Finite Group Databasebeta

← C2×C4×C3⋊(C3⋊S4) #13181394

Subgroups — C2×C4×C3⋊(C3⋊S4) #13181394

Order 1728.

2262 conjugacy classes of subgroups; only the 291 normal subgroups are shown.

normal subgroups only

This group has 2262 conjugacy classes of subgroups — too many to draw. Only its 291 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×C3⋊(C3⋊S4)C211yesmaximal, characteristic
2C2×C32×C4×A4C211yesmaximal, characteristic, complemented
3C2×C3⋊(C3⋊SL(2,ℤ/4))C211yesmaximal, characteristic, complemented
4C4×C3⋊(C3⋊S4)C214yesmaximal, direct product factor
5C2×C4×C3⋊S4—312nomaximal
6C2×C4×#638—31nomaximal
7C22×C32×A4C2211yescharacteristic
8C2×C3⋊(C3⋊S4)C414yesdirect product factor
9C2×C3⋊(C3⋊S4)C2212yes—
10C2×C4×C3⋊(C3⋊D3)—41nomaximal
11C32×C4×A4C2212yescomplemented
12C3⋊(C3⋊SL(2,ℤ/4))C2212yescomplemented
13C22×C3⋊S4—312no—
14C23×C32×C4D311yescharacteristic, complemented
15C22×#638—31no—
16C22×C4×C3⋊D3—31no—
17C2×C3×C4×A4D3112yescomplemented
18C2×C3⋊SL(2,ℤ/4)—312no—
19C4×C3⋊S4—348no—
20C2×#3757—31no—
21C4×#638—34no—
22C4×#638—34no—
23C2×#3752—31no—
24C2×#3750—31no—
25C2×C4×C3⋊Dic3—31no—
26C22×C3⋊(C3⋊D3)—41no—
27C2×C32×A4C2×C412yescomplemented
28C2×C32×A4C2311yescharacteristic
29C3⋊(C3⋊S4)C2×C414yesdirect product factor
30C2×C33×C4—41no—
31C2×C3⋊(C3⋊Dic3)—41no—
32C4×C3⋊(C3⋊D3)—44no—
33C2×C4×S4—99no—
34C2×C4×#67—94no—
35C24×C32C2×D311yescharacteristic
36C23×C3⋊D3—31no—
37C22×C3×A4C2×D3112yes—
38C2×C3⋊S4—348no—
39C2×C3⋊S4—324no—
40C22×C32×C4—31no—
41C22×C32×C4—31no—
42C22×C32×C4—32no—
43C22×C32×C4C2×D312yescomplemented
44C2×#638—34no—
45C2×#638—32no—
46C2×#638—34no—
47C2×#638—32no—
48C22×C3⋊Dic3—31no—
49C22×C3⋊Dic3—31no—
50C22×C3⋊Dic3—31no—
51C2×C4×C3⋊D3—34no—
52C2×C4×C3⋊D3—61no—
53C2×C4×C3⋊D3—62no—
54C2×C4×C3⋊D3—61no—
55C2×C4×C3⋊D3—1212no—
56C3×C4×A4C2×D3124yescomplemented
57C3⋊SL(2,ℤ/4)—324no—
58#3757—32no—
59#3757—32no—
60#3752—34no—
61#3750—34no—
62C4×C3⋊Dic3—34no—
63C22×C33—41no—
64C2×C3⋊(C3⋊D3)—44no—
65C2×C3⋊(C3⋊D3)—42no—
66C32×A4C22×C411yescharacteristic, complemented
67C33×C4—42no—
68C3⋊(C3⋊Dic3)—42no—
69C22×S4—99no—
70C23×C3×C4C3⋊D314yescomplemented
71C22×#67—94no—
72C22×C4×D3—94no—
73C2×C4×A4C3⋊D319yescomplemented
74C2×SL(2,ℤ/4)—99no—
75C4×S4—936no—
76C2×#217—94no—
77C4×#67—916no—
78C4×#67—916no—
