Finite Group Databasebeta

← C22×C3×C3⋊SL(2,ℤ/4) #13181391

Subgroups — C22×C3×C3⋊SL(2,ℤ/4) #13181391

Order 1728.

1694 conjugacy classes of subgroups; only the 278 normal subgroups are shown.

normal subgroups only

This group has 1694 conjugacy classes of subgroups — too many to draw. Only its 278 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×C32×A4C211yesmaximal, characteristic
2C2×C3×C3⋊SL(2,ℤ/4)C216yesmaximal, direct product factor
3C22×C3⋊SL(2,ℤ/4)C311yesmaximal, characteristic, direct product factor
4C22×C3×SL(2,ℤ/4)—33nomaximal
5C22×C3×#217—31nomaximal
6C22×C32×A4C414yescomplemented
7C22×C32×A4C2213yes—
8C22×C3×C3⋊Dic3—41nomaximal
9C3×C3⋊SL(2,ℤ/4)C2214yesdirect product factor
10C25×C32D311yescharacteristic
11C23×C3×A4—23no—
12C23×C3×A4—21no—
13C23×C3×A4D313yes—
14C23×C3×A4C2×C311yescharacteristic
15C23×C3×Dic3—31no—
16C23×C3×Dic3—31no—
17C2×C3⋊SL(2,ℤ/4)C2×C316yesdirect product factor
18C2×C3×SL(2,ℤ/4)—318no—
19C2×C3×#217—36no—
20C2×C3×#217—36no—
21C23×C33—41no—
22C2×C32×A4C2×C416yescomplemented
23C2×C32×A4C2311yescharacteristic
24C2×C3×C3⋊Dic3—46no—
25C22×SL(2,ℤ/4)—33no—
26C22×C3×C22⋊C4—91no—
27C22×#217—31no—
28C24×C32—31no—
29C24×C32—33no—
30C24×C32—34no—
31C24×C32Dic314yescomplemented
32C24×C32C2×D313yes—
33C22×C3×A4—24no—
34C22×C3×A4—23no—
35C22×C3×A4—212no—
36C22×C3×A4—29no—
37C22×C3×A4Dic3112yescomplemented
38C22×C3×A4C3×C414yescomplemented
39C22×C3×A4C2×D319yes—
40C22×C3×A4C22×C313yes—
41C22×C3⋊Dic3—41no—
42C22×C3×Dic3—36no—
43C22×C3×Dic3—36no—
44C22×C3×Dic3—64no—
45C22×C3×Dic3—63no—
46C22×C3×Dic3—61no—
47C22×C3×Dic3—123no—
48C3⋊SL(2,ℤ/4)C22×C314yesdirect product factor
49C3×SL(2,ℤ/4)—312no—
50C3×#217—312no—
51C3×#217—34no—
52C22×C33—44no—
53C22×C33—43no—
54C32×A4C22×C411yescharacteristic, complemented
55C3×C3⋊Dic3—44no—
56C25×C3—21no—
57C25×C3C3×D311yescharacteristic
58C25×C3C3⋊D311yescharacteristic
59C23×A4—23no—
60C23×A4C3×D313yes—
61C23×C3×C4—91no—
62C23×C3×C4—91no—
63C23×Dic3—31no—
64C23×Dic3—31no—
65C2×SL(2,ℤ/4)—318no—
66C2×C3×C22⋊C4—96no—
67C2×C3×C22⋊C4—96no—
68C2×#217—36no—
69C2×#217—36no—
70C23×C32—34no—
71C23×C32—34no—
72C23×C32—33no—
73C23×C32—33no—
74C23×C32—312no—
75C23×C32—36no—
76C23×C32—33no—
77C23×C32—41no—
78C23×C32—43no—
79C23×C32—66no—
80C23×C32—61no—
81C23×C32—83no—
82C23×C32—81no—
83C23×C32C2×Dic316yescomplemented
84C23×C32S411yescharacteristic
85C23×C32C22×D311yescharacteristic
86C2×C3×A4—26no—
87C2×C3×A4—21no—
