Finite Group Databasebeta

← C22×#88431 #13150591

Subgroups — C22×#88431 #13150591

Order 1728.

1119 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1119 conjugacy classes of subgroups — too many to draw. Only its 219 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×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C22×#634⋊C3C211yesmaximal, characteristic, complemented
3C22×#6180C211yesmaximal, characteristic, complemented
4C2×#88431C2112yesmaximal, direct product factor
5C22×#3717—31nomaximal
6C22×#3716—31nomaximal
7C23×(C3⋊D3)⋊C3—21no—
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3—21no—
10C2×C4⋊((C3⋊D3)⋊C3)—23no—
11C2×C4⋊((C3⋊D3)⋊C3)C2216yes—
12C2×#634⋊C3—23no—
13C2×#634⋊C3C2216yescomplemented
14C2×#6180C2216yescomplemented
15#88431C22116yesdirect product factor
16C22×C4⋊(C3⋊D3)D311yescharacteristic
17C22×#634D311yescharacteristic, complemented
18C22×C3×D12—31no—
19C22×C3×Dic6—31no—
20C22×C3×C3⋊C8—31no—
21C22×C3×C3⋊C8—31no—
22C22×C3×C3⋊C8—61no—
23C2×#3717—312no—
24C2×#3716—312no—
25C23×H(3,𝔽3)D411yescharacteristic
26C22×(C3⋊D3)⋊C3—26no—
27C22×(C3⋊D3)⋊C3—44no—
28C2×C4×H(3,𝔽3)D413yes—
29C2×C4×H(3,𝔽3)C2313yes—
30C2×(C3⋊Dic3)⋊C3—26no—
31C4⋊((C3⋊D3)⋊C3)—26no—
32C4⋊((C3⋊D3)⋊C3)C2314yes—
33#634⋊C3—26no—
34#634⋊C3C2314yescomplemented
35#6180C2314yescomplemented
36C22×#215—91no—
37C22×#214—91no—
38C22×#204—91no—
39C23×C3⋊D3—21no—
40C23×C3×D3—61no—
41C22×C32×C4—21no—
42C22×C32×C4C2×D311yescharacteristic
43C22×C32×C4C2×D311yescharacteristic
44C22×C3⋊Dic3—21no—
45C2×C4⋊(C3⋊D3)—23no—
46C2×C4⋊(C3⋊D3)C2×D316yes—
47C2×#634—23no—
48C2×#634C2×D316yescomplemented
49C22×C3×Dic3—61no—
50C2×C3×D12—36no—
51C2×C3×D12—63no—
52C2×C3×Dic6—36no—
53C2×C3×Dic6—63no—
54C2×C3×C3⋊C8—36no—
55C2×C3×C3⋊C8—36no—
56C2×C3×C3⋊C8—66no—
57#3717—316no—
58#3716—316no—
59C22×H(3,𝔽3)SD1614yescomplemented
60C22×H(3,𝔽3)C2×D413yes—
61C2×(C3⋊D3)⋊C3—24no—
62C2×(C3⋊D3)⋊C3—412no—
63C4×H(3,𝔽3)C2×D413yes—
64C4×H(3,𝔽3)C2411yescharacteristic
65(C3⋊Dic3)⋊C3—24no—
66C22×C3×Q8—91no—
67C22×C3×D4—91no—
68C22×D12—31no—
69C22×D12—91no—
70C22×Dic6—31no—
71C22×Dic6—91no—
72C22×C3×C8—91no—
73C2×#215—912no—
74C2×#214—912no—
75C22×C3⋊C8—31no—
76C22×C3⋊C8—31no—
77C22×C3⋊C8—61no—
78C2×#204—912no—
79C23×C32—21no—
80C23×C32#6711yescharacteristic
81C23×C32#6711yescharacteristic
82C22×C3⋊D3—26no—
83C22×C3⋊D3—44no—
84C22×C3×D3—66no—
85C22×C3×D3—124no—
86C2×C32×C4—23no—
87C2×C32×C4—23no—
88C2×C32×C4#6713yes—
89C2×C32×C4#6713yes—
90C2×C32×C4C22×D313yes—
91C2×C32×C4C22×D313yes—
92C2×C3⋊Dic3—26no—
93C4⋊(C3⋊D3)—26no—
94C4⋊(C3⋊D3)C22×D314yes—
95#634—26no—
96#634C22×D314yescomplemented
97C2×C3×Dic3—66no—
98C3×D12—34no—
99C3×D12—66no—
100C3×Dic6—34no—
101C3×Dic6—66no—
102C3×C3⋊C8—34no—
103C3×C3⋊C8—34no—
104C3×C3⋊C8—64no—
105C22×SD16—271noSylow
106C2×H(3,𝔽3)C2×SD1616yescomplemented
107C2×H(3,𝔽3)C22×D411yescharacteristic
108(C3⋊D3)⋊C3—44no—
109C24×C3—181no—
110C23×D3—61no—
111C23×D3—181no—
112C2×C3×Q8—96no—
113C2×C3×Q8—183no—
114C2×C3×D4—96no—
115C2×C3×D4—183no—
116C22×C3×C4—31no—
117C22×C3×C4—31no—
118C22×C3×C4—61no—
119C22×C3×C4—181no—
120C22×C3×C4D3211yescharacteristic
121C22×Dic3—61no—
122C22×Dic3—181no—
123C2×D12—36no—
124C2×D12—63no—
125C2×D12—96no—
126C2×D12—183no—
127C2×Dic6—36no—
128C2×Dic6—63no—
129C2×Dic6—96no—
130C2×Dic6—183no—
131C2×C3×C8—96no—
132#215—916no—
133#214—916no—
134C2×C3⋊C8—36no—
135C2×C3⋊C8—36no—
136C2×C3⋊C8—66no—
137#204—916no—
138C22×C32—24no—
139C22×C32—23no—
140C22×C32#21414yescomplemented
141C22×C32#21514yescomplemented
142C22×C32C2×#6713yes—
143C22×C32C2×#6713yes—
144C2×C3⋊D3—24no—
