Finite Group Databasebeta

← C22×#88613 #13164510

Subgroups — C22×#88613 #13164510

Order 1728.

1384 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1384 conjugacy classes of subgroups — too many to draw. Only its 171 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×#88613C216yesmaximal, direct product factor
3C22×C3×SL(2,ℤ/4)—33nomaximal
4C22×C3×#217—31nomaximal
5C22×C22⋊H(3,𝔽3)C414yescomplemented
6C22×C22⋊H(3,𝔽3)C2213yes—
7C22×#1084—41nomaximal
8#88613C2214yesdirect product factor
9C25×C32D311yescharacteristic
10C23×C3×A4D313yes—
11C23×C3×Dic3—31no—
12C23×C3×Dic3—31no—
13C2×C3×SL(2,ℤ/4)—318no—
14C2×C3×#217—36no—
15C2×C3×#217—36no—
16C23×H(3,𝔽3)—41no—
17C2×C22⋊H(3,𝔽3)C2×C416yescomplemented
18C2×C22⋊H(3,𝔽3)C2311yescharacteristic
19C2×#1084—46no—
20C22×SL(2,ℤ/4)—33no—
21C22×C3×C22⋊C4—91no—
22C22×#217—31no—
23C24×C32—31no—
24C24×C32—33no—
25C24×C32—34no—
26C24×C32Dic314yescomplemented
27C24×C32C2×D313yes—
28C22×C3×A4Dic3112yescomplemented
29C22×C3×A4C2×D319yes—
30C22×C3×Dic3—36no—
31C22×C3×Dic3—36no—
32C22×C3×Dic3—64no—
33C22×C3×Dic3—63no—
34C22×C3×Dic3—61no—
35C22×C3×Dic3—123no—
36C3×SL(2,ℤ/4)—312no—
37C3×#217—312no—
38C3×#217—34no—
39C22×H(3,𝔽3)—44no—
40C22×H(3,𝔽3)—43no—
41C22⋊H(3,𝔽3)C22×C411yescharacteristic, complemented
42#1084—44no—
43C25×C3—31no—
44C25×C3C3⋊D311yescharacteristic
45C23×A4—33no—
46C23×C3×C4—91no—
47C23×C3×C4—91no—
48C23×Dic3—31no—
49C23×Dic3—31no—
50C2×SL(2,ℤ/4)—318no—
51C2×C3×C22⋊C4—96no—
52C2×C3×C22⋊C4—96no—
53C2×#217—36no—
54C2×#217—36no—
55C23×C32—34no—
56C23×C32—34no—
57C23×C32—33no—
58C23×C32—33no—
59C23×C32—312no—
60C23×C32—36no—
61C23×C32—33no—
62C23×C32—43no—
63C23×C32—66no—
64C23×C32—61no—
65C23×C32C2×Dic316yescomplemented
66C23×C32S411yescharacteristic
67C23×C32C22×D311yescharacteristic
68C2×C3×A4C2×Dic3118yescomplemented
69C2×C3×A4C22×D313yes—
70C2×C3×Dic3—34no—
71C2×C3×Dic3—34no—
72C2×C3×Dic3—612no—
73C2×C3×Dic3—66no—
74C2×C3×Dic3—66no—
75C2×C3×Dic3—1218no—
76C22×C22⋊C4—91noSylow
77C2×H(3,𝔽3)—46no—
78C2×H(3,𝔽3)—41no—
79C24×C3—31no—
80C24×C3—31no—
81C24×C3—34no—
82C24×C3—33no—
83C24×C3—34no—
84C24×C3—33no—
85C24×C3—34no—
86C24×C3—33no—
87C24×C3—61no—
88C24×C3—63no—
89C24×C3—64no—
90C24×C3C3⋊Dic314yes—
91C24×C3C2×C3⋊D313yes—
92C22×A4—312no—
93C22×A4—39no—
94C22×C3×C4—96no—
95C22×C3×C4—96no—
96C22×C3×C4—184no—
97C22×C3×C4—183no—
98C22×C3×C4—181no—
99C22×Dic3—36no—
100C22×Dic3—36no—
101C22×Dic3—64no—
102C22×Dic3—63no—
103C22×Dic3—61no—
104C22×Dic3—123no—
105SL(2,ℤ/4)—312no—
106C3×C22⋊C4—912no—
107C3×C22⋊C4—94no—
108#217—312no—
109#217—34no—
110C22×C32—36no—
111C22×C32—31no—
112C22×C32—36no—
113C22×C32—36no—
114C22×C32—312no—
115C22×C32—31no—
116C22×C32—33no—
117C22×C32—49no—
118C22×C32—412no—
119C22×C32—63no—
120C22×C32—64no—
121C22×C32SL(2,ℤ/4)14yescomplemented
122C22×C32C22×Dic311yescharacteristic, complemented
123C22×C32C2×S413yes—
124C3×A4C22×Dic313yescomplemented
125C3×Dic3—64no—
126C3×Dic3—64no—
127C3×Dic3—1212no—
128C25#27211yescharacteristic
129C23×C4—91no—
130C23×C4—91no—
131C2×C22⋊C4—96no—
132C2×C22⋊C4—96no—
133H(3,𝔽3)—41noSylow
134C23×C3—33no—
135C23×C3—34no—
136C23×C3—33no—
137C23×C3—34no—
138C23×C3—33no—
139C23×C3—33no—
140C23×C3—34no—
141C23×C3—34no—
142C23×C3—33no—
143C23×C3—312no—
144C23×C3—36no—
145C23×C3—33no—
146C23×C3—312no—
147C23×C3—36no—
148C23×C3—36no—
149C23×C3—31no—
150C23×C3—31no—
151C23×C3—64no—
152C23×C3—64no—
153C23×C3—63no—
154C23×C3—63no—
155C23×C3—63no—
156C23×C3—612no—
157C23×C3—66no—
158C23×C3—66no—
