Finite Group Databasebeta

← C3×D3×#859 #13165892

Subgroups — C3×D3×#859 #13165892

Order 1728.

494 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 78 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⋊#859C211yesmaximal, characteristic, complemented
3C32×#859C211yesmaximal, characteristic, complemented
4C2×C3×#859—31nomaximal
5D3×#859C311yesmaximal, characteristic, direct product factor
6C3×D3×#103—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×D3×Dic3—41nomaximal
9C3×D3×SL(2,𝔽3)C412yescomplemented
10C2×C3×D3×Q8D311yescharacteristic
11C22×C3×SL(2,𝔽3)—31no—
12C2×D3×SL(2,𝔽3)—21no—
13C2×D3×SL(2,𝔽3)C2×C311yescharacteristic
14C2×C3×D3×C8—31no—
15C3×D3×C4⋊C4—31no—
16C3⋊#859C2×C311yescharacteristic, complemented
17C3×#859—21no—
18C3×#859—31no—
19C3×#859—61no—
20C3×#859D311yescharacteristic, direct product factor
21C3×#859C2×C311yescharacteristic, complemented
22C32×#103—31no—
23C3×#835—31no—
24C3×#816—31no—
25C3×#810—31no—
26C22×C32×D3—41no—
27C2×C3×C3⋊Dic3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)C2×C411yescharacteristic, complemented
30C3×D3×Dic3—44no—
31C2×#859—31no—
32C2×C3×#103—91no—
33D3×#103—31no—
34C2×C32×Q8C2×D311yescharacteristic
35C22×C3×Dic3—121no—
36C3×D3×Q8—32no—
37C3×D3×Q8Dic312yescomplemented
38C2×C3×C4×D3—31no—
39C2×C3×C4×D3—61no—
40C2×C3×Dic6—31no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)—21no—
43C2×C3×SL(2,𝔽3)—21no—
44C2×C3×SL(2,𝔽3)—32no—
45C2×C3×SL(2,𝔽3)—41no—
46C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
47C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
48C2×D3×Dic3—41no—
49D3×SL(2,𝔽3)—22no—
50D3×SL(2,𝔽3)C3×C412yescomplemented
51C2×C32×C8—31no—
52C32×C4⋊C4—31no—
53C3×C4⋊Dic3—31no—
54C3×Dic3⋊C4—61no—
55C2×C3×C3⋊C8—31no—
56C3×D3×C8—62no—
57C22×C33—41no—
58C2×C32×D3—42no—
59C2×C32×D3—42no—
60C2×C32×D3—42no—
61C3×C3⋊Dic3—42no—
62C32×Dic3—42no—
63C22×C3×Q8—31no—
64C2×D3×Q8C3×D311yescharacteristic
65C22×SL(2,𝔽3)—31no—
66C22×SL(2,𝔽3)—61no—
67C22×C3×C8—91no—
68C2×C3×C4⋊C4—91no—
69C2×D3×C8—31no—
70D3×C4⋊C4—31no—
71#859—31no—
72#859—61no—
73#859C3×D311yescharacteristic, direct product factor
74C3×#103—31no—
75C3×#103—31no—
76C3×#103—61no—
77C3×#103—91no—
78C3×#103—91no—
79C3×#103—91no—
80#835—31no—
81#816—31no—
82#810—31no—
83C23×C32—121no—
84C22×C3×D3—41no—
85C22×C3×D3—81no—
86C22×C3×D3S411yescharacteristic
87C32×Q8—31no—
88C32×Q8C2×Dic311yescharacteristic, complemented
89C2×C32×C4—31no—
90C2×C32×C4—61no—
91C2×C3⋊Dic3—41no—
92C2×C3×Dic3—31no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—81no—
97C2×C3×Dic3—121no—
98C2×C3×Dic3—124no—
99C2×C3×Dic3—241no—
100C3×C4×D3—32no—
101C3×C4×D3—32no—
102C3×C4×D3—122no—
103C3×Dic6—32no—
104C3×Dic6—61no—
105C3×SL(2,𝔽3)—21no—
106C3×SL(2,𝔽3)—21no—
107C3×SL(2,𝔽3)—21no—
108C3×SL(2,𝔽3)—41no—
109C3×SL(2,𝔽3)C4×D311yescharacteristic, complemented
110C3×SL(2,𝔽3)C2×C3×C411yescharacteristic, complemented
111D3×Dic3—44no—
112C32×C8—61no—
113C3×C3⋊C8—61no—
114C2×#103—91noSylow
115C2×C33—41no—
116C2×C33—41no—
117C2×C33—41no—
118C32×D3—44no—