79C2×(C2×D3)⋊C4—94no—
80C2×Dic3⋊C4—94no—
81C2×C4×Dic3—94no—
82C23×C32—31no—
83C23×C32—31no—
84C23×C32—32no—
85C23×C32C4×D312yescomplemented
86C23×C32C22×D311yescharacteristic
87C22×C3⋊D3—32no—
88C22×C3⋊D3—34no—
89C22×C3⋊D3—62no—
90C22×C3⋊D3—61no—
91C22×C3⋊D3—61no—
92C22×C3⋊D3—1212no—
93C2×C3×A4C4×D3124yescomplemented
94C2×C3×A4C22×D3112yes—
95C3⋊S4—348no—
96C2×C32×C4—31no—
97C2×C32×C4—32no—
98C2×C32×C4—32no—
99C2×C32×C4—32no—
100C2×C32×C4—412no—
101C2×C32×C4—61no—
102C2×C32×C4S411yescharacteristic, complemented
103#638—38no—
104#638—34no—
105#638—34no—
106C2×C3⋊Dic3—32no—
107C2×C3⋊Dic3—34no—
108C2×C3⋊Dic3—32no—
109C2×C3⋊Dic3—32no—
110C2×C3⋊Dic3—61no—
111C2×C3⋊Dic3—62no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—1212no—
114C4×C3⋊D3—64no—
115C4×C3⋊D3—64no—
116C4×C3⋊D3—1248no—
117C2×C4×D4—271noSylow
118C2×C33—42no—
119C2×C33—41no—
120C3⋊(C3⋊D3)—44no—
121C24×C3C2×C3⋊D314yes—
122C23×D3—94no—
123C22×A4C2×C3⋊D319yes—
124C2×S4—936no—
125C2×S4—918no—
126C22×C3×C4—34no—
127C22×C3×C4—34no—
128C22×C3×C4—38no—
129C22×C3×C4C2×C3⋊D318yescomplemented
130C2×#67—98no—
131C2×#67—916no—
132C2×#67—916no—
133C2×#67—98no—
134C22×Dic3—94no—
135C22×Dic3—94no—
136C22×Dic3—94no—
137C2×C4×D3—916no—
138C2×C4×D3—184no—
139C2×C4×D3—188no—
140C2×C4×D3—184no—
141C2×C4×D3—369no—
142C4×A4C2×C3⋊D3118yescomplemented
143SL(2,ℤ/4)—918no—
144#217—98no—
145#217—98no—
146(C2×D3)⋊C4—916no—
147Dic3⋊C4—916no—
148C4×Dic3—916no—
149C22×C32—32no—
150C22×C32—31no—
151C22×C32—32no—
152C22×C32—32no—
153C22×C32—31no—
154C22×C32—31no—
155C22×C32—412no—
156C22×C32—61no—
157C22×C32C2×C4×D311yescharacteristic, complemented
158C22×C32C2×S411yescharacteristic
159C2×C3⋊D3—34no—
160C2×C3⋊D3—62no—
161C2×C3⋊D3—64no—
162C2×C3⋊D3—64no—
163C2×C3⋊D3—62no—
164C2×C3⋊D3—1224no—
165C2×C3⋊D3—1248no—
166C3×A4C2×C4×D3112yescomplemented
167C32×C4—32no—
168C32×C4—424no—
169C32×C4C2×S412yescomplemented
170C3⋊Dic3—34no—
171C3⋊Dic3—62no—
172C3⋊Dic3—62no—
173C3⋊Dic3—1224no—
174C22×D4—271no—
175C23×C4—271no—
176C23×C4C3⋊(C3⋊D3)11yescharacteristic, complemented
177C4×D4—274no—
178C4×D4—274no—
179C2×C4⋊C4—271no—
180C2×C22⋊C4—271no—
181C2×C22⋊C4—271no—
182C2×C42—271no—
183C33—41noSylow
184C23×C3—34no—
185C23×C3—34no—
186C23×C3—38no—
187C23×C3C4×C3⋊D318yescomplemented
188C23×C3C22×C3⋊D314yes—
