Finite Group Databasebeta

← (((C3⋊D3)⋊C3)⋊D3)⋊C4 #13048326

Subgroups — (((C3⋊D3)⋊C3)⋊D3)⋊C4 #13048326

Order 1296.

524 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 69 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)⋊C3)⋊D3C211yesmaximal, characteristic
2#10619901C211yesmaximal, characteristic, complemented
3#10619890C211yesmaximal, characteristic, complemented
4#10619885C211yesmaximal, characteristic, complemented
5C3×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
6#6289⋊C3C211yesmaximal, characteristic, complemented
7Dic3×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
8#88949—31nomaximal
9D3×#624—31nomaximal
10C4×((C3⋊D3)⋊C3)⋊C2—31nomaximal
11C2×C3⋊#272C2211yescharacteristic
12C2×C3⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
14((C3⋊D3)⋊C3)⋊D3C414yescomplemented
15C3×#1084C2211yescharacteristic, complemented
16C3⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
17Dic3×H(3,𝔽3)C2211yescharacteristic, complemented
18C3×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
19C2×#1113—31no—
20C2×D3×C3⋊D3—31no—
21#6294—31no—
22#6289—31no—
23#6289D311yescharacteristic, complemented
24Dic3×C3⋊D3—31no—
25Dic3×C3⋊D3—31no—
26Dic3×C3⋊D3D311yescharacteristic, complemented
27C3×#624—31no—
28C3×#624—31no—
29C3×#624—31no—
30C3×#624—31no—
31C3×#624—61no—
32C3×D3×Dic3—31no—
33C3×D3×Dic3—31no—
34C2×((C3⋊D3)⋊C3)⋊C2—31no—
35C2×((C3⋊D3)⋊C3)⋊C2—61no—
36C4×#272—31no—
37C4×(C3⋊D3)⋊C3—31no—
38C4×(C3⋊D3)⋊C3—31no—
39#6199—31no—
40#6199—61no—
41((C3⋊D3)⋊C3)⋊C4—31no—
42((C3⋊D3)⋊C3)⋊C4D311yescharacteristic, complemented
43C2×C3×H(3,𝔽3)C2311yescharacteristic
44C3⋊#272C2×C412yescomplemented
45C3⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
46C3×(C3⋊D3)⋊C3C2×C412yescomplemented
47C2×#624—91no—
48C4×D32—91no—
49C4×D32—91no—
50C4×D32—91no—
51C4×D32—91no—
52C2×C3⋊(C3⋊D3)C2×D311yescharacteristic
53C2×C3×C3⋊D3—31no—
54C2×C3×C3⋊D3—31no—
55C2×C3×C3⋊D3—31no—
56C2×C3×C3⋊D3—61no—
57C2×C3×C3⋊D3C2×D311yescharacteristic
58C2×C32×D3—31no—
59#1113—34no—
60D3×C3⋊D3—34no—
61C3⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
62C3×C3⋊Dic3—31no—
63C3×C3⋊Dic3C2×D311yescharacteristic, complemented
64C32×Dic3—21no—
65C32×Dic3—31no—
66C32×Dic3—31no—
67C32×Dic3—31no—
68C32×Dic3—61no—
69C32×Dic3C2×D311yescharacteristic, complemented
70C32×Dic3C2×D311yescharacteristic, complemented
71C2×#272—31no—
72C2×#272—61no—
73C2×#272—61no—
74C2×#272—121no—
75C2×(C3⋊D3)⋊C3—21no—
76C2×(C3⋊D3)⋊C3—31no—
77C2×(C3⋊D3)⋊C3—61no—
78C2×(C3⋊D3)⋊C3C2×D311yescharacteristic
