Finite Group Databasebeta

← C3×#10593394 #13166742

Subgroups — C3×#10593394 #13166742

Order 1728.

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

normal subgroups only

This group has 460 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×#3788C211yesmaximal, characteristic
2C3×#63900C211yesmaximal, characteristic
3C3×#63874C211yesmaximal, characteristic
4C3×#63743C212yesmaximal, complemented
5C3×#63742C212yesmaximal, complemented
6#10593394C311yesmaximal, characteristic, direct product factor
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C3×#624—21no—
9C2×C3×#622C2212yes—
10C3×#3788C414yescomplemented
11C32×C4⋊Dic3—21no—
12C32×C4×Dic3—21no—
13C3×#3724C2211yescharacteristic
14C3×#3724C2211yescharacteristic
15C3×#3724C2212yes—
16C3×#3722—21no—
17C2×#3788C2×C311yescharacteristic
18C3×D3×C4⋊C4—61no—
19C3×#872—61no—
20#63900C2×C311yescharacteristic
21#63874C2×C311yescharacteristic
22#63743C2×C312yescomplemented
23#63742C2×C312yescomplemented
24C22×C3×C3⋊D3D411yescharacteristic
25C2×C33×C4D411yescharacteristic
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3×C4×C3⋊D3C2×C412yes—
28C3×C4×C3⋊D3Q812yes—
29C2×C32×Dic3—22no—
30C2×C32×Dic3—21no—
31C2×C32×Dic3—21no—
32C3×#624—42no—
33C3×#622C2×C414yescomplemented
34C3×#428—91nomaximal
35C2×C4×C3⋊D3C22×C311yescharacteristic
36C2×C3×C4×D3—61no—
37C2×C3×C4×D3—61no—
38C2×C3×C4×D3—62no—
39C2×#624—21no—
40C2×#622C22×C312yes—
41#3788C3×C414yescomplemented
42C32×C4⋊C4—61no—
43C32×C42—61no—
44C3×(C2×D3)⋊C4—61no—
45C3×(C2×D3)⋊C4—61no—
46C3×C4⋊Dic3—21no—
47C3×C4⋊Dic3—21no—
48C3×C4⋊Dic3—41no—
49C3×Dic3⋊C4—62no—
50C3×Dic3⋊C4—61no—
51C3×Dic3⋊C4—61no—
52C3×C4×Dic3—21no—
53C3×C4×Dic3—21no—
54C3×C4×Dic3—41no—
55#3724C22×C311yescharacteristic
56#3724C22×C311yescharacteristic
57#3724C22×C312yes—
58#3722—21no—
59C22×C33C2×D411yescharacteristic
60C2×C3×C3⋊D3—21no—
61C2×C3×C3⋊D3—21no—
62C2×C3×C3⋊D3C4⋊C412yes—
63C33×C4C4⋊C412yes—
64C3×C3⋊Dic3C22×C411yescharacteristic
65C3×C3⋊Dic3C2×Q811yescharacteristic
66C32×Dic3—41no—
67C32×Dic3—42no—
68C32×Dic3—41no—
69C2×C3×M16—91no—
70C3×#117—91no—
71C2×C3×C4⋊C4—91no—
72D3×C4⋊C4—61no—
73#872—61no—
74C3×#107—92no—
75C3×#106—92no—
76C22×C3⋊D3C3×D411yescharacteristic
77C22×C3×D3—61no—
78C22×C3×D3—61no—
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4—61no—
84C2×C32×C4—61no—
85C2×C32×C4—62no—
86C2×C32×C4C3×D411yescharacteristic
87C2×C3⋊Dic3C23×C311yescharacteristic
88C4×C3⋊D3C2×C3×C412yes—
89C4×C3⋊D3C3×Q812yes—
90C2×C3×Dic3—21no—
91C2×C3×Dic3—21no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—21no—
94C2×C3×Dic3—21no—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—61no—
101C3×C4×D3—62no—
102C3×C4×D3—62no—
103C3×C4×D3—62no—
104C3×C4×D3—62no—
105C3×C4×D3—124no—
106#624—42no—
107#622C2×C3×C414yescomplemented
108#428—91noSylow
109C2×C33C2×C4⋊C411yescharacteristic
110C2×C33AGL(1,ℤ/8)11yescharacteristic
111C2×C33#13711yescharacteristic
112C3×C3⋊D3—22no—
113C22×C3×C4—91no—
114C22×C3×C4—181no—
115C2×C4×D3—61no—
116C2×C4×D3—61no—
117C2×C4×D3—62no—
118C3×M16—94no—
119C2×C3×C8—92no—
120C3×C4⋊C4—61no—
121C3×C4⋊C4—61no—
122C3×C4⋊C4—91no—
123C3×C4⋊C4—92no—
124C3×C4⋊C4—91no—
125C3×C4⋊C4—121no—
126C3×C22⋊C4—181no—
127C3×C42—61no—
128C3×C42—61no—
129C3×C42—121no—
130(C2×D3)⋊C4—61no—
131(C2×D3)⋊C4—61no—
132C4⋊Dic3—21no—
133Dic3⋊C4—61no—
134Dic3⋊C4—61no—
135Dic3⋊C4—62no—
136C4×Dic3—21no—
137C22×C32—21no—
138C22×C32—21no—
139C22×C32—41no—
140C22×C32—41no—
141C22×C32C2×C3×D411yescharacteristic
142C2×C3⋊D3—21no—
143C2×C3⋊D3—21no—
144C2×C3⋊D3C3×C4⋊C412yes—
145C2×C3×D3—62no—
146C2×C3×D3—62no—
147C2×C3×D3—62no—
148C2×C3×D3—62no—
149C2×C3×D3—121no—
