Finite Group Databasebeta

← C3×#10593209 #13165885

Subgroups — C3×#10593209 #13165885

Order 1728.

430 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 62 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
4C3×#5420—31nomaximal
5#10593209C311yesmaximal, characteristic, direct product factor
6C3×#4802—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C3×D3×SL(2,𝔽3)C412yescomplemented
9C3×(C2×D3)⋊Dic3—41nomaximal
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×ΓL(1,𝔽25)—31no—
15C3×#892—31no—
16C3⋊#859C2×C311yescharacteristic, complemented
17C3×#859—21no—
18C3×#859—31no—
19C3×#859—61no—
20C3×#859D311yescharacteristic, complemented
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
30#5420—31no—
31C3×#419—91no—
32#4802—31no—
33C2×C32×Q8C2×D311yescharacteristic
34C3×D3×Q8—32no—
35C3×D3×Q8Dic312yescomplemented
36C2×C3×C4×D3—31no—
37C2×C3×Dic6—31no—
38C2×C3×SL(2,𝔽3)—21no—
39C2×C3×SL(2,𝔽3)—21no—
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)—32no—
42C2×C3×SL(2,𝔽3)—41no—
43C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
44C2×C3×SL(2,𝔽3)C22×C311yescharacteristic
45D3×SL(2,𝔽3)—22no—
46D3×SL(2,𝔽3)C3×C412yescomplemented
47C2×C32×C8—31no—
48C32×C4⋊C4—31no—
49C3×#217—121no—
50C3×(C2×D3)⋊C4—61no—
51C3×C4⋊Dic3—31no—
52C3×C4×Dic3—61no—
53C2×C3×C3⋊C8—31no—
54C3×ΓL(1,𝔽25)—62no—
55(C2×D3)⋊Dic3—41no—
56C22×C33—41no—
57C2×C32×D3—42no—
58C2×C32×D3—81no—
59C2×C32×D3—81no—
60C3×C3⋊Dic3—81no—
61C32×Dic3—81no—
62C22×C3×Q8—31no—
63C2×D3×Q8C3×D311yescharacteristic
64C22×SL(2,𝔽3)—31no—
65C22×SL(2,𝔽3)—61no—
66C2×C3×M16—91no—
67C3×#117—91no—
68C2×ΓL(1,𝔽25)—31no—
69#892—31no—
70#859—31no—
71#859—61no—
72#859C3×D311yescharacteristic, complemented
73C3×#103—31no—
74C3×#103—31no—
75C3×#103—61no—
76C3×#103—91no—
77C3×#103—91no—
78C3×#103—91no—
79#835—31no—
80#816—31no—
81#810—31no—
82C23×C32—121no—
83C22×C3×D3—41no—
84C22×C3×D3—81no—
85C22×C3×D3S411yescharacteristic
86C32×Q8—31no—
87C32×Q8C2×Dic311yescharacteristic
88C2×C32×C4—31no—
89C2×C32×C4—61no—
90C2×C3⋊Dic3—41no—
91C2×C3×Dic3—31no—
92C2×C3×Dic3—41no—
93C2×C3×Dic3—41no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—81no—
96C2×C3×Dic3—121no—
97C2×C3×Dic3—241no—
98C3×C4×D3—32no—
99C3×C4×D3—32no—
100C3×Dic6—32no—
101C3×Dic6—61no—
102C3×SL(2,𝔽3)—21no—
103C3×SL(2,𝔽3)—21no—
104C3×SL(2,𝔽3)—21no—
105C3×SL(2,𝔽3)—41no—
106C3×SL(2,𝔽3)C4×D311yescharacteristic
107C3×SL(2,𝔽3)C2×C3×C411yescharacteristic
108C32×C8—61no—
109C3×C3⋊C8—61no—
110#419—91noSylow
111C2×C33—41no—
112C2×C33—41no—
113C2×C33—41no—
114C32×D3—82no—