189C22×D3—98no—
190C22×D3—916no—
191C22×D3—184no—
192C22×D3—188no—
193C22×D3—184no—
194C22×D3—369no—
195C2×A4C4×C3⋊D3118yescomplemented
196C2×A4C22×C3⋊D319yes—
197S4—936no—
198C2×C3×C4—34no—
199C2×C3×C4—38no—
200C2×C3×C4—38no—
201C2×C3×C4—38no—
202C2×C3×C4—49no—
203C2×C3×C4—64no—
204C2×C3×C4C3⋊S414yescomplemented
205#67—932no—
206#67—916no—
207#67—916no—
208C2×Dic3—98no—
209C2×Dic3—916no—
210C2×Dic3—98no—
211C2×Dic3—98no—
212C2×Dic3—184no—
213C2×Dic3—188no—
214C2×Dic3—184no—
215C2×Dic3—369no—
216C4×D3—1816no—
217C4×D3—1816no—
218C4×D3—3636no—
219C2×C32—31no—
220C2×C32—31no—
221C2×C32—32no—
222C2×C32—424no—
223C2×C32—412no—
224C2×C32C4×S412yescomplemented
225C2×C32C22×S411yescharacteristic
226C3⋊D3—64no—
227C3⋊D3—1248no—
228C24—271no—
229C24C2×C3⋊(C3⋊D3)11yescharacteristic
230C2×D4—272no—
231C2×D4—274no—
232C2×D4—274no—
233C2×D4—272no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—32no—
237C22×C4—274no—
238C22×C4—271no—
239C22×C4—271no—
240C22×C4—271no—
241C22×C4—541no—
242C22×C4—542no—
243C22×C4—541no—
244C22×C4C2×C3⋊(C3⋊D3)12yescomplemented
245C4⋊C4—274no—
246C22⋊C4—274no—
247C22⋊C4—272no—
248C22⋊C4—272no—
249C42—274no—
250C22×C3—38no—
251C22×C3—34no—
252C22×C3—38no—
253C22×C3—38no—
254C22×C3—34no—
255C22×C3—34no—
256C22×C3—49no—
257C22×C3—64no—
258C22×C3C2×C4×C3⋊D314yescomplemented
259C22×C3C2×C3⋊S414yes—
260C2×D3—916no—
261C2×D3—188no—
262C2×D3—1816no—
263C2×D3—1816no—
264C2×D3—188no—
265C2×D3—3618no—
266C2×D3—3636no—
267A4C2×C4×C3⋊D319yescomplemented
268C3×C4—38no—
269C3×C4—418no—
270C3×C4C2×C3⋊S418yescomplemented
271Dic3—916no—
272Dic3—188no—
273Dic3—188no—
274Dic3—3618no—
275C32—412no—
276C32C2×C4×S411yescharacteristic, complemented
277C23—31no—
278C23—31no—
279C23—32no—
280C23—272no—
281C23—274no—
282C23—541no—
283C23—542no—
284C23—541no—
285C23C4×C3⋊(C3⋊D3)12yescomplemented
286C23C22×C3⋊(C3⋊D3)11yescharacteristic
287D4—278no—
288D4—274no—
289D4—274no—
290C2×C4—31no—
291C2×C4—32no—
292C2×C4—32no—
293C2×C4—32no—
294C2×C4—61no—
295C2×C4—272no—
296C2×C4—274no—
297C2×C4—272no—
298C2×C4—272no—
299C2×C4—544no—
300C2×C4—541no—
301C2×C4—542no—
302C2×C4—541no—
303C2×C4—544no—
304C2×C4C3⋊(C3⋊S4)11yescharacteristic, direct product factor
305C2×C3—34no—
306C2×C3—34no—
307C2×C3—38no—
308C2×C3—418no—
