Finite Group Databasebeta

← C22×Dic3×S4 #11923618

Subgroups — C22×Dic3×S4 #11923618

Order 1152.

2636 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 2636 conjugacy classes of subgroups — too many to draw. Only its 271 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
1C23×C3×S4C211yesmaximal, characteristic
2C22×Dic3×A4C211yesmaximal, characteristic, complemented
3C22×C3⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C2×Dic3×S4C2112yesmaximal, direct product factor
5C22×C4×S4—31nomaximal
6C22×D4×Dic3—31nomaximal
7C23×C3×A4C2211yescharacteristic
8C22×C3×S4C418yescomplemented
9C22×C3×S4C2216yes—
10C22×D3×Dic3—41nomaximal
11C2×Dic3×A4C2216yescomplemented
12C2×C3⋊SL(2,ℤ/4)C2216yescomplemented
13Dic3×S4C22116yesdirect product factor
14C23×S4D311yescharacteristic
15C23×C3×D4—31no—
16C24×Dic3—31no—
17C24×Dic3D311yescharacteristic, complemented
18C22×C4×A4—31no—
19C22×SL(2,ℤ/4)—31no—
20C22×SL(2,ℤ/4)—61no—
21C2×C4×S4—312no—
22C22×#217—31no—
23C22×#217—31no—
24C2×D4×Dic3—312no—
25C2×D4×Dic3—312no—
26C22×C4⋊Dic3—31no—
27C22×C4×Dic3—31no—
28C23×C3×D3—41no—
29C22×C3×A4C2×C414yescomplemented
30C22×C3×A4C2313yes—
31C2×C3×S4C2×C4124yescomplemented
32C2×C3×S4C2314yes—
33C22×C3⋊Dic3—41no—
34C22×C3×Dic3—41no—
35C2×D3×Dic3—412no—
36Dic3×A4C2314yescomplemented
37C3⋊SL(2,ℤ/4)C2314yescomplemented
38C22×C4×D4—91noSylow
39C25×C3—31no—
40C25×C3C2×D311yescharacteristic
41C23×A4—21no—
42C23×A4C2×D311yescharacteristic
43C22×S4Dic318yesdirect product factor
44C22×S4C2×D316yes—
45C22×C3×D4—38no—
46C22×C3×D4—36no—
47C22×C3×D4—38no—
48C22×C3×D4—36no—
49C23×C3×C4—31no—
50C23×Dic3—31no—
51C23×Dic3—31no—
52C23×Dic3—36no—
53C23×Dic3—31no—
54C23×Dic3—312no—
55C23×Dic3—31no—
56C23×Dic3—61no—
57C23×Dic3—66no—
58C23×Dic3—61no—
59C23×Dic3C2×D316yescomplemented
60C22×C4×D3—121no—
61C2×C4×A4—36no—
62C2×SL(2,ℤ/4)—36no—
63C2×SL(2,ℤ/4)—66no—
64C4×S4—316no—
65C2×#217—312no—
66C2×#217—36no—
67C2×#217—36no—
68D4×Dic3—348no—
69D4×Dic3—316no—
70C2×C4⋊Dic3—312no—
71C2×C4×Dic3—312no—
72C23×C32—41no—
73C22×C3×D3—48no—
74C22×C3×D3—46no—
75C2×C3×A4C22×C416yescomplemented
76C2×C3×A4C2411yescharacteristic
77C3×S4C22×C418yescomplemented
78C2×C3⋊Dic3—46no—
79C2×C3×Dic3—46no—
80D3×Dic3—416no—
81C23×D4—31no—
82C24×C4—31no—
83C24×C4—91no—
84C2×C4×D4—912no—
85C2×C4×D4—912no—
86C22×C4⋊C4—91no—
87C22×C22⋊C4—91no—
88C22×C22⋊C4—91no—
89C22×C42—91no—
90C24×C3—31no—
91C24×C3—33no—
92C24×C3—34no—
93C24×C3—36no—
94C24×C3—38no—
95C24×C3—64no—
96C24×C3—63no—
97C24×C3—61no—
98C24×C3C4×D314yescomplemented
99C24×C3C22×D313yes—
100C23×D3—41no—
101C22×A4—24no—
102C22×A4—23no—
103C22×A4C2×Dic314yescomplemented
104C22×A4C22×D313yes—
105C2×S4C2×Dic3124yesdirect product factor
106C2×S4C22×D314yes—
107C2×C3×D4—312no—
108C2×C3×D4—348no—
109C2×C3×D4—324no—
110C2×C3×D4—324no—
111C2×C3×D4—34no—
112C22×C3×C4—36no—
113C22×C3×C4—38no—
114C22×C3×C4—121no—
115C22×Dic3—31no—
116C22×Dic3—36no—
117C22×Dic3—36no—
118C22×Dic3—36no—
119C22×Dic3—312no—
120C22×Dic3—36no—
121C22×Dic3—36no—
122C22×Dic3—316no—
123C22×Dic3—64no—
124C22×Dic3—63no—