88C2×C3×A4—218no—
89C2×C3×A4—23no—
90C2×C3×A4C2×Dic3118yescomplemented
91C2×C3×A4C2×C3×C416yescomplemented
92C2×C3×A4C22×D313yes—
93C2×C3×A4C23×C311yescharacteristic
94C2×C3⋊Dic3—46no—
95C2×C3×Dic3—34no—
96C2×C3×Dic3—34no—
97C2×C3×Dic3—612no—
98C2×C3×Dic3—66no—
99C2×C3×Dic3—66no—
100C2×C3×Dic3—1218no—
101C22×C22⋊C4—91noSylow
102C2×C33—46no—
103C2×C33—41no—
104C24×C3—24no—
105C24×C3—23no—
106C24×C3—31no—
107C24×C3—31no—
108C24×C3—34no—
109C24×C3—33no—
110C24×C3—34no—
111C24×C3—33no—
112C24×C3—61no—
113C24×C3—63no—
114C24×C3—64no—
115C24×C3C3×Dic314yescomplemented
116C24×C3C3⋊Dic314yescomplemented
117C24×C3C2×C3×D313yes—
118C24×C3C2×C3⋊D313yes—
119C22×A4—212no—
120C22×A4—29no—
121C22×A4C3×Dic3112yescomplemented
122C22×A4C2×C3×D319yes—
123C22×C3×C4—96no—
124C22×C3×C4—96no—
125C22×C3×C4—184no—
126C22×C3×C4—183no—
127C22×C3×C4—181no—
128C22×Dic3—36no—
129C22×Dic3—36no—
130C22×Dic3—64no—
131C22×Dic3—63no—
132C22×Dic3—61no—
133C22×Dic3—123no—
134SL(2,ℤ/4)—312no—
135C3×C22⋊C4—912no—
136C3×C22⋊C4—94no—
137#217—312no—
138#217—34no—
139C22×C32—36no—
140C22×C32—31no—
141C22×C32—36no—
142C22×C32—36no—
143C22×C32—312no—
144C22×C32—31no—
145C22×C32—33no—
146C22×C32—43no—
147C22×C32—44no—
148C22×C32—49no—
149C22×C32—412no—
150C22×C32—63no—
151C22×C32—64no—
152C22×C32—89no—
153C22×C32—812no—
154C22×C32—83no—
155C22×C32—84no—
156C22×C32SL(2,ℤ/4)14yescomplemented
157C22×C32C22×Dic311yescharacteristic, complemented
158C22×C32C2×S413yes—
159C3×A4—21no—
160C3×A4—23no—
161C3×A4C22×Dic313yescomplemented
162C3×A4C22×C3×C411yescharacteristic, complemented
163C3⋊Dic3—44no—
164C3×Dic3—64no—
165C3×Dic3—64no—
166C3×Dic3—1212no—
167C25C3×C3⋊D311yescharacteristic
168C23×C4—91no—
169C23×C4—91no—
170C2×C22⋊C4—96no—
171C2×C22⋊C4—96no—
172C33—41noSylow
173C23×C3—26no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—33no—
177C23×C3—33no—
178C23×C3—33no—
179C23×C3—33no—
180C23×C3—34no—
181C23×C3—34no—
182C23×C3—34no—
183C23×C3—34no—
184C23×C3—33no—
185C23×C3—33no—
186C23×C3—312no—
187C23×C3—312no—
188C23×C3—36no—
189C23×C3—36no—
190C23×C3—43no—
191C23×C3—63no—
192C23×C3—64no—
193C23×C3—63no—
194C23×C3—64no—
195C23×C3—63no—
196C23×C3—612no—
197C23×C3—66no—
198C23×C3—66no—
199C23×C3—66no—
200C23×C3—61no—
201C23×C3—61no—
202C23×C3—612no—
203C23×C3—62no—
204C23×C3—83no—
205C23×C3C2×C3×Dic316yescomplemented
206C23×C3C2×C3⋊Dic316yescomplemented
207C23×C3C3×S411yescharacteristic
208C23×C3C3⋊S411yescharacteristic
