Finite Group Databasebeta

← C2×#11721702 #13153764

Subgroups — C2×#11721702 #13153764

Order 1728.

3065 conjugacy classes of subgroups; only the 372 normal subgroups are shown.

normal subgroups only

This group has 3065 conjugacy classes of subgroups — too many to draw. Only its 372 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
1C25×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C22×#638⋊C3C213yesmaximal, complemented
3#11721702C218yesmaximal, direct product factor
4C2×#3757⋊C3C213yesmaximal, complemented
5C2×#64162C311yesmaximal, characteristic, complemented
6C2×C3×#953—31nomaximal
7C24×H(3,𝔽3)—212no—
8C24×H(3,𝔽3)C2213yes—
9C24×H(3,𝔽3)C2214yescomplemented
10C23×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
11C2×#638⋊C3—212no—
12C2×#638⋊C3C22112yescomplemented
13C22×(C3⋊Dic3)⋊C3C2213yescomplemented
14#3757⋊C3C22112yescomplemented
15C25×C32—21no—
16C25×C32D311yescharacteristic, complemented
17C25×C32C2×C311yescharacteristic, complemented
18C22×#638C2×C313yescomplemented
19C22×C3×#67—33no—
20#64162C2×C318yescomplemented
21C2×#3757C2×C313yescomplemented
22C3×#953—38no—
23C2×C3×#217—33no—
24C23×H(3,𝔽3)—232no—
25C23×H(3,𝔽3)—212no—
26C23×H(3,𝔽3)—224no—
27C23×H(3,𝔽3)D4112yescomplemented
28C23×H(3,𝔽3)C2316yes—
29C23×H(3,𝔽3)C2311yescharacteristic
30C22×(C3⋊D3)⋊C3—26no—
31C22×(C3⋊D3)⋊C3C2312yescomplemented
32#638⋊C3—224no—
33C2×(C3⋊Dic3)⋊C3—26no—
34C2×(C3⋊Dic3)⋊C3C2316yescomplemented
35C2×C3×#120—91no—
36C2×#953—31no—
37C2×#953—91no—
38C24×C32—224no—
39C24×C32—23no—
40C24×C32—24no—
41C24×C32—212no—
42C24×C32—212no—
43C24×C32C2×D313yes—
44C24×C32C2×D314yescomplemented
45C24×C32C22×C313yes—
46C24×C32C22×C314yescomplemented
47C23×C3⋊D3C22×C311yescharacteristic, complemented
48C23×C3×D3—31no—
49C2×#638—212no—
50C2×#638C22×C3112yescomplemented
51C22×C3⋊Dic3C22×C313yescomplemented
52C2×C3×#67—312no—
53C2×C3×#67—612no—
54C22×C3×Dic3—33no—
55#3757C22×C3112yescomplemented
56C3×#217—312no—
57C22×H(3,𝔽3)—232no—
58C22×H(3,𝔽3)—224no—
59C22×H(3,𝔽3)—212no—
60C22×H(3,𝔽3)C2×D416yes—
61C22×H(3,𝔽3)C2×D4112yescomplemented
62C22×H(3,𝔽3)C2411yescharacteristic
63C2×(C3⋊D3)⋊C3—26no—
64C2×(C3⋊D3)⋊C3—44no—
65(C3⋊Dic3)⋊C3—26no—
66C25×C3—31no—
67C25×C3—31no—
68C25×C3—61no—
69C25×C3C3×D311yescharacteristic
70C22×C3×D4—93no—
71C22×#67—33no—
72C22×#67—93no—
73C3×#120—98no—
74C2×C3×C22⋊C4—93no—
75#953—38no—
76#953—98no—
77C2×#217—33no—
78C2×#217—93no—
79C23×C32—264no—
80C23×C32—232no—
81C23×C32—232no—
82C23×C32—224no—
83C23×C32—248no—
84C23×C32—212no—
85C23×C32—224no—
86C23×C32—212no—
87C23×C32—224no—
88C23×C32—212no—
89C23×C32—26no—
90C23×C32—21no—
91C23×C32#67112yescomplemented
92C23×C32C3×D4112yescomplemented
93C23×C32C22×D316yes—
94C23×C32C22×D311yescharacteristic
95C23×C32C23×C316yes—
96C23×C32C23×C311yescharacteristic
97C22×C3⋊D3—26no—
98C22×C3⋊D3C23×C312yescomplemented
99C22×C3×D3—32no—
100C22×C3×D3—66no—
101#638—224no—
102C2×C3⋊Dic3—26no—
103C2×C3⋊Dic3C23×C316yescomplemented
104C3×#67—624no—
105C2×C3×Dic3—36no—
106C2×C3×Dic3—66no—
107C2×#120—91noSylow
108C2×H(3,𝔽3)—212no—
109C2×H(3,𝔽3)#12014yescomplemented
110C2×H(3,𝔽3)C22×D413yes—
111(C3⋊D3)⋊C3—42no—
112C24×C3—212no—
113C24×C3—33no—
114C24×C3—33no—
115C24×C3—34no—
116C24×C3—34no—
117C24×C3—612no—
118C24×C3—612no—
119C24×C3—624no—
120C24×C3—63no—
121C24×C3—64no—
122C24×C3—91no—
123C24×C3C2×C3×D313yes—
124C24×C3C2×C3×D314yes—
125C23×D3—31no—
126C23×D3—91no—
127C2×C3×D4—912no—
128C2×C3×D4—1812no—
129C22×C3×C4—93no—
130C2×#67—312no—
131C2×#67—612no—
132C2×#67—912no—
133C2×#67—1812no—
134C22×Dic3—33no—
135C22×Dic3—93no—
