Finite Group Databasebeta

← C2×C3×C3⋊#857 #13172987

Subgroups — C2×C3×C3⋊#857 #13172987

Order 1728.

460 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 46 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×C32×C42⋊C3C211yesmaximal, characteristic, complemented
2C3×C3⋊#857C212yesmaximal, direct product factor
3C2×C3⋊#857C311yesmaximal, characteristic, direct product factor
4C2×C3×#857—33nomaximal
5C2×C3×#805—31nomaximal
6C2×C3×C3⋊S4—41nomaximal
7C32×C42⋊C3C2211yescharacteristic, complemented
8C2×C3×#235—31no—
9C2×C32×C42D311yescharacteristic, complemented
10C2×C3×#208—31no—
11C2×C3×C42⋊C3—21no—
12C2×C3×C42⋊C3—23no—
13C2×C3×C42⋊C3D313yescomplemented
14C2×C3×C42⋊C3C2×C311yescharacteristic, complemented
15C3⋊#857C2×C312yesdirect product factor
16C3×#857—36no—
17C3×#805—32no—
18C3×#805—32no—
19C2×C32×A4—41no—
20C3×C3⋊S4—42no—
21C2×#857—33no—
22C2×C3×#104—91no—
23C2×#805—31no—
24C2×C3⋊S4—41no—
25C2×C3×S4—123no—
26C22×C32×C4—31no—
27C2×C3×#67—61no—
28C3×#235—32no—
29C3×#235—32no—
30C3×#235—61no—
31C3×#235—61no—
32C2×C3×D12—31no—
33C2×C3×C4×D3—61no—
34C2×C3×Dic6—31no—
35C32×C42—31no—
36C32×C42C2×D311yescharacteristic, complemented
37C3×#208—32no—
38C3×#208—61no—
39C2×C3×C3⋊C8—61no—
40C3×C42⋊C3—21no—
41C3×C42⋊C3—23no—
42C3×C42⋊C3C2×D313yescomplemented
43C3×C42⋊C3C22×C311yescharacteristic, complemented
44C2×C3×C3⋊D3—161no—
45C32×A4—41no—
46C2×C3×#41—91no—
47C2×#235—31no—
48C2×C3×M16—91no—
49C2×C3×C42—21no—
50C2×C3×C42C3×D311yescharacteristic, complemented
51C2×C3×C42C3⋊D311yescharacteristic, complemented
52C2×#208—31no—
53C2×C42⋊C3—23no—
54C2×C42⋊C3C3×D313yescomplemented
55#857—36no—
56C3×#104—92no—
57C3×#104—92no—
58#805—32no—
59#805—32no—
60C23×C32S411yescharacteristic
61C22×C3×D3—61no—
62C2×C3×A4—41no—
63C2×C3×A4—43no—
64C2×C3×A4—83no—
65C2×C3×A4—81no—
66C3⋊S4—42no—
67C3×S4—126no—
68C2×C32×C4—31no—
69C2×C32×C4—31no—
70C2×C32×C4—31no—
71C2×C32×C4—31no—
72C2×C32×C4—31no—
73C2×C32×C4—31no—
74C3×#67—62no—
75C3×#67—62no—
76C2×C3×Dic3—61no—
77C3×D12—32no—
78C3×D12—61no—
79C3×C4×D3—62no—
80C3×C4×D3—62no—
81C3×Dic6—32no—
82C3×Dic6—61no—
83C3×C3⋊C8—62no—
84C2×#104—91noSylow
85C2×C33—161no—
86C3×C3⋊D3—162no—
87C2×S4—123no—
88C3×#41—92no—
89C3×#41—92no—
90C3×#41—181no—
91C3×#41—181no—
92C2×C3×Q8—91no—
93C2×C3×D4—91no—
94C2×C3×D4—181no—
95C22×C3×C4—31no—
96C22×C3×C4—31no—
97C22×C3×C4—61no—
98C22×C3×C4—181no—
99C2×#67—61no—
100#235—32no—
101#235—32no—
102#235—61no—
103#235—61no—
104C2×D12—31no—
105C2×C4×D3—61no—
106C2×Dic6—31no—
107C3×M16—92no—
108C3×M16—181no—
109C2×C3×C8—181no—
110C3×C42—21no—
111C3×C42—31no—
112C3×C42—31no—
113C3×C42—61no—
114C3×C42C2×C3×D311yescharacteristic, complemented
115C3×C42C2×C3⋊D311yescharacteristic, complemented
116#208—32no—
117#208—61no—
118C2×C3⋊C8—61no—
119C42⋊C3—23no—
120C42⋊C3C2×C3×D313yescomplemented
121C22×C32—31no—
122C22×C32—31no—
123C22×C32C2×S411yescharacteristic
124C2×C3⋊D3—161no—
125C2×C3×D3—62no—
126C2×C3×D3—121no—
127C2×C3×D3—121no—
128C2×C3×D3—483no—
129C3×A4—41no—
130C3×A4—43no—
131C3×A4—83no—
132C3×A4—81no—
133C32×C4—31no—
134C32×C4—31no—
135C32×C4—31no—
136C32×C4—31no—
137C3×Dic3—62no—
138C2×#41—91no—
139C2×M16—91no—
140C2×C42C3×C3⋊D311yescharacteristic, complemented
141#104—92no—
142#104—92no—
143C33—161noSylow
144C23×C3—21no—
145C23×C3—181no—
146C23×C3C3×S411yescharacteristic
147C23×C3C3⋊S411yescharacteristic
148C22×D3—61no—
149C2×A4—43no—
150C2×A4—83no—
151S4—126no—
152C3×Q8—92no—
153C3×Q8—181no—
154C3×D4—92no—
155C3×D4—181no—
156C3×D4—182no—
