Finite Group Databasebeta

← C3×#10593374 #13166716

Subgroups — C3×#10593374 #13166716

Order 1728.

600 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C2×C3×#3797C211yesmaximal, characteristic
2C2×C3×#3788C211yesmaximal, characteristic
3C3×#63874C211yesmaximal, characteristic
4C3×#63740C214yesmaximal, complemented
5#10593374C311yesmaximal, characteristic, direct product factor
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C32×Dic6—21no—
8C2×C3×#627C2212yes—
9C2×C3×#624—21no—
10C3×#3797—22no—
11C3×#3797C414yescomplemented
12C2×C3×#622C2212yes—
13C3×#3788—22no—
14C32×C4×Dic3—21no—
15C3×#3724C2212yes—
16C3×#3722—21no—
17C2×C3×D3×Q8—61no—
18C2×#3797C2×C311yescharacteristic
19C2×#3788C2×C311yescharacteristic
20C3×#872—61no—
21#63874C2×C311yescharacteristic
22#63740C2×C314yescomplemented
23C22×C3×C3⋊D3D411yescharacteristic
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C3×C4×C3⋊D3C2×C412yes—
27C3×C4×C3⋊D3D412yes—
28C2×C32×Dic3—22no—
29C2×C32×Dic3—22no—
30C32×Dic6—24no—
31C3×#627—22no—
32C3×#627C2×C414yes—
33C3×#624—24no—
34C3×#622—22no—
35C3×#419—91nomaximal
36C2×C32×Q8—61no—
37C2×C4×C3⋊D3C22×C311yescharacteristic
38C3×D3×Q8—64no—
39C3×D3×Q8—64no—
40C2×C3×C4×D3—62no—
41C2×C3×C4×D3—61no—
42C2×C3×C4×D3—61no—
43C2×C3×Dic6—21no—
44C2×C3×Dic6—21no—
45C2×C3×Dic6—41no—
46C2×C3×Dic6—62no—
47C2×#627C22×C312yes—
48C2×#624—21no—
49#3797—22no—
50#3797C3×C414yescomplemented
51C2×#622C22×C312yes—
52#3788—22no—
53C32×C42—61no—
54C3×(C2×D3)⋊C4—62no—
55C3×Dic3⋊C4—62no—
56C3×C4×Dic3—21no—
57C3×C4×Dic3—21no—
58C3×C4×Dic3—41no—
59#3724C22×C312yes—
60#3722—21no—
61C22×C33C2×D411yescharacteristic
62C2×C3×C3⋊D3—22no—
63C2×C3×C3⋊D3C22⋊C412yes—
64C33×C4C22⋊C412yes—
65C3×C3⋊Dic3C22×C411yescharacteristic
66C3×C3⋊Dic3C2×D411yescharacteristic
67C32×Dic3—24no—
68C32×Dic3—42no—
69C22×C3×Q8—91no—
70C2×D3×Q8—61no—
71C2×C3×M16—91no—
72C3×#117—91no—
73#872—61no—
74C3×#103—94no—
75C22×C3⋊D3C3×D411yescharacteristic
76C22×C3×D3—61no—
77C22×C3×D3—61no—
78C32×Q8—64no—
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4—62no—
84C2×C32×C4—62no—
85C2×C32×C4C3×D411yescharacteristic
86C2×C3⋊Dic3C23×C311yescharacteristic
87C4×C3⋊D3C2×C3×C412yes—
88C4×C3⋊D3C3×D412yes—
89C2×C3×Dic3—22no—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—42no—
94C2×C3×Dic3—42no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—61no—
97C3×C4×D3—62no—
98C3×C4×D3—62no—
99C3×C4×D3—62no—
100C3×C4×D3—62no—
101C3×C4×D3—68no—
102C3×Dic6—24no—
103C3×Dic6—24no—
104C3×Dic6—44no—
105C3×Dic6—64no—
106C3×Dic6—64no—
107#627—22no—
108#627C2×C3×C414yes—
109#624—24no—
110#622—22no—
111#419—91noSylow
112C2×C33C2×C22⋊C411yescharacteristic
113C2×C33#13712yes—
114C3×C3⋊D3—22no—
115C2×C3×Q8—61no—
116C2×C3×Q8—61no—
117C2×C3×Q8—92no—
118C2×C3×Q8—94no—
119C2×C3×Q8—121no—
120C2×C3×Q8—182no—
121C22×C3×C4—91no—
122C22×C3×C4—181no—
123D3×Q8—64no—
124D3×Q8—64no—
125C2×C4×D3—61no—
126C2×C4×D3—61no—
127C2×C4×D3—62no—
128C2×Dic6—21no—
129C2×Dic6—62no—
130C3×M16—182no—
131C2×C3×C8—92no—
132C3×C4⋊C4—92no—
133C3×C22⋊C4—181no—
134C3×C42—61no—
135C3×C42—61no—
136C3×C42—121no—
137(C2×D3)⋊C4—62no—
138Dic3⋊C4—62no—
139C4×Dic3—21no—
140C22×C32—21no—
141C22×C32—21no—
142C22×C32—41no—
143C22×C32—41no—
144C22×C32C2×C3×D411yescharacteristic
145C2×C3⋊D3—22no—
146C2×C3⋊D3C3×C22⋊C412yes—
147C2×C3×D3—62no—
148C2×C3×D3—62no—
