Finite Group Databasebeta

← C2×#11721766 #13153861

Subgroups — C2×#11721766 #13153861

Order 1728.

1353 conjugacy classes of subgroups; only the 261 normal subgroups are shown.

normal subgroups only

This group has 1353 conjugacy classes of subgroups — too many to draw. Only its 261 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
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88542C211yesmaximal, characteristic
3C2×#88542C211yesmaximal, characteristic
4C2×#88542C211yesmaximal, characteristic
5C2×C4⋊#1084C211yesmaximal, characteristic, complemented
6C2×#88540C211yesmaximal, characteristic, complemented
7C2×C4×#1084C211yesmaximal, characteristic, complemented
8#11721766C218yesmaximal, direct product factor
9C2×C3×#898—34nomaximal
10C23×#272C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×#1084C2211yescharacteristic
15C22×#1084C2211yescharacteristic
16C22×#1084C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2214yescomplemented
18#88542C2214yes—
19#88542C2214yes—
20#88542C2214yes—
21C4⋊#1084C2214yescomplemented
22#88540C2214yescomplemented
23C4×#1084C2214yescomplemented
24C2×C32×C4⋊C4D314yescomplemented
25C2×C3×(C2×D3)⋊C4—34no—
26C2×C3×(C2×D3)⋊C4—34no—
27C2×C3×(C2×D3)⋊C4—34no—
28C2×C3×C4⋊Dic3—34no—
29C2×C3×Dic3⋊C4—34no—
30C2×C3×C4×Dic3—34no—
31C3×#898—332no—
32C23×H(3,𝔽3)C2311yescharacteristic
33C22×#272—22no—
34C22×#272—22no—
35C22×#272—22no—
36C22×#272C2312yes—
37C2×C4×H(3,𝔽3)—22no—
38C2×C4×H(3,𝔽3)—22no—
39C2×C4×H(3,𝔽3)—22no—
40C2×C4×H(3,𝔽3)C2312yes—
41C2×C4×H(3,𝔽3)C2312yes—
42C2×C4×H(3,𝔽3)C2312yes—
43C2×#1084—22no—
44C2×#1084—22no—
45C2×#1084—22no—
46C2×#1084C2312yes—
47C2×#1084C2312yes—
48C2×#1084C2312yes—
49C2×C3×#126—91no—
50C2×#898—34no—
51C23×C3×D3—34no—
52C22×C32×C4C2×D314yes—
53C22×C32×C4C2×D314yes—
54C22×C32×C4C2×D314yes—
55C22×C3×Dic3—34no—
56C22×C3×Dic3—34no—
57C22×C3×Dic3—34no—
58C32×C4⋊C4C2×D3116yescomplemented
59C3×(C2×D3)⋊C4—316no—
60C3×(C2×D3)⋊C4—316no—
61C3×(C2×D3)⋊C4—316no—
62C3×C4⋊Dic3—316no—
63C3×Dic3⋊C4—316no—
64C3×C4×Dic3—316no—
65C22×H(3,𝔽3)#4112yes—
66C22×H(3,𝔽3)#4112yes—
67C22×H(3,𝔽3)#4112yes—
68C22×H(3,𝔽3)C2411yescharacteristic
69C2×#272—22no—
70C2×#272—22no—
71C2×#272—22no—
72C2×#272—44no—
73C4×H(3,𝔽3)—22no—
74C4×H(3,𝔽3)—22no—
75C4×H(3,𝔽3)—22no—
76#1084—22no—
77#1084—22no—
78#1084—22no—
79C3×#126—98no—
80C2×C3×C4⋊C4—34no—
81C2×C3×C4⋊C4—91no—
82C2×C3×C4⋊C4—91no—
83C2×C3×C4⋊C4C3⋊D311yescharacteristic
84C2×C3×C22⋊C4—91no—
85C2×C3×C22⋊C4—91no—
86C2×C3×C22⋊C4—91no—
87C2×C3×C42—91no—
88C2×(C2×D3)⋊C4—34no—
89C2×(C2×D3)⋊C4—34no—
90C2×(C2×D3)⋊C4—34no—
