Finite Group Databasebeta

← C22×(C3⋊SL(2,ℤ/4))⋊C3 #13164447

Subgroups — C22×(C3⋊SL(2,ℤ/4))⋊C3 #13164447

Order 1728.

1232 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 150 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×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
2C2×(C3⋊SL(2,ℤ/4))⋊C3C216yesmaximal, direct product factor
3C22×C3⋊SL(2,ℤ/4)C311yesmaximal, characteristic, complemented
4C22×C3×#217—31nomaximal
5C22×C22⋊H(3,𝔽3)C414yescomplemented
6C22×C22⋊H(3,𝔽3)C2213yes—
7C22×(C3⋊Dic3)⋊C3—41nomaximal
8(C3⋊SL(2,ℤ/4))⋊C3C2214yesdirect product factor
9C25×C32D311yescharacteristic
10C23×C3×A4—21no—
11C23×C3×A4C2×C311yescharacteristic
12C23×C3×Dic3—31no—
13C23×C3×Dic3—31no—
14C2×C3⋊SL(2,ℤ/4)C2×C316yescomplemented
15C2×C3×#217—36no—
16C2×C3×#217—36no—
17C23×H(3,𝔽3)—41no—
18C2×C22⋊H(3,𝔽3)C2×C416yescomplemented
19C2×C22⋊H(3,𝔽3)C2311yescharacteristic
20C2×(C3⋊Dic3)⋊C3—46no—
21C22×SL(2,ℤ/4)—91no—
22C22×C3×C22⋊C4—91no—
23C22×#217—31no—
24C24×C32—31no—
25C24×C32—33no—
26C24×C32—34no—
27C24×C32Dic314yescomplemented
28C24×C32C2×D313yes—
29C22×C3×A4—24no—
30C22×C3×A4—23no—
31C22×C3×A4C3×C414yescomplemented
32C22×C3×A4C22×C313yes—
33C22×C3⋊Dic3—41no—
34C22×C3×Dic3—36no—
35C22×C3×Dic3—36no—
36C22×C3×Dic3—64no—
37C22×C3×Dic3—63no—
38C22×C3×Dic3—61no—
39C3⋊SL(2,ℤ/4)C22×C314yescomplemented
40C3×#217—312no—
41C3×#217—34no—
42C22×H(3,𝔽3)—44no—
43C22×H(3,𝔽3)—43no—
44C22⋊H(3,𝔽3)C22×C411yescharacteristic, complemented
45(C3⋊Dic3)⋊C3—44no—
46C25×C3—31no—
47C25×C3C3×D311yescharacteristic
48C23×A4—31no—
49C23×A4—61no—
50C23×C3×C4—91no—
51C23×C3×C4—91no—
52C23×Dic3—31no—
53C23×Dic3—31no—
54C2×SL(2,ℤ/4)—96no—
55C2×C3×C22⋊C4—96no—
56C2×C3×C22⋊C4—96no—
57C2×#217—36no—
58C2×#217—36no—
59C23×C32—34no—
60C23×C32—34no—
61C23×C32—33no—
62C23×C32—33no—
63C23×C32—312no—
64C23×C32—36no—
65C23×C32—33no—
66C23×C32—41no—
67C23×C32—66no—
68C23×C32—61no—
69C23×C32—81no—
70C23×C32C2×Dic316yescomplemented
71C23×C32S411yescharacteristic
72C23×C32C22×D311yescharacteristic
73C2×C3×A4—26no—
74C2×C3×A4—21no—
75C2×C3×A4C2×C3×C416yescomplemented
76C2×C3×A4C23×C311yescharacteristic
77C2×C3⋊Dic3—46no—
78C2×C3×Dic3—34no—
79C2×C3×Dic3—34no—
80C2×C3×Dic3—612no—
81C2×C3×Dic3—66no—
82C2×C3×Dic3—66no—
83C22×C22⋊C4—91noSylow
84C2×H(3,𝔽3)—46no—
85C2×H(3,𝔽3)—41no—
86C24×C3—31no—
87C24×C3—31no—
88C24×C3—34no—
89C24×C3—33no—
90C24×C3—34no—
91C24×C3—33no—
92C24×C3—34no—
93C24×C3—33no—
94C24×C3—61no—
95C24×C3—63no—
96C24×C3—64no—
97C24×C3C3×Dic314yes—
98C24×C3C2×C3×D313yes—
99C22×A4—34no—
100C22×A4—33no—
101C22×A4—64no—
102C22×A4—63no—
103C22×C3×C4—96no—
