Finite Group Databasebeta

← C22×(C3⋊S4)⋊C4 #11923620

Subgroups — C22×(C3⋊S4)⋊C4 #11923620

Order 1152.

2716 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 2716 conjugacy classes of subgroups — too many to draw. Only its 231 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×(C3⋊S4)⋊C4C2112yesmaximal, direct product factor
5C22×C4×S4—31nomaximal
6C22×#67⋊C4—31nomaximal
7C23×C3×A4C2211yescharacteristic
8C22×C3⋊S4C418yescomplemented
9C22×C3⋊S4C2216yes—
10C22×#624—41nomaximal
11C2×Dic3×A4C2216yescomplemented
12C2×C3×SL(2,ℤ/4)C2216yescomplemented
13(C3⋊S4)⋊C4C22116yesdirect product factor
14C23×S4—31no—
15C23×S4—61no—
16C23×#67—31no—
17C24×Dic3D311yescharacteristic, complemented
18C23×C4×D3—31no—
19C22×C4×A4—31no—
20C22×SL(2,ℤ/4)D311yescharacteristic, complemented
21C2×C4×S4—312no—
22C22×C3×C22⋊C4—31no—
23C22×(C2×D3)⋊C4—31no—
24C22×Dic3⋊C4—31no—
25C22×C4×Dic3—31no—
26C2×#67⋊C4—312no—
27C2×#67⋊C4—312no—
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×#624—412no—
36Dic3×A4C2314yescomplemented
37C3×SL(2,ℤ/4)C2314yescomplemented
38C22×C4×D4—91noSylow
39C25×C3C2×D311yescharacteristic
40C24×D3—31no—
41C23×A4—21no—
42C23×A4C2×D311yescharacteristic
43C22×S4—38no—
44C22×S4—36no—
45C22×S4—68no—
46C22×S4—66no—
47C23×C3×C4—31no—
48C23×C3×C4—31no—
49C22×#67—38no—
50C22×#67—36no—
51C22×#67—38no—
52C22×#67—36no—
53C23×Dic3—31no—
54C23×Dic3—31no—
55C23×Dic3—36no—
56C23×Dic3—31no—
57C23×Dic3C2×D316yescomplemented
58C22×C4×D3—312no—
59C22×C4×D3—61no—
60C22×C4×D3—66no—
61C22×C4×D3—61no—
62C22×C4×D3—121no—
63C2×C4×A4—36no—
64C2×SL(2,ℤ/4)C2×D316yescomplemented
65C4×S4—316no—
66C2×C3×C22⋊C4—36no—
67C2×C3×C22⋊C4—36no—
68C2×(C2×D3)⋊C4—312no—
69C2×Dic3⋊C4—312no—
70C2×C4×Dic3—312no—
71#67⋊C4—348no—
72#67⋊C4—316no—
73C23×C32—41no—
74C22×C3⋊D3—48no—
75C22×C3⋊D3—46no—
76C2×C3×A4C22×C416yescomplemented
77C2×C3×A4C2411yescharacteristic
78C3⋊S4C22×C418yescomplemented
79C2×C3×Dic3—46no—
80C2×C3×Dic3—46no—
81#624—416no—
82C23×D4—91no—
83C24×C4—31no—
84C24×C4—91no—
85C2×C4×D4—912no—
86C2×C4×D4—912no—
87C22×C4⋊C4—91no—
88C22×C22⋊C4—31no—
89C22×C22⋊C4—91no—
90C22×C42—91no—
91C24×C3—31no—
92C24×C3—33no—
93C24×C3—34no—
94C24×C3C4×D314yescomplemented
95C24×C3C22×D313yes—
96C23×D3—36no—
97C23×D3—38no—
98C23×D3—63no—
99C23×D3—64no—
100C23×D3—61no—
101C23×D3—121no—
102C23×D3—241no—
103C22×A4—24no—
104C22×A4—23no—
105C22×A4C4×D314yescomplemented
106C22×A4C22×D313yes—
107C2×S4—324no—
108C2×S4—34no—
109C2×S4—624no—
110C2×S4—64no—
111C22×C3×C4—36no—
112C22×C3×C4—36no—
113C22×C3×C4—63no—
114C22×C3×C4—64no—
115C22×C3×C4—61no—
116C22×C3×C4—121no—
117C2×#67—312no—
118C2×#67—348no—
119C2×#67—324no—
120C2×#67—324no—
121C2×#67—34no—
122C22×Dic3—36no—
123C22×Dic3—38no—
124C22×Dic3—31no—
125C22×Dic3—36no—
126C22×Dic3—36no—
127C22×Dic3—312no—
128C22×Dic3—36no—
129C22×Dic3—41no—
130C22×Dic3—64no—
131C22×Dic3—63no—
