Finite Group Databasebeta

← C2×S4×C3⋊Dic3 #13180776

Subgroups — C2×S4×C3⋊Dic3 #13180776

Order 1728.

1526 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 1526 conjugacy classes of subgroups — too many to draw. Only its 173 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
4S4×C3⋊Dic3C214yesmaximal, direct product factor
5C2×Dic3×S4—34nomaximal
6C2×D4×C3⋊Dic3—31nomaximal
7C22×C32×A4C2211yescharacteristic
8C2×C32×S4C414yescomplemented
9C2×C32×S4C2212yes—
10C2×D3×C3⋊Dic3—41nomaximal
11A4×C3⋊Dic3C2212yescomplemented
12C3⋊(C3⋊SL(2,ℤ/4))C2212yescomplemented
13C22×C3×S4D314yes—
14C22×C32×D4—31no—
15C23×C3⋊Dic3—31no—
16C23×C3⋊Dic3D311yescharacteristic, complemented
17C2×Dic3×A4—34no—
18C2×C3⋊SL(2,ℤ/4)—34no—
19C2×C3⋊SL(2,ℤ/4)—64no—
20Dic3×S4—316no—
21C2×#3757—31no—
22C2×#3757—31no—
23D4×C3⋊Dic3—34no—
24D4×C3⋊Dic3—34no—
25C2×C4⋊(C3⋊Dic3)—31no—
26C2×C4×C3⋊Dic3—31no—
27C22×C32×D3—41no—
28C2×C32×A4C2×C412yescomplemented
29C2×C32×A4C2311yescharacteristic
30C32×S4C2×C414yescomplemented
31C2×C3⋊(C3⋊Dic3)—41no—
32C2×C3×C3⋊Dic3—41no—
33D3×C3⋊Dic3—44no—
34C2×C4×S4—91no—
35C2×D4×Dic3—94no—
36C24×C32—31no—
37C24×C32C2×D311yescharacteristic
38C22×C3×A4—24no—
39C22×C3×A4C2×D314yes—
40C2×C3×S4Dic3116yescomplemented
41C2×C3×S4C2×D318yes—
42C2×C32×D4—34no—
43C2×C32×D4—32no—
44C2×C32×D4—34no—
45C2×C32×D4—32no—
46C22×C32×C4—31no—
47C22×C3⋊Dic3—31no—
48C22×C3⋊Dic3—31no—
49C22×C3⋊Dic3—32no—
50C22×C3⋊Dic3—31no—
51C22×C3⋊Dic3—34no—
52C22×C3⋊Dic3—31no—
53C22×C3⋊Dic3—61no—
54C22×C3⋊Dic3—62no—
55C22×C3⋊Dic3—61no—
56C22×C3⋊Dic3C2×D312yescomplemented
57C2×D3×Dic3—124no—
58Dic3×A4—38no—
59C3⋊SL(2,ℤ/4)—38no—
60C3⋊SL(2,ℤ/4)—68no—
61#3757—34no—
62#3757—32no—
63#3757—32no—
64C4⋊(C3⋊Dic3)—34no—
65C4×C3⋊Dic3—34no—
66C22×C33—41no—
67C2×C32×D3—44no—
68C2×C32×D3—42no—
69C32×A4C22×C411yescharacteristic, complemented
70C3⋊(C3⋊Dic3)—42no—
71C3×C3⋊Dic3—42no—
72C22×S4C3⋊D311yescharacteristic
73C22×C3×D4—34no—
74C23×Dic3—34no—
75C23×Dic3—94no—
76C2×C4×A4—91no—
77C2×SL(2,ℤ/4)—91no—
78C2×SL(2,ℤ/4)—184no—
79C4×S4—94no—
80C2×#217—94no—
81C2×#217—94no—
82D4×Dic3—916no—
83D4×Dic3—916no—
84C2×C4⋊Dic3—94no—
85C2×C4×Dic3—94no—
86C23×C32—31no—
87C23×C32—31no—
88C23×C32—32no—
89C23×C32—32no—
90C23×C32—34no—
91C23×C32—62no—
92C23×C32—61no—
93C23×C32—61no—
94C23×C32C4×D312yescomplemented
95C23×C32C22×D311yescharacteristic
96C22×C3×D3—44no—
97C2×C3×A4—28no—
98C2×C3×A4—24no—
99C2×C3×A4C2×Dic318yescomplemented
100C2×C3×A4C22×D314yes—
101C3×S4C2×Dic3116yescomplemented
102C32×D4—38no—
103C32×D4—34no—
104C32×D4—34no—
105C2×C32×C4—32no—
106C2×C32×C4—34no—
107C2×C3⋊Dic3—31no—
108C2×C3⋊Dic3—32no—
109C2×C3⋊Dic3—32no—
110C2×C3⋊Dic3—32no—
111C2×C3⋊Dic3—32no—
112C2×C3⋊Dic3—32no—
113C2×C3⋊Dic3—61no—
114C2×C3⋊Dic3—64no—
115C2×C3⋊Dic3—62no—
116C2×C3⋊Dic3—61no—
117C2×C3⋊Dic3—61no—
118C2×C3⋊Dic3—64no—
119C2×C3⋊Dic3—124no—
120C2×C3⋊Dic3—244no—