149C2×C3×D3—64no—
150C2×C3×D3—122no—
151C32×C4—22no—
152C32×C4—22no—
153C32×C4—42no—
154C32×C4—42no—
155C32×C4—64no—
156C32×C4—122no—
157C32×C4C3×C22⋊C412yes—
158C3⋊Dic3C22×C3×C411yescharacteristic
159C3⋊Dic3C2×C3×D411yescharacteristic
160C3×Dic3—24no—
161C3×Dic3—24no—
162C3×Dic3—44no—
163C3×Dic3—42no—
164C3×Dic3—44no—
165C3×Dic3—42no—
166C3×Dic3—61no—
167C3×Dic3—61no—
168C3×Dic3—61no—
169C3×Dic3—61no—
170C22×Q8—91no—
171C2×M16—91no—
172#117—91no—
173#103—94no—
174C33#41911yescharacteristic, complemented, Sylow
175C23×C3—91no—
176C22×D3—61no—
177C22×D3—61no—
178C3×Q8—64no—
179C3×Q8—64no—
180C3×Q8—94no—
181C3×Q8—124no—
182C3×Q8—182no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—62no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4—62no—
191C2×C3×C4—91no—
192C2×C3×C4—92no—
193C2×C3×C4—92no—
194C2×C3×C4—122no—
195C2×C3×C4—122no—
196C2×C3×C4—184no—
197C2×C3×C4GO+(4,𝔽2)11yescharacteristic
198C2×Dic3—22no—
199C2×Dic3—22no—
200C2×Dic3—61no—
201C2×Dic3—61no—
202C4×D3—62no—
203C4×D3—62no—
204C4×D3—62no—
205C4×D3—62no—
206C4×D3—68no—
207Dic6—24no—
208Dic6—64no—
209Dic6—64no—
210C3×C8—182no—
211C2×C32—22no—
212C2×C32—22no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—42no—
218C2×C32—42no—
219C2×C32C2×C3×C22⋊C411yescharacteristic
220C2×C32C3×#13712yes—
221C3⋊D3—22no—
222C3×D3—64no—
223C3×D3—122no—
224C2×Q8—61no—
225C2×Q8—92no—
226C2×Q8—94no—
227C2×Q8—182no—
228C22×C4—91no—
229C22×C4—181no—
230M16—182no—
231C2×C8—92no—
232C4⋊C4—92no—
233C22⋊C4—181no—
234C42—61no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—92no—
240C22×C3—182no—
241C22×C3C2×GO+(4,𝔽2)11yescharacteristic
242C2×D3—62no—
243C2×D3—62no—
244C2×D3—64no—
245C2×D3—122no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—42no—
249C3×C4—42no—
250C3×C4—64no—
251C3×C4—64no—
252C3×C4—91no—
253C3×C4—91no—
254C3×C4—124no—
255C3×C4—122no—
256C3×C4—124no—
257C3×C4—122no—
258C3×C4D32⋊C412yes—
259Dic3—24no—
260Dic3—42no—
261Dic3—61no—
262Dic3—61no—
263Dic3—61no—
264Dic3—61no—
265C32—21no—
266C32—21no—
267C32—41no—
268C32—41no—
269C32C3×#41911yescharacteristic, complemented
270C23—91no—
271Q8—64no—
272Q8—94no—
273Q8—182no—
274C2×C4—62no—
275C2×C4—62no—
276C2×C4—91no—
277C2×C4—92no—
278C2×C4—92no—
279C2×C4—184no—
280C2×C4C3×GO+(4,𝔽2)11yescharacteristic
281C8—182no—
282C2×C3—22no—
283C2×C3—22no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—42no—
287C2×C3—42no—
288C2×C3—41no—
289C2×C3—41no—
290C2×C3—182no—
291C2×C3#6422612yes—
292C2×C3C2×D32⋊C411yescharacteristic
293D3—64no—
294D3—122no—
295C22—92no—
296C22—182no—
297C22C2×C3×GO+(4,𝔽2)11yescharacteristic
298C4—64no—
299C4—91no—
300C4—91no—
301C4—122no—
302C4C3×D32⋊C412yes—
303C3—21no—
304C3—21no—
305C3—41no—
306C3—41no—
307C3#1059337411yescharacteristic, direct product factor
308C2—182no—
309C2C3×#6422612yes—
310C2C2×C3×D32⋊C411yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
cosocleC2×C3⋊Dic3C23×C3
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C3×#419
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#63740C2semidirect
5. #10593374C3direct (N × H)
11. C3×#3797C4semidirect
22. #63740C2×C3semidirect
50. #3797C3×C4semidirect
174. C33#419semidirect
269. C32C3×#419semidirect
307. C3#10593374direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37972
C2×C3×#37882
C3×#638742
C3×#637402
#105933743
C3×#4199