Finite Group Databasebeta

← Dic3×#859 #11920666

Subgroups — Dic3×#859 #11920666

Order 1152.

429 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 79 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×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3⋊#859C211yesmaximal, characteristic
3C2×C3×#859C211yesmaximal, characteristic
4C4×#859—31nomaximal
5Dic3×#103—31nomaximal
6C22×C3×SL(2,𝔽3)C2211yescharacteristic
7C2×Dic32—41nomaximal
8Dic3×SL(2,𝔽3)C412yescomplemented
9C3⋊#859C412yescomplemented
10C3×#859C412yescomplemented
11C2×Q8×Dic3D311yescharacteristic
12C2×C4×SL(2,𝔽3)—31no—
13C2×#859—31no—
14C2×#859—61no—
15C2×#859D311yescharacteristic
16C2×C3×#103—31no—
17C2×#835—31no—
18C2×C8×Dic3—31no—
19Dic3×C4⋊C4—31no—
20#4551—31no—
21#4529—31no—
22C22×C3⋊Dic3—41no—
23C22×C3×Dic3—41no—
24C22×C3×Dic3—41no—
25C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
26C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
27C2×C3×SL(2,𝔽3)C2×C411yescharacteristic
28Dic32—44no—
29C4×#103—91noSylow
30C22×C3×Q8C2×D311yescharacteristic
31C22×SL(2,𝔽3)—21no—
32C22×SL(2,𝔽3)C2×D311yescharacteristic
33C22×C3×C8—31no—
34C2×C3×C4⋊C4—31no—
35Q8×Dic3—32no—
36Q8×Dic3Dic312yescomplemented
37C2×C4⋊Dic3—31no—
38C2×C4⋊Dic3—31no—
39C2×C4×Dic3—31no—
40C2×C4×Dic3—61no—
41C2×C4×Dic3—121no—
42C22×C3⋊C8—31no—
43C4×SL(2,𝔽3)—32no—
44#859—32no—
45#859—62no—
46#859Dic312yesdirect product factor
47C3×#103—32no—
48C3×#103—32no—
49#835—32no—
50#835—32no—
51C3⋊#95—61no—
52C8×Dic3—62no—
53C23×C32—41no—
54C2×C3⋊Dic3—42no—
55C2×C3⋊Dic3—42no—
56C2×C3⋊Dic3—42no—
57C2×C3×Dic3—42no—
58C2×C3×Dic3—42no—
59C2×C3×Dic3—42no—
60C2×C3×Dic3—42no—
61C2×C3×Dic3—42no—
62C2×C3×Dic3—42no—
63C3×SL(2,𝔽3)C4211yescharacteristic, complemented
64C2×C4×Q8—31no—
65C2×#103—31no—
66C2×#103—91no—
67C2×C4×C8—91no—
68C4×C4⋊C4—91no—
69#340—91no—
70#340—91no—
71C2×C3×Q8—31no—
72C2×C3×Q8—31no—
73C2×C3×Q8—31no—
74C2×C3×Q8C4×D311yescharacteristic
75C2×C3×Q8C4×D311yescharacteristic
76C2×C3×Q8C2×Dic311yescharacteristic
77C22×C3×C4—31no—
78C22×C3×C4—61no—
79C22×C3×C4—121no—
80C22×Dic3—31no—
81C22×Dic3—41no—
82C22×Dic3—61no—
83C22×Dic3—121no—
84C22×Dic3—241no—
85C22×Dic3S411yescharacteristic
86C2×SL(2,𝔽3)—21no—
87C2×SL(2,𝔽3)—21no—
88C2×SL(2,𝔽3)—21no—
89C2×SL(2,𝔽3)C4×D311yescharacteristic
90C2×SL(2,𝔽3)C4×D311yescharacteristic
91C2×SL(2,𝔽3)C2×Dic311yescharacteristic
92C2×C3×C8—32no—
93C2×C3×C8—61no—
94C2×C3×C8—61no—
95C3×C4⋊C4—32no—
96C3×C4⋊C4—61no—
97C4⋊Dic3—32no—
98C4⋊Dic3—32no—
99C4⋊Dic3—61no—
100C4⋊Dic3—61no—
101C4×Dic3—32no—
102C4×Dic3—32no—
103C4×Dic3—124no—
104C4×Dic3—122no—
105C2×C3⋊C8—32no—
106C2×C3⋊C8—61no—
107C2×C3⋊C8—61no—
108C22×C32—41no—
109C22×C32—41no—
110C22×C32—41no—
111C22×C32—41no—
112C22×C32—41no—
113C22×C32—41no—
114C22×C32—41no—
115C3⋊Dic3—44no—
116C3×Dic3—44no—
117C3×Dic3—44no—
118C22×Q8D3211yescharacteristic
119C22×C8—31no—
120C22×C8—91no—
121C4×Q8—32no—
122C4×Q8—92no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—91no—
125C2×C4⋊C4—91no—
126C2×C42—91no—
127C2×C42—181no—
128#103—32no—
129#103—32no—
130#103—92no—
131#103—92no—
132C4×C8—182no—
133#95—181no—
134C23×C3—41no—
135C23×C3—81no—
136C23×C3C2×S411yescharacteristic
137C3×Q8—31no—
138C3×Q8—31no—
139C3×Q8—31no—
140C3×Q8—61no—
141C3×Q8C4×Dic311yescharacteristic, complemented
142C2×C3×C4—31no—
143C2×C3×C4—31no—
144C2×C3×C4—31no—
145C2×C3×C4—31no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C3×C4—62no—
149C2×C3×C4—122no—
150C2×C3×C4—122no—
151C2×C3×C4—122no—
152C2×C3×C4—121no—
153C2×C3×C4—121no—
154C2×Dic3—32no—
155C2×Dic3—42no—
156C2×Dic3—42no—
157C2×Dic3—42no—
158C2×Dic3—61no—
159C2×Dic3—61no—
160C2×Dic3—62no—
161C2×Dic3—122no—
162C2×Dic3—122no—