104C22×C3×C4—96no—
105C22×C3×C4—184no—
106C22×C3×C4—183no—
107C22×C3×C4—181no—
108C22×Dic3—36no—
109C22×Dic3—36no—
110C22×Dic3—64no—
111C22×Dic3—63no—
112C22×Dic3—61no—
113C22×Dic3—361no—
114SL(2,ℤ/4)—94no—
115C3×C22⋊C4—912no—
116C3×C22⋊C4—94no—
117#217—312no—
118#217—34no—
119C22×C32—36no—
120C22×C32—31no—
121C22×C32—36no—
122C22×C32—36no—
123C22×C32—312no—
124C22×C32—31no—
125C22×C32—33no—
126C22×C32—43no—
127C22×C32—44no—
128C22×C32—63no—
129C22×C32—64no—
130C22×C32—83no—
131C22×C32—84no—
132C22×C32SL(2,ℤ/4)14yescomplemented
133C22×C32C22×Dic311yescharacteristic, complemented
134C22×C32C2×S413yes—
135C3×A4—21no—
136C3×A4C22×C3×C411yescharacteristic, complemented
137C3⋊Dic3—44no—
138C3×Dic3—64no—
139C3×Dic3—64no—
140C25(C3⋊D3)⋊C311yescharacteristic
141C23×C4—91no—
142C23×C4—91no—
143C2×C22⋊C4—96no—
144C2×C22⋊C4—96no—
145H(3,𝔽3)—41noSylow
146C23×C3—33no—
147C23×C3—34no—
148C23×C3—33no—
149C23×C3—34no—
150C23×C3—33no—
151C23×C3—33no—
152C23×C3—34no—
153C23×C3—34no—
154C23×C3—33no—
155C23×C3—312no—
156C23×C3—36no—
157C23×C3—33no—
158C23×C3—312no—
159C23×C3—36no—
160C23×C3—36no—
161C23×C3—31no—
162C23×C3—31no—
163C23×C3—64no—
164C23×C3—64no—
165C23×C3—63no—
166C23×C3—63no—
167C23×C3—63no—
168C23×C3—612no—
169C23×C3—66no—
170C23×C3—66no—
171C23×C3—61no—
172C23×C3—63no—
173C23×C3—66no—
174C23×C3—612no—
175C23×C3—121no—
176C23×C3—241no—
177C23×C3C2×C3×Dic316yes—
178C23×C3C3×S411yescharacteristic
179C23×C3C22×C3×D311yescharacteristic
180C2×A4—36no—
181C2×A4—31no—
182C2×A4—66no—
183C2×A4—61no—
184C2×C3×C4—94no—
185C2×C3×C4—94no—
186C2×C3×C4—1812no—
187C2×C3×C4—186no—
188C2×C3×C4—186no—
189C2×Dic3—34no—
190C2×Dic3—34no—
191C2×Dic3—612no—
192C2×Dic3—66no—
193C2×Dic3—66no—
194C2×Dic3—366no—
195C2×C32—31no—
196C2×C32—31no—
197C2×C32—36no—
198C2×C32—46no—
199C2×C32—41no—
200C2×C32—86no—
201C2×C32—81no—
202C2×C32C2×SL(2,ℤ/4)16yescomplemented
203C2×C32C22×S411yescharacteristic
204C24—31no—
205C24—33no—
206C24—34no—
207C24(C3⋊Dic3)⋊C314yescomplemented
208C24C2×(C3⋊D3)⋊C313yes—
209C22×C4—96no—
210C22×C4—96no—
211C22×C4—184no—
212C22×C4—183no—
213C22×C4—181no—
214C22⋊C4—912no—
215C22⋊C4—94no—
216C22×C3—36no—
217C22×C3—31no—
218C22×C3—31no—
219C22×C3—33no—
220C22×C3—36no—
221C22×C3—36no—
222C22×C3—312no—
223C22×C3—36no—
224C22×C3—36no—
225C22×C3—312no—
226C22×C3—31no—
227C22×C3—33no—
228C22×C3—36no—
229C22×C3—31no—
230C22×C3—34no—
231C22×C3—33no—
232C22×C3—31no—
233C22×C3—66no—
234C22×C3—61no—
235C22×C3—61no—
236C22×C3—63no—
237C22×C3—66no—
238C22×C3—66no—
239C22×C3—612no—
240C22×C3—63no—
241C22×C3—64no—
242C22×C3—63no—
243C22×C3—66no—
