Finite Group Databasebeta

← C3×#10592438 #13162777

Subgroups — C3×#10592438 #13162777

Order 1728.

1088 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 222 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
1C22×C3×#634C211yesmaximal, characteristic
2C3×#64133C211yesmaximal, characteristic
3C2×C3×#3746C212yesmaximal, complemented
4C33×C22⋊C8C211yesmaximal, characteristic
5C3×#63643C212yesmaximal, complemented
6#10592438C311yesmaximal, characteristic, direct product factor
7C3×#4739—34nomaximal
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3—21no—
10C2×C3×#634—21no—
11C2×C3×#634—24no—
12C2×C3×#634C2211yescharacteristic
13C2×C3×#634C2212yes—
14C2×C33×C8C2212yes—
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C3×#3746—24no—
19C22×#634C2×C311yescharacteristic
20C22×C3×Dic6—34no—
21#64133C2×C311yescharacteristic
22C2×#3746C2×C312yescomplemented
23C3×#924—34no—
24C2×C3×Dic12—38no—
25C32×C22⋊C8—24no—
26C32×C22⋊C8D314yes—
27C32×C22⋊C8C2×C311yescharacteristic
28#63643C2×C312yescomplemented
29C3×#816—38no—
30C23×C33D411yescharacteristic
31C2×C33×C4—22no—
32C2×C33×C4D411yescharacteristic
33C2×C33×C4C2311yescharacteristic
34C2×C3×C3⋊Dic3—24no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3—21no—
38C3×#634—24no—
39C3×#634—22no—
40C3×#634—22no—
41C3×#634D414yes—
42C33×C8—22no—
43C3×#451—91no—
44#4739—34no—
45C22×C32×C4—24no—
46C22×C32×C4C2×D314yes—
47C22×C32×C4C22×C311yescharacteristic
48C22×C3⋊Dic3—21no—
49C2×#634—21no—
50C2×#634—24no—
51C2×#634C22×C311yescharacteristic
52C2×#634C22×C312yes—
53C22×C3×Dic3—64no—
54C2×C3×Dic6—38no—
55C2×C3×Dic6—34no—
56C2×C3×Dic6—616no—
57C2×C3×Dic6—64no—
58C2×C32×C8—28no—
59C2×C32×C8C2×D318yes—
60C2×C32×C8C22×C312yes—
61#3757—21no—
62C4⋊(C3⋊Dic3)C22×C311yescharacteristic
63#3750—21no—
64#3746—24no—
65C3×#217—64no—
66C3×C4⋊Dic3—34no—
67C3×Dic3⋊C4—64no—
68C3×Dic12—616no—
69C22×C33—21no—
70C22×C33—21no—
71C22×C33Q1612yescomplemented
72C22×C33C2×D411yescharacteristic
73C33×C4—21no—
74C33×C4C2×D412yes—
75C3×C3⋊Dic3—24no—
76C3×C3⋊Dic3—41no—
77C3×C3⋊Dic3—41no—
78C22×C3×Q8—91no—
79C22×Dic6—34no—
80C2×C3×Q16—92no—
81C3×C22⋊Q8—91no—
82#924—34no—
83C2×Dic12—38no—
84C3×#103—92no—
85C3×C22⋊C8—24no—
86C3×C22⋊C8C3×D314yes—
87C3×C22⋊C8C3⋊D311yescharacteristic
88#816—38no—
89C23×C32—24no—
90C23×C32D1214yes—
91C23×C32C3×D411yescharacteristic
92C2×C32×C4—22no—
93C2×C32×C4—24no—
94C2×C32×C4—24no—
95C2×C32×C4—28no—
96C2×C32×C4—48no—
97C2×C32×C4D1214yes—
98C2×C32×C4C3×D411yescharacteristic
99C2×C32×C4C22×D314yes—
100C2×C32×C4C23×C311yescharacteristic
101C2×C3⋊Dic3—24no—
102C2×C3⋊Dic3—22no—
103C2×C3⋊Dic3—21no—
104C2×C3⋊Dic3—21no—
105#634—24no—
106#634—22no—
107#634—22no—
108#634C3×D414yes—
109C2×C3×Dic3—616no—
110C2×C3×Dic3—68no—
111C2×C3×Dic3—64no—
112C2×C3×Dic3—64no—
113C3×Dic6—316no—
114C3×Dic6—616no—
115C3×Dic6—68no—
116C3×Dic6—68no—
117C32×C8—22no—
118C32×C8—28no—
119C32×C8—48no—
120#451—91noSylow
121C2×C33—22no—
122C2×C33#12011yescharacteristic
123C2×C33C2×Q1611yescharacteristic
124C2×C33#13711yescharacteristic
125C2×C3×Q8—92no—
126C2×C3×Q8—91no—
127C2×C3×Q8—184no—
128C2×C3×Q8—181no—
129C22×C3×C4—24no—
130C22×C3×C4—181no—
131C22×C3×C4C2×C3×D314yes—
132C22×C3×C4C2×C3⋊D311yescharacteristic
133C22×Dic3—64no—
134C2×Dic6—38no—
135C2×Dic6—34no—
136C2×Dic6—616no—
137C2×Dic6—64no—
138C3×Q16—184no—
139C2×C3×C8—28no—
140C2×C3×C8C2×C3×D318yes—
141C2×C3×C8C2×C3⋊D312yes—
142C3×C4⋊C4—91no—
143C3×C4⋊C4—181no—
144C3×C22⋊C4—181no—
145#217—64no—
146C4⋊Dic3—34no—
147Dic3⋊C4—64no—
148Dic12—616no—
149C22×C32—21no—
150C22×C32—21no—
151C22×C32—28no—
152C22×C32—24no—
153C22×C32—24no—
154C22×C32—24no—
155C22×C32—44no—
156C22×C32—44no—
157C22×C32Dic1218yescomplemented
158C22×C32C3×Q1612yescomplemented
159C22×C32C2×D1214yes—
160C22×C32C2×C3×D411yescharacteristic