136C3×C22⋊C4—912no—
137#217—312no—
138#217—912no—
139C22×C32—264no—
140C22×C32—232no—
141C22×C32—232no—
142C22×C32—248no—
143C22×C32—224no—
144C22×C32—224no—
145C22×C32—224no—
146C22×C32—212no—
147C22×C32—212no—
148C22×C32—21no—
149C22×C32—26no—
150C22×C32—212no—
151C22×C32C2×#6716yes—
152C22×C32C2×#67112yescomplemented
153C22×C32C2×C3×D416yes—
154C22×C32C2×C3×D4112yescomplemented
155C22×C32C23×D311yescharacteristic
156C22×C32C24×C311yescharacteristic
157C2×C3⋊D3—26no—
158C2×C3⋊D3—44no—
159C2×C3×D3—66no—
160C2×C3×D3—124no—
161C3⋊Dic3—26no—
162C3×Dic3—66no—
163C25(C3⋊D3)⋊C311yescharacteristic, complemented
164C22×D4—93no—
165#120—98no—
166C2×C22⋊C4—93no—
167H(3,𝔽3)C2×#12011yescharacteristic, complemented, Sylow
168C23×C3—232no—
169C23×C3—212no—
170C23×C3—224no—
171C23×C3—312no—
172C23×C3—312no—
173C23×C3—36no—
174C23×C3—36no—
175C23×C3—31no—
176C23×C3—31no—
177C23×C3—612no—
178C23×C3—612no—
179C23×C3—624no—
180C23×C3—624no—
181C23×C3—632no—
182C23×C3—632no—
183C23×C3—664no—
184C23×C3—624no—
185C23×C3—648no—
186C23×C3—612no—
187C23×C3—66no—
188C23×C3—61no—
189C23×C3—92no—
190C23×C3—186no—
191C23×C3C3×#67112yes—
192C23×C3C22×C3×D316yes—
193C23×C3C22×C3×D311yescharacteristic
194C22×D3—32no—
195C22×D3—66no—
196C22×D3—92no—
197C22×D3—186no—
198C3×D4—1824no—
199C2×C3×C4—96no—
200C2×C3×C4—186no—
201#67—624no—
202#67—1824no—
203C2×Dic3—36no—
204C2×Dic3—66no—
205C2×Dic3—96no—
206C2×Dic3—186no—
207C2×C32—224no—
208C2×C32—212no—
209C2×C32—212no—
210C2×C32—24no—
211C2×C32—23no—
212C2×C32#95314yescomplemented
213C2×C32C3×#12014yescomplemented
214C2×C32C22×#6713yes—
215C2×C32C22×C3×D413yes—
216C3⋊D3—42no—
217C3×D3—122no—
218C24—212no—
219C24—91no—
220C24C2×(C3⋊D3)⋊C313yes—
221C24C2×(C3⋊D3)⋊C314yescomplemented
222C2×D4—912no—
223C2×D4—1812no—
224C22×C4—93no—
225C22⋊C4—912no—
226C22×C3—232no—
227C22×C3—224no—
228C22×C3—212no—
229C22×C3—31no—
230C22×C3—31no—
231C22×C3—36no—
232C22×C3—36no—
233C22×C3—312no—
234C22×C3—312no—
235C22×C3—612no—
236C22×C3—612no—
237C22×C3—624no—
238C22×C3—624no—
239C22×C3—632no—
240C22×C3—632no—
241C22×C3—664no—
242C22×C3—624no—
243C22×C3—648no—
244C22×C3—61no—
245C22×C3—66no—
246C22×C3—612no—
247C22×C3—186no—
248C22×C3—364no—
249C22×C3C2×C3×#6716yes—
250C22×C3C2×C3×#67112yes—
251C22×C3C23×C3×D311yescharacteristic
252C2×D3—66no—
253C2×D3—124no—
254C2×D3—186no—
255C2×D3—364no—
256C3×C4—186no—
257Dic3—66no—
258Dic3—186no—
259C32—21no—
260C32C2×#95311yescharacteristic, complemented
261C32C2×C3×#12011yescharacteristic, complemented
262C23—232no—
263C23—212no—
264C23—224no—
265C23—92no—
266C23—186no—
267C23#638⋊C3112yescomplemented
268C23C22×(C3⋊D3)⋊C316yes—
269C23C22×(C3⋊D3)⋊C311yescharacteristic
270D4—1824no—
271C2×C4—96no—
272C2×C4—186no—
273C2×C3—212no—
274C2×C3—33no—
275C2×C3—33no—
276C2×C3—34no—
277C2×C3—34no—
278C2×C3—612no—
279C2×C3—612no—
280C2×C3—624no—
281C2×C3—63no—
282C2×C3—64no—
283C2×C3—362no—
284C2×C3C3×#95314yes—
285C2×C3C22×C3×#6713yes—
286D3—122no—
287D3—362no—
288C22—232no—
289C22—212no—
290C22—224no—
291C22—186no—
292C22—364no—
293C22C2×#638⋊C316yes—
294C22C2×#638⋊C3112yescomplemented
295C22C23×(C3⋊D3)⋊C311yescharacteristic
296C4—186no—
297C3—31no—
298C3—31no—
299C3—61no—
300C3C2×C3×#95311yescharacteristic
301C2—212no—
302C2—362no—
303C2#1172170214yesdirect product factor
304C2C22×#638⋊C313yes—
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
FrattiniC22×C3C23×C3×D3
FittingC25×H(3,𝔽3)C2
socleC23×C3C22×C3×D3
hypercenterC25(C3⋊D3)⋊C3
hypocenterC32C2×C3×#120
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C25