132C22×Dic3S411yescharacteristic, complemented
133C22×Dic3C22×D314yescomplemented
134C2×C4×D3—316no—
135C2×C4×D3—612no—
136C2×C4×D3—624no—
137C2×C4×D3—612no—
138C2×C4×D3—1212no—
139C4×A4—34no—
140SL(2,ℤ/4)C22×D314yescomplemented
141C3×C22⋊C4—312no—
142C3×C22⋊C4—34no—
143(C2×D3)⋊C4—316no—
144Dic3⋊C4—316no—
145C4×Dic3—316no—
146C22×C32—44no—
147C22×C32—43no—
148C2×C3⋊D3—424no—
149C2×C3⋊D3—44no—
150C3×A4C23×C411yescharacteristic, complemented
151C3×Dic3—44no—
152C3×Dic3—44no—
153C25—91no—
154C25D3211yescharacteristic
155C22×D4—96no—
156C22×D4—98no—
157C22×D4—98no—
158C22×D4—96no—
159C23×C4—31no—
160C23×C4—31no—
161C23×C4—36no—
162C23×C4—91no—
163C23×C4—912no—
164C23×C4—91no—
165C23×C4—91no—
166C23×C4—96no—
167C23×C4—181no—
168C23×C4—186no—
169C23×C4—181no—
170C4×D4—916no—
171C4×D4—948no—
172C2×C4⋊C4—912no—
173C2×C22⋊C4—36no—
174C2×C22⋊C4—36no—
175C2×C22⋊C4—912no—
176C2×C42—912no—
177C23×C3—34no—
178C23×C3—34no—
179C23×C3—33no—
180C23×C3—33no—
181C23×C3—33no—
182C23×C3—312no—
183C23×C3—36no—
184C23×C3—41no—
185C23×C3—61no—
186C23×C3—66no—
187C23×C3—81no—
188C23×C3C2×C4×D316yescomplemented
189C23×C3C2×S411yescharacteristic
190C23×C3C23×D311yescharacteristic
191C22×D3—34no—
192C22×D3—324no—
193C22×D3—66no—
194C22×D3—612no—
195C22×D3—624no—
196C22×D3—68no—
197C22×D3—66no—
198C22×D3—126no—
199C22×D3—128no—
200C22×D3—246no—
201C22×D3—248no—
202C2×A4—21no—
203C2×A4—26no—
204C2×A4C2×C4×D316yescomplemented
205C2×A4C23×D311yescharacteristic
206S4—38no—
207S4—68no—
208C2×C3×C4—34no—
209C2×C3×C4—34no—
210C2×C3×C4—66no—
211C2×C3×C4—612no—
212C2×C3×C4—66no—
213C2×C3×C4—126no—
214#67—348no—
215#67—38no—
216#67—38no—
217C2×Dic3—34no—
218C2×Dic3—324no—
219C2×Dic3—36no—
220C2×Dic3—34no—
221C2×Dic3—312no—
222C2×Dic3—34no—
223C2×Dic3—46no—
224C2×Dic3—66no—
225C2×Dic3C2×S416yescomplemented
226C4×D3—616no—
227C4×D3—616no—
228C4×D3—1216no—
229C2×C32—46no—
230C2×C32—41no—
231C3⋊D3—48no—
232C24—31no—
233C24—33no—
234C24—34no—
235C24—96no—
236C24—98no—
237C24—183no—
238C24—184no—
239C24—181no—
240C24#62414yescomplemented
241C24C2×D3213yes—
242C2×D4—912no—
243C2×D4—948no—
244C2×D4—924no—
245C2×D4—94no—
246C2×D4—924no—
247C22×C4—31no—
248C22×C4—36no—
249C22×C4—36no—
250C22×C4—34no—
251C22×C4—63no—
252C22×C4—64no—
253C22×C4—61no—
254C22×C4—96no—
255C22×C4—96no—
256C22×C4—916no—
257C22×C4—96no—
258C22×C4—98no—
259C22×C4—91no—
260C22×C4—96no—
261C22×C4—912no—
262C22×C4—1812no—
263C22×C4—1824no—
264C22×C4—1812no—
265C22×C4—184no—
266C22×C4—183no—
267C4⋊C4—916no—
268C22⋊C4—312no—
269C22⋊C4—34no—
270C22⋊C4—916no—
271C42—916no—
272C22×C3—36no—
273C22×C3—31no—
274C22×C3—36no—
275C22×C3—36no—
276C22×C3—312no—
277C22×C3—31no—
278C22×C3—33no—
279C22×C3—43no—
280C22×C3—44no—
281C22×C3—63no—
282C22×C3—64no—
283C22×C3—83no—
284C22×C3—84no—
285C22×C3C4×S414yescomplemented
286C22×C3C22×C4×D311yescharacteristic, complemented