161C32×C4—21no—
162C32×C4—28no—
163C32×C4—24no—
164C32×C4—44no—
165C32×C4D3×D418yes—
166C32×C4C2×C3×D412yes—
167C3⋊Dic3—24no—
168C3⋊Dic3—41no—
169C3⋊Dic3—41no—
170C3×Dic3—616no—
171C3×Dic3—124no—
172C3×Dic3—124no—
173C22×Q8—91no—
174C2×Q16—92no—
175C22⋊Q8—91no—
176#103—92no—
177C22⋊C8C3×C3⋊D311yescharacteristic
178C33#45111yescharacteristic, complemented, Sylow
179C23×C3—24no—
180C23×C3C3×D1214yes—
181C23×C3C4⋊(C3⋊D3)11yescharacteristic
182C3×Q8—94no—
183C3×Q8—184no—
184C3×Q8—182no—
185C3×Q8—182no—
186C2×C3×C4—28no—
187C2×C3×C4—22no—
188C2×C3×C4—24no—
189C2×C3×C4—24no—
190C2×C3×C4—48no—
191C2×C3×C4—184no—
192C2×C3×C4—182no—
193C2×C3×C4—181no—
194C2×C3×C4—181no—
195C2×C3×C4C3×D1214yes—
196C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
197C2×C3×C4C22×C3×D314yes—
198C2×C3×C4C22×C3⋊D311yescharacteristic
199C2×Dic3—616no—
200C2×Dic3—68no—
201C2×Dic3—64no—
202C2×Dic3—64no—
203Dic6—316no—
204Dic6—616no—
205Dic6—68no—
206Dic6—68no—
207C3×C8—28no—
208C3×C8—22no—
209C3×C8—48no—
210C2×C32—22no—
211C2×C32—24no—
212C2×C32—24no—
213C2×C32—24no—
214C2×C32—28no—
215C2×C32—48no—
216C2×C32#88214yes—
217C2×C32C2×Dic1214yes—
218C2×C32#90914yes—
219C2×C32C3×#12011yescharacteristic
220C2×C32C2×C3×Q1611yescharacteristic
221C2×C32C3×#13711yescharacteristic
222C2×Q8—92no—
223C2×Q8—91no—
224C2×Q8—184no—
225C2×Q8—181no—
226C22×C4—181no—
227C22×C4C2×C3×C3⋊D311yescharacteristic
228Q16—184no—
229C2×C8C2×C3×C3⋊D312yes—
230C4⋊C4—91no—
231C4⋊C4—181no—
232C22⋊C4—181no—
233C22×C3—24no—
234C22×C3—24no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—28no—
238C22×C3—24no—
239C22×C3—44no—
240C22×C3—44no—
241C22×C3C3×Dic1218yescomplemented
242C22×C3#374612yescomplemented
243C22×C3C2×C3×D1214yes—
244C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
245C3×C4—24no—
246C3×C4—21no—
247C3×C4—28no—
248C3×C4—44no—
249C3×C4—184no—
250C3×C4—361no—
251C3×C4—361no—
252C3×C4C3×D3×D418yes—
253C3×C4D4×C3⋊D312yes—
254Dic3—616no—
255Dic3—124no—
256Dic3—124no—
257C32—24no—
258C32#473914yescomplemented
259C32C3×#45111yescharacteristic, complemented
260C23C3×C4⋊(C3⋊D3)11yescharacteristic
261Q8—94no—
262Q8—184no—
263Q8—182no—
264Q8—182no—
265C2×C4—22no—
266C2×C4—184no—
267C2×C4—182no—
268C2×C4—181no—
269C2×C4—181no—
270C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
271C2×C4C22×C3×C3⋊D311yescharacteristic
272C8—22no—
273C2×C3—28no—
274C2×C3—22no—
275C2×C3—24no—
276C2×C3—24no—
277C2×C3—24no—
278C2×C3—48no—
279C2×C3C3×#88214yes—
280C2×C3C2×C3×Dic1214yes—
281C2×C3C3×#90914yes—
282C2×C3#6409111yescharacteristic
283C2×C3C2×#374611yescharacteristic
284C2×C3#6411811yescharacteristic
285C22—21no—
286C22—21no—
287C22C3×#374612yescomplemented
288C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
289C4—21no—
290C4—184no—
291C4—361no—
292C4—361no—
293C4C3×D4×C3⋊D312yes—
294C3—24no—
295C3C3×#473914yescomplemented
296C3#1059243811yescharacteristic, direct product factor
297C2—22no—
298C2C3×#6409111yescharacteristic
299C2C2×C3×#374611yescharacteristic
300C2C3×#6411811yescharacteristic
subgroupG/H
centerC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊C8C2
socleC22×C33C2×D4
prime radicalC23×C33D4
hypercenterC3×C22⋊C8C3⋊D3
hypocenterC32C3×#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3746C2semidirect
5. C3×#63643C2semidirect
6. #10592438C3direct (N × H)
22. C2×#3746C2×C3semidirect
28. #63643C2×C3semidirect
71. C22×C33Q16semidirect
157. C22×C32Dic12semidirect
158. C22×C32C3×Q16semidirect
178. C33#451semidirect
241. C22×C3C3×Dic12semidirect
242. C22×C3#3746semidirect
258. C32#4739semidirect
259. C32C3×#451semidirect
287. C22C3×#3746semidirect
295. C3C3×#4739semidirect
296. C3#10592438direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6342
C3×#641332
C2×C3×#37462
C33×C22⋊C82
C3×#636432
#105924383
C3×#47393