Finite Group Databasebeta

← D3×C3⋊#859 #13166040

Subgroups — D3×C3⋊#859 #13166040

Order 1728.

622 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 88 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×D3×SL(2,𝔽3)C211yesmaximal, characteristic
2C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
3C3×C3⋊#859C211yesmaximal, characteristic, complemented
4C2×C3⋊#859—31nomaximal
5D3×#859—33nomaximal
6D3×#835—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×D3×C3⋊Dic3—41nomaximal
9C3×D3×SL(2,𝔽3)C412yescomplemented
10C2×C3×D3×Q8D311yescharacteristic
11C22×C3×SL(2,𝔽3)—31no—
12C2×D3×SL(2,𝔽3)D313yes—
13D3×C4⋊Dic3—31no—
14C2×D3×C3⋊C8—31no—
15C3⋊#859—31no—
16C3⋊#859—33no—
17C3⋊#859—61no—
18C3⋊#859—63no—
19C3⋊#859D311yescharacteristic, direct product factor
20C3×#859—33no—
21#63662—31no—
22C3×#835—31no—
23#63566—31no—
24#63565—31no—
25C22×C32×D3—41no—
26C2×C3⋊(C3⋊Dic3)—41no—
27C2×C3×C3⋊Dic3—41no—
28C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
29D3×C3⋊Dic3—44no—
30C2×#859—93no—
31C2×#835—91no—
32D3×#103—91no—
33C2×C32×Q8C2×D311yescharacteristic
34C22×C3⋊Dic3—121no—
35C3×D3×Q8—32no—
36C3×D3×Q8Dic312yescomplemented
37C2×C3×C4×D3—31no—
38C2×C3×Dic6—31no—
39C2×C3×SL(2,𝔽3)—21no—
40C2×C3×SL(2,𝔽3)—23no—
41C2×C3×SL(2,𝔽3)—32no—
42C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
43C2×C3×SL(2,𝔽3)C2×D313yes—
44C2×D3×Dic3—61no—
45C2×D3×Dic3—123no—
46D3×SL(2,𝔽3)Dic316yescomplemented
47C4⋊(C3⋊Dic3)—31no—
48C2×C3⋊(C3⋊C8)—31no—
49C3×C4⋊Dic3—31no—
50C2×C3×C3⋊C8—31no—
51Dic3⋊Dic3—61no—
52D3×C3⋊C8—62no—
53C22×C33—41no—
54C2×C32×D3—42no—
55C2×C32×D3—42no—
56C2×C32×D3—42no—
57C3⋊(C3⋊Dic3)—42no—
58C3×C3⋊Dic3—42no—
59C22×C3×Q8—31no—
60C2×D3×Q8C3⋊D311yescharacteristic
61C22×SL(2,𝔽3)—33no—
62C2×C4⋊Dic3—91no—
63C22×C3⋊C8—91no—
64C2×D3×C8—91no—
65D3×C4⋊C4—91no—
66#859—33no—
67#859—93no—
68#859—183no—
69C3×#103—91no—
70#835—31no—
71#835—91no—
72#835—91no—
73#835—91no—
74#835—91no—
75#835—181no—
76#816—91no—
77#810—91no—
78C23×C32—121no—
79C22×C3×D3—43no—
80C22×C3×D3S411yescharacteristic
81C32×Q8—31no—
82C32×Q8C2×Dic311yescharacteristic, complemented
83C2×C32×C4—31no—
84C2×C3⋊Dic3—41no—
85C2×C3⋊Dic3—61no—
86C2×C3⋊Dic3—121no—
87C2×C3⋊Dic3—123no—
88C2×C3⋊Dic3—124no—
89C2×C3⋊Dic3—241no—
90C2×C3⋊Dic3—243no—
91C2×C3×Dic3—31no—
92C2×C3×Dic3—61no—
93C2×C3×Dic3—123no—
94C3×C4×D3—32no—
95C3×C4×D3—32no—
96C3×Dic6—32no—
97C3×Dic6—61no—
98C3×SL(2,𝔽3)—21no—
99C3×SL(2,𝔽3)—23no—
100C3×SL(2,𝔽3)C4×D311yescharacteristic, complemented
101C3×SL(2,𝔽3)C2×Dic313yescomplemented
102D3×Dic3—1212no—
103D3×Dic3—122no—
104C3⋊(C3⋊C8)—61no—
105C3×C3⋊C8—61no—
106C2×#103—271noSylow
107C2×C33—41no—
108C2×C33—41no—
