Finite Group Databasebeta

← C3×#10593520 #13166887

Subgroups — C3×#10593520 #13166887

Order 1728.

564 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 74 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×#622C211yesmaximal, characteristic, complemented
2C2×C3×#3774C211yesmaximal, characteristic, complemented
3C3×#63980C212yesmaximal, complemented
4C3×#63743C211yesmaximal, characteristic, complemented
5C3×#63739C212yesmaximal, complemented
6#10593520C311yesmaximal, characteristic, direct product factor
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C32×#67—22no—
9C2×C3×#625C2212yes—
10C2×C3×#622—22no—
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C3×#3774—22no—
14C3×#3724C2212yes—
15C3×(C2×D3)⋊Dic3—22no—
16C22×#622C2×C311yescharacteristic, complemented
17C2×#3774C2×C311yescharacteristic, complemented
18C3×#939—62no—
19#63980C2×C312yescomplemented
20#63743C2×C311yescharacteristic, complemented
21#63739C2×C312yescomplemented
22C23×C33D411yescharacteristic
23C22×C32×D3—22no—
24C2×C3×C3⋊Dic3—21no—
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3D411yescharacteristic
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C32×#67—44no—
29C2×C32×Dic3—22no—
30C3×#625—22no—
31C3×#625—22no—
32C3×#622—21no—
33C3×#622D412yescomplemented
34C3×#466—91nomaximal
35C2×C32×D4—62no—
36C22×C3⋊Dic3C22×C311yescharacteristic
37C2×C3×#67—22no—
38C2×C3×#67—22no—
39C2×C3×#67—42no—
40C2×C3×#67—62no—
41C22×C3×Dic3—62no—
42C2×#625C22×C312yes—
43C2×#622—22no—
44C2×#622C22×C311yescharacteristic
45C2×#622C22×C311yescharacteristic
46#3774—22no—
47C3×#217—62no—
48C3×(C2×D3)⋊C4—62no—
49C3×Dic3⋊C4—62no—
50#3724C22×C312yes—
51(C2×D3)⋊Dic3—22no—
52C22×C33—21no—
53C22×C33—21no—
54C22×C33D812yescomplemented
55C22×C33C2×D411yescharacteristic
56C2×C32×D3—42no—
57C2×C32×D3—42no—
58C2×C32×D3—42no—
59C3×C3⋊Dic3—21no—
60C3×C3⋊Dic3C2×D411yescharacteristic
61C3×C3⋊Dic3#4111yescharacteristic
62C32×Dic3—42no—
63C2×C3×D8—91no—
64C22×C3×C8—91no—
65C3×#121—92no—
66#939—62no—
67C3×#107—91no—
68C3×#102—92no—
69C23×C32—22no—
70C23×C32—42no—
71C23×C32—62no—
72C23×C32C3×D411yescharacteristic
73C22×C3×D3—22no—
74C22×C3×D3—22no—
75C22×C3×D3—42no—
76C32×D4—124no—
77C2×C32×C4—62no—
78C2×C3⋊Dic3—21no—
79C2×C3⋊Dic3—21no—
80C2×C3⋊Dic3C3×D411yescharacteristic
81C2×C3⋊Dic3C23×C311yescharacteristic
82C3×#67—44no—
83C3×#67—48no—
84C3×#67—44no—
85C3×#67—122no—
86C3×#67—122no—
87C2×C3×Dic3—22no—
88C2×C3×Dic3—22no—
89C2×C3×Dic3—42no—
90C2×C3×Dic3—62no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—122no—
93C2×C3×Dic3—122no—
94#625—22no—
95#625—22no—
96#622—21no—
97#622C3×D412yescomplemented
98#466—91noSylow
99C2×C33—22no—
100C2×C33#12111yescharacteristic
101C2×C33C2×D811yescharacteristic
102C2×C33#13511yescharacteristic
103C32×D3—82no—
104C2×C3×D4—62no—
105C2×C3×D4—62no—
106C2×C3×D4—92no—
107C2×C3×D4—122no—
108C22×C3×C4—91no—
109C2×#67—22no—
110C2×#67—62no—
111C22×Dic3—62no—
112C3×D8—182no—
113C2×C3×C8—91no—
114C2×C3×C8—91no—
115C2×C3×C8—182no—
116C3×C4⋊C4—92no—
117C3×C22⋊C4—182no—
118#217—62no—
119(C2×D3)⋊C4—62no—
120Dic3⋊C4—62no—
121C22×C32—22no—
122C22×C32—24no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—42no—
126C22×C32—44no—
127C22×C32—42no—
128C22×C32—42no—
129C22×C32—82no—
130C22×C32—82no—
131C22×C32—122no—
132C22×C32—122no—
133C22×C32—122no—
134C22×C32C3×D812yescomplemented
135C22×C32C2×C3×D411yescharacteristic
136C2×C3×D3—42no—
137C2×C3×D3—42no—
138C2×C3×D3—42no—
139C2×C3×D3—44no—
140C2×C3×D3—42no—
141C2×C3×D3—42no—
142C2×C3×D3—42no—
143C2×C3×D3—82no—
144C2×C3×D3—82no—
145C32×C4—122no—
146C3⋊Dic3—21no—
147C3⋊Dic3C2×C3×D411yescharacteristic
