Finite Group Databasebeta

← C2×#11721765 #13153858

Subgroups — C2×#11721765 #13153858

Order 1728.

1771 conjugacy classes of subgroups; only the 325 normal subgroups are shown.

normal subgroups only

This group has 1771 conjugacy classes of subgroups — too many to draw. Only its 325 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×#272C211yesmaximal, characteristic
2C22×#6229C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#88542C212yesmaximal
5C2×C4⋊#1084C212yesmaximal, complemented
6#11721765C218yesmaximal, direct product factor
7C2×C3×#897—34nomaximal
8C23×#272C2211yescharacteristic
9C22×C4×H(3,𝔽3)C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×#1084C2211yescharacteristic
12C22×#1084C2212yes—
13C2×C4×#272—24no—
14C2×C4×#272C2214yes—
15C2×#6229—24no—
16C2×#6229C2214yescomplemented
17C4⋊C4×H(3,𝔽3)C2214yescomplemented
18#88542C2218yes—
19C4⋊#1084C2218yescomplemented
20C22×C3×C4×D3—34no—
21C22×C3×Dic6—34no—
22C2×C32×C4⋊C4D314yescomplemented
23C2×C3×(C2×D3)⋊C4—38no—
24C2×C3×C4⋊Dic3—38no—
25C3×#897—332no—
26C23×H(3,𝔽3)C2311yescharacteristic
27C22×#272—22no—
28C22×#272—22no—
29C22×#272Q814yescomplemented
30C22×#272C2312yes—
31C2×C4×H(3,𝔽3)—24no—
32C2×C4×H(3,𝔽3)D414yes—
33C2×C4×H(3,𝔽3)C2314yes—
34C2×C4×H(3,𝔽3)C2312yes—
35C2×#1084—24no—
36C2×#1084—22no—
37C2×#1084C2314yes—
38C2×#1084C2312yes—
39C4×#272—28no—
40#6229—28no—
41C2×C3×C22⋊Q8—91no—
42C2×#897—34no—
43C23×C3×D3—34no—
44C22×C32×C4C2×D318yes—
45C22×C32×C4C2×D314yes—
46C22×C3×Dic3—34no—
47C22×C3×Dic3—38no—
48C2×C3×C4×D3—316no—
49C2×C3×C4×D3—616no—
50C2×C3×Dic6—316no—
51C2×C3×Dic6—616no—
52C32×C4⋊C4C2×D3116yescomplemented
53C3×(C2×D3)⋊C4—332no—
54C3×C4⋊Dic3—332no—
55C22×H(3,𝔽3)C2×D412yes—
56C22×H(3,𝔽3)C2×Q812yes—
57C22×H(3,𝔽3)#4112yes—
58C22×H(3,𝔽3)C2411yescharacteristic
59C2×#272—28no—
60C2×#272—22no—
61C2×#272—22no—
62C2×#272C2×Q814yescomplemented
63C4×H(3,𝔽3)—24no—
64C4×H(3,𝔽3)C2×D414yes—
65#1084—24no—
66#1084—22no—
67C22×C3×Q8—91no—
68C23×C3×C4—91no—
69C22×C4×D3—34no—
70C22×Dic6—34no—
71C3×C22⋊Q8—98no—
72C2×C3×C4⋊C4—34no—
73C2×C3×C4⋊C4—92no—
74C2×C3×C4⋊C4C3⋊D311yescharacteristic
75C2×C3×C22⋊C4—92no—
76C2×(C2×D3)⋊C4—38no—
77C2×C4⋊Dic3—38no—
78#897—332no—
79C23×C32C22×D314yes—
80C22×C3×D3—316no—
81C22×C3×D3—38no—
82C22×C3×D3—68no—
83C22×C3×D3—68no—
84C2×C32×C4—216no—
85C2×C32×C4D12116yes—
86C2×C32×C4C22×D3116yes—
87C2×C32×C4C22×D318yes—
88C2×C3×Dic3—38no—
89C2×C3×Dic3—316no—
90C2×C3×Dic3—616no—
91C2×C3×Dic3—68no—
92C3×C4×D3—632no—
93C3×Dic6—632no—
94C2×C22⋊Q8—91noSylow
95C2×H(3,𝔽3)C22⋊Q814yescomplemented
96C2×H(3,𝔽3)C22×D411yescharacteristic
97C2×H(3,𝔽3)C22×Q811yescharacteristic
98C2×H(3,𝔽3)C2×#4111yescharacteristic
99#272—24no—
100C24×C3—91no—
101C23×D3—34no—
102C2×C3×Q8—94no—
103C2×C3×Q8—184no—
104C22×C3×C4—34no—
105C22×C3×C4—38no—
106C22×C3×C4—92no—
107C22×C3×C4—91no—
108C22×C3×C4—94no—
109C22×C3×C4—184no—
110C22×C3×C4C2×C3⋊D312yes—
111C22×C3×C4C2×C3⋊D311yescharacteristic
112C22×Dic3—34no—
113C22×Dic3—38no—
114C2×C4×D3—316no—
115C2×C4×D3—616no—
116C2×Dic6—316no—
117C2×Dic6—616no—
118C3×C4⋊C4—316no—
119C3×C4⋊C4—98no—
120C3×C4⋊C4C2×C3⋊D314yes—
121C3×C22⋊C4—98no—
122(C2×D3)⋊C4—332no—
123C4⋊Dic3—332no—
124C22×C32C2×D1218yes—
