Finite Group Databasebeta

← C2×C3×C3⋊#859 #13173020

Subgroups — C2×C3×C3⋊#859 #13173020

Order 1728.

715 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 130 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×C32×SL(2,𝔽3)C211yesmaximal, characteristic
2C3×C3⋊#859C212yesmaximal, direct product factor
3C2×C3⋊#859C311yesmaximal, characteristic, direct product factor
4C2×C3×#859—33nomaximal
5C2×C3×#835—31nomaximal
6C22×C3×C3⋊Dic3—41nomaximal
7C2×C32×SL(2,𝔽3)C412yescomplemented
8C2×C32×SL(2,𝔽3)C2211yescharacteristic
9C22×C32×Q8D311yescharacteristic
10C22×C3×SL(2,𝔽3)—21no—
11C22×C3×SL(2,𝔽3)—23no—
12C22×C3×SL(2,𝔽3)D313yes—
13C22×C3×SL(2,𝔽3)C2×C311yescharacteristic
14C2×C3×C4⋊Dic3—31no—
15C22×C3×C3⋊C8—31no—
16C3⋊#859C2×C312yesdirect product factor
17C3×#859—36no—
18C3×#835—32no—
19C3×#835—32no—
20C23×C33—41no—
21C2×C3×C3⋊Dic3—42no—
22C2×C3×C3⋊Dic3—44no—
23C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
24C2×#859—33no—
25C2×C3×#103—91no—
26C2×#835—31no—
27C2×C32×Q8—32no—
28C2×C32×Q8—31no—
29C2×C32×Q8Dic312yescomplemented
30C2×C32×Q8C2×D311yescharacteristic
31C22×C32×C4—31no—
32C22×C3⋊Dic3—41no—
33C22×C3×Dic3—61no—
34C22×C3×Dic3—123no—
35C2×C3×SL(2,𝔽3)—22no—
36C2×C3×SL(2,𝔽3)—21no—
37C2×C3×SL(2,𝔽3)—26no—
38C2×C3×SL(2,𝔽3)—23no—
39C2×C3×SL(2,𝔽3)Dic316yescomplemented
40C2×C3×SL(2,𝔽3)C3×C412yescomplemented
41C2×C3×SL(2,𝔽3)C2×D313yes—
42C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
43C3×C4⋊Dic3—32no—
44C3×C4⋊Dic3—61no—
45C2×C3×C3⋊C8—32no—
46C2×C3×C3⋊C8—62no—
47C22×C33—42no—
48C22×C33—42no—
49C22×C33—42no—
50C22×C33—41no—
51C3×C3⋊Dic3—44no—
52C22×C3×Q8—21no—
53C22×C3×Q8C3×D311yescharacteristic
54C22×C3×Q8C3⋊D311yescharacteristic
55C22×SL(2,𝔽3)—23no—
56C22×SL(2,𝔽3)C3×D313yes—
57C22×C3×C8—91no—
58C2×C3×C4⋊C4—91no—
59C2×C4⋊Dic3—31no—
60C22×C3⋊C8—31no—
61#859—36no—
62C3×#103—92no—
63C3×#103—92no—
64#835—32no—
65#835—32no—
66C23×C32—41no—
67C23×C32—43no—
68C23×C32—83no—
69C23×C32—81no—
70C23×C32S411yescharacteristic
71C32×Q8—32no—
72C32×Q8—31no—
73C32×Q8—61no—
74C32×Q8C2×Dic311yescharacteristic, complemented
75C2×C32×C4—32no—
76C2×C32×C4—32no—
77C2×C32×C4—31no—
78C2×C32×C4—31no—
79C2×C3⋊Dic3—42no—
80C2×C3⋊Dic3—44no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—126no—
83C2×C3×Dic3—1212no—
84C2×C3×Dic3—122no—
85C3×SL(2,𝔽3)—21no—
86C3×SL(2,𝔽3)—23no—
87C3×SL(2,𝔽3)C2×Dic313yescomplemented
88C3×SL(2,𝔽3)C2×C3×C411yescharacteristic, complemented
89C3×C3⋊C8—62no—
90C2×#103—91noSylow
91C2×C33—44no—
92C2×C33—41no—
93C2×C33—41no—
94C2×C33—41no—
95C2×C3×Q8—22no—
96C2×C3×Q8—21no—
97C2×C3×Q8—31no—
98C2×C3×Q8—31no—
99C2×C3×Q8—32no—
100C2×C3×Q8—32no—
101C2×C3×Q8—61no—
102C2×C3×Q8—62no—
103C2×C3×Q8C3×Dic312yescomplemented
104C2×C3×Q8C3⋊Dic312yescomplemented
105C2×C3×Q8C2×C3×D311yescharacteristic
