Finite Group Databasebeta

← C22×#88438 #13150648

Subgroups — C22×#88438 #13150648

Order 1728.

1199 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 1199 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×#6229C211yesmaximal, characteristic, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4C2×#88438C2112yesmaximal, direct product factor
5C22×#3717—31nomaximal
6C22×#3715—31nomaximal
7C23×(C3⋊D3)⋊C3—21no—
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084—21no—
10C2×#6229—23no—
11C2×#6229C2216yescomplemented
12C2×C4⋊((C3⋊D3)⋊C3)—23no—
13C2×C4⋊((C3⋊D3)⋊C3)C2216yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2216yescomplemented
15#88438C22116yesdirect product factor
16C22×C4⋊(C3⋊D3)D311yescharacteristic
17C22×C3×D12—31no—
18C22×C3×Dic6—31no—
19C22×C3×Dic6—31no—
20C22×C3×Dic6—61no—
21C22×C3⋊(C3⋊C8)D311yescharacteristic, complemented
22C22×C3×C3⋊C8—31no—
23C2×#3717—312no—
24C2×#3715—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×#1084—26no—
31#6229—26no—
32#6229C2314yescomplemented
33C4⋊((C3⋊D3)⋊C3)—26no—
34C4⋊((C3⋊D3)⋊C3)C2314yes—
35(C3⋊(C3⋊C8))⋊C3C2314yescomplemented
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
44C2×C4⋊(C3⋊D3)—23no—
45C2×C4⋊(C3⋊D3)C2×D316yes—
46C22×C3×Dic3—62no—
47C22×C3×Dic3—61no—
48C22×C3×Dic3—61no—
49C2×C3×D12—36no—
50C2×C3×D12—63no—
51C2×C3×Dic6—36no—
52C2×C3×Dic6—36no—
53C2×C3×Dic6—66no—
54C2×C3×Dic6—66no—
55C2×C3×Dic6—63no—
56C2×C3×Dic6—63no—
57C2×C3⋊(C3⋊C8)C2×D316yescomplemented
58C2×C3×C3⋊C8—36no—
59#3717—316no—
60#3715—316no—
61C22×H(3,𝔽3)SD1614yescomplemented
62C22×H(3,𝔽3)C2×D413yes—
63C2×(C3⋊D3)⋊C3—24no—
64C2×(C3⋊D3)⋊C3—412no—
65C4×H(3,𝔽3)C2×D413yes—
66C4×H(3,𝔽3)C2411yescharacteristic
67#1084—24no—
68C22×C3×Q8—91no—
69C22×C3×D4—91no—
70C22×D12—31no—
71C22×D12—91no—
72C22×Dic6—31no—
73C22×Dic6—31no—
74C22×Dic6—61no—
75C22×C3×C8—91no—
76C2×#215—912no—
77C2×#214—912no—
78C22×C3⋊C8—31no—
79C22×C3⋊C8—91no—
80C2×#204—912no—
81C23×C32—21no—
82C23×C32D1211yescharacteristic
83C23×C32#6711yescharacteristic
84C22×C3⋊D3—26no—
85C22×C3⋊D3—44no—
86C22×C3×D3—66no—
87C22×C3×D3—124no—
88C2×C32×C4—23no—
89C2×C32×C4—23no—
90C2×C32×C4D1213yes—
91C2×C32×C4#6713yes—
92C2×C32×C4C22×D313yes—
93C2×C32×C4C22×D313yes—
94C4⋊(C3⋊D3)—26no—
95C4⋊(C3⋊D3)C22×D314yes—
96C2×C3×Dic3—612no—
97C2×C3×Dic3—66no—
98C2×C3×Dic3—66no—
99C3×D12—34no—
100C3×D12—66no—
101C3×Dic6—34no—
102C3×Dic6—34no—
103C3×Dic6—64no—
104C3×Dic6—612no—
105C3×Dic6—66no—
106C3×Dic6—66no—
107C3⋊(C3⋊C8)C22×D314yescomplemented
108C3×C3⋊C8—34no—
109C22×SD16—271noSylow
110C2×H(3,𝔽3)C2×SD1616yescomplemented
111C2×H(3,𝔽3)C22×D411yescharacteristic
112(C3⋊D3)⋊C3—44no—
113C24×C3—181no—
114C23×D3—61no—
115C23×D3—181no—
116C2×C3×Q8—96no—
117C2×C3×Q8—183no—
118C2×C3×D4—96no—
119C2×C3×D4—183no—
120C22×C3×C4—31no—
121C22×C3×C4—31no—
122C22×C3×C4—61no—
123C22×C3×C4—181no—
124C22×C3×C4D3211yescharacteristic
125C22×Dic3—61no—
126C22×Dic3—61no—
127C22×Dic3—62no—
128C2×D12—36no—
129C2×D12—63no—
130C2×D12—96no—
131C2×D12—183no—
132C2×Dic6—36no—
133C2×Dic6—36no—
134C2×Dic6—66no—
135C2×Dic6—63no—
136C2×Dic6—63no—
137C2×Dic6—66no—
138C2×C3×C8—96no—
139#215—916no—
140#214—916no—
141C2×C3⋊C8—36no—
142C2×C3⋊C8—96no—
143#204—916no—
144C22×C32—24no—
145C22×C32—23no—
146C22×C32#20414yescomplemented
147C22×C32#21414yescomplemented
148C22×C32C2×D1213yes—
149C22×C32C2×#6713yes—
150C2×C3⋊D3—24no—
151C2×C3⋊D3—412no—
152C2×C3×D3—64no—
153C2×C3×D3—1212no—
154C32×C4—21no—
155C32×C4—23no—
156C32×C4C2×D1213yes—
157C32×C4C2×#6713yes—
158C32×C4C23×D311yescharacteristic