209C23×C3C22×C3×D311yescharacteristic
210C23×C3C22×C3⋊D311yescharacteristic
211C2×A4—218no—
212C2×A4—23no—
213C2×A4C2×C3×Dic3118yescomplemented
214C2×A4C22×C3×D313yes—
215C2×C3×C4—94no—
216C2×C3×C4—94no—
217C2×C3×C4—1812no—
218C2×C3×C4—186no—
219C2×C3×C4—186no—
220C2×Dic3—34no—
221C2×Dic3—34no—
222C2×Dic3—612no—
223C2×Dic3—66no—
224C2×Dic3—66no—
225C2×Dic3—1218no—
226C2×C32—31no—
227C2×C32—31no—
228C2×C32—36no—
229C2×C32—46no—
230C2×C32—41no—
231C2×C32—418no—
232C2×C32—43no—
233C2×C32—818no—
234C2×C32—83no—
235C2×C32—86no—
236C2×C32—81no—
237C2×C32C2×SL(2,ℤ/4)16yescomplemented
238C2×C32C22×S411yescharacteristic
239C24—31no—
240C24—33no—
241C24—34no—
242C24C3×C3⋊Dic314yescomplemented
243C24C2×C3×C3⋊D313yes—
244C22×C4—96no—
245C22×C4—96no—
246C22×C4—184no—
247C22×C4—183no—
248C22×C4—181no—
249C22⋊C4—912no—
250C22⋊C4—94no—
251C22×C3—23no—
252C22×C3—24no—
253C22×C3—21no—
254C22×C3—36no—
255C22×C3—36no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—36no—
259C22×C3—36no—
260C22×C3—312no—
261C22×C3—36no—
262C22×C3—36no—
263C22×C3—312no—
264C22×C3—31no—
265C22×C3—33no—
266C22×C3—31no—
267C22×C3—33no—
268C22×C3—49no—
269C22×C3—412no—
270C22×C3—66no—
271C22×C3—61no—
272C22×C3—66no—
273C22×C3—612no—
274C22×C3—63no—
275C22×C3—66no—
276C22×C3—61no—
277C22×C3—63no—
278C22×C3—64no—
279C22×C3—63no—
280C22×C3—64no—
281C22×C3—66no—
282C22×C3—68no—
283C22×C3—89no—
284C22×C3—812no—
285C22×C3C3×SL(2,ℤ/4)14yescomplemented
286C22×C3C3⋊SL(2,ℤ/4)14yesdirect product factor
287C22×C3C22×C3×Dic311yescharacteristic, complemented
288C22×C3C22×C3⋊Dic311yescharacteristic, complemented
289C22×C3C2×C3×S413yes—
290C22×C3C2×C3⋊S413yes—
291A4—23no—
292A4C22×C3×Dic313yescomplemented
293C3×C4—184no—
294C3×C4—184no—
295Dic3—64no—
296Dic3—64no—
297Dic3—1212no—
298C32—41no—
299C32—43no—
300C32—83no—
301C32—81no—
302C32C22×SL(2,ℤ/4)11yescharacteristic, complemented
303C23—33no—
304C23—33no—
305C23—34no—
306C23—34no—
307C23—33no—
308C23—312no—
309C23—36no—
310C23—66no—
311C23—61no—
312C23C2×C3×C3⋊Dic316yescomplemented
313C23C3×C3⋊S411yescharacteristic
314C23C22×C3×C3⋊D311yescharacteristic
315C2×C4—94no—
316C2×C4—94no—
317C2×C4—1812no—
318C2×C4—186no—
319C2×C4—186no—
320C2×C3—26no—
321C2×C3—21no—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—36no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—36no—
328C2×C3—418no—
329C2×C3—43no—
330C2×C3—61no—
331C2×C3—61no—
332C2×C3—66no—
333C2×C3—818no—