109C2×C33—41no—
110C32×D3—44no—
111C2×C3×Q8—21no—
112C2×C3×Q8—32no—
113C2×C3×Q8—91no—
114C2×C3×Q8—92no—
115C2×C3×Q8D3211yescharacteristic
116C2×C3×Q8C2×C3⋊D311yescharacteristic
117C22×C3×C4—91no—
118C22×Dic3—181no—
119C22×Dic3—363no—
120D3×Q8—32no—
121D3×Q8C3⋊Dic312yescomplemented
122C2×C4×D3—31no—
123C2×C4×D3—181no—
124C2×Dic6—31no—
125C2×SL(2,𝔽3)—23no—
126C2×SL(2,𝔽3)—36no—
127C2×SL(2,𝔽3)D3213yes—
128C2×C3×C8—91no—
129C3×C4⋊C4—91no—
130C4⋊Dic3—31no—
131C4⋊Dic3—91no—
132C4⋊Dic3—91no—
133C4⋊Dic3—181no—
134C4⋊Dic3—181no—
135Dic3⋊C4—181no—
136C2×C3⋊C8—31no—
137C2×C3⋊C8—91no—
138C2×C3⋊C8—91no—
139C2×C3⋊C8—181no—
140C2×C3⋊C8—182no—
141D3×C8—182no—
142C22×C32—41no—
143C22×C32—43no—
144C22×C32—83no—
145C22×C32—81no—
146C22×C32—122no—
147C22×C32—122no—
148C22×C32—122no—
149C22×C32C2×S411yescharacteristic
150C2×C3×D3—46no—
151C2×C3×D3—46no—
152C2×C3×D3—46no—
153C2×C3×D3#22612yes—
154C2×C3×D3GL(2,𝔽3)12yes—
155C2×C3×D3SL(2,ℤ/4)12yes—
156C32×C4—31no—
157C32×C4—31no—
158C3⋊Dic3—42no—
159C3⋊Dic3—122no—
160C3⋊Dic3—126no—
161C3⋊Dic3—121no—
162C3⋊Dic3—246no—
163C3⋊Dic3—242no—
164C3×Dic3—32no—
165C3×Dic3—126no—
166C3×Dic3—121no—
167C22×Q8—31no—
168C22×C8—271no—
169C2×C4⋊C4—271no—
170#103—91no—
171#103—271no—
172#103—271no—
173#103—271no—
174C33—41noSylow
175C23×C3—31no—
176C23×C3—123no—
177C22×D3C3⋊S411yescharacteristic
178C3×Q8—21no—
179C3×Q8—31no—
180C3×Q8—31no—
181C3×Q8—61no—
182C3×Q8—92no—
183C3×Q8—181no—
184C3×Q8D3×Dic311yescharacteristic, complemented
185C3×Q8C2×C3⋊Dic311yescharacteristic, complemented
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—61no—
189C2×C3×C4—92no—
190C2×C3×C4—92no—
191C2×C3×C4—91no—
192C2×C3×C4—181no—
193C2×Dic3—31no—
194C2×Dic3—61no—
195C2×Dic3—123no—
196C2×Dic3—181no—
197C2×Dic3—181no—
198C2×Dic3—363no—
199C2×Dic3—361no—
200C2×Dic3—362no—
201C2×Dic3—3612no—
202C2×Dic3—723no—
203C4×D3—32no—
204C4×D3—32no—
205C4×D3—362no—
206Dic6—32no—
207Dic6—61no—
208SL(2,𝔽3)—23no—
209SL(2,𝔽3)D3×Dic313yescomplemented
210C3×C8—181no—
211C3⋊C8—61no—
212C3⋊C8—181no—
213C3⋊C8—181no—
214C3⋊C8—361no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—43no—
219C2×C32—43no—
220C2×C32—43no—
221C2×C32—83no—
222C2×C32—83no—
223C2×C32—83no—
224C2×C32—81no—
225C2×C32—81no—
226C2×C32—81no—
227C2×C32—124no—
228C2×C32C2×#22611yescharacteristic
229C2×C32C2×GL(2,𝔽3)11yescharacteristic
230C2×C32C2×SL(2,ℤ/4)11yescharacteristic
231C3×D3—412no—
232C3×D3#85914yescomplemented
233C2×Q8—32no—
234C2×Q8—91no—
235C2×Q8—92no—
236C2×Q8D3×C3⋊D311yescharacteristic
237C22×C4—91no—
238C22×C4—541no—
239C2×C8—91no—
240C2×C8—271no—
241C2×C8—542no—
242C4⋊C4—91no—
243C4⋊C4—271no—
244C4⋊C4—541no—