79((C3⋊D3)⋊C3)⋊C2—34no—
80((C3⋊D3)⋊C3)⋊C2—64no—
81C4×H(3,𝔽3)—31no—
82#1084C2×D311yescharacteristic, complemented
83(C3⋊Dic3)⋊C3—21no—
84(C3⋊Dic3)⋊C3—31no—
85(C3⋊Dic3)⋊C3—61no—
86(C3⋊Dic3)⋊C3C2×D311yescharacteristic, complemented
87C3×H(3,𝔽3)C22×C411yescharacteristic, complemented, Sylow
88C22×C3⋊D3—91no—
89C2×D32—91no—
90C2×D32—91no—
91C2×D32—91no—
92C2×D32—91no—
93C2×D32—181no—
94C4×C3⋊D3—31no—
95C4×C3⋊D3—31no—
96C4×C3⋊D3—91no—
97C4×C3⋊D3—91no—
98C2×C3×Dic3—91no—
99C2×C3×Dic3—91no—
100C3×C4×D3—91no—
101C3×C4×D3—91no—
102C3×C4×D3—91no—
103C3×C4×D3—91no—
104C3×C4×D3—91no—
105C3×C4×D3—91no—
106C3×C4×D3—91no—
107C3×C4×D3—91no—
108C3×C4×D3—181no—
109#624—31no—
110#624—31no—
111#624—31no—
112#624—31no—
113#624—61no—
114#624—91no—
115#624—91no—
116#624—91no—
117#624—91no—
118#624—181no—
119D3×Dic3—31no—
120D3×Dic3—31no—
121D3×Dic3—91no—
122D3×Dic3—91no—
123D3×Dic3—91no—
124D3×Dic3—91no—
125D3×Dic3—91no—
126D3×Dic3—91no—
127D3×Dic3—181no—
128D3×Dic3—181no—
129C2×C33—21no—
130C2×C33C22×D311yescharacteristic
131C2×C33C22×D311yescharacteristic
132C3⋊(C3⋊D3)C4×D312yescomplemented
133C3×C3⋊D3—32no—
134C3×C3⋊D3—32no—
135C3×C3⋊D3—32no—
136C3×C3⋊D3—62no—
137C3×C3⋊D3C4×D312yescomplemented
138C32×D3—32no—
139C2×H(3,𝔽3)—21no—
140C2×H(3,𝔽3)—21no—
141C2×H(3,𝔽3)—41no—
142C2×H(3,𝔽3)C22×D311yescharacteristic
143#272—32no—
144#272—62no—
145#272—62no—
146#272—122no—
147(C3⋊D3)⋊C3—22no—
148(C3⋊D3)⋊C3—32no—
149(C3⋊D3)⋊C3—62no—
150(C3⋊D3)⋊C3C4×D312yescomplemented
151C2×C4×D3—271no—
152C2×C4×D3—271no—
153C2×C4×D3—271no—
154C2×C4×D3—271no—
155C22×C32—91no—
156C2×C3⋊D3—31no—
157C2×C3⋊D3—31no—
158C2×C3⋊D3—31no—
159C2×C3⋊D3—31no—
160C2×C3⋊D3—61no—
161C2×C3⋊D3—61no—
162C2×C3⋊D3—91no—
163C2×C3⋊D3—94no—
164C2×C3⋊D3—181no—
165C2×C3⋊D3D3211yescharacteristic
166C2×C3×D3—31no—
167C2×C3×D3—31no—
168C2×C3×D3—61no—
169C2×C3×D3—91no—
170C2×C3×D3—91no—
171C2×C3×D3—91no—
172C2×C3×D3—91no—
173C2×C3×D3—91no—
174C2×C3×D3—91no—
175C2×C3×D3—181no—
176C2×C3×D3—181no—
177C2×C3×D3—181no—
178C2×C3×D3—181no—
179C2×C3×D3—361no—
180D32—94no—
181D32—94no—
182D32—94no—
183D32—94no—
184D32—184no—
185C32×C4—31no—
186C32×C4—31no—
187C32×C4—61no—
188C32×C4—91no—
189C32×C4—91no—
190C3⋊Dic3—31no—
191C3⋊Dic3—31no—
192C3⋊Dic3—61no—
193C3⋊Dic3—91no—
194C3⋊Dic3—181no—
195C3⋊Dic3D3211yescharacteristic, complemented
196C3×Dic3—31no—
197C3×Dic3—31no—
198C3×Dic3—31no—
199C3×Dic3—31no—
200C3×Dic3—31no—
201C3×Dic3—31no—
202C3×Dic3—31no—
203C3×Dic3—31no—
204C3×Dic3—61no—
205C3×Dic3—61no—
206C3×Dic3—61no—
207C3×Dic3—61no—
