2×C32×D4C22×C311yescharacteristic
52C2×#638—22no—
53#3830—22no—
54D4×C3⋊D3—22no—
55#3828C22×C312yes—
56C2×C3×#67—68no—
57C3×#237—68no—
58C3×D3×D4—68no—
59C3×#235—38no—
60C32×C22⋊C4—216no—
61C32×C22⋊C4—28no—
62C32×C22⋊C4—22no—
63C32×C42—24no—
64C32×C42C2×D314yescomplemented
65C32×C42C22×C311yescharacteristic, complemented
66#3756—22no—
67#3755—22no—
68#3748C22×C312yescomplemented
69C3×#216—68no—
70C3×#215—68no—
71C3×#208—38no—
72C22×C33—22no—
73C22×C33—42no—
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3—82no—
77C33×C4—41no—
78C33×C4—41no—
79C33×C4C2×D412yes—
80C3×C3⋊Dic3—22no—
81C3×H(5,𝔽2)—91no—
82#1004—34no—
83C3×#137—92no—
84C3×#124—24no—
85C3×#124C3×D314yescomplemented
86C3×#124C3⋊D311yescharacteristic, complemented
87#942—38no—
88C3×#104—92no—
89C3×#100—91no—
90#833—34no—
91#805—38no—
92C23×C32—28no—
93C23×C32#6718yes—
94C23×C32C3×D412yes—
95C22×C3⋊D3—22no—
96C22×C3×D3—68no—
97C32×Q8—22no—
98C32×Q8—28no—
99C32×Q8—48no—
100C32×D4—22no—
101C32×D4—28no—
102C32×D4—48no—
103C2×C32×C4—28no—
104C2×C32×C4—28no—
105C2×C32×C4—24no—
106C2×C32×C4—24no—
107C2×C32×C4—21no—
108C2×C32×C4—21no—
109C2×C32×C4—24no—
110C2×C32×C4C22×D314yes—
111C2×C32×C4C23×C311yescharacteristic
112#638—24no—
113#638—22no—
114C2×C3⋊Dic3—22no—
115C4⋊(C3⋊D3)C3×D412yes—
116C4×C3⋊D3—22no—
117#634C3×D412yes—
118C3×#67—616no—
119C3×#67—68no—
120C2×C3×Dic3—68no—
121C3×D12—38no—
122C3×C4×D3—68no—
123C3×Dic6—38no—
124C3⋊(C3⋊C8)—22no—
125C3×C3⋊C8—68no—
126#455—91noSylow
127C2×C33—21no—
128C2×C33—42no—
129C2×C33#12011yescharacteristic
130C3×C3⋊D3—42no—
131C3×#41—92no—
132C3×#41—182no—
133C2×C3×Q8—24no—
134C2×C3×Q8C2×C3×D314yescomplemented
135C2×C3×Q8C2×C3⋊D311yescharacteristic, complemented
136C2×C3×D4—24no—
137C2×C3×D4—182no—
138C2×C3×D4—182no—
139C2×C3×D4C2×C3×D314yes—
140C2×C3×D4C2×C3⋊D311yescharacteristic
141C2×#67—68no—
142#237—68no—
143D3×D4—68no—
144#235—38no—
145C3×Q16—182no—
146C3×SD16—182no—
147C3×M16—92no—
148C3×C22⋊C4—216no—
149C3×C22⋊C4—28no—
150C3×C22⋊C4—22no—
151C3×C42—24no—
152C3×C42C2×C3×D314yescomplemented
153C3×C42C2×C3⋊D311yescharacteristic, complemented
154#216—68no—
155#215—68no—
156#208—38no—
157C22×C32—22no—
158C22×C32—24no—
159C22×C32—28no—
160C22×C32—48no—
161C22×C32—48no—
162C22×C32—42no—
163C22×C32—416no—
164C22×C32C2×#6714yes—
165C22×C32C2×C3×D411yescharacteristic
166C2×C3⋊D3—22no—
167C2×C3⋊D3—82no—
168C2×C3×D3—68no—
169C2×C3×D3—248no—
170C32×C4—28no—
171C32×C4—48no—
172C32×C4—44no—
173C32×C4—41no—
174C32×C4—48no—
175C32×C4—44no—
176C32×C4—41no—
177C32×C4D3×D418yes—
178C32×C4C2×C3×D412yes—
179C3⋊Dic3—22no—
180C3×Dic3—68no—
181H(5,𝔽2)—91no—
182#137—92no—
183#124C3×C3⋊D311yescharacteristic, complemented
184#104—92no—
185#100—91no—
186C33#45511yescharacteristic, complemented, Sylow
187C23×C3—28no—
188C23×C3—182no—
189C23×C3C3×#6718yes—
190C23×C3#63812yes—
191C22×D3—68no—
192C3×Q8—28no—
193C3×Q8—22no—
194C3×Q8—48no—
195C3×Q8—92no—
196C3×D4—28no—
197C3×D4—22no—
198C3×D4—48no—
199C3×D4—92no—
200C3×D4—184no—
201C3×D4—182no—
202C2×C3×C4—28no—
203C2×C3×C4—28no—
204C2×C3×C4—24no—
205C2×C3×C4—24no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—24no—
209C2×C3×C4—182no—
210C2×C3×C4—182no—