245C22×C3—21no—
246C22×C3—32no—
247C22×C3—32no—
248C22×C3—32no—
249C22×C3—43no—
250C22×C3—83no—
251C22×C3—126no—
252C22×C3—126no—
253C22×C3—126no—
254C22×C3D3×S411yescharacteristic
255C22×C3C2×C3⋊S411yescharacteristic
256C2×D3C3⋊#22612yes—
257C2×D3C3⋊GL(2,𝔽3)12yes—
258C2×D3C3⋊SL(2,ℤ/4)12yes—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—31no—
263C3×C4—61no—
264C3×C4—61no—
265C3×C4—92no—
266C3×C4—361no—
267Dic3—32no—
268Dic3—126no—
269Dic3—121no—
270Dic3—366no—
271Dic3—361no—
272Dic3—361no—
273Dic3—721no—
274Dic3—726no—
275C32—41no—
276C32—43no—
277C32—83no—
278C32—81no—
279C32C2×#85911yescharacteristic, complemented
280C23—31no—
281Q8—31no—
282Q8—92no—
283Q8—181no—
284Q8D3×C3⋊Dic311yescharacteristic, complemented
285C2×C4—31no—
286C2×C4—92no—
287C2×C4—92no—
288C2×C4—91no—
289C2×C4—181no—
290C2×C4—541no—
291C2×C4—1082no—
292C8—181no—
293C8—541no—
294C2×C3—21no—
295C2×C3—21no—
296C2×C3—21no—
297C2×C3—34no—
298C2×C3—43no—
299C2×C3—43no—
300C2×C3—43no—
301C2×C3—83no—
302C2×C3—83no—
303C2×C3—83no—
304C2×C3—1212no—
305C2×C3D3×#22611yescharacteristic
306C2×C3D3×GL(2,𝔽3)11yescharacteristic
307C2×C3D3×SL(2,ℤ/4)11yescharacteristic
308C2×C3C2×C3⋊#22611yescharacteristic
309C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
310C2×C3C2×C3⋊SL(2,ℤ/4)11yescharacteristic
311D3C3⋊#85914yesdirect product factor
312C22—32no—
313C22—32no—
314C22—32no—
315C22D3×C3⋊S411yescharacteristic
316C4—31no—
317C4—31no—
318C4—92no—
319C4—361no—
320C4—1081no—
321C3—21no—
322C3—43no—
323C3—83no—
324C3D3×#85911yescharacteristic, complemented
325C3C2×C3⋊#85911yescharacteristic, complemented
326C2—34no—
327C2D3×C3⋊#22611yescharacteristic
328C2D3×C3⋊GL(2,𝔽3)11yescharacteristic
329C2D3×C3⋊SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, hypocenterC32×SL(2,𝔽3)C2×C4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×D3×SL(2,𝔽3)C2
cosocleC2×C32×SL(2,𝔽3)C22
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C3⋊(C3⋊#859)C2semidirect
3. C3×C3⋊#859C2semidirect
9. C3×D3×SL(2,𝔽3)C4semidirect
19. C3⋊#859D3direct (N × H)
28. C32×SL(2,𝔽3)C2×C4semidirect
36. C3×D3×Q8Dic3semidirect
46. D3×SL(2,𝔽3)Dic3semidirect
82. C32×Q8C2×Dic3semidirect
100. C3×SL(2,𝔽3)C4×D3semidirect
101. C3×SL(2,𝔽3)C2×Dic3semidirect
121. D3×Q8C3⋊Dic3semidirect
184. C3×Q8D3×Dic3semidirect
185. C3×Q8C2×C3⋊Dic3semidirect
209. SL(2,𝔽3)D3×Dic3semidirect
232. C3×D3#859semidirect
279. C32C2×#859semidirect
284. Q8D3×C3⋊Dic3semidirect
311. D3C3⋊#859direct (N × H)
324. C3D3×#859semidirect
325. C3C2×C3⋊#859semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×SL(2,𝔽3)2
C3⋊(C3⋊#859)2
C3×C3⋊#8592
C2×C3⋊#8593
D3×#8593
D3×#8353
C2×D3×C3⋊Dic34