148C3⋊Dic3C3×#4111yescharacteristic
149C3×Dic3—42no—
150C3×Dic3—44no—
151C3×Dic3—42no—
152C3×Dic3—62no—
153C3×Dic3—62no—
154C3×Dic3—122no—
155C2×D8—91no—
156C22×C8—91no—
157#121—92no—
158#107—91no—
159#102—92no—
160C33#46611yescharacteristic, complemented, Sylow
161C23×C3—22no—
162C23×C3—42no—
163C23×C3—62no—
164C23×C3—62no—
165C23×C3—122no—
166C23×C3GO+(4,𝔽2)11yescharacteristic
167C22×D3—22no—
168C3×D4—124no—
169C3×D4—128no—
170C3×D4—124no—
171C3×D4—182no—
172C3×D4—182no—
173C2×C3×C4—62no—
174C2×C3×C4—62no—
175C2×C3×C4—91no—
176C2×C3×C4—91no—
177C2×C3×C4—122no—
178C2×C3×C4—181no—
179C2×C3×C4—181no—
180#67—44no—
181#67—122no—
182#67—122no—
183C2×Dic3—22no—
184C2×Dic3—62no—
185C2×Dic3—62no—
186C2×Dic3—122no—
187C2×Dic3—122no—
188C3×C8—92no—
189C3×C8—181no—
190C2×C32—22no—
191C2×C32—22no—
192C2×C32—22no—
193C2×C32—22no—
194C2×C32—42no—
195C2×C32—42no—
196C2×C32—42no—
197C2×C32—44no—
198C2×C32—84no—
199C2×C32—242no—
200C2×C32C3×#12111yescharacteristic
201C2×C32C2×C3×D811yescharacteristic
202C2×C32C3×#13511yescharacteristic
203C3×D3—82no—
204C3×D3—84no—
205C3×D3—82no—
206C2×D4—62no—
207C2×D4—92no—
208C22×C4—91no—
209D8—182no—
210C2×C8—91no—
211C2×C8—91no—
212C2×C8—182no—
213C4⋊C4—92no—
214C22⋊C4—182no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—22no—
218C22×C3—24no—
219C22×C3—42no—
220C22×C3—42no—
221C22×C3—42no—
222C22×C3—44no—
223C22×C3—82no—
224C22×C3—82no—
225C22×C3—122no—
226C22×C3—122no—
227C22×C3—122no—
228C22×C3—124no—
229C22×C3—122no—
230C22×C3—122no—
231C22×C3—122no—
232C22×C3—242no—
233C22×C3—242no—
234C22×C3#377412yescomplemented
235C22×C3C2×GO+(4,𝔽2)11yescharacteristic
236C2×D3—42no—
237C2×D3—42no—
238C2×D3—42no—
239C3×C4—91no—
240C3×C4—91no—
241C3×C4—122no—
242C3×C4—124no—
243C3×C4—122no—
244C3×C4—181no—
245Dic3—42no—
246Dic3—62no—
247Dic3—62no—
248Dic3—122no—
249C32—22no—
250C32—42no—
251C32C3×#46611yescharacteristic, complemented
252C23—62no—
253C23C3×GO+(4,𝔽2)11yescharacteristic
254D4—124no—
255D4—182no—
256D4—182no—
257C2×C4—62no—
258C2×C4—91no—
259C2×C4—91no—
260C2×C4—181no—
261C2×C4—181no—
262C8—92no—
263C8—181no—
264C2×C3—22no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—22no—
268C2×C3—44no—
269C2×C3—42no—
270C2×C3—42no—
271C2×C3—42no—
272C2×C3—84no—
273C2×C3—242no—
274C2×C3—244no—
275C2×C3—242no—
276C2×C3C2×#377411yescharacteristic
277C2×C3#6423711yescharacteristic
278C2×C3#6424211yescharacteristic
279D3—82no—
280C22—21no—
281C22—21no—
282C22—122no—
283C22—122no—
284C22—122no—
285C22C3×#377412yescomplemented
286C22C2×C3×GO+(4,𝔽2)11yescharacteristic
287C4—91no—
288C4—91no—
289C4—122no—
290C4—181no—
291C3—22no—
292C3—42no—
293C3#1059352011yescharacteristic, direct product factor
294C2—22no—
295C2—242no—
296C2C2×C3×#377411yescharacteristic
297C2C3×#6423711yescharacteristic
298C2C3×#6424211yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC22×C33C2×D4
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#466
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×#622C2semidirect
2. C2×C3×#3774C2semidirect
3. C3×#63980C2semidirect
4. C3×#63743C2semidirect
5. C3×#63739C2semidirect
6. #10593520C3direct (N × H)
16. C22×#622C2×C3semidirect
17. C2×#3774C2×C3semidirect
19. #63980C2×C3semidirect
20. #63743C2×C3semidirect
21. #63739C2×C3semidirect
33. C3×#622D4semidirect
54. C22×C33D8semidirect
97. #622C3×D4semidirect
134. C22×C32C3×D8semidirect
160. C33#466semidirect
234. C22×C3#3774semidirect
251. C32C3×#466semidirect
285. C22C3×#3774semidirect
293. C3#10593520direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#6222
C2×C3×#37742
C3×#639802
C3×#637432
C3×#637392
#105935203
C3×#4669