106C2×C3×Q8C2×C3⋊D311yescharacteristic
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C22×C3×C4—61no—
110C22×C3×C4—181no—
111C22×Dic3—61no—
112C22×Dic3—123no—
113C2×SL(2,𝔽3)—26no—
114C2×SL(2,𝔽3)—23no—
115C2×SL(2,𝔽3)C3×Dic316yescomplemented
116C2×SL(2,𝔽3)C2×C3×D313yes—
117C2×C3×C8—92no—
118C2×C3×C8—182no—
119C3×C4⋊C4—92no—
120C3×C4⋊C4—181no—
121C4⋊Dic3—32no—
122C4⋊Dic3—61no—
123C2×C3⋊C8—32no—
124C2×C3⋊C8—62no—
125C22×C32—41no—
126C22×C32—42no—
127C22×C32—42no—
128C22×C32—42no—
129C22×C32—43no—
130C22×C32—46no—
131C22×C32—46no—
132C22×C32—46no—
133C22×C32—86no—
134C22×C32—86no—
135C22×C32—83no—
136C22×C32—86no—
137C22×C32—81no—
138C22×C32—82no—
139C22×C32—82no—
140C22×C32—82no—
141C22×C32#22612yes—
142C22×C32GL(2,𝔽3)12yes—
143C22×C32SL(2,ℤ/4)12yes—
144C22×C32C2×S411yescharacteristic
145C32×C4—31no—
146C32×C4—31no—
147C32×C4—32no—
148C3⋊Dic3—44no—
149C3×Dic3—1212no—
150C3×Dic3—122no—
151C22×Q8C3×C3⋊D311yescharacteristic
152C22×C8—91no—
153C2×C4⋊C4—91no—
154#103—92no—
155#103—92no—
156C33—41noSylow
157C23×C3—21no—
158C23×C3—43no—
159C23×C3—83no—
160C23×C3C3×S411yescharacteristic
161C23×C3C3⋊S411yescharacteristic
162C3×Q8—21no—
163C3×Q8—32no—
164C3×Q8—32no—
165C3×Q8—31no—
166C3×Q8—31no—
167C3×Q8—62no—
168C3×Q8—61no—
169C3×Q8—61no—
170C3×Q8—61no—
171C3×Q8—62no—
172C3×Q8C2×C3×Dic311yescharacteristic, complemented
173C3×Q8C2×C3⋊Dic311yescharacteristic, complemented
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—32no—
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—32no—
182C2×C3×C4—61no—
183C2×C3×C4—62no—
184C2×C3×C4—61no—
185C2×C3×C4—62no—
186C2×C3×C4—182no—
187C2×C3×C4—362no—
188C2×Dic3—62no—
189C2×Dic3—126no—
190C2×Dic3—1212no—
191C2×Dic3—122no—
192SL(2,𝔽3)—23no—
193SL(2,𝔽3)C2×C3×Dic313yescomplemented
194C3×C8—182no—
195C3⋊C8—62no—
196C2×C32—41no—
197C2×C32—44no—
198C2×C32—41no—
199C2×C32—41no—
200C2×C32—43no—
201C2×C32—43no—
202C2×C32—412no—
203C2×C32—43no—
204C2×C32—83no—
205C2×C32—812no—
206C2×C32—83no—
207C2×C32—83no—
208C2×C32—81no—
209C2×C32—84no—
210C2×C32—81no—
211C2×C32—81no—
212C2×C32#85914yescomplemented
213C2×C32C2×#22611yescharacteristic
214C2×C32C2×GL(2,𝔽3)11yescharacteristic
215C2×C32C2×SL(2,ℤ/4)11yescharacteristic
216C2×Q8—31no—
217C2×Q8—32no—
218C2×Q8C3×C3⋊Dic312yescomplemented
219C2×Q8C2×C3×C3⋊D311yescharacteristic
220C22×C4—31no—
221C22×C4—181no—
222C2×C8—92no—
223C2×C8—182no—
224C4⋊C4—92no—
225C4⋊C4—181no—
226C22×C3—21no—
227C22×C3—22no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—43no—
231C22×C3—46no—
232C22×C3—46no—
233C22×C3—46no—
234C22×C3—83no—
235C22×C3—86no—
236C22×C3—86no—
237C22×C3—86no—
238C22×C3C3×#22612yes—
239C22×C3C3×GL(2,𝔽3)12yes—
240C22×C3C3×SL(2,ℤ/4)12yes—
241C22×C3C3⋊#22612yes—
242C22×C3C3⋊GL(2,𝔽3)12yes—
243C22×C3C3⋊SL(2,ℤ/4)12yes—