208C3×Dic3—61no—
209C3×Dic3—61no—
210C3×Dic3—91no—
211C3×Dic3—91no—
212C3×Dic3—91no—
213C3×Dic3—121no—
214C3×Dic3—181no—
215C3×Dic3D3211yescharacteristic
216C33—21no—
217C33C2×C4×D311yescharacteristic, complemented
218C33C2×C4×D311yescharacteristic, complemented
219H(3,𝔽3)—21no—
220H(3,𝔽3)—21no—
221H(3,𝔽3)—41no—
222H(3,𝔽3)C2×C4×D311yescharacteristic, complemented
223C22×D3—271no—
224C22×D3—271no—
225C22×D3—271no—
226C22×D3—271no—
227C22×D3—541no—
228C2×C3×C4—271no—
229C2×C3×C4—271no—
230C2×C3×C4—271no—
231C2×C3×C4—271no—
232C2×Dic3—91no—
233C2×Dic3—91no—
234C2×Dic3—271no—
235C2×Dic3—271no—
236C4×D3—91no—
237C4×D3—91no—
238C4×D3—91no—
239C4×D3—91no—
240C4×D3—91no—
241C4×D3—91no—
242C4×D3—91no—
243C4×D3—91no—
244C4×D3—181no—
245C4×D3—271no—
246C4×D3—271no—
247C4×D3—271no—
248C4×D3—271no—
249C4×D3—271no—
250C4×D3—271no—
251C4×D3—271no—
252C4×D3—271no—
253C4×D3—541no—
254C4×D3—541no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32—31no—
259C2×C32—31no—
260C2×C32—41no—
261C2×C32—61no—
262C2×C32—61no—
263C2×C32—61no—
264C2×C32—92no—
265C2×C32—121no—
266C2×C32C2×D3211yescharacteristic
267C2×C32C2×D3211yescharacteristic
268C2×C32C2×D3211yescharacteristic
269C3⋊D3—32no—
270C3⋊D3—32no—
271C3⋊D3—32no—
272C3⋊D3—32no—
273C3⋊D3—62no—
274C3⋊D3—62no—
275C3⋊D3—92no—
276C3⋊D3—182no—
277C3⋊D3#62412yescomplemented
278C3×D3—32no—
279C3×D3—32no—
280C3×D3—62no—
281C3×D3—92no—
282C3×D3—92no—
283C3×D3—92no—
284C3×D3—92no—
285C3×D3—92no—
286C3×D3—92no—
287C3×D3—182no—
288C3×D3—182no—
289C3×D3—182no—
290C3×D3—182no—
291C3×D3—362no—
292C22×C4—811noSylow
293C22×C3—91no—
294C22×C3—91no—
295C22×C3—181no—
296C22×C3—271no—
297C22×C3—271no—
298C2×D3—31no—
299C2×D3—91no—
300C2×D3—91no—
301C2×D3—91no—
302C2×D3—91no—
303C2×D3—91no—
304C2×D3—91no—
305C2×D3—181no—
306C2×D3—181no—
307C2×D3—181no—
308C2×D3—181no—
309C2×D3—271no—
310C2×D3—274no—
311C2×D3—274no—
312C2×D3—274no—
313C2×D3—274no—
314C2×D3—361no—
315C2×D3—541no—
316C2×D3—544no—
317C3×C4—31no—
318C3×C4—91no—
319C3×C4—91no—
320C3×C4—91no—
321C3×C4—91no—
322C3×C4—91no—
323C3×C4—91no—
324C3×C4—181no—
325C3×C4—181no—
326C3×C4—181no—
327C3×C4—271no—
328Dic3—31no—
329Dic3—31no—
330Dic3—31no—
331Dic3—61no—
332Dic3—91no—
333Dic3—91no—
334Dic3—91no—
335Dic3—181no—
336Dic3—271no—
337Dic3—541no—
338Dic3((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
339C32—21no—
340C32—21no—
341C32—21no—
342C32—31no—
343C32—31no—
344C32—41no—
345C32—61no—
346C32—61no—
347C32—61no—
348C32—121no—
349C32C4×D3211yescharacteristic
350C32C4×D3211yescharacteristic, complemented