125C22×Dic3—612no—
126C22×Dic3—624no—
127C22×Dic3—64no—
128C22×Dic3—63no—
129C22×Dic3—61no—
130C22×Dic3—612no—
131C22×Dic3—121no—
132C22×Dic3—241no—
133C22×Dic3S411yescharacteristic, direct product factor
134C22×Dic3C22×D314yescomplemented
135C2×C4×D3—1212no—
136C4×A4—34no—
137SL(2,ℤ/4)—34no—
138SL(2,ℤ/4)—64no—
139#217—316no—
140#217—312no—
141#217—34no—
142C4⋊Dic3—316no—
143C4×Dic3—316no—
144C22×C32—44no—
145C22×C32—43no—
146C2×C3×D3—424no—
147C2×C3×D3—44no—
148C3×A4C23×C411yescharacteristic, complemented
149C3⋊Dic3—44no—
150C3×Dic3—44no—
151C25—31no—
152C25D3211yescharacteristic
153C22×D4—36no—
154C22×D4—38no—
155C22×D4—38no—
156C22×D4—36no—
157C23×C4—31no—
158C23×C4—36no—
159C23×C4—91no—
160C23×C4—91no—
161C23×C4—96no—
162C23×C4—91no—
163C23×C4—91no—
164C23×C4—912no—
165C23×C4—181no—
166C23×C4—186no—
167C23×C4—181no—
168C4×D4—916no—
169C4×D4—948no—
170C2×C4⋊C4—912no—
171C2×C22⋊C4—912no—
172C2×C22⋊C4—96no—
173C2×C22⋊C4—96no—
174C2×C42—912no—
175C23×C3—34no—
176C23×C3—34no—
177C23×C3—33no—
178C23×C3—33no—
179C23×C3—33no—
180C23×C3—312no—
181C23×C3—36no—
182C23×C3—34no—
183C23×C3—324no—
184C23×C3—41no—
185C23×C3—61no—
186C23×C3—66no—
187C23×C3—66no—
188C23×C3—612no—
189C23×C3—624no—
190C23×C3—68no—
191C23×C3—66no—
192C23×C3—81no—
193C23×C3C2×C4×D316yescomplemented
194C23×C3C2×S411yescharacteristic
195C23×C3C23×D311yescharacteristic
196C22×D3—46no—
197C22×D3—48no—
198C2×A4—21no—
199C2×A4—26no—
200C2×A4C22×Dic316yescomplemented
201C2×A4C23×D311yescharacteristic
202S4C22×Dic318yesdirect product factor
203C3×D4—348no—
204C3×D4—38no—
205C3×D4—38no—
206C2×C3×C4—34no—
207C2×C3×C4—324no—
208C2×C3×C4—126no—
209C2×Dic3—36no—
210C2×Dic3—34no—
211C2×Dic3—312no—
212C2×Dic3—34no—
213C2×Dic3—34no—
214C2×Dic3—34no—
215C2×Dic3—66no—
216C2×Dic3—616no—
217C2×Dic3—612no—
218C2×Dic3—66no—
219C2×Dic3—66no—
220C2×Dic3—616no—
221C2×Dic3—126no—
222C2×Dic3—246no—
223C2×Dic3C2×S416yesdirect product factor
224C4×D3—1216no—
225C2×C32—46no—
226C2×C32—41no—
227C3×D3—48no—
228C24—31no—
229C24—33no—
230C24—34no—
231C24—36no—
232C24—38no—
233C24—63no—
234C24—64no—
235C24—61no—
236C24D3×Dic314yescomplemented
237C24C2×D3213yes—
238C2×D4—312no—
239C2×D4—348no—
240C2×D4—324no—
241C2×D4—34no—
242C2×D4—324no—
243C22×C4—36no—
244C22×C4—38no—
245C22×C4—31no—
246C22×C4—34no—
247C22×C4—96no—
248C22×C4—96no—
249C22×C4—91no—
250C22×C4—96no—
251C22×C4—912no—
252C22×C4—96no—
253C22×C4—96no—
254C22×C4—916no—
255C22×C4—1812no—
256C22×C4—184no—
257C22×C4—183no—
258C22×C4—1824no—
259C22×C4—1812no—
260C22×C4—183no—
261C22×C4—184no—
262C22×C4—181no—
263C4⋊C4—916no—
264C22⋊C4—916no—
265C22⋊C4—912no—
266C22⋊C4—94no—
267C42—916no—
268C22×C3—36no—
269C22×C3—31no—
270C22×C3—36no—
271C22×C3—36no—
272C22×C3—312no—
273C22×C3—31no—
274C22×C3—33no—
275C22×C3—38no—
276C22×C3—43no—
277C22×C3—44no—
278C22×C3—63no—
279C22×C3—64no—
280C22×C3—64no—
281C22×C3—624no—
282C22×C3—624no—
283C22×C3—64no—
284C22×C3—83no—
285C22×C3—84no—
286C22×C3C4×S414yescomplemented
287C22×C3C22×C4×D311yescharacteristic, complemented
288C22×C3C22×S413yes—
