Finite Group Databasebeta

← C3×#10592733 #13163072

Subgroups — C3×#10592733 #13163072

Order 1728.

1060 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1060 conjugacy classes of subgroups — too many to draw. Only its 214 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×#121C211yesmaximal, characteristic
2C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C3×#63659C212yesmaximal, complemented
4C3×#63657C211yesmaximal, characteristic, complemented
5C3×#63635C212yesmaximal, complemented
6#10592733C311yesmaximal, characteristic, direct product factor
7C3×#5034—34nomaximal
8C2×C33×D4—21no—
9C2×C33×D4C2211yescharacteristic
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3—21no—
12C33×C4⋊C4C2211yescharacteristic
13C33×C22⋊C4—21no—
14C3×C4⋊(C3⋊Dic3)—21no—
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C2×C3×C3⋊(C3⋊C8)C2212yes—
17C32×#121—24no—
18C32×#121D314yes—
19C32×#121C2×C311yescharacteristic
20C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
21C2×C3×C4⋊Dic3—34no—
22#63659C2×C312yescomplemented
23#63657C2×C311yescharacteristic, complemented
24#63635C2×C312yescomplemented
25C3×#832—38no—
26C3×#830—34no—
27C3×#808—38no—
28C23×C33—21no—
29C23×C33D411yescharacteristic
30C33×D4—22no—
31C33×D4—42no—
32C2×C33×C4—22no—
33C2×C33×C4—21no—
34C2×C33×C4D411yescharacteristic
35C2×C33×C4C2311yescharacteristic
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—42no—
38C3×C3⋊(C3⋊C8)—22no—
39C3×#481—91no—
40#5034—34no—
41C2×C32×D4—28no—
42C2×C32×D4—24no—
43C2×C32×D4—24no—
44C2×C32×D4—21no—
45C2×C32×D4C2×D314yes—
46C2×C32×D4C22×C311yescharacteristic
47C22×C32×C4—24no—
48C22×C32×C4C2×D314yes—
49C22×C32×C4C22×C311yescharacteristic
50C22×C3⋊Dic3—21no—
51C22×C3×Dic3—64no—
52C32×C4⋊C4—24no—
53C32×C4⋊C4C2×D314yes—
54C32×C4⋊C4C22×C311yescharacteristic
55C32×C22⋊C4—28no—
56C32×C22⋊C4—24no—
57C32×C22⋊C4—21no—
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)C22×C312yes—
60C2×C3⋊(C3⋊C8)C22×C312yes—
61C3×C4⋊Dic3—38no—
62C3×C4⋊Dic3—64no—
63C2×C3×C3⋊C8—38no—
64C22×C33—21no—
65C22×C33—21no—
66C22×C33—41no—
67C22×C33—41no—
68C22×C33—41no—
69C22×C33SD1612yes—
70C22×C33C2×D411yescharacteristic
71C33×C4—21no—
72C33×C4—41no—
73C33×C4#4112yes—
74C3×C3⋊Dic3—42no—
75C3×#121—24no—
76C3×#121C3×D314yes—
77C3×#121C3⋊D311yescharacteristic
78C2×C3×C4⋊C4—91no—
79C2×C4⋊Dic3—34no—
80C3×#106—92no—
81C3×#102—92no—
82C3×C22⋊C8—91no—
83#832—38no—
84#830—34no—
85#808—38no—
86C23×C32—28no—
87C23×C32—24no—
88C23×C32—21no—
89C23×C32—24no—
90C23×C32#6714yes—
91C23×C32C3×D411yescharacteristic
92C32×D4—22no—
93C32×D4—28no—
94C32×D4—48no—
95C32×D4—416no—
96C32×D4—48no—
97C32×D4—42no—
98C2×C32×C4—22no—
99C2×C32×C4—28no—
100C2×C32×C4—24no—
101C2×C32×C4—21no—