244C22×C3C2×C3×S411yescharacteristic
245C22×C3C2×C3⋊S411yescharacteristic
246C3×C4—31no—
247C3×C4—31no—
248C3×C4—32no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—32no—
252C3×C4—61no—
253C3×C4—61no—
254C3×C4—62no—
255C3×C4—362no—
256Dic3—1212no—
257Dic3—122no—
258C32—41no—
259C32—43no—
260C32—83no—
261C32—81no—
262C32C2×#85911yescharacteristic, complemented
263C23C3×C3⋊S411yescharacteristic
264Q8—32no—
265Q8—31no—
266Q8—61no—
267Q8C2×C3×C3⋊Dic311yescharacteristic, complemented
268C2×C4—31no—
269C2×C4—31no—
270C2×C4—32no—
271C2×C4—32no—
272C2×C4—182no—
273C2×C4—362no—
274C8—182no—
275C2×C3—24no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—412no—
280C2×C3—43no—
281C2×C3—43no—
282C2×C3—43no—
283C2×C3—812no—
284C2×C3—83no—
285C2×C3—83no—
286C2×C3—83no—
287C2×C3C3×#85914yescomplemented
288C2×C3C3⋊#85914yesdirect product factor
289C2×C3C2×C3×#22611yescharacteristic
290C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
291C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
292C2×C3C2×C3⋊#22611yescharacteristic
293C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
294C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
295C22C3×C3⋊#22612yes—
296C22C3×C3⋊GL(2,𝔽3)12yes—
297C22C3×C3⋊SL(2,ℤ/4)12yes—
298C22C2×C3×C3⋊S411yescharacteristic
299C4—31no—
300C4—31no—
301C4—32no—
302C4—362no—
303C3—21no—
304C3—43no—
305C3—83no—
306C3C2×C3×#85911yescharacteristic, complemented
307C3C2×C3⋊#85911yescharacteristic, direct product factor
308C2C3×C3⋊#85914yesdirect product factor
309C2C2×C3×C3⋊#22611yescharacteristic
310C2C2×C3×C3⋊GL(2,𝔽3)11yescharacteristic
311C2C2×C3×C3⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC23×C3C3⋊S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
FrattiniC22C2×C3×C3⋊S4
FittingC22×C32×Q8D3
socleC23×C32S4
prime radicalC22×C32×SL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊#859C2direct (N × H)
3. C2×C3⋊#859C3direct (N × H)
7. C2×C32×SL(2,𝔽3)C4semidirect
16. C3⋊#859C2×C3direct (N × H)
23. C32×SL(2,𝔽3)C2×C4semidirect
29. C2×C32×Q8Dic3semidirect
39. C2×C3×SL(2,𝔽3)Dic3semidirect
40. C2×C3×SL(2,𝔽3)C3×C4semidirect
74. C32×Q8C2×Dic3semidirect
87. C3×SL(2,𝔽3)C2×Dic3semidirect
88. C3×SL(2,𝔽3)C2×C3×C4semidirect
103. C2×C3×Q8C3×Dic3semidirect
104. C2×C3×Q8C3⋊Dic3semidirect
115. C2×SL(2,𝔽3)C3×Dic3semidirect
172. C3×Q8C2×C3×Dic3semidirect
173. C3×Q8C2×C3⋊Dic3semidirect
193. SL(2,𝔽3)C2×C3×Dic3semidirect
212. C2×C32#859semidirect
218. C2×Q8C3×C3⋊Dic3semidirect
262. C32C2×#859semidirect
267. Q8C2×C3×C3⋊Dic3semidirect
287. C2×C3C3×#859semidirect
288. C2×C3C3⋊#859direct (N × H)
306. C3C2×C3×#859semidirect
307. C3C2×C3⋊#859direct (N × H)
308. C2C3×C3⋊#859direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×SL(2,𝔽3)2
C3×C3⋊#8592
C2×C3⋊#8593
C2×C3×#8593
C2×C3×#8353
C22×C3×C3⋊Dic34