211C2×C3×C4C22×C3×D314yes—
212C2×C3×C4C22×C3⋊D311yescharacteristic
213#67—616no—
214#67—68no—
215C2×Dic3—68no—
216D12—38no—
217C4×D3—68no—
218Dic6—38no—
219C3×C8—182no—
220C3⋊C8—68no—
221C2×C32—28no—
222C2×C32—24no—
223C2×C32—21no—
224C2×C32—24no—
225C2×C32—42no—
226C2×C32—48no—
227C2×C32—88no—
228C2×C32#93714yes—
229C2×C32C3×#12011yescharacteristic
230C3⋊D3—42no—
231C3×D3—128no—
232#41—92no—
233#41—182no—
234C2×Q8C2×C3×C3⋊D311yescharacteristic, complemented
235C2×D4—182no—
236C2×D4—182no—
237C2×D4C2×C3×C3⋊D311yescharacteristic
238Q16—182no—
239SD16—182no—
240M16—92no—
241C22⋊C4—22no—
242C42C2×C3×C3⋊D311yescharacteristic, complemented
243C22×C3—28no—
244C22×C3—22no—
245C22×C3—24no—
246C22×C3—48no—
247C22×C3—48no—
248C22×C3—42no—
249C22×C3—416no—
250C22×C3—182no—
251C22×C3—722no—
252C22×C3C2×C3×#6714yes—
253C22×C3C2×#63811yescharacteristic
254C2×D3—68no—
255C2×D3—248no—
256C3×C4—28no—
257C3×C4—48no—
258C3×C4—48no—
259C3×C4—44no—
260C3×C4—41no—
261C3×C4—44no—
262C3×C4—41no—
263C3×C4—182no—
264C3×C4C3×D3×D418yes—
265C3×C4D4×C3⋊D312yes—
266Dic3—68no—
267C32—24no—
268C32#506114yescomplemented
269C32C3×#45511yescharacteristic, complemented
270C23—182no—
271C23C3×#63812yes—
272Q8—22no—
273Q8—92no—
274D4—22no—
275D4—92no—
276D4—184no—
277D4—182no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—182no—
281C2×C4—182no—
282C2×C4C22×C3×C3⋊D311yescharacteristic
283C8—182no—
284C2×C3—28no—
285C2×C3—24no—
286C2×C3—21no—
287C2×C3—24no—
288C2×C3—48no—
289C2×C3—42no—
290C2×C3—88no—
291C2×C3—362no—
292C2×C3C3×#93714yes—
293C2×C3#6414611yescharacteristic
294D3—128no—
295C22—22no—
296C22—42no—
297C22—182no—
298C22—722no—
299C22C2×C3×#63811yescharacteristic
300C4—41no—
301C4—41no—
302C4—182no—
303C4C3×D4×C3⋊D312yes—
304C3—24no—
305C3C3×#506114yescomplemented
306C3#1059276011yescharacteristic, direct product factor
307C2—21no—
308C2—42no—
309C2—362no—
310C2C3×#6414611yescharacteristic
subgroupG/H
centerC2×C3#64146
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#124C2
socleC2×C33#120
prime radicalC3×#64361C2
hypercenterC3×#124C3⋊D3
hypocenterC32C3×#455
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
2. C33×#124C2semidirect
3. C3×#64151C2semidirect
4. C3×#63660C2semidirect
5. C3×#63632C2semidirect
6. #10592760C3direct (N × H)
8. C2×C33×Q8C22semidirect
15. C33×C42C22semidirect
18. C3×#3748C22semidirect
22. C32×#124D3semidirect
23. C32×#124C2×C3semidirect
24. #64151C2×C3semidirect
26. #63660C2×C3semidirect
27. #63632C2×C3semidirect
47. C2×C32×Q8C2×D3semidirect
48. C2×C32×Q8C22×C3semidirect
64. C32×C42C2×D3semidirect
65. C32×C42C22×C3semidirect
68. #3748C22×C3semidirect
85. C3×#124C3×D3semidirect
86. C3×#124C3⋊D3semidirect
134. C2×C3×Q8C2×C3×D3semidirect
135. C2×C3×Q8C2×C3⋊D3semidirect
152. C3×C42C2×C3×D3semidirect
153. C3×C42C2×C3⋊D3semidirect
183. #124C3×C3⋊D3semidirect
186. C33#455semidirect
234. C2×Q8C2×C3×C3⋊D3semidirect
242. C42C2×C3×C3⋊D3semidirect
268. C32#5061semidirect
269. C32C3×#455semidirect
305. C3C3×#5061semidirect
306. C3#10592760direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643612
C33×#1242
C3×#641512
C3×#636602
C3×#636322
#105927603
C3×#50613