351C32C2×#62411yescharacteristic, complemented
352C23—811no—
353C2×C4—271no—
354C2×C4—271no—
355C2×C4—271no—
356C2×C4—271no—
357C2×C4—811no—
358C2×C4—811no—
359C2×C3—21no—
360C2×C3—31no—
361C2×C3—31no—
362C2×C3—61no—
363C2×C3—61no—
364C2×C3—61no—
365C2×C3—92no—
366C2×C3—92no—
367C2×C3—121no—
368C2×C3—182no—
369C2×C3—272no—
370C2×C3—272no—
371C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
372C2×C3D3311yescharacteristic
373D3—32no—
374D3—92no—
375D3—92no—
376D3—92no—
377D3—92no—
378D3—92no—
379D3—92no—
380D3—182no—
381D3—182no—
382D3—182no—
383D3—182no—
384D3—272no—
385D3—362no—
386D3—542no—
387C22—91no—
388C22—271no—
389C22—271no—
390C22—814no—
391C4—31no—
392C4—91no—
393C4—91no—
394C4—271no—
395C3—21no—
396C3—31no—
397C3—31no—
398C3—61no—
399C3—61no—
400C3—61no—
401C3—121no—
402C3C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
403C3D3×#62411yescharacteristic
404C2—92no—
405C2—272no—
406C2—272no—
407C2D3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2D3×((C3⋊D3)⋊C3)⋊C2
commutator, hypocenterC3×H(3,𝔽3)C22×C4
FrattiniC2×C3D33
Fitting, cosocleC2×C3×H(3,𝔽3)C23
socleC2×C32C2×D32
prime radicalC2×((C3⋊D3)⋊C3)⋊D3C2
LCSG ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. #10619901C2semidirect
3. #10619890C2semidirect
4. #10619885C2semidirect
5. C3×((C3⋊D3)⋊C3)⋊C4C2semidirect
6. #6289⋊C3C2semidirect
7. Dic3×(C3⋊D3)⋊C3C2semidirect
14. ((C3⋊D3)⋊C3)⋊D3C4semidirect
15. C3×#1084C22semidirect
16. C3⋊((C3⋊Dic3)⋊C3)C22semidirect
17. Dic3×H(3,𝔽3)C22semidirect
18. C3×(C3⋊Dic3)⋊C3C22semidirect
23. #6289D3semidirect
26. Dic3×C3⋊D3D3semidirect
42. ((C3⋊D3)⋊C3)⋊C4D3semidirect
44. C3⋊#272C2×C4semidirect
45. C3⋊((C3⋊D3)⋊C3)C2×C4semidirect
46. C3×(C3⋊D3)⋊C3C2×C4semidirect
61. C3⋊(C3⋊Dic3)C2×D3semidirect
63. C3×C3⋊Dic3C2×D3semidirect
69. C32×Dic3C2×D3semidirect
70. C32×Dic3C2×D3semidirect
82. #1084C2×D3semidirect
86. (C3⋊Dic3)⋊C3C2×D3semidirect
87. C3×H(3,𝔽3)C22×C4semidirect
132. C3⋊(C3⋊D3)C4×D3semidirect
137. C3×C3⋊D3C4×D3semidirect
150. (C3⋊D3)⋊C3C4×D3semidirect
195. C3⋊Dic3D32semidirect
217. C33C2×C4×D3semidirect
218. C33C2×C4×D3semidirect
222. H(3,𝔽3)C2×C4×D3semidirect
277. C3⋊D3#624semidirect
338. Dic3((C3⋊D3)⋊C3)⋊C2semidirect
350. C32C4×D32semidirect
351. C32C2×#624semidirect
402. C3C4×((C3⋊D3)⋊C3)⋊C2semidirect

Maximal subgroups

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

subgroupindex
C2×((C3⋊D3)⋊C3)⋊D32
#106199012
#106198902
#106198852
C3×((C3⋊D3)⋊C3)⋊C42
#6289⋊C32
Dic3×(C3⋊D3)⋊C32
#889493
D3×#6243
C4×((C3⋊D3)⋊C3)⋊C23