115C2×C3×Q8—21no—
116C2×C3×Q8—32no—
117C2×C3×Q8—91no—
118C2×C3×Q8—92no—
119C2×C3×Q8D3211yescharacteristic
120C2×C3×Q8C2×C3×D311yescharacteristic
121C22×C3×C4—91no—
122D3×Q8—32no—
123D3×Q8C3×Dic312yescomplemented
124C2×C4×D3—31no—
125C2×Dic6—31no—
126C2×SL(2,𝔽3)—21no—
127C2×SL(2,𝔽3)—21no—
128C2×SL(2,𝔽3)—32no—
129C2×SL(2,𝔽3)—41no—
130C2×SL(2,𝔽3)—62no—
131C2×SL(2,𝔽3)C2×C3×D311yescharacteristic
132C3×M16—182no—
133C2×C3×C8—31no—
134C2×C3×C8—31no—
135C2×C3×C8—61no—
136C2×C3×C8—91no—
137C3×C4⋊C4—31no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—91no—
141C3×C22⋊C4—181no—
142C3×C42—181no—
143#217—121no—
144(C2×D3)⋊C4—61no—
145C4⋊Dic3—31no—
146C4×Dic3—61no—
147C2×C3⋊C8—31no—
148ΓL(1,𝔽25)—62no—
149C22×C32—41no—
150C22×C32—41no—
151C22×C32—81no—
152C22×C32—81no—
153C22×C32—81no—
154C22×C32—122no—
155C22×C32—161no—
156C22×C32—241no—
157C22×C32—241no—
158C22×C32C2×S411yescharacteristic
159C2×C3×D3—21no—
160C2×C3×D3—21no—
161C2×C3×D3—42no—
162C2×C3×D3—81no—
163C2×C3×D3—81no—
164C2×C3×D3—82no—
165C2×C3×D3—82no—
166C2×C3×D3—82no—
167C2×C3×D3SL(2,ℤ/4)12yes—
168C32×C4—31no—
169C32×C4—31no—
170C32×C4—121no—
171C3⋊Dic3—81no—
172C3×Dic3—32no—
173C3×Dic3—81no—
174C3×Dic3—81no—
175C3×Dic3—82no—
176C3×Dic3—121no—
177C3×Dic3—241no—
178C3×Dic3—242no—
179C22×Q8—31no—
180C2×M16—91no—
181#117—91no—
182#103—31no—
183#103—91no—
184#103—91no—
185#103—91no—
186C33—41noSylow
187C23×C3—31no—
188C23×C3—121no—
189C23×C3—241no—
190C22×D3C3×S411yescharacteristic
191C3×Q8—21no—
192C3×Q8—31no—
193C3×Q8—31no—
194C3×Q8—61no—
195C3×Q8—92no—
196C3×Q8—181no—
197C3×Q8D3×Dic311yescharacteristic
198C3×Q8C2×C3×Dic311yescharacteristic
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—92no—
205C2×C3×C4—92no—
206C2×C3×C4—91no—
207C2×C3×C4—121no—
208C2×C3×C4—181no—
209C2×Dic3—31no—
210C2×Dic3—41no—
211C2×Dic3—61no—
212C2×Dic3—121no—
213C2×Dic3—241no—
214C4×D3—32no—
215C4×D3—32no—
216Dic6—32no—
217Dic6—61no—
218SL(2,𝔽3)—21no—
219SL(2,𝔽3)—21no—
220SL(2,𝔽3)—41no—
221SL(2,𝔽3)C3×C4×D311yescharacteristic
222C3×C8—61no—
223C3×C8—61no—
224C3×C8—121no—
225C3×C8—181no—
226C3⋊C8—61no—
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230C2×C32—41no—
231C2×C32—41no—
232C2×C32—41no—
233C2×C32—81no—
234C2×C32—81no—
235C2×C32—81no—
236C2×C32—81no—
237C2×C32—81no—
238C2×C32—81no—
239C2×C32—81no—
240C2×C32—81no—
241C2×C32—81no—
242C2×C32—161no—
243C2×C32—161no—
244C2×C32—161no—
245C2×C32—242no—
246C2×C32#98311yescharacteristic
247C2×C32#98311yescharacteristic
248C2×C32C2×SL(2,ℤ/4)11yescharacteristic
249C3×D3—22no—