287C22×C3C22×S413yes—
288C2×D3—38no—
289C2×D3—64no—
290C2×D3—624no—
291C2×D3—624no—
292C2×D3—64no—
293C2×D3—124no—
294C2×D3—1224no—
295C2×D3—2424no—
296C2×D3—244no—
297A4—21no—
298A4C22×C4×D311yescharacteristic, complemented
299C3×C4—64no—
300C3×C4—64no—
301C3×C4—124no—
302Dic3—38no—
303Dic3—34no—
304Dic3—44no—
305Dic3C22×S414yescomplemented
306C32—41noSylow
307C23—33no—
308C23—33no—
309C23—34no—
310C23—34no—
311C23—33no—
312C23—312no—
313C23—36no—
314C23—61no—
315C23—66no—
316C23—94no—
317C23—924no—
318C23—186no—
319C23—1812no—
320C23—1824no—
321C23—186no—
322C23—188no—
323C23C2×#62416yescomplemented
324C23D3×S411yescharacteristic
325C23C22×D3211yescharacteristic
326D4—98no—
327D4—948no—
328D4—98no—
329C2×C4—36no—
330C2×C4—34no—
331C2×C4—34no—
332C2×C4—66no—
333C2×C4—612no—
334C2×C4—66no—
335C2×C4—94no—
336C2×C4—912no—
337C2×C4—94no—
338C2×C4—924no—
339C2×C4—96no—
340C2×C4—94no—
341C2×C4—1816no—
342C2×C4—1816no—
343C2×C4—186no—
344C2×C3—31no—
345C2×C3—31no—
346C2×C3—36no—
347C2×C3—46no—
348C2×C3—41no—
349C2×C3—86no—
350C2×C3—81no—
351C2×C3C2×C4×S416yescomplemented
352C2×C3C23×S411yescharacteristic
353D3—68no—
354D3—128no—
355D3—248no—
356C22—31no—
357C22—36no—
358C22—36no—
359C22—36no—
360C22—312no—
361C22—31no—
362C22—33no—
363C22—63no—
364C22—64no—
365C22—98no—
366C22—184no—
367C22—1824no—
368C22—1824no—
369C22—184no—
370C22(C3⋊S4)⋊C414yesdirect product factor
371C22C22×#62411yescharacteristic, complemented
372C22C2×D3×S413yes—
373C4—34no—
374C4—64no—
375C4—64no—
376C4—98no—
377C4—94no—
378C3—41no—
379C3—81no—
380C3C22×C4×S411yescharacteristic, complemented
381C2—31no—
382C2—31no—
383C2—36no—
384C2—188no—
385C2C2×(C3⋊S4)⋊C416yesdirect product factor
386C2C22×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×(C3⋊S4)⋊C4C2direct (N × H)
8. C22×C3⋊S4C4semidirect
11. C2×Dic3×A4C22semidirect
12. C2×C3×SL(2,ℤ/4)C22semidirect
13. (C3⋊S4)⋊C4C22direct (N × H)
17. C24×Dic3D3semidirect
20. C22×SL(2,ℤ/4)D3semidirect
29. C22×C3×A4C2×C4semidirect
31. C2×C3⋊S4C2×C4semidirect
36. Dic3×A4C23semidirect
37. C3×SL(2,ℤ/4)C23semidirect
57. C23×Dic3C2×D3semidirect
64. C2×SL(2,ℤ/4)C2×D3semidirect
76. C2×C3×A4C22×C4semidirect
78. C3⋊S4C22×C4semidirect
94. C24×C3C4×D3semidirect
105. C22×A4C4×D3semidirect
132. C22×Dic3S4semidirect
133. C22×Dic3C22×D3semidirect
140. SL(2,ℤ/4)C22×D3semidirect
150. C3×A4C23×C4semidirect
188. C23×C3C2×C4×D3semidirect
204. C2×A4C2×C4×D3semidirect
225. C2×Dic3C2×S4semidirect
240. C24#624semidirect
285. C22×C3C4×S4semidirect
286. C22×C3C22×C4×D3semidirect
298. A4C22×C4×D3semidirect
305. Dic3C22×S4semidirect
323. C23C2×#624semidirect
351. C2×C3C2×C4×S4semidirect
370. C22(C3⋊S4)⋊C4direct (N × H)
371. C22C22×#624semidirect
380. C3C22×C4×S4semidirect
385. C2C2×(C3⋊S4)⋊C4direct (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×(C3⋊S4)⋊C42
C22×C4×S43
C22×#67⋊C43
C22×#6244