309C2×C3—49no—
310C2×C3C4×C3⋊S418yescomplemented
311C2×C3C22×C3⋊S414yes—
312D3—1816no—
313D3—3636no—
314C22—32no—
315C22—31no—
316C22—32no—
317C22—32no—
318C22—31no—
319C22—31no—
320C22—61no—
321C22—274no—
322C22—542no—
323C22—544no—
324C22—544no—
325C22—542no—
326C22C2×C4×C3⋊(C3⋊D3)11yescharacteristic, complemented
327C22C2×C3⋊(C3⋊S4)11yescharacteristic
328C4—32no—
329C4—274no—
330C4—542no—
331C4—542no—
332C4C2×C3⋊(C3⋊S4)12yesdirect product factor
333C3—49no—
334C3C2×C4×C3⋊S414yescomplemented
335C2—31no—
336C2—31no—
337C2—32no—
338C2—544no—
339C2C4×C3⋊(C3⋊S4)12yesdirect product factor
340C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
center, hypercenterC2×C4C3⋊(C3⋊S4)
commutator, hypocenterC32×A4C22×C4
FrattiniC2C22×C3⋊(C3⋊S4)
FittingC23×C32×C4D3
socleC24×C32C2×D3
prime radicalC22×C3⋊(C3⋊S4)C2
cosocleC2×C32×A4C23
LCSG ▷ C32×A4
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C32×C4×A4C2semidirect
3. C2×C3⋊(C3⋊SL(2,ℤ/4))C2semidirect
4. C4×C3⋊(C3⋊S4)C2direct (N × H)
8. C2×C3⋊(C3⋊S4)C4direct (N × H)
11. C32×C4×A4C22semidirect
12. C3⋊(C3⋊SL(2,ℤ/4))C22semidirect
14. C23×C32×C4D3semidirect
17. C2×C3×C4×A4D3semidirect
27. C2×C32×A4C2×C4semidirect
29. C3⋊(C3⋊S4)C2×C4direct (N × H)
43. C22×C32×C4C2×D3semidirect
56. C3×C4×A4C2×D3semidirect
66. C32×A4C22×C4semidirect
70. C23×C3×C4C3⋊D3semidirect
73. C2×C4×A4C3⋊D3semidirect
85. C23×C32C4×D3semidirect
93. C2×C3×A4C4×D3semidirect
102. C2×C32×C4S4semidirect
129. C22×C3×C4C2×C3⋊D3semidirect
142. C4×A4C2×C3⋊D3semidirect
157. C22×C32C2×C4×D3semidirect
166. C3×A4C2×C4×D3semidirect
169. C32×C4C2×S4semidirect
176. C23×C4C3⋊(C3⋊D3)semidirect
187. C23×C3C4×C3⋊D3semidirect
195. C2×A4C4×C3⋊D3semidirect
204. C2×C3×C4C3⋊S4semidirect
224. C2×C32C4×S4semidirect
244. C22×C4C2×C3⋊(C3⋊D3)semidirect
258. C22×C3C2×C4×C3⋊D3semidirect
267. A4C2×C4×C3⋊D3semidirect
270. C3×C4C2×C3⋊S4semidirect
276. C32C2×C4×S4semidirect
285. C23C4×C3⋊(C3⋊D3)semidirect
304. C2×C4C3⋊(C3⋊S4)direct (N × H)
310. C2×C3C4×C3⋊S4semidirect
326. C22C2×C4×C3⋊(C3⋊D3)semidirect
332. C4C2×C3⋊(C3⋊S4)direct (N × H)
334. C3C2×C4×C3⋊S4semidirect
339. C2C4×C3⋊(C3⋊S4)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊(C3⋊S4)2
C2×C32×C4×A42
C2×C3⋊(C3⋊SL(2,ℤ/4))2
C4×C3⋊(C3⋊S4)2
C2×C4×C3⋊S43
C2×C4×#6383
C2×C4×C3⋊(C3⋊D3)4