Finite Group Databasebeta

← (C2×S4)⋊(C3⋊Dic3) #13165991

Subgroups — (C2×S4)⋊(C3⋊Dic3) #13165991

Order 1728.

986 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C22×C32×S4C211yesmaximal, characteristic
2C2×A4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C2×C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
4(C2×S4)⋊Dic3—34nomaximal
5#10592919—31nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×C32×S4—21no—
8C2×C32×S4—21no—
9C2×C32×S4C412yescomplemented
10A4×C3⋊Dic3—21no—
11C3⋊(C3⋊SL(2,ℤ/4))—21no—
12(C2×D3)⋊(C3⋊Dic3)—41nomaximal
13C22×C3×S4D314yes—
14C22×C32×D4—31no—
15C23×C3⋊Dic3D311yescharacteristic, complemented
16C2×Dic3×A4—34no—
17C2×C3⋊SL(2,ℤ/4)—34no—
18C2×C3⋊SL(2,ℤ/4)—64no—
19C2×#3757—31no—
20C2×#3757—31no—
21C2×#3757—31no—
22#63658—31no—
23#63658—31no—
24C22×C32×D3—41no—
25C2×C32×A4C2×C411yescharacteristic
26C2×C32×A4D411yescharacteristic
27C2×C32×A4D411yescharacteristic
28C32×S4—22no—
29C2×C3⋊(C3⋊Dic3)—41no—
30C2×C3×C3⋊Dic3—41no—
31(C2×S4)⋊C4—91no—
32#5220—94no—
33C24×C32—31no—
34C24×C32C2×D311yescharacteristic
35C22×C3×A4—24no—
36C22×C3×A4C2×D314yes—
37C2×C3×S4—24no—
38C2×C3×S4—24no—
39C2×C3×S4Dic318yescomplemented
40C2×C32×D4—32no—
41C2×C32×D4—32no—
42C2×C32×D4—61no—
43C2×C32×D4—61no—
44C2×C32×D4—61no—
45C2×C32×D4—61no—
46C22×C32×C4—31no—
47C22×C3⋊Dic3—21no—
48C22×C3⋊Dic3—31no—
49C22×C3⋊Dic3—31no—
50C22×C3⋊Dic3—31no—
51C22×C3⋊Dic3—31no—
52C22×C3⋊Dic3—61no—
53Dic3×A4—64no—
54C3⋊SL(2,ℤ/4)—68no—
55C3⋊SL(2,ℤ/4)—64no—
56#3757—62no—
57#3757—61no—
58#3757—61no—
59#3757—61no—
60#3757—61no—
61(C2×D3)⋊Dic3—124no—
62C22×C33—41no—
63C2×C32×D3—42no—
64C2×C32×D3—81no—
65C2×C32×D3—81no—
66C32×A4C22⋊C411yescharacteristic, complemented
67C3⋊(C3⋊Dic3)—81no—
68C3×C3⋊Dic3—81no—
69C22×S4C3⋊D311yescharacteristic
70C22×C3×D4—34no—
71C23×Dic3—34no—
72C2×C4×A4—91no—
73C2×SL(2,ℤ/4)—91no—
74C2×SL(2,ℤ/4)—184no—
75C2×#217—94no—
76C2×#217—94no—
77C2×#217—94no—
78C3⋊#95—94no—
79C3⋊#95—94no—
80C23×C32—31no—
81C23×C32—31no—
82C23×C32—31no—
83C23×C32—31no—
84C23×C32—32no—
85C23×C32—61no—
86C23×C32—61no—
87C23×C32—61no—
88C23×C32—61no—
89C23×C32—61no—
90C23×C32—61no—
91C23×C32C4×D311yescharacteristic
92C23×C32D1211yescharacteristic
93C23×C32#6711yescharacteristic
94C22×C3×D3—44no—
95C2×C3×A4—24no—
96C2×C3×A4—24no—
97C2×C3×A4—24no—
98C2×C3×A4C2×Dic314yes—
99C2×C3×A4#6714yes—
100C2×C3×A4#6714yes—
101C3×S4—28no—
102C32×D4—62no—
103C32×D4—62no—
104C32×D4—62no—
105C32×D4—62no—
106C2×C32×C4—32no—
107C2×C32×C4—61no—
108C2×C32×C4—61no—
109C2×C3⋊Dic3—32no—
110C2×C3⋊Dic3—31no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—61no—
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—61no—
120C2×C3⋊Dic3—124no—
121C2×C3⋊Dic3—244no—
122C2×C3⋊Dic3S411yescharacteristic, complemented
123C2×C3×Dic3—124no—
124#386—271noSylow
125C2×C33—41no—
126C2×C33—41no—
127C2×C33—41no—
128C32×D3—82no—