289C2×D3—44no—
290C2×D3—424no—
291A4—21no—
292A4C23×Dic311yescharacteristic, complemented
293C3×C4—38no—
294C3×C4—124no—
295Dic3—34no—
296Dic3—64no—
297Dic3—64no—
298Dic3—124no—
299Dic3—244no—
300Dic3C22×S414yesdirect product factor
301C32—41noSylow
302C23—33no—
303C23—33no—
304C23—34no—
305C23—34no—
306C23—33no—
307C23—312no—
308C23—36no—
309C23—34no—
310C23—324no—
311C23—61no—
312C23—66no—
313C23—66no—
314C23—612no—
315C23—624no—
316C23—66no—
317C23—68no—
318C23C2×D3×Dic316yescomplemented
319C23D3×S411yescharacteristic
320C23C22×D3211yescharacteristic
321D4—38no—
322D4—348no—
323D4—38no—
324C2×C4—34no—
325C2×C4—324no—
326C2×C4—36no—
327C2×C4—94no—
328C2×C4—912no—
329C2×C4—96no—
330C2×C4—94no—
331C2×C4—94no—
332C2×C4—94no—
333C2×C4—1816no—
334C2×C4—186no—
335C2×C4—1816no—
336C2×C4—186no—
337C2×C4—1812no—
338C2×C4—186no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—36no—
342C2×C3—46no—
343C2×C3—41no—
344C2×C3—68no—
345C2×C3—86no—
346C2×C3—81no—
347C2×C3C2×C4×S416yescomplemented
348C2×C3C23×S411yescharacteristic
349D3—48no—
350C22—31no—
351C22—36no—
352C22—36no—
353C22—36no—
354C22—312no—
355C22—31no—
356C22—33no—
357C22—38no—
358C22—63no—
359C22—64no—
360C22—64no—
361C22—624no—
362C22—624no—
363C22—64no—
364C22Dic3×S414yesdirect product factor
365C22C22×D3×Dic311yescharacteristic, complemented
366C22C2×D3×S413yes—
367C4—38no—
368C4—34no—
369C4—94no—
370C4—184no—
371C4—184no—
372C3—41no—
373C3—81no—
374C3C22×C4×S411yescharacteristic, complemented
375C2—31no—
376C2—31no—
377C2—36no—
378C2—68no—
379C2C2×Dic3×S416yesdirect product factor
380C2C22×D3×S411yescharacteristic
subgroupG/H
center, hypercenterC23D3×S4
commutator, hypocenterC3×A4C23×C4
FrattiniC2C22×D3×S4
Fitting, socleC25×C3C2×D3
prime radicalC23×C3×S4C2
cosocleC2×C3×A4C24
LCSG ▷ C3×A4
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C22×Dic3×A4C2semidirect
3. C22×C3⋊SL(2,ℤ/4)C2semidirect
4. C2×Dic3×S4C2direct (N × H)
8. C22×C3×S4C4semidirect
11. C2×Dic3×A4C22semidirect
12. C2×C3⋊SL(2,ℤ/4)C22semidirect
13. Dic3×S4C22direct (N × H)
17. C24×Dic3D3semidirect
29. C22×C3×A4C2×C4semidirect
31. C2×C3×S4C2×C4semidirect
36. Dic3×A4C23semidirect
37. C3⋊SL(2,ℤ/4)C23semidirect
43. C22×S4Dic3direct (N × H)
59. C23×Dic3C2×D3semidirect
75. C2×C3×A4C22×C4semidirect
77. C3×S4C22×C4semidirect
98. C24×C3C4×D3semidirect
103. C22×A4C2×Dic3semidirect
105. C2×S4C2×Dic3direct (N × H)
133. C22×Dic3S4direct (N × H)
134. C22×Dic3C22×D3semidirect
148. C3×A4C23×C4semidirect
193. C23×C3C2×C4×D3semidirect
200. C2×A4C22×Dic3semidirect
202. S4C22×Dic3direct (N × H)
223. C2×Dic3C2×S4direct (N × H)
236. C24D3×Dic3semidirect
286. C22×C3C4×S4semidirect
287. C22×C3C22×C4×D3semidirect
292. A4C23×Dic3semidirect
300. Dic3C22×S4direct (N × H)
318. C23C2×D3×Dic3semidirect
347. C2×C3C2×C4×S4semidirect
364. C22Dic3×S4direct (N × H)
365. C22C22×D3×Dic3semidirect
374. C3C22×C4×S4semidirect
379. C2C2×Dic3×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×S42
C22×Dic3×A42
C22×C3⋊SL(2,ℤ/4)2
C2×Dic3×S42
C22×C4×S43
C22×D4×Dic33
C22×D3×Dic34