121C2×C3⋊Dic3S411yescharacteristic, direct product factor
122C2×C3×Dic3—124no—
123D3×Dic3—1216no—
124C2×C4×D4—271noSylow
125C2×C33—42no—
126C2×C33—41no—
127C32×D3—44no—
128C24×C3—34no—
129C24×C3D3214yes—
130C22×A4—24no—
131C22×A4C2×C3⋊D311yescharacteristic
132C2×S4C3⋊Dic314yesdirect product factor
133C2×S4C2×C3⋊D312yes—
134C2×C3×D4—38no—
135C2×C3×D4—316no—
136C2×C3×D4—316no—
137C2×C3×D4—38no—
138C22×C3×C4—34no—
139C22×Dic3—38no—
140C22×Dic3—94no—
141C22×Dic3—94no—
142C22×Dic3—98no—
143C22×Dic3—94no—
144C22×Dic3—94no—
145C22×Dic3—916no—
146C22×Dic3—184no—
147C22×Dic3—188no—
148C22×Dic3—184no—
149C2×C4×D3—361no—
150C4×A4—92no—
151SL(2,ℤ/4)—92no—
152SL(2,ℤ/4)—188no—
153#217—916no—
154#217—98no—
155#217—98no—
156C4⋊Dic3—916no—
157C4×Dic3—916no—
158C22×C32—32no—
159C22×C32—31no—
160C22×C32—32no—
161C22×C32—32no—
162C22×C32—31no—
163C22×C32—31no—
164C22×C32—34no—
165C22×C32—44no—
166C22×C32—61no—
167C22×C32—62no—
168C22×C32—64no—
169C22×C32—64no—
170C22×C32—62no—
171C22×C32—84no—
172C22×C32C2×C4×D311yescharacteristic, complemented
173C22×C32C2×S411yescharacteristic
174C2×C3×D3—416no—
175C2×C3×D3—48no—
176C3×A4—24no—
177C3×A4C22×Dic314yescomplemented
178C32×C4—34no—
179C3⋊Dic3—32no—
180C3⋊Dic3—62no—
181C3⋊Dic3—62no—
182C3⋊Dic3—128no—
183C3⋊Dic3—248no—
184C3⋊Dic3C2×S412yesdirect product factor
185C3×Dic3—128no—
186C22×D4—31no—
187C23×C4—91no—
188C23×C4—271no—
189C4×D4—274no—
190C4×D4—274no—
191C2×C4⋊C4—271no—
192C2×C22⋊C4—271no—
193C2×C22⋊C4—271no—
194C2×C42—271no—
195C33—41noSylow
196C23×C3—34no—
197C23×C3—34no—
198C23×C3—38no—
199C23×C3—38no—
200C23×C3—316no—
201C23×C3—64no—
202C23×C3—68no—
203C23×C3—64no—
204C23×C3D3×Dic318yescomplemented
205C23×C3C2×D3214yes—
206C22×D3—41no—
207C2×A4—28no—
208C2×A4—24no—
209C2×A4C2×C3⋊Dic312yescomplemented
210C2×A4C22×C3⋊D311yescharacteristic
211S4C2×C3⋊Dic314yesdirect product factor
212C3×D4—332no—
213C3×D4—316no—
214C3×D4—316no—
215C2×C3×C4—38no—
216C2×C3×C4—316no—
217C2×C3×C4—361no—
218C2×Dic3—34no—
219C2×Dic3—98no—
220C2×Dic3—98no—
221C2×Dic3—94no—
222C2×Dic3—98no—
223C2×Dic3—98no—
224C2×Dic3—98no—
225C2×Dic3—1816no—
226C2×Dic3—184no—
227C2×Dic3—1816no—
228C2×Dic3—188no—
229C2×Dic3—184no—
230C2×Dic3—184no—
231C2×Dic3—361no—
232C2×Dic3—724no—
233C4×D3—364no—
234C2×C32—31no—
235C2×C32—31no—
236C2×C32—32no—
237C2×C32—44no—
238C2×C32—48no—
239C2×C32—64no—
240C2×C32—84no—
241C2×C32—88no—
242C2×C32C4×S412yescomplemented
243C2×C32C22×S411yescharacteristic
244C3×D3—416no—
245C24—31no—
246C24D3×C3⋊D311yescharacteristic
247C2×D4—32no—
248C2×D4—34no—
249C2×D4—34no—
250C2×D4—32no—
251C22×C4—31no—
252C22×C4—92no—
253C22×C4—271no—
254C22×C4—271no—
255C22×C4—272no—
256C22×C4—271no—
257C22×C4—271no—
258C22×C4—274no—
259C22×C4—541no—
260C22×C4—542no—
261C22×C4—541no—
262C4⋊C4—274no—
263C22⋊C4—274no—
264C22⋊C4—272no—
265C22⋊C4—272no—
266C42—274no—
267C22×C3—38no—
268C22×C3—34no—
269C22×C3—34no—
270C22×C3—34no—