129C24×C3—34no—
130C24×C3D3214yes—
131C22×A4—24no—
132C22×A4C2×C3⋊D311yescharacteristic
133C2×S4—21no—
134C2×S4—21no—
135C2×S4C3⋊Dic312yescomplemented
136C2×C3×D4—38no—
137C2×C3×D4—38no—
138C2×C3×D4—64no—
139C2×C3×D4—64no—
140C2×C3×D4—64no—
141C2×C3×D4—64no—
142C22×C3×C4—34no—
143C22×Dic3—64no—
144C22×Dic3—94no—
145C22×Dic3—94no—
146C22×Dic3—94no—
147C22×Dic3—94no—
148C22×Dic3—184no—
149C4×A4—181no—
150SL(2,ℤ/4)—188no—
151SL(2,ℤ/4)—181no—
152#217—184no—
153#217—188no—
154#217—184no—
155#217—184no—
156#217—184no—
157(C2×D3)⋊C4—361no—
158C22×C32—31no—
159C22×C32—31no—
160C22×C32—31no—
161C22×C32—31no—
162C22×C32—31no—
163C22×C32—31no—
164C22×C32—31no—
165C22×C32—31no—
166C22×C32—31no—
167C22×C32—44no—
168C22×C32—61no—
169C22×C32—62no—
170C22×C32—62no—
171C22×C32—62no—
172C22×C32—84no—
173C22×C32—121no—
174C22×C32—121no—
175C22×C32—121no—
176C22×C32—121no—
177C22×C32(C2×D3)⋊C411yescharacteristic, complemented
178C22×C32C2×S411yescharacteristic
179C2×C3×D3—48no—
180C2×C3×D3—84no—
181C2×C3×D3—84no—
182C3×A4—24no—
183C3×A4#21714yescomplemented
184C32×C4—62no—
185C3⋊Dic3—21no—
186C3⋊Dic3—61no—
187C3⋊Dic3—121no—
188C3⋊Dic3—121no—
189C3⋊Dic3—244no—
190C3⋊Dic3—248no—
191C3×Dic3—244no—
192C22×D4—31no—
193C23×C4—91no—
194C2×C22⋊C4—271no—
195C2×C22⋊C4—271no—
196C2×C22⋊C4—271no—
197#95—271no—
198#95—271no—
199C33—41noSylow
200C23×C3—34no—
201C23×C3—34no—
202C23×C3—34no—
203C23×C3—34no—
204C23×C3—38no—
205C23×C3—64no—
206C23×C3—64no—
207C23×C3—64no—
208C23×C3—64no—
209C23×C3—64no—
210C23×C3—64no—
211C23×C3D3×Dic314yes—
212C23×C3#62514yes—
213C23×C3#62614yes—
214C22×D3—41no—
215C2×A4—24no—
216C2×A4—24no—
217C2×A4—24no—
218C2×A4C2×C3⋊Dic311yescharacteristic
219C2×A4#63811yescharacteristic
220C2×A4#63811yescharacteristic
221S4—22no—
222C3×D4—68no—
223C3×D4—68no—
224C3×D4—68no—
225C3×D4—68no—
226C2×C3×C4—38no—
227C2×C3×C4—64no—
228C2×C3×C4—64no—
229C2×C3×C4—361no—
230C2×Dic3—34no—
231C2×Dic3—98no—
232C2×Dic3—94no—
233C2×Dic3—184no—
234C2×Dic3—184no—
235C2×Dic3—184no—
236C2×Dic3—184no—
237C2×Dic3—184no—
238C2×Dic3—184no—
239C2×Dic3—184no—
240C2×Dic3—184no—
241C2×Dic3—184no—
242C2×Dic3—361no—
243C2×Dic3—724no—
244C2×C32—31no—
245C2×C32—31no—
246C2×C32—31no—
247C2×C32—31no—
248C2×C32—44no—
249C2×C32—44no—
250C2×C32—44no—
251C2×C32—84no—
252C2×C32—84no—
253C2×C32—84no—
254C2×C32—122no—
255C2×C32C4×S411yescharacteristic
256C2×C32C4⋊S411yescharacteristic
257C2×C32GL(2,ℤ/4)11yescharacteristic
258C3×D3—88no—
259C24—31no—
260C24D3×C3⋊D311yescharacteristic
261C2×D4—32no—
262C2×D4—32no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C22×C4—31no—
268C22×C4—181no—
269C22×C4—271no—
270C22×C4—271no—
271C22×C4—271no—
272C22×C4—271no—
273C22×C4—541no—
274C22⋊C4—541no—
275C22⋊C4—542no—
276C22⋊C4—541no—
277C22⋊C4—541no—
278C22⋊C4—541no—
279C22×C3—34no—
280C22×C3—34no—
281C22×C3—34no—
282C22×C3—34no—