244C22×C3—612no—
245C22×C3—124no—
246C22×C3—123no—
247C22×C3—244no—
248C22×C3—243no—
249C22×C3C3×SL(2,ℤ/4)14yes—
250C22×C3C22×C3×Dic311yescharacteristic
251C22×C3C2×C3×S413yes—
252A4—31no—
253A4—61no—
254C3×C4—184no—
255C3×C4—184no—
256Dic3—64no—
257Dic3—64no—
258Dic3—364no—
259C32—41no—
260C32—81no—
261C32C22×SL(2,ℤ/4)11yescharacteristic, complemented
262C23—34no—
263C23—34no—
264C23—33no—
265C23—33no—
266C23—33no—
267C23—312no—
268C23—36no—
269C23—61no—
270C23—66no—
271C23C2×(C3⋊Dic3)⋊C316yescomplemented
272C23(C3⋊S4)⋊C311yescharacteristic
273C23C22×(C3⋊D3)⋊C311yescharacteristic
274C2×C4—94no—
275C2×C4—94no—
276C2×C4—1812no—
277C2×C4—186no—
278C2×C4—186no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—36no—
282C2×C3—31no—
283C2×C3—31no—
284C2×C3—36no—
285C2×C3—31no—
286C2×C3—36no—
287C2×C3—61no—
288C2×C3—61no—
289C2×C3—66no—
290C2×C3—121no—
291C2×C3—126no—
292C2×C3—241no—
293C2×C3—246no—
294C2×C3C2×C3×SL(2,ℤ/4)16yes—
295C2×C3C22×C3×S411yescharacteristic
296C22—36no—
297C22—31no—
298C22—31no—
299C22—33no—
300C22—36no—
301C22—36no—
302C22—312no—
303C22—63no—
304C22—64no—
305C22(C3⋊SL(2,ℤ/4))⋊C314yesdirect product factor
306C22C22×(C3⋊Dic3)⋊C311yescharacteristic, complemented
307C22C2×(C3⋊S4)⋊C313yes—
308C4—184no—
309C4—184no—
310C3—31no—
311C3—121no—
312C3—241no—
313C3C22×C3×SL(2,ℤ/4)11yescharacteristic
314C2—31no—
315C2—31no—
316C2—36no—
317C2C2×(C3⋊SL(2,ℤ/4))⋊C316yesdirect product factor
318C2C22×(C3⋊S4)⋊C311yescharacteristic
subgroupG/H
center, hypercenterC23(C3⋊S4)⋊C3
commutator, hypocenterC3×A4C22×C3×C4
FrattiniC2×C3C22×C3×S4
FittingC25×C32D3
socleC25×C3C3×D3
prime radicalC23×C22⋊H(3,𝔽3)C2
cosocleC2×C3×A4C23×C3
LCSG ▷ C3×A4
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×(C3⋊SL(2,ℤ/4))⋊C3C2direct (N × H)
3. C22×C3⋊SL(2,ℤ/4)C3semidirect
5. C22×C22⋊H(3,𝔽3)C4semidirect
8. (C3⋊SL(2,ℤ/4))⋊C3C22direct (N × H)
14. C2×C3⋊SL(2,ℤ/4)C2×C3semidirect
18. C2×C22⋊H(3,𝔽3)C2×C4semidirect
27. C24×C32Dic3semidirect
31. C22×C3×A4C3×C4semidirect
39. C3⋊SL(2,ℤ/4)C22×C3semidirect
44. C22⋊H(3,𝔽3)C22×C4semidirect
70. C23×C32C2×Dic3semidirect
75. C2×C3×A4C2×C3×C4semidirect
132. C22×C32SL(2,ℤ/4)semidirect
133. C22×C32C22×Dic3semidirect
136. C3×A4C22×C3×C4semidirect
202. C2×C32C2×SL(2,ℤ/4)semidirect
207. C24(C3⋊Dic3)⋊C3semidirect
261. C32C22×SL(2,ℤ/4)semidirect
271. C23C2×(C3⋊Dic3)⋊C3semidirect
305. C22(C3⋊SL(2,ℤ/4))⋊C3direct (N × H)
306. C22C22×(C3⋊Dic3)⋊C3semidirect
317. C2C2×(C3⋊SL(2,ℤ/4))⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊H(3,𝔽3)2
C2×(C3⋊SL(2,ℤ/4))⋊C32
C22×C3⋊SL(2,ℤ/4)3
C22×C3×#2173
C22×(C3⋊Dic3)⋊C34