119C2×C3×Q8—21no—
120C2×C3×Q8—32no—
121C2×C3×Q8—91no—
122C2×C3×Q8—92no—
123C2×C3×Q8D3211yescharacteristic
124C2×C3×Q8C2×C3×D311yescharacteristic
125C22×C3×C4—91no—
126C22×C3×C4—181no—
127C22×Dic3—121no—
128D3×Q8—32no—
129D3×Q8C3×Dic312yescomplemented
130C2×C4×D3—31no—
131C2×C4×D3—61no—
132C2×Dic6—31no—
133C2×SL(2,𝔽3)—21no—
134C2×SL(2,𝔽3)—21no—
135C2×SL(2,𝔽3)—32no—
136C2×SL(2,𝔽3)—41no—
137C2×SL(2,𝔽3)—62no—
138C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
139C2×C3×C8—31no—
140C2×C3×C8—31no—
141C2×C3×C8—61no—
142C2×C3×C8—91no—
143C2×C3×C8—182no—
144C3×C4⋊C4—31no—
145C3×C4⋊C4—31no—
146C3×C4⋊C4—61no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4—181no—
149C4⋊Dic3—31no—
150Dic3⋊C4—61no—
151C2×C3⋊C8—31no—
152D3×C8—62no—
153C22×C32—41no—
154C22×C32—41no—
155C22×C32—81no—
156C22×C32—81no—
157C22×C32—81no—
158C22×C32—122no—
159C22×C32—122no—
160C22×C32—122no—
161C22×C32—161no—
162C22×C32C2×S411yescharacteristic
163C2×C3×D3—42no—
164C2×C3×D3—42no—
165C2×C3×D3—42no—
166C2×C3×D3—82no—
167C2×C3×D3—82no—
168C2×C3×D3—82no—
169C2×C3×D3#22612yes—
170C2×C3×D3GL(2,𝔽3)12yes—
171C2×C3×D3SL(2,ℤ/4)12yes—
172C32×C4—31no—
173C32×C4—31no—
174C32×C4—121no—
175C3⋊Dic3—42no—
176C3×Dic3—32no—
177C3×Dic3—42no—
178C3×Dic3—42no—
179C3×Dic3—82no—
180C3×Dic3—122no—
181C3×Dic3—121no—
182C3×Dic3—242no—
183C22×Q8—31no—
184C22×C8—91no—
185C2×C4⋊C4—91no—
186#103—31no—
187#103—91no—
188#103—91no—
189#103—91no—
190C33—41noSylow
191C23×C3—31no—
192C23×C3—121no—
193C23×C3—241no—
194C22×D3C3×S411yescharacteristic
195C3×Q8—21no—
196C3×Q8—31no—
197C3×Q8—31no—
198C3×Q8—61no—
199C3×Q8—92no—
200C3×Q8—181no—
201C3×Q8D3×Dic311yescharacteristic, complemented
202C3×Q8C2×C3×Dic311yescharacteristic, complemented
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—61no—
206C2×C3×C4—61no—
207C2×C3×C4—61no—
208C2×C3×C4—92no—
209C2×C3×C4—92no—
210C2×C3×C4—91no—
211C2×C3×C4—121no—
212C2×C3×C4—181no—
213C2×C3×C4—362no—
214C2×Dic3—31no—
215C2×Dic3—41no—
216C2×Dic3—61no—
217C2×Dic3—121no—
218C2×Dic3—124no—
219C2×Dic3—241no—
220C4×D3—32no—
221C4×D3—32no—
222C4×D3—122no—
223Dic6—32no—
224Dic6—61no—
225SL(2,𝔽3)—21no—
226SL(2,𝔽3)—21no—
227SL(2,𝔽3)—41no—
228SL(2,𝔽3)C3×C4×D311yescharacteristic, complemented
229C3×C8—61no—
230C3×C8—61no—
231C3×C8—121no—
232C3×C8—181no—
233C3⋊C8—61no—
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32—41no—
239C2×C32—41no—
240C2×C32—81no—
241C2×C32—81no—
242C2×C32—81no—
243C2×C32—81no—
244C2×C32—81no—
245C2×C32—81no—
246C2×C32—81no—
247C2×C32—81no—
248C2×C32—81no—
249C2×C32—124no—
250C2×C32—161no—
251C2×C32—161no—
252C2×C32—161no—
253C2×C32C2×#22611yescharacteristic
254C2×C32C2×GL(2,𝔽3)11yescharacteristic
255C2×C32C2×SL(2,ℤ/4)11yescharacteristic
256C3×D3—44no—
257C3×D3—84no—
258C3×D3#85914yesdirect product factor
259C2×Q8—32no—
260C2×Q8—91no—
261C2×Q8—92no—
262C2×Q8C3×D3211yescharacteristic
263C22×C4—91no—
264C22×C4—181no—
265C2×C8—31no—
266C2×C8—91no—
267C2×C8—182no—
268C4⋊C4—31no—
269C4⋊C4—91no—
270C4⋊C4—181no—
271C22×C3—21no—
272C22×C3—32no—
273C22×C3—32no—