102C2×C32×C4—24no—
103C2×C32×C4—24no—
104C2×C32×C4—28no—
105C2×C32×C4—48no—
106C2×C32×C4#6714yes—
107C2×C32×C4C3×D411yescharacteristic
108C2×C32×C4C22×D314yes—
109C2×C32×C4C23×C311yescharacteristic
110C2×C3⋊Dic3—22no—
111C2×C3⋊Dic3—42no—
112C2×C3×Dic3—68no—
113C2×C3×Dic3—128no—
114C3⋊(C3⋊C8)—22no—
115C3×C3⋊C8—68no—
116#481—91noSylow
117C2×C33—22no—
118C2×C33—81no—
119C2×C33#12311yescharacteristic
120C2×C33C2×SD1611yescharacteristic
121C2×C33AGL(1,ℤ/8)11yescharacteristic
122C2×C3×D4—28no—
123C2×C3×D4—24no—
124C2×C3×D4—21no—
125C2×C3×D4—24no—
126C2×C3×D4C2×C3×D314yes—
127C2×C3×D4C2×C3⋊D311yescharacteristic
128C22×C3×C4—24no—
129C22×C3×C4—181no—
130C22×C3×C4C2×C3×D314yes—
131C22×C3×C4C2×C3⋊D311yescharacteristic
132C22×Dic3—64no—
133C2×C3×C8—92no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—92no—
136C3×C4⋊C4—181no—
137C3×C4⋊C4C2×C3×D314yes—
138C3×C4⋊C4C2×C3⋊D311yescharacteristic
139C3×C22⋊C4—28no—
140C3×C22⋊C4—24no—
141C3×C22⋊C4—21no—
142C4⋊Dic3—38no—
143C4⋊Dic3—64no—
144C2×C3⋊C8—38no—
145C22×C32—21no—
146C22×C32—21no—
147C22×C32—28no—
148C22×C32—24no—
149C22×C32—24no—
150C22×C32—24no—
151C22×C32—44no—
152C22×C32—44no—
153C22×C32—48no—
154C22×C32—48no—
155C22×C32—48no—
156C22×C32—44no—
157C22×C32—44no—
158C22×C32—44no—
159C22×C32—41no—
160C22×C32—41no—
161C22×C32—41no—
162C22×C32#21418yes—
163C22×C32C3×SD1612yes—
164C22×C32C2×#6714yes—
165C22×C32C2×C3×D411yescharacteristic
166C32×C4—21no—
167C32×C4—28no—
168C32×C4—24no—
169C32×C4—44no—
170C32×C4—48no—
171C32×C4—44no—
172C32×C4—41no—
173C32×C4#23718yes—
174C32×C4C3×#4112yes—
175C3⋊Dic3—42no—
176C3×Dic3—128no—
177#121C3×C3⋊D311yescharacteristic
178C2×C4⋊C4—91no—
179#106—92no—
180#102—92no—
181C22⋊C8—91no—
182C33#48111yescharacteristic, complemented, Sylow
183C23×C3—28no—
184C23×C3—24no—
185C23×C3—21no—
186C23×C3—24no—
187C23×C3C3×#6714yes—
188C23×C3#63811yescharacteristic
189C3×D4—28no—
190C3×D4—22no—
191C3×D4—48no—
192C3×D4—416no—
193C3×D4—48no—
194C3×D4—42no—
195C2×C3×C4—28no—
196C2×C3×C4—22no—
197C2×C3×C4—28no—
198C2×C3×C4—24no—
199C2×C3×C4—21no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—48no—
203C2×C3×C4—182no—
204C2×C3×C4—362no—
205C2×C3×C4C3×#6714yes—
206C2×C3×C4#63811yescharacteristic
207C2×C3×C4C22×C3×D314yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209C2×Dic3—68no—
210C2×Dic3—128no—
211C3×C8—182no—
212C3⋊C8—68no—
213C2×C32—22no—
214C2×C32—24no—
215C2×C32—24no—
216C2×C32—24no—
217C2×C32—28no—
218C2×C32—48no—
219C2×C32—88no—
220C2×C32—84no—
221C2×C32—81no—
222C2×C32C2×#21414yes—
223C2×C32#93514yes—
224C2×C32#94914yes—
225C2×C32C3×#12311yescharacteristic
226C2×C32C2×C3×SD1611yescharacteristic
227C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