159C23×C3—61no—
160C23×C3—63no—
161C23×C3—66no—
162C23×C3—612no—
163C23×C3—123no—
164C23×C3C2×C3⋊Dic316yes—
165C23×C3C3⋊S411yescharacteristic
166C23×C3C22×C3⋊D311yescharacteristic
167C2×A4—318no—
168C2×A4—33no—
169C2×C3×C4—94no—
170C2×C3×C4—94no—
171C2×C3×C4—1812no—
172C2×C3×C4—186no—
173C2×C3×C4—186no—
174C2×Dic3—34no—
175C2×Dic3—34no—
176C2×Dic3—612no—
177C2×Dic3—66no—
178C2×Dic3—66no—
179C2×Dic3—1218no—
180C2×C32—31no—
181C2×C32—31no—
182C2×C32—36no—
183C2×C32—418no—
184C2×C32—43no—
185C2×C32C2×SL(2,ℤ/4)16yescomplemented
186C2×C32C22×S411yescharacteristic
187C24—31no—
188C24—33no—
189C24—34no—
190C24#108414yescomplemented
191C24C2×#27213yes—
192C22×C4—96no—
193C22×C4—96no—
194C22×C4—184no—
195C22×C4—183no—
196C22×C4—181no—
197C22⋊C4—912no—
198C22⋊C4—94no—
199C22×C3—36no—
200C22×C3—31no—
201C22×C3—31no—
202C22×C3—33no—
203C22×C3—36no—
204C22×C3—36no—
205C22×C3—312no—
206C22×C3—36no—
207C22×C3—36no—
208C22×C3—312no—
209C22×C3—31no—
210C22×C3—33no—
211C22×C3—36no—
212C22×C3—31no—
213C22×C3—34no—
214C22×C3—33no—
215C22×C3—31no—
216C22×C3—66no—
217C22×C3—61no—
218C22×C3—61no—
219C22×C3—63no—
220C22×C3—66no—
221C22×C3—66no—
222C22×C3—612no—
223C22×C3—63no—
224C22×C3—64no—
225C22×C3—63no—
226C22×C3—66no—
227C22×C3—612no—
228C22×C3—1212no—
229C22×C3—129no—
230C22×C3C3⋊SL(2,ℤ/4)14yes—
231C22×C3C22×C3⋊Dic311yescharacteristic
232C22×C3C2×C3⋊S413yes—
233A4—33no—
234C3×C4—184no—
235C3×C4—184no—
236Dic3—64no—
237Dic3—64no—
238Dic3—1212no—
239C32—43no—
240C32C22×SL(2,ℤ/4)11yescharacteristic, complemented
241C23—34no—
242C23—34no—
243C23—33no—
244C23—33no—
245C23—33no—
246C23—312no—
247C23—36no—
248C23—61no—
249C23—66no—
250C23C2×#108416yescomplemented
251C23#625811yescharacteristic
252C23C22×#27211yescharacteristic
253C2×C4—94no—
254C2×C4—94no—
255C2×C4—1812no—
256C2×C4—186no—
257C2×C4—186no—
258C2×C3—31no—
259C2×C3—31no—
260C2×C3—36no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—36no—
264C2×C3—31no—
265C2×C3—36no—
266C2×C3—61no—
267C2×C3—61no—
268C2×C3—66no—
269C2×C3—123no—
270C2×C3—1218no—
271C2×C3C2×C3⋊SL(2,ℤ/4)16yes—
272C2×C3C22×C3⋊S411yescharacteristic
273C22—36no—
274C22—31no—
275C22—31no—
276C22—33no—
277C22—36no—
278C22—36no—
279C22—312no—
280C22—63no—
281C22—64no—
282C22#8861314yesdirect product factor
283C22C22×#108411yescharacteristic, complemented
284C22C2×#625813yes—
285C4—184no—
286C4—184no—
287C3—31no—
288C3—123no—
289C3C22×C3⋊SL(2,ℤ/4)11yescharacteristic
290C2—31no—
291C2—31no—
292C2—36no—
293C2C2×#8861316yesdirect product factor
294C2C22×#625811yescharacteristic
subgroupG/H
center, hypercenterC23×C3C3⋊S4
commutator, hypocenterC22⋊H(3,𝔽3)C22×C4
FrattiniC2×C3C22×C3⋊S4
FittingC25×C32D3
socleC25×C3C3⋊D3
prime radicalC23×C22⋊H(3,𝔽3)C2
cosocleC2×C22⋊H(3,𝔽3)C23
LCSG ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C2×#88613C2direct (N × H)
5. C22×C22⋊H(3,𝔽3)C4semidirect
8. #88613C22direct (N × H)
17. C2×C22⋊H(3,𝔽3)C2×C4semidirect
26. C24×C32Dic3semidirect
28. C22×C3×A4Dic3semidirect
41. C22⋊H(3,𝔽3)C22×C4semidirect
65. C23×C32C2×Dic3semidirect
68. C2×C3×A4C2×Dic3semidirect
121. C22×C32SL(2,ℤ/4)semidirect
122. C22×C32C22×Dic3semidirect
124. C3×A4C22×Dic3semidirect
185. C2×C32C2×SL(2,ℤ/4)semidirect
190. C24#1084semidirect
240. C32C22×SL(2,ℤ/4)semidirect
250. C23C2×#1084semidirect
282. C22#88613direct (N × H)
283. C22C22×#1084semidirect
293. C2C2×#88613direct (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×#886132
C22×C3×SL(2,ℤ/4)3
C22×C3×#2173
C22×#10844