163C2×Dic3—122no—
164C2×Dic3—121no—
165C2×Dic3—121no—
166C2×Dic3—242no—
167C2×Dic3—242no—
168C2×Dic3—242no—
169C2×Dic3#22612yes—
170C2×Dic3GL(2,𝔽3)12yes—
171C2×Dic3SL(2,ℤ/4)12yes—
172SL(2,𝔽3)—21no—
173SL(2,𝔽3)C4×Dic311yescharacteristic, complemented
174C3×C8—62no—
175C3⋊C8—62no—
176C2×C32—41no—
177C2×C32—41no—
178C2×C32—41no—
179C2×C32—41no—
180C2×C32—41no—
181C2×C32—41no—
182C2×C32—41no—
183C2×Q8—31no—
184C2×Q8—31no—
185C2×Q8—31no—
186C2×Q8D3×Dic311yescharacteristic
187C2×Q8D3×Dic311yescharacteristic
188C2×Q8#62411yescharacteristic
189C22×C4—31no—
190C22×C4—31no—
191C22×C4—61no—
192C22×C4—91no—
193C22×C4—181no—
194C2×C8—32no—
195C2×C8—61no—
196C2×C8—61no—
197C2×C8—92no—
198C2×C8—181no—
199C2×C8—181no—
200C4⋊C4—32no—
201C4⋊C4—61no—
202C4⋊C4—92no—
203C4⋊C4—92no—
204C4⋊C4—181no—
205C4⋊C4—181no—
206C42—92no—
207C42—92no—
208C42—362no—
209C22×C3—41no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3—81no—
217C22×C3—81no—
218C22×C3—81no—
219C22×C3—81no—
220C22×C3—81no—
221C22×C3—81no—
222C22×C3—81no—
223C22×C3C4×S411yescharacteristic
224C22×C3C4×S411yescharacteristic
225C22×C3C2×#22611yescharacteristic
226C22×C3C2×GL(2,𝔽3)11yescharacteristic
227C22×C3#98511yescharacteristic
228C22×C3#98511yescharacteristic
229C22×C3C2×SL(2,ℤ/4)11yescharacteristic
230C3×C4—31no—
231C3×C4—31no—
232C3×C4—31no—
233C3×C4—31no—
234C3×C4—124no—
235C3×C4—122no—
236Dic3—44no—
237Dic3—62no—
238Dic3—124no—
239Dic3—122no—
240Dic3—244no—
241Dic3#85914yesdirect product factor
242C32—41noSylow
243C23D3×S411yescharacteristic
244Q8—31no—
245Q8—31no—
246Q8—31no—
247Q8—61no—
248Q8Dic3211yescharacteristic, complemented
249C2×C4—32no—
250C2×C4—32no—
251C2×C4—31no—
252C2×C4—31no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—32no—
258C2×C4—62no—
259C2×C4—92no—
260C2×C4—121no—
261C2×C4—121no—
262C2×C4—181no—
263C2×C4—181no—
264C2×C4—182no—
265C2×C4—361no—
266C2×C4—361no—
267C8—62no—
268C8—182no—
269C2×C3—41no—
270C2×C3—41no—
271C2×C3—41no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—81no—
277C2×C3—81no—
278C2×C3—81no—
279C2×C3—81no—
280C2×C3—81no—
281C2×C3—81no—
282C2×C3—81no—
283C2×C3C4×#22611yescharacteristic
284C2×C3C4×#22611yescharacteristic
285C2×C3C4×GL(2,𝔽3)11yescharacteristic
286C2×C3C4×GL(2,𝔽3)11yescharacteristic
287C2×C3C4×SL(2,ℤ/4)11yescharacteristic
288C2×C3C2×#85911yescharacteristic
289C2×C3#542411yescharacteristic
290C22#6419711yescharacteristic
291C22#6419811yescharacteristic
292C22D3×#22611yescharacteristic
293C22D3×GL(2,𝔽3)11yescharacteristic
294C22Dic3×S411yescharacteristic
295C22(C3⋊S4)⋊C411yescharacteristic
296C22D3×SL(2,ℤ/4)11yescharacteristic
297C4—34no—
298C4—31no—
299C4—31no—
300C4—31no—
301C4—31no—
302C4—122no—
303C4—182no—
304C4—362no—
305C3—41no—
306C3—81no—
307C3C4×#85911yescharacteristic, complemented
308C2#1059319111yescharacteristic
309C2Dic3×#22611yescharacteristic
310C2#1059319311yescharacteristic
311C2Dic3×GL(2,𝔽3)11yescharacteristic
312C2#1059321411yescharacteristic
313C2D3×#85911yescharacteristic
314C2Dic3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC23D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C42
FittingC22×C3×Q8C2×D3
socleC23×C3C2×S4
prime radical, cosocleC22×C3×SL(2,𝔽3)C22
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
8. Dic3×SL(2,𝔽3)C4semidirect
9. C3⋊#859C4semidirect
10. C3×#859C4semidirect
36. Q8×Dic3Dic3semidirect
46. #859Dic3direct (N × H)
63. C3×SL(2,𝔽3)C42semidirect
141. C3×Q8C4×Dic3semidirect
173. SL(2,𝔽3)C4×Dic3semidirect
241. Dic3#859direct (N × H)
248. Q8Dic32semidirect
307. C3C4×#859semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×SL(2,𝔽3)2
C2×C3⋊#8592
C2×C3×#8592
C4×#8593
Dic3×#1033
C2×Dic324