125C22×C32#23718yes—
126C22×C32D3×Q818yes—
127C22×C32C23×D314yes—
128C2×C3×D3—316no—
129C2×C3×D3—632no—
130C2×C3×D3—68no—
131C2×C3×D3—68no—
132C32×C4—216no—
133C32×C4C2×D12116yes—
134C3×Dic3—616no—
135C3×Dic3—68no—
136C22×Q8—91no—
137C23×C4—91no—
138C22⋊Q8—98no—
139C2×C4⋊C4—92no—
140C2×C4⋊C4#27211yescharacteristic, complemented
141C2×C22⋊C4—92no—
142H(3,𝔽3)C2×C22⋊Q811yescharacteristic, complemented, Sylow
143C23×C3—34no—
144C23×C3—92no—
145C23×C3—94no—
146C23×C3—182no—
147C23×C3—182no—
148C23×C3C22×C3⋊D311yescharacteristic
149C22×D3—38no—
150C22×D3—316no—
151C22×D3—68no—
152C22×D3—68no—
153C3×Q8—188no—
154C2×C3×C4—24no—
155C2×C3×C4—316no—
156C2×C3×C4—38no—
157C2×C3×C4—316no—
158C2×C3×C4—616no—
159C2×C3×C4—94no—
160C2×C3×C4—92no—
161C2×C3×C4—182no—
162C2×C3×C4—184no—
163C2×C3×C4—188no—
164C2×C3×C4C4⋊(C3⋊D3)14yes—
165C2×C3×C4C22×C3⋊D314yes—
166C2×C3×C4C22×C3⋊D312yes—
167C2×Dic3—38no—
168C2×Dic3—316no—
169C2×Dic3—616no—
170C2×Dic3—68no—
171C4×D3—632no—
172Dic6—632no—
173C2×C32#897116yescomplemented
174C2×C32C22×D1214yes—
175C2×C32C2×#23714yes—
176C2×C32C2×D3×Q814yes—
177C3×D3—616no—
178C24—91no—
179C2×Q8—94no—
180C2×Q8—184no—
181C22×C4—92no—
182C22×C4—91no—
183C22×C4—94no—
184C22×C4—184no—
185C22×C4C2×#27212yes—
186C22×C4C2×#27211yescharacteristic
187C4⋊C4—98no—
188C4⋊C4C2×#27214yescomplemented
189C22⋊C4—98no—
190C22×C3—34no—
191C22×C3—38no—
192C22×C3—38no—
193C22×C3—38no—
194C22×C3—94no—
195C22×C3—182no—
196C22×C3—182no—
197C22×C3—188no—
198C22×C3C2×C4⋊(C3⋊D3)12yes—
199C22×C3#383012yes—
200C22×C3Q8×C3⋊D312yes—
201C22×C3C23×C3⋊D311yescharacteristic
202C2×D3—316no—
203C2×D3—68no—
204C2×D3—632no—
205C2×D3—68no—
206C3×C4—24no—
207C3×C4—316no—
208C3×C4—616no—
209C3×C4—182no—
210C3×C4—184no—
211C3×C4C2×C4⋊(C3⋊D3)14yes—
212Dic3—616no—
213Dic3—68no—
214C32C2×#89714yescomplemented
215C23—92no—
216C23—94no—
217C23—182no—
218C23—182no—
219C23C22×#27211yescharacteristic
220Q8—188no—
221C2×C4—24no—
222C2×C4—94no—
223C2×C4—92no—
224C2×C4—182no—
225C2×C4—184no—
226C2×C4—188no—
227C2×C4C4⋊#27214yes—
228C2×C4C22×#27214yes—
229C2×C4C22×#27212yes—
230C2×C3—316no—
231C2×C3—34no—
232C2×C3—34no—
233C2×C3—34no—
234C2×C3—184no—
235C2×C3#6410614yes—
236C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
237C2×C3C2×#383011yescharacteristic
238C2×C3C2×Q8×C3⋊D311yescharacteristic
239D3—616no—
240C22—94no—
241C22—188no—
242C22—182no—
243C22—182no—
244C22C2×C4⋊#27212yes—
245C22#8874412yes—
246C22Q8×#27212yes—
247C22C23×#27211yescharacteristic
248C4—24no—
249C4—182no—
250C4—184no—
251C4C2×C4⋊#27214yes—
252C3—34no—
253C3C2×#6410611yescharacteristic
254C2—184no—
255C2#1172176514yesdirect product factor
256C2C22×C4⋊#27211yescharacteristic
257C2C2×#8874411yescharacteristic
258C2C2×Q8×#27211yescharacteristic
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×C22⋊Q8
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
2. C22×#6229C2semidirect
3. C2×C4⋊C4×H(3,𝔽3)C2semidirect
5. C2×C4⋊#1084C2semidirect
6. #11721765C2direct (N × H)
16. C2×#6229C22semidirect
17. C4⋊C4×H(3,𝔽3)C22semidirect
19. C4⋊#1084C22semidirect
22. C2×C32×C4⋊C4D3semidirect
29. C22×#272Q8semidirect
52. C32×C4⋊C4C2×D3semidirect
62. C2×#272C2×Q8semidirect
95. C2×H(3,𝔽3)C22⋊