334C2×C3—83no—
335C2×C3C2×C3×SL(2,ℤ/4)16yescomplemented
336C2×C3C2×C3⋊SL(2,ℤ/4)16yesdirect product factor
337C2×C3C22×C3×S411yescharacteristic
338C2×C3C22×C3⋊S411yescharacteristic
339C22—36no—
340C22—31no—
341C22—36no—
342C22—36no—
343C22—312no—
344C22—31no—
345C22—33no—
346C22—63no—
347C22—64no—
348C22C3×C3⋊SL(2,ℤ/4)14yesdirect product factor
349C22C22×C3×C3⋊Dic311yescharacteristic, complemented
350C22C2×C3×C3⋊S413yes—
351C4—184no—
352C4—184no—
353C3—21no—
354C3—43no—
355C3—83no—
356C3C22×C3×SL(2,ℤ/4)11yescharacteristic, complemented
357C3C22×C3⋊SL(2,ℤ/4)11yescharacteristic, direct product factor
358C2—31no—
359C2—31no—
360C2—36no—
361C2C2×C3×C3⋊SL(2,ℤ/4)16yesdirect product factor
362C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
center, hypercenterC23×C3C3⋊S4
commutator, hypocenterC3×A4C22×C3×C4
FrattiniC2C22×C3×C3⋊S4
Fitting, socleC25×C32D3
prime radicalC23×C32×A4C2
cosocleC2×C3×A4C23×C3
LCSG ▷ C3×A4
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×C3×C3⋊SL(2,ℤ/4)C2direct (N × H)
3. C22×C3⋊SL(2,ℤ/4)C3direct (N × H)
6. C22×C32×A4C4semidirect
9. C3×C3⋊SL(2,ℤ/4)C22direct (N × H)
17. C2×C3⋊SL(2,ℤ/4)C2×C3direct (N × H)
22. C2×C32×A4C2×C4semidirect
31. C24×C32Dic3semidirect
37. C22×C3×A4Dic3semidirect
38. C22×C3×A4C3×C4semidirect
48. C3⋊SL(2,ℤ/4)C22×C3direct (N × H)
54. C32×A4C22×C4semidirect
83. C23×C32C2×Dic3semidirect
90. C2×C3×A4C2×Dic3semidirect
91. C2×C3×A4C2×C3×C4semidirect
115. C24×C3C3×Dic3semidirect
116. C24×C3C3⋊Dic3semidirect
121. C22×A4C3×Dic3semidirect
156. C22×C32SL(2,ℤ/4)semidirect
157. C22×C32C22×Dic3semidirect
161. C3×A4C22×Dic3semidirect
162. C3×A4C22×C3×C4semidirect
205. C23×C3C2×C3×Dic3semidirect
206. C23×C3C2×C3⋊Dic3semidirect
213. C2×A4C2×C3×Dic3semidirect
237. C2×C32C2×SL(2,ℤ/4)semidirect
242. C24C3×C3⋊Dic3semidirect
285. C22×C3C3×SL(2,ℤ/4)semidirect
286. C22×C3C3⋊SL(2,ℤ/4)direct (N × H)
287. C22×C3C22×C3×Dic3semidirect
288. C22×C3C22×C3⋊Dic3semidirect
292. A4C22×C3×Dic3semidirect
302. C32C22×SL(2,ℤ/4)semidirect
312. C23C2×C3×C3⋊Dic3semidirect
335. C2×C3C2×C3×SL(2,ℤ/4)semidirect
336. C2×C3C2×C3⋊SL(2,ℤ/4)direct (N × H)
348. C22C3×C3⋊SL(2,ℤ/4)direct (N × H)
349. C22C22×C3×C3⋊Dic3semidirect
356. C3C22×C3×SL(2,ℤ/4)semidirect
357. C3C22×C3⋊SL(2,ℤ/4)direct (N × H)
361. C2C2×C3×C3⋊SL(2,ℤ/4)direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×A42
C2×C3×C3⋊SL(2,ℤ/4)2
C22×C3⋊SL(2,ℤ/4)3
C22×C3×SL(2,ℤ/4)3
C22×C3×#2173
C22×C3×C3⋊Dic34