283C22×C3—34no—
284C22×C3—34no—
285C22×C3—34no—
286C22×C3—34no—
287C22×C3—34no—
288C22×C3—41no—
289C22×C3—64no—
290C22×C3—68no—
291C22×C3—68no—
292C22×C3—68no—
293C22×C3—84no—
294C22×C3—124no—
295C22×C3—124no—
296C22×C3—124no—
297C22×C3—124no—
298C22×C3(C2×D3)⋊Dic314yescomplemented
299C22×C3D3×S414yes—
300C2×D3—42no—
301C2×D3—81no—
302C2×D3—81no—
303A4—24no—
304A4#375711yescharacteristic, complemented
305C3×C4—68no—
306C3×C4—721no—
307Dic3—64no—
308Dic3—184no—
309Dic3—364no—
310Dic3—364no—
311Dic3—721no—
312Dic3—728no—
313C32—44no—
314C32—84no—
315C32(C2×S4)⋊C411yescharacteristic, complemented
316C23—31no—
317C23—31no—
318C23—31no—
319C23—31no—
320C23—32no—
321C23—61no—
322C23—61no—
323C23—61no—
324C23—61no—
325C23—61no—
326C23—61no—
327C23D3×C3⋊Dic311yescharacteristic
328C23#629011yescharacteristic
329C23#629111yescharacteristic
330D4—62no—
331D4—62no—
332D4—62no—
333D4—62no—
334C2×C4—32no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—91no—
338C2×C4—272no—
339C2×C4—271no—
340C2×C4—541no—
341C2×C4—541no—
342C2×C4—541no—
343C2×C4—541no—
344C2×C4—541no—
345C2×C4—541no—
346C2×C4—541no—
347C2×C4—541no—
348C2×C4—541no—
349C2×C3—34no—
350C2×C3—34no—
351C2×C3—34no—
352C2×C3—34no—
353C2×C3—41no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—84no—
357C2×C3—84no—
358C2×C3—84no—
359C2×C3—128no—
360C2×C3Dic3×S414yes—
361C2×C3#6420714yes—
362C2×C3#6420914yes—
363D3—82no—
364C22—31no—
365C22—31no—
366C22—31no—
367C22—31no—
368C22—31no—
369C22—31no—
370C22—31no—
371C22—31no—
372C22—31no—
373C22—61no—
374C22—62no—
375C22—62no—
376C22—62no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22(C2×D3)⋊(C3⋊Dic3)11yescharacteristic, complemented
382C22C3⋊D3×S411yescharacteristic
383C4—62no—
384C4—181no—
385C4—541no—
386C4—1081no—
387C4—1081no—
388C3—41no—
389C3—84no—
390C3(C2×S4)⋊Dic314yescomplemented
391C2—31no—
392C2—31no—
393C2—31no—
394C2—31no—
395C2—122no—
396C2S4×C3⋊Dic311yescharacteristic
397C2#1172418711yescharacteristic
398C2#1172418911yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22C3⋊D3×S4
commutatorC2×C32×A4C2×C4
Fitting, socleC24×C32C2×D3
prime radicalC22×C32×S4C2
cosocleC22×C32×A4C22
hypocenterC32×A4C22⋊C4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C2×A4×C3⋊Dic3C2semidirect
3. C2×C3⋊(C3⋊SL(2,ℤ/4))C2semidirect
9. C2×C32×S4C4semidirect
15. C23×C3⋊Dic3D3semidirect
39. C2×C3×S4Dic3semidirect
66. C32×A4C22⋊C4semidirect
122. C2×C3⋊Dic3S4semidirect
135. C2×S4C3⋊Dic3semidirect
177. C22×C32(C2×D3)⋊C4semidirect
183. C3×A4#217semidirect
298. C22×C3(C2×D3)⋊Dic3semidirect
304. A4#3757semidirect
315. C32(C2×S4)⋊C4semidirect
381. C22(C2×D3)⋊(C3⋊Dic3)semidirect
390. C3(C2×S4)⋊Dic3semidirect

Maximal subgroups

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

subgroupindex
C22×C32×S42
C2×A4×C3⋊Dic32
C2×C3⋊(C3⋊SL(2,ℤ/4))2
(C2×S4)⋊Dic33
#105929193
(C2×D3)⋊(C3⋊Dic3)4