Complements

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

NH ≅ G/Ntype
1. C25×H(3,𝔽3)C2semidirect
2. C22×#638⋊C3C2semidirect
3. #11721702C2direct (N × H)
4. C2×#3757⋊C3C2semidirect
5. C2×#64162C3semidirect
9. C24×H(3,𝔽3)C22semidirect
10. C23×(C3⋊D3)⋊C3C22semidirect
12. C2×#638⋊C3C22semidirect
13. C22×(C3⋊Dic3)⋊C3C22semidirect
14. #3757⋊C3C22semidirect
16. C25×C32D3semidirect
17. C25×C32C2×C3semidirect
18. C22×#638C2×C3semidirect
20. #64162C2×C3semidirect
21. C2×#3757C2×C3semidirect
27. C23×H(3,𝔽3)D4semidirect
31. C22×(C3⋊D3)⋊C3C23semidirect
34. C2×(C3⋊Dic3)⋊C3C23semidirect
44. C24×C32C2×D3semidirect
46. C24×C32C22×C3semidirect
47. C23×C3⋊D3C22×C3semidirect
50. C2×#638C22×C3semidirect
51. C22×C3⋊Dic3C22×C3semidirect
55. #3757C22×C3semidirect
61. C22×H(3,𝔽3)C2×D4semidirect
91. C23×C32#67semidirect
92. C23×C32C3×D4semidirect
98. C22×C3⋊D3C23×C3semidirect
103. C2×C3⋊Dic3C23×C3semidirect
109. C2×H(3,𝔽3)#120semidirect
152. C22×C32C2×#67semidirect
154. C22×C32C2×C3×D4semidirect
163. C25(C3⋊D3)⋊C3semidirect
167. H(3,𝔽3)C2×#120semidirect
212. C2×C32#953semidirect
213. C2×C32C3×#120semidirect
221. C24C2×(C3⋊D3)⋊C3semidirect
260. C32C2×#953semidirect
261. C32C2×C3×#120semidirect
267. C23#638⋊C3semidirect
294. C22C2×#638⋊C3semidirect
303. C2#11721702direct (N × H)

Maximal subgroups

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

subgroupindex
C25×H(3,𝔽3)2
C22×#638⋊C32
#117217022
C2×#3757⋊C32
C2×#641623
C2×C3×#9533