91C2×C4⋊Dic3—34no—
92C2×Dic3⋊C4—34no—
93C2×C4×Dic3—34no—
94#898—332no—
95C23×C32C22×D314yes—
96C22×C3×D3—38no—
97C22×C3×D3—68no—
98C22×C3×D3—68no—
99C22×C3×D3—68no—
100C2×C32×C4—28no—
101C2×C32×C4—28no—
102C2×C32×C4—28no—
103C2×C32×C4C22×D318yes—
104C2×C32×C4C22×D318yes—
105C2×C32×C4C22×D318yes—
106C2×C3×Dic3—38no—
107C2×C3×Dic3—38no—
108C2×C3×Dic3—38no—
109C2×C3×Dic3—68no—
110C2×C3×Dic3—68no—
111C2×C3×Dic3—68no—
112C2×#126—91noSylow
113C2×H(3,𝔽3)#12614yescomplemented
114C2×H(3,𝔽3)C2×#4111yescharacteristic
115C2×H(3,𝔽3)C2×#4111yescharacteristic
116C2×H(3,𝔽3)C2×#4111yescharacteristic
117#272—42no—
118C24×C3—91no—
119C23×D3—34no—
120C22×C3×C4—34no—
121C22×C3×C4—34no—
122C22×C3×C4—34no—
123C22×C3×C4—91no—
124C22×C3×C4—91no—
125C22×C3×C4—91no—
126C22×C3×C4C2×C3⋊D311yescharacteristic
127C22×C3×C4C2×C3⋊D311yescharacteristic
128C22×C3×C4C2×C3⋊D311yescharacteristic
129C22×Dic3—34no—
130C22×Dic3—34no—
131C22×Dic3—34no—
132C3×C4⋊C4—316no—
133C3×C4⋊C4—94no—
134C3×C4⋊C4—94no—
135C3×C4⋊C4C2×C3⋊D314yes—
136C3×C22⋊C4—94no—
137C3×C22⋊C4—94no—
138C3×C22⋊C4—94no—
139C3×C42—94no—
140(C2×D3)⋊C4—316no—
141(C2×D3)⋊C4—316no—
142(C2×D3)⋊C4—316no—
143C4⋊Dic3—316no—
144Dic3⋊C4—316no—
145C4×Dic3—316no—
146C22×C32#23518yes—
147C22×C32#23718yes—
148C22×C32#23918yes—
149C22×C32C23×D314yes—
150C2×C3×D3—68no—
151C2×C3×D3—68no—
152C2×C3×D3—68no—
153C2×C3×D3—1216no—
154C32×C4—28no—
155C32×C4—28no—
156C32×C4—28no—
157C3×Dic3—68no—
158C3×Dic3—68no—
159C3×Dic3—68no—
160#126—98no—
161C2×C4⋊C4—91no—
162C2×C4⋊C4—91no—
163C2×C4⋊C4#27211yescharacteristic, complemented
164C2×C22⋊C4—91no—
165C2×C22⋊C4—91no—
166C2×C22⋊C4—91no—
167C2×C42—91no—
168H(3,𝔽3)C2×#12611yescharacteristic, complemented, Sylow
169C23×C3—34no—
170C23×C3—92no—
171C23×C3—182no—
172C23×C3—182no—
173C23×C3—182no—
174C23×C3C22×C3⋊D311yescharacteristic
175C22×D3—38no—
176C22×D3—68no—
177C22×D3—68no—
178C22×D3—68no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—38no—
183C2×C3×C4—38no—
184C2×C3×C4—38no—
185C2×C3×C4—68no—
186C2×C3×C4—68no—
187C2×C3×C4—68no—
188C2×C3×C4—92no—
189C2×C3×C4—92no—
190C2×C3×C4—92no—
191C2×C3×C4—182no—
192C2×C3×C4—182no—
193C2×C3×C4—182no—
194C2×C3×C4C22×C3⋊D312yes—
195C2×C3×C4C22×C3⋊D312yes—
196C2×C3×C4C22×C3⋊D312yes—
197C2×Dic3—38no—
198C2×Dic3—38no—
199C2×Dic3—38no—
200C2×Dic3—68no—
201C2×Dic3—68no—
202C2×Dic3—68no—
203C2×C32#898116yescomplemented
204C2×C32C2×#23514yes—
205C2×C32C2×#23714yes—
206C2×C32C2×#23914yes—
207C3×D3—128no—
208C24—91no—
209C22×C4—91no—
210C22×C4—91no—
211C22×C4—91no—
212C22×C4C2×#27211yescharacteristic