228C2×D4—21no—
229C2×D4C2×C3×C3⋊D311yescharacteristic
230C22×C4—181no—
231C22×C4C2×C3×C3⋊D311yescharacteristic
232C2×C8—92no—
233C4⋊C4—92no—
234C4⋊C4—181no—
235C4⋊C4C2×C3×C3⋊D311yescharacteristic
236C22⋊C4—21no—
237C22×C3—24no—
238C22×C3—24no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—28no—
242C22×C3—24no—
243C22×C3—44no—
244C22×C3—44no—
245C22×C3—48no—
246C22×C3—48no—
247C22×C3—48no—
248C22×C3—44no—
249C22×C3—44no—
250C22×C3—44no—
251C22×C3—41no—
252C22×C3—41no—
253C22×C3—41no—
254C22×C3C3×#21418yes—
255C22×C3#375412yes—
256C22×C3C2×C3×#6714yes—
257C22×C3C2×#63811yescharacteristic
258C3×C4—24no—
259C3×C4—21no—
260C3×C4—28no—
261C3×C4—44no—
262C3×C4—48no—
263C3×C4—44no—
264C3×C4—41no—
265C3×C4—362no—
266C3×C4C3×#23718yes—
267C3×C4#383012yes—
268Dic3—128no—
269C32—24no—
270C32#503414yescomplemented
271C32C3×#48111yescharacteristic, complemented
272C23—21no—
273C23C3×#63811yescharacteristic
274D4—22no—
275D4—42no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—182no—
279C2×C4—362no—
280C2×C4C3×#63811yescharacteristic
281C2×C4C22×C3×C3⋊D311yescharacteristic
282C8—182no—
283C2×C3—28no—
284C2×C3—22no—
285C2×C3—24no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3—48no—
289C2×C3—88no—
290C2×C3—84no—
291C2×C3—81no—
292C2×C3C2×C3×#21414yes—
293C2×C3C3×#93514yes—
294C2×C3C3×#94914yes—
295C2×C3C2×#375411yescharacteristic
296C2×C3#6414411yescharacteristic
297C2×C3#6415811yescharacteristic
298C22—21no—
299C22—21no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22C3×#375412yes—
304C22C2×C3×#63811yescharacteristic
305C4—21no—
306C4—41no—
307C4—362no—
308C4C3×#383012yes—
309C3—24no—
310C3C3×#503414yescomplemented
311C3#1059273311yescharacteristic, direct product factor
312C2—22no—
313C2—81no—
314C2C2×C3×#375411yescharacteristic
315C2C3×#6414411yescharacteristic
316C2C3×#6415811yescharacteristic
subgroupG/H
centerC22×C3C2×#638
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
Fitting, prime radicalC33×#121C2
socleC22×C33C2×D4
hypercenterC3×#121C3⋊D3
hypocenterC32C3×#481
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#121

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
3. C3×#63659C2semidirect
4. C3×#63657C2semidirect
5. C3×#63635C2semidirect
6. #10592733C3direct (N × H)
20. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
22. #63659C2×C3semidirect
23. #63657C2×C3semidirect
24. #63635C2×C3semidirect
182. C33#481semidirect
270. C32#5034semidirect
271. C32C3×#481semidirect
310. C3C3×#5034semidirect
311. C3#10592733direct (N × H)

Maximal subgroups

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

subgroupindex
C33×#1212
C2×C3×C4⋊(C3⋊Dic3)2
C3×#636592
C3×#636572
C3×#636352
#105927333
C3×#50343