150C2×C3×D3—121no—
151C32×C4—22no—
152C32×C4—22no—
153C32×C4—42no—
154C32×C4—42no—
155C32×C4—121no—
156C32×C4—121no—
157C32×C4—122no—
158C32×C4C3×C4⋊C412yes—
159C3⋊Dic3C22×C3×C411yescharacteristic
160C3⋊Dic3C2×C3×Q811yescharacteristic
161C3×Dic3—42no—
162C3×Dic3—41no—
163C3×Dic3—41no—
164C3×Dic3—42no—
165C3×Dic3—42no—
166C3×Dic3—42no—
167C3×Dic3—41no—
168C3×Dic3—41no—
169C3×Dic3—61no—
170C3×Dic3—61no—
171C3×Dic3—61no—
172C3×Dic3—61no—
173C3×Dic3—82no—
174C2×M16—91no—
175#117—91no—
176C2×C4⋊C4—91no—
177#107—92no—
178#106—92no—
179C33#42811yescharacteristic, complemented, Sylow
180C23×C3—91no—
181C22×D3—61no—
182C22×D3—61no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—61no—
188C2×C3×C4—61no—
189C2×C3×C4—62no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—62no—
193C2×C3×C4—91no—
194C2×C3×C4—92no—
195C2×C3×C4—92no—
196C2×C3×C4—121no—
197C2×C3×C4—121no—
198C2×C3×C4—122no—
199C2×C3×C4—362no—
200C2×C3×C4GO+(4,𝔽2)11yescharacteristic
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—22no—
204C2×Dic3—61no—
205C2×Dic3—61no—
206C4×D3—62no—
207C4×D3—62no—
208C4×D3—62no—
209C4×D3—62no—
210C4×D3—124no—
211C3×C8—94no—
212C2×C32—21no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—41no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32C2×C3×C4⋊C411yescharacteristic
225C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
226C2×C32C3×#13711yescharacteristic
227C3⋊D3—22no—
228C3×D3—64no—
229C3×D3—122no—
230C22×C4—91no—
231C22×C4—181no—
232M16—94no—
233C2×C8—92no—
234C4⋊C4—61no—
235C4⋊C4—91no—
236C4⋊C4—92no—
237C4⋊C4—91no—
238C22⋊C4—181no—
239C42—61no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—92no—
245C22×C3—181no—
246C22×C3—181no—
247C22×C3C2×GO+(4,𝔽2)11yescharacteristic
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—62no—
251C2×D3—62no—
252C2×D3—121no—
253C2×D3—121no—
254C3×C4—22no—
255C3×C4—22no—
256C3×C4—42no—
257C3×C4—42no—
258C3×C4—91no—
259C3×C4—91no—
260C3×C4—122no—
261C3×C4—121no—
262C3×C4—121no—
263C3×C4—122no—
264C3×C4—122no—
265C3×C4—122no—
266C3×C4—121no—
267C3×C4—121no—
268C3×C4—242no—
269C3×C4#377312yes—
270Dic3—42no—
271Dic3—41no—
272Dic3—41no—
273Dic3—61no—
274Dic3—61no—
275Dic3—61no—
276Dic3—61no—
277C32—21no—
278C32—21no—
279C32—41no—
280C32—41no—
281C32C3×#42811yescharacteristic, complemented
282C23—91no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—62no—
286C2×C4—91no—
287C2×C4—92no—
288C2×C4—92no—
289C2×C4—362no—
290C2×C4C3×GO+(4,𝔽2)11yescharacteristic
291C8—94no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—41no—
304C2×C3—182no—
305C2×C3#6422411yescharacteristic
306C2×C3#6422611yescharacteristic
307C2×C3C2×#377311yescharacteristic
308D3—64no—
309D3—122no—
310C22—92no—
311C22—181no—
312C22—181no—
313C22C2×C3×GO+(4,𝔽2)11yescharacteristic
314C4—91no—
315C4—91no—
316C4—122no—
317C4—121no—
318C4—121no—
319C4C3×#377312yes—
320C3—21no—
321C3—21no—
322C3—41no—
323C3—41no—
324C3#1059339411yescharacteristic, direct product factor
325C2—182no—
326C2C3×#6422411yescharacteristic
327C2C3×#6422611yescharacteristic
328C2C2×C3×#377311yescharacteristic
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×#428
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×#63743C2semidirect
5. C3×#63742C2semidirect
6. #10593394C3direct (N × H)
10. C3×#3788C4semidirect
22. #63743C2×C3semidirect
23. #63742C2×C3semidirect
33. C3×#622C2×C4semidirect
41. #3788C3×C4semidirect
107. #622C2×C3×C4semidirect
179. C33#428semidirect
281. C32C3×#428semidirect
324. C3#10593394direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37882
C3×#639002
C3×#638742
C3×#637432
C3×#637422
#105933943
C3×#4289