145C2×C3⋊D3—412no—
146C2×C3×D3—64no—
147C2×C3×D3—1212no—
148C32×C4—21no—
149C32×C4—23no—
150C32×C4C2×#6713yes—
151C32×C4C2×#6713yes—
152C32×C4C23×D311yescharacteristic
153C32×C4C23×D311yescharacteristic
154C3⋊Dic3—24no—
155C3×Dic3—64no—
156C22×Q8—91no—
157C22×D4—91no—
158C2×SD16—2712no—
159C22×C8—91no—
160H(3,𝔽3)C22×SD1611yescharacteristic, complemented, Sylow
161C23×C3—31no—
162C23×C3—31no—
163C23×C3—61no—
164C23×C3—186no—
165C23×C3—364no—
166C23×C3#62511yescharacteristic
167C22×D3—66no—
168C22×D3—124no—
169C22×D3—186no—
170C22×D3—364no—
171C3×Q8—94no—
172C3×Q8—186no—
173C3×D4—94no—
174C3×D4—186no—
175C2×C3×C4—33no—
176C2×C3×C4—33no—
177C2×C3×C4—33no—
178C2×C3×C4—33no—
179C2×C3×C4—63no—
180C2×C3×C4—63no—
181C2×C3×C4—186no—
182C2×C3×C4#62513yes—
183C2×C3×C4C2×D3213yes—
184C2×Dic3—66no—
185C2×Dic3—186no—
186D12—34no—
187D12—66no—
188D12—94no—
189D12—186no—
190Dic6—34no—
191Dic6—66no—
192Dic6—94no—
193Dic6—186no—
194C3×C8—94no—
195C3⋊C8—34no—
196C3⋊C8—34no—
197C3⋊C8—64no—
198C2×C32—21no—
199C2×C32—26no—
200C2×C32C2×#21416yescomplemented
201C2×C32C2×#21516yescomplemented
202C2×C32C22×#6711yescharacteristic
203C2×C32C22×#6711yescharacteristic
204C3⋊D3—44no—
205C3×D3—124no—
206C24—181no—
207C2×Q8—96no—
208C2×Q8—183no—
209C2×D4—96no—
210C2×D4—183no—
211C22×C4—181no—
212C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
213SD16—2716no—
214C2×C8—96no—
215C22×C3—33no—
216C22×C3—33no—
217C22×C3—34no—
218C22×C3—34no—
219C22×C3—63no—
220C22×C3—64no—
221C22×C3—184no—
222C22×C3—3612no—
223C22×C3#371514yes—
224C22×C3C2×#62513yes—
225C2×D3—64no—
226C2×D3—1212no—
227C2×D3—184no—
228C2×D3—3612no—
229C3×C4—31no—
230C3×C4—31no—
231C3×C4—33no—
232C3×C4—33no—
233C3×C4—61no—
234C3×C4—63no—
235C3×C4—184no—
236C3×C4C2×#62513yes—
237C3×C4C22×D3211yescharacteristic
238Dic3—64no—
239Dic3—184no—
240C32—21no—
241C32C22×#21411yescharacteristic, complemented
242C32C22×#21511yescharacteristic, complemented
243C23—186no—
244C23—364no—
245C23#619811yescharacteristic
246Q8—94no—
247Q8—186no—
248D4—94no—
249D4—186no—
250C2×C4—186no—
251C2×C4#619813yes—
252C2×C4C2×((C3⋊D3)⋊C3)⋊C213yes—
253C8—94no—
254C2×C3—31no—
255C2×C3—31no—
256C2×C3—36no—
257C2×C3—36no—
258C2×C3—61no—
259C2×C3—66no—
260C2×C3—364no—
261C2×C3C2×#371516yes—
262C2×C3C22×#62511yescharacteristic
263D3—124no—
264D3—364no—
265C22—184no—
266C22—3612no—
267C22#8843114yesdirect product factor
268C22C2×#619813yes—
269C4—184no—
270C4C2×#619813yes—
271C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
272C3—31no—
273C3—31no—
274C3—61no—
275C3C22×#371511yescharacteristic
276C2—364no—
277C2C2×#8843116yesdirect product factor
278C2C22×#619811yescharacteristic
subgroupG/H
centerC23#6198
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3#625
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C22×SD16
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#634⋊C3C2semidirect
3. C22×#6180C2semidirect
4. C2×#88431C2direct (N × H)
13. C2×#634⋊C3C22semidirect
14. C2×#6180C22semidirect
15. #88431C22direct (N × H)
17. C22×#634D3semidirect
34. #634⋊C3C23semidirect
35. #6180C23semidirect
48. C2×#634C2×D3semidirect
59. C22×H(3,𝔽3)SD16semidirect
96. #634C22×D3semidirect
106. C2×H(3,𝔽3)C2×SD16semidirect
140. C22×C32#214semidirect
141. C22×C32#215semidirect
160. H(3,𝔽3)C22×SD16semidirect
200. C2×C32C2×#214semidirect
201. C2×C32C2×#215semidirect
241. C32C22×#214semidirect
242. C32C22×#215semidirect
267. C22#88431direct (N × H)
277. C2C2×#88431direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
C22×#634⋊C32
C22×#61802
C2×#884312
C22×#37173
C22×#37163