159C32×C4C23×D311yescharacteristic
160C3×Dic3—68no—
161C3×Dic3—64no—
162C3×Dic3—64no—
163C22×Q8—91no—
164C22×D4—91no—
165C2×SD16—2712no—
166C22×C8—91no—
167H(3,𝔽3)C22×SD1611yescharacteristic, complemented, Sylow
168C23×C3—31no—
169C23×C3—31no—
170C23×C3—61no—
171C23×C3—186no—
172C23×C3—364no—
173C23×C3#62611yescharacteristic
174C22×D3—66no—
175C22×D3—124no—
176C22×D3—186no—
177C22×D3—364no—
178C3×Q8—94no—
179C3×Q8—186no—
180C3×D4—94no—
181C3×D4—186no—
182C2×C3×C4—33no—
183C2×C3×C4—33no—
184C2×C3×C4—33no—
185C2×C3×C4—33no—
186C2×C3×C4—63no—
187C2×C3×C4—63no—
188C2×C3×C4—186no—
189C2×C3×C4#62613yes—
190C2×C3×C4C2×D3213yes—
191C2×Dic3—66no—
192C2×Dic3—66no—
193C2×Dic3—612no—
194D12—34no—
195D12—66no—
196D12—94no—
197D12—186no—
198Dic6—34no—
199Dic6—34no—
200Dic6—64no—
201Dic6—66no—
202Dic6—66no—
203Dic6—612no—
204C3×C8—94no—
205C3⋊C8—34no—
206C3⋊C8—94no—
207C2×C32—21no—
208C2×C32—26no—
209C2×C32C2×#20416yescomplemented
210C2×C32C2×#21416yescomplemented
211C2×C32C22×D1211yescharacteristic
212C2×C32C22×#6711yescharacteristic
213C3⋊D3—44no—
214C3×D3—124no—
215C24—181no—
216C2×Q8—96no—
217C2×Q8—183no—
218C2×D4—96no—
219C2×D4—183no—
220C22×C4—181no—
221C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
222SD16—2716no—
223C2×C8—96no—
224C22×C3—33no—
225C22×C3—33no—
226C22×C3—34no—
227C22×C3—34no—
228C22×C3—63no—
229C22×C3—64no—
230C22×C3—184no—
231C22×C3—3612no—
232C22×C3#371614yes—
233C22×C3C2×#62613yes—
234C2×D3—64no—
235C2×D3—1212no—
236C2×D3—184no—
237C2×D3—3612no—
238C3×C4—31no—
239C3×C4—31no—
240C3×C4—33no—
241C3×C4—33no—
242C3×C4—61no—
243C3×C4—63no—
244C3×C4—184no—
245C3×C4C2×#62613yes—
246C3×C4C22×D3211yescharacteristic
247Dic3—64no—
248Dic3—64no—
249Dic3—68no—
250C32—21no—
251C32C22×#20411yescharacteristic, complemented
252C32C22×#21411yescharacteristic, complemented
253C23—186no—
254C23—364no—
255C23#620011yescharacteristic
256Q8—94no—
257Q8—186no—
258D4—94no—
259D4—186no—
260C2×C4—186no—
261C2×C4#620013yes—
262C2×C4C2×((C3⋊D3)⋊C3)⋊C213yes—
263C8—94no—
264C2×C3—31no—
265C2×C3—31no—
266C2×C3—36no—
267C2×C3—36no—
268C2×C3—61no—
269C2×C3—66no—
270C2×C3—364no—
271C2×C3C2×#371616yes—
272C2×C3C22×#62611yescharacteristic
273D3—124no—
274D3—364no—
275C22—184no—
276C22—3612no—
277C22#8843814yesdirect product factor
278C22C2×#620013yes—
279C4—184no—
280C4C2×#620013yes—
281C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
282C3—31no—
283C3—31no—
284C3—61no—
285C3C22×#371611yescharacteristic
286C2—364no—
287C2C2×#8843816yesdirect product factor
288C2C22×#620011yescharacteristic
subgroupG/H
centerC23#6200
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3#626
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
1. C22×#6229C2semidirect
3. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. C2×#88438C2direct (N × H)
11. C2×#6229C22semidirect
14. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
15. #88438C22direct (N × H)
21. C22×C3⋊(C3⋊C8)D3semidirect
32. #6229C23semidirect
35. (C3⋊(C3⋊C8))⋊C3C23semidirect
57. C2×C3⋊(C3⋊C8)C2×D3semidirect
61. C22×H(3,𝔽3)SD16semidirect
107. C3⋊(C3⋊C8)C22×D3semidirect
110. C2×H(3,𝔽3)C2×SD16semidirect
146. C22×C32#204semidirect
147. C22×C32#214semidirect
167. H(3,𝔽3)C22×SD16semidirect
209. C2×C32C2×#204semidirect
210. C2×C32C2×#214semidirect
251. C32C22×#204semidirect
252. C32C22×#214semidirect
277. C22#88438direct (N × H)
287. C2C2×#88438direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62292
C22×C4⋊((C3⋊D3)⋊C3)2
C22×(C3⋊(C3⋊C8))⋊C32
C2×#884382
C22×#37173
C22×#37153