271C22×C3—38no—
272C22×C3—38no—
273C22×C3—316no—
274C22×C3—41no—
275C22×C3—64no—
276C22×C3—68no—
277C22×C3—616no—
278C22×C3—68no—
279C22×C3—616no—
280C22×C3—84no—
281C22×C3C2×D3×Dic314yescomplemented
282C22×C3D3×S414yes—
283C2×D3—42no—
284C2×D3—44no—
285A4—24no—
286A4C22×C3⋊Dic311yescharacteristic, complemented
287C3×C4—316no—
288C3×C4—362no—
289Dic3—38no—
290Dic3—98no—
291Dic3—188no—
292Dic3—188no—
293Dic3—362no—
294Dic3—728no—
295C32—44no—
296C32—84no—
297C32C2×C4×S411yescharacteristic, complemented
298C23—31no—
299C23—31no—
300C23—32no—
301C23—32no—
302C23—34no—
303C23—61no—
304C23—62no—
305C23—61no—
306C23D3×C3⋊Dic312yescomplemented
307C23C2×D3×C3⋊D311yescharacteristic
308D4—38no—
309D4—34no—
310D4—34no—
311C2×C4—32no—
312C2×C4—34no—
313C2×C4—91no—
314C2×C4—272no—
315C2×C4—272no—
316C2×C4—271no—
317C2×C4—272no—
318C2×C4—272no—
319C2×C4—272no—
320C2×C4—544no—
321C2×C4—541no—
322C2×C4—544no—
323C2×C4—541no—
324C2×C4—542no—
325C2×C4—541no—
326C2×C3—34no—
327C2×C3—34no—
328C2×C3—38no—
329C2×C3—42no—
330C2×C3—41no—
331C2×C3—616no—
332C2×C3—88no—
333C2×C3—84no—
334C2×C3Dic3×S418yescomplemented
335C2×C3C2×D3×S414yes—
336D3—44no—
337C22—32no—
338C22—31no—
339C22—32no—
340C22—32no—
341C22—31no—
342C22—31no—
343C22—34no—
344C22—61no—
345C22—62no—
346C22—64no—
347C22—64no—
348C22—62no—
349C22C2×D3×C3⋊Dic311yescharacteristic, complemented
350C22C3⋊D3×S411yescharacteristic
351C4—34no—
352C4—92no—
353C4—272no—
354C4—542no—
355C4—542no—
356C3—41no—
357C3—84no—
358C3C2×Dic3×S414yescomplemented
359C2—31no—
360C2—31no—
361C2—32no—
362C2—64no—
363C2S4×C3⋊Dic312yesdirect product factor
364C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊D3×S4
commutator, hypocenterC32×A4C22×C4
FrattiniC2C2×C3⋊D3×S4
Fitting, socleC24×C32C2×D3
prime radicalC22×C32×S4C2
cosocleC2×C32×A4C23
LCSG ▷ 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
4. S4×C3⋊Dic3C2direct (N × H)
8. C2×C32×S4C4semidirect
11. A4×C3⋊Dic3C22semidirect
12. C3⋊(C3⋊SL(2,ℤ/4))C22semidirect
16. C23×C3⋊Dic3D3semidirect
28. C2×C32×A4C2×C4semidirect
30. C32×S4C2×C4semidirect
40. C2×C3×S4Dic3semidirect
56. C22×C3⋊Dic3C2×D3semidirect
69. C32×A4C22×C4semidirect
94. C23×C32C4×D3semidirect
99. C2×C3×A4C2×Dic3semidirect
101. C3×S4C2×Dic3semidirect
121. C2×C3⋊Dic3S4direct (N × H)
132. C2×S4C3⋊Dic3direct (N × H)
172. C22×C32C2×C4×D3semidirect
177. C3×A4C22×Dic3semidirect
184. C3⋊Dic3C2×S4direct (N × H)
204. C23×C3D3×Dic3semidirect
209. C2×A4C2×C3⋊Dic3semidirect
211. S4C2×C3⋊Dic3direct (N × H)
242. C2×C32C4×S4semidirect
281. C22×C3C2×D3×Dic3semidirect
286. A4C22×C3⋊Dic3semidirect
297. C32C2×C4×S4semidirect
306. C23D3×C3⋊Dic3semidirect
334. C2×C3Dic3×S4semidirect
349. C22C2×D3×C3⋊Dic3semidirect
358. C3C2×Dic3×S4semidirect
363. C2S4×C3⋊Dic3direct (N × H)

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
S4×C3⋊Dic32
C2×Dic3×S43
C2×D4×C3⋊Dic33
C2×D3×C3⋊Dic34