157C3×D4—182no—
158C2×C3×C4—31no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—61no—
171C2×C3×C4—61no—
172C2×C3×C4—61no—
173C2×C3×C4—61no—
174C2×C3×C4—61no—
175C2×C3×C4—61no—
176C2×C3×C4—182no—
177C2×C3×C4—182no—
178C2×C3×C4—181no—
179#67—62no—
180#67—62no—
181C2×Dic3—61no—
182D12—32no—
183D12—61no—
184C4×D3—62no—
185C4×D3—62no—
186Dic6—32no—
187Dic6—61no—
188C3×C8—182no—
189C3⋊C8—62no—
190C2×C32—31no—
191C2×C32—31no—
192C2×C32—161no—
193C2×C32—163no—
194C2×C32—323no—
195C2×C32—321no—
196C2×C32#85711yescharacteristic, complemented
197C3⋊D3—162no—
198C3×D3—122no—
199C3×D3—486no—
200#41—92no—
201#41—92no—
202#41—181no—
203#41—181no—
204C2×Q8—91no—
205C2×D4—91no—
206C2×D4—181no—
207C22×C4—31no—
208C22×C4—181no—
209M16—92no—
210M16—181no—
211C2×C8—181no—
212C42—31no—
213C42C2×C3×C3⋊D311yescharacteristic, complemented
214C22×C3—21no—
215C22×C3—31no—
216C22×C3—31no—
217C22×C3—31no—
218C22×C3—31no—
219C22×C3—61no—
220C22×C3—61no—
221C22×C3—182no—
222C22×C3—361no—
223C22×C3—361no—
224C22×C3C2×C3×S411yescharacteristic
225C22×C3C2×C3⋊S411yescharacteristic
226C2×D3—62no—
227C2×D3—121no—
228C2×D3—121no—
229C2×D3—483no—
230A4—43no—
231A4—83no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—31no—
235C3×C4—31no—
236C3×C4—31no—
237C3×C4—31no—
238C3×C4—31no—
239C3×C4—31no—
240C3×C4—61no—
241C3×C4—61no—
242C3×C4—61no—
243C3×C4—61no—
244C3×C4—182no—
245Dic3—62no—
246C32—161no—
247C32—163no—
248C32—323no—
249C32—321no—
250C32C2×#85711yescharacteristic, complemented
251C23—181no—
252C23C3×C3⋊S411yescharacteristic
253Q8—92no—
254Q8—181no—
255D4—92no—
256D4—181no—
257D4—182no—
258D4—182no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—182no—
266C2×C4—182no—
267C2×C4—181no—
268C8—182no—
269C2×C3—21no—
270C2×C3—31no—
271C2×C3—31no—
272C2×C3—31no—
273C2×C3—31no—
274C2×C3—61no—
275C2×C3—61no—
276C2×C3—163no—
277C2×C3—323no—
278C2×C3—362no—
279C2×C3C3×#85711yescharacteristic, complemented
280C2×C3C3⋊#85711yescharacteristic, direct product factor
281D3—122no—
282D3—486no—
283C22—31no—
284C22—31no—
285C22—182no—
286C22—361no—
287C22—361no—
288C22C2×C3×C3⋊S411yescharacteristic
289C4—31no—
290C4—31no—
291C4—31no—
292C4—31no—
293C4—182no—
294C3—21no—
295C3—163no—
296C3—323no—
297C3C2×C3×#85711yescharacteristic, complemented
298C3C2×C3⋊#85711yescharacteristic, direct product factor
299C2—31no—
300C2—31no—
301C2—362no—
302C2C3×C3⋊#85711yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3C3⋊#857
commutator, cosocle, hypocenterC3×C42⋊C3C22×C3
FrattiniC22C2×C3×C3⋊S4
FittingC2×C32×C42D3
socleC23×C32S4
LCSG ▷ C3×C42⋊C3
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C32×C42⋊C3C2semidirect
2. C3×C3⋊#857C2direct (N × H)
3. C2×C3⋊#857C3direct (N × H)
7. C32×C42⋊C3C22semidirect
9. C2×C32×C42D3semidirect
13. C2×C3×C42⋊C3D3semidirect
14. C2×C3×C42⋊C3C2×C3semidirect
15. C3⋊#857C2×C3direct (N × H)
36. C32×C42C2×D3semidirect
42. C3×C42⋊C3C2×D3semidirect
43. C3×C42⋊C3C22×C3semidirect
50. C2×C3×C42C3×D3semidirect
51. C2×C3×C42C3⋊D3semidirect
54. C2×C42⋊C3C3×D3semidirect
114. C3×C42C2×C3×D3semidirect
115. C3×C42C2×C3⋊D3semidirect
120. C42⋊C3C2×C3×D3semidirect
140. C2×C42C3×C3⋊D3semidirect
196. C2×C32#857semidirect
213. C42C2×C3×C3⋊D3semidirect
250. C32C2×#857semidirect
279. C2×C3C3×#857semidirect
280. C2×C3C3⋊#857direct (N × H)
297. C3C2×C3×#857semidirect
298. C3C2×C3⋊#857direct (N × H)
302. C2C3×C3⋊#857direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C42⋊C32
C3×C3⋊#8572
C2×C3⋊#8573
C2×C3×#8573
C2×C3×#8053
C2×C3×C3⋊S44