274C22×C3—32no—
275C22×C3—41no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3—122no—
279C22×C3—122no—
280C22×C3—122no—
281C22×C3—161no—
282C22×C3—242no—
283C22×C3—242no—
284C22×C3—242no—
285C22×C3D3×S411yescharacteristic
286C22×C3C2×C3×S411yescharacteristic
287C2×D3C3×#22612yes—
288C2×D3C3×GL(2,𝔽3)12yes—
289C2×D3C3×SL(2,ℤ/4)12yes—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—31no—
293C3×C4—31no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—92no—
297C3×C4—121no—
298C3×C4—121no—
299C3×C4—241no—
300C3×C4—361no—
301Dic3—32no—
302Dic3—42no—
303Dic3—122no—
304Dic3—121no—
305Dic3—242no—
306C32—41no—
307C32—41no—
308C32—81no—
309C32—81no—
310C32—81no—
311C32—161no—
312C32C2×#85911yescharacteristic, complemented
313C23—31no—
314Q8—31no—
315Q8—92no—
316Q8—181no—
317Q8C3×D3×Dic311yescharacteristic, complemented
318C2×C4—31no—
319C2×C4—61no—
320C2×C4—92no—
321C2×C4—92no—
322C2×C4—91no—
323C2×C4—181no—
324C2×C4—362no—
325C8—61no—
326C8—181no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—34no—
331C2×C3—41no—
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—81no—
335C2×C3—81no—
336C2×C3—81no—
337C2×C3—81no—
338C2×C3—81no—
339C2×C3—81no—
340C2×C3—124no—
341C2×C3—161no—
342C2×C3—161no—
343C2×C3—161no—
344C2×C3—244no—
345C2×C3D3×#22611yescharacteristic
346C2×C3D3×GL(2,𝔽3)11yescharacteristic
347C2×C3D3×SL(2,ℤ/4)11yescharacteristic
348C2×C3C2×C3×#22611yescharacteristic
349C2×C3C2×C3×GL(2,𝔽3)11yescharacteristic
350C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
351D3C3×#85914yesdirect product factor
352C22—32no—
353C22—32no—
354C22—32no—
355C22C3×D3×S411yescharacteristic
356C4—31no—
357C4—31no—
358C4—92no—
359C4—121no—
360C4—361no—
361C3—21no—
362C3—41no—
363C3—81no—
364C3—81no—
365C3—161no—
366C3D3×#85911yescharacteristic, direct product factor
367C3C2×C3×#85911yescharacteristic, complemented
368C2—34no—
369C2C3×D3×#22611yescharacteristic
370C2C3×D3×GL(2,𝔽3)11yescharacteristic
371C2C3×D3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C2×C3×C4
FrattiniC22C3×D3×S4
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
prime radicalC2×C3×D3×SL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C22×C3
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
2. C3×C3⋊#859C2semidirect
3. C32×#859C2semidirect
5. D3×#859C3direct (N × H)
9. C3×D3×SL(2,𝔽3)C4semidirect
16. C3⋊#859C2×C3semidirect
20. C3×#859D3direct (N × H)
21. C3×#859C2×C3semidirect
29. C32×SL(2,𝔽3)C2×C4semidirect
37. C3×D3×Q8Dic3semidirect
50. D3×SL(2,𝔽3)C3×C4semidirect
73. #859C3×D3direct (N × H)
88. C32×Q8C2×Dic3semidirect
109. C3×SL(2,𝔽3)C4×D3semidirect
110. C3×SL(2,𝔽3)C2×C3×C4semidirect
129. D3×Q8C3×Dic3semidirect
201. C3×Q8D3×Dic3semidirect
202. C3×Q8C2×C3×Dic3semidirect
228. SL(2,𝔽3)C3×C4×D3semidirect
258. C3×D3#859direct (N × H)
312. C32C2×#859semidirect
317. Q8C3×D3×Dic3semidirect
351. D3C3×#859direct (N × H)
366. C3D3×#859direct (N × H)
367. 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⋊#8592
C32×#8592
C2×C3×#8593
D3×#8593
C3×D3×#1033
C2×C3×D3×Dic34