250C3×D3—82no—
251C3×D3—84no—
252C2×Q8—32no—
253C2×Q8—91no—
254C2×Q8—92no—
255C2×Q8C3×D3211yescharacteristic
256C22×C4—91no—
257M16—182no—
258C2×C8—31no—
259C2×C8—91no—
260C4⋊C4—31no—
261C4⋊C4—91no—
262C22⋊C4—181no—
263C42—181no—
264C22×C3—21no—
265C22×C3—32no—
266C22×C3—41no—
267C22×C3—61no—
268C22×C3—61no—
269C22×C3—81no—
270C22×C3—81no—
271C22×C3—122no—
272C22×C3—161no—
273C22×C3—242no—
274C22×C3—241no—
275C22×C3—241no—
276C22×C3—242no—
277C22×C3—242no—
278C22×C3D3×S411yescharacteristic
279C22×C3C2×C3×S411yescharacteristic
280C2×D3—21no—
281C2×D3—21no—
282C2×D3C3×SL(2,ℤ/4)12yes—
283C3×C4—31no—
284C3×C4—31no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—61no—
288C3×C4—61no—
289C3×C4—92no—
290C3×C4—121no—
291C3×C4—121no—
292C3×C4—241no—
293C3×C4—361no—
294Dic3—32no—
295Dic3—81no—
296Dic3—121no—
297Dic3—241no—
298Dic3—242no—
299C32—41no—
300C32—41no—
301C32—81no—
302C32—81no—
303C32—81no—
304C32—161no—
305C32#542011yescharacteristic, complemented
306C23—31no—
307Q8—31no—
308Q8—92no—
309Q8—181no—
310Q8C3×D3×Dic311yescharacteristic
311C2×C4—31no—
312C2×C4—61no—
313C2×C4—92no—
314C2×C4—92no—
315C2×C4—91no—
316C2×C4—181no—
317C8—61no—
318C8—181no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—41no—
325C2×C3—62no—
326C2×C3—81no—
327C2×C3—81no—
328C2×C3—81no—
329C2×C3—81no—
330C2×C3—81no—
331C2×C3—81no—
332C2×C3—161no—
333C2×C3—161no—
334C2×C3—161no—
335C2×C3—242no—
336C2×C3—244no—
337C2×C3#6420111yescharacteristic
338C2×C3#6420211yescharacteristic
339C2×C3D3×SL(2,ℤ/4)11yescharacteristic
340C2×C3C3×#98311yescharacteristic
341C2×C3C3×#98311yescharacteristic
342C2×C3C2×C3×SL(2,ℤ/4)11yescharacteristic
343D3—22no—
344C22—32no—
345C22—61no—
346C22—61no—
347C22C3×D3×S411yescharacteristic
348C4—31no—
349C4—31no—
350C4—92no—
351C4—121no—
352C4—361no—
353C3—21no—
354C3—41no—
355C3—81no—
356C3—81no—
357C3—161no—
358C3#1059320911yescharacteristic, direct product factor
359C3C3×#542011yescharacteristic, complemented
360C2—62no—
361C2C3×#6420111yescharacteristic
362C2C3×#6420211yescharacteristic
363C2C3×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. #10593209C3direct (N × H)
8. C3×D3×SL(2,𝔽3)C4semidirect
16. C3⋊#859C2×C3semidirect
20. C3×#859D3semidirect
21. C3×#859C2×C3semidirect
35. C3×D3×Q8Dic3semidirect
46. D3×SL(2,𝔽3)C3×C4semidirect
72. #859C3×D3semidirect
123. D3×Q8C3×Dic3semidirect
305. C32#5420semidirect
358. C3#10593209direct (N × H)
359. C3C3×#5420semidirect

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
C3×#54203
#105932093
C3×#48023
C3×(C2×D3)⋊Dic34