213C22×C4C2×#27211yescharacteristic
214C22×C4C2×#27211yescharacteristic
215C4⋊C4—94no—
216C4⋊C4—94no—
217C4⋊C4C2×#27214yescomplemented
218C22⋊C4—94no—
219C22⋊C4—94no—
220C22⋊C4—94no—
221C42—94no—
222C22×C3—34no—
223C22×C3—38no—
224C22×C3—38no—
225C22×C3—38no—
226C22×C3—182no—
227C22×C3—182no—
228C22×C3—182no—
229C22×C3—364no—
230C22×C3#382812yes—
231C22×C3#383012yes—
232C22×C3#383212yes—
233C22×C3C23×C3⋊D311yescharacteristic
234C2×D3—68no—
235C2×D3—68no—
236C2×D3—68no—
237C2×D3—1216no—
238C3×C4—22no—
239C3×C4—22no—
240C3×C4—22no—
241C3×C4—68no—
242C3×C4—68no—
243C3×C4—68no—
244C3×C4—182no—
245C3×C4—182no—
246C3×C4—182no—
247Dic3—68no—
248Dic3—68no—
249Dic3—68no—
250C32C2×#89814yescomplemented
251C23—92no—
252C23—182no—
253C23—182no—
254C23—182no—
255C23C22×#27211yescharacteristic
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—92no—
260C2×C4—92no—
261C2×C4—92no—
262C2×C4—182no—
263C2×C4—182no—
264C2×C4—182no—
265C2×C4C22×#27212yes—
266C2×C4C22×#27212yes—
267C2×C4C22×#27212yes—
268C2×C3—316no—
269C2×C3—34no—
270C2×C3—34no—
271C2×C3—34no—
272C2×C3—362no—
273C2×C3#6410714yes—
274C2×C3C2×#382811yescharacteristic
275C2×C3C2×#383011yescharacteristic
276C2×C3C2×#383211yescharacteristic
277D3—128no—
278C22—182no—
279C22—182no—
280C22—182no—
281C22—364no—
282C22#8874012yes—
283C22#8874412yes—
284C22#8874812yes—
285C22C23×#27211yescharacteristic
286C4—22no—
287C4—22no—
288C4—22no—
289C4—182no—
290C4—182no—
291C4—182no—
292C3—34no—
293C3C2×#6410711yescharacteristic
294C2—362no—
295C2#1172176614yesdirect product factor
296C2C2×#8874011yescharacteristic
297C2C2×#8874411yescharacteristic
298C2C2×#8874811yescharacteristic
subgroupG/H
center, socleC23×C3C22×C3⋊D3
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C23×C3⋊D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC23×#272C22
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterH(3,𝔽3)C2×#126
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×H(3,𝔽3)C2semidirect
5. C2×C4⋊#1084C2semidirect
6. C2×#88540C2semidirect
7. C2×C4×#1084C2semidirect
8. #11721766C2direct (N × H)
17. C4⋊C4×H(3,𝔽3)C22semidirect
21. C4⋊#1084C22semidirect
22. #88540C22semidirect
23. C4×#1084C22semidirect
24. C2×C32×C4⋊C4D3semidirect
58. C32×C4⋊C4C2×D3semidirect
113. C2×H(3,𝔽3)#126semidirect
163. C2×C4⋊C4#272semidirect
168. H(3,𝔽3)C2×#126semidirect
203. C2×C32#898semidirect
217. C4⋊C4C2×#272semidirect
250. C32C2×#898semidirect
295. C2#11721766direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
C2×#885422
C2×C4⋊#10842
C2×#885402
C2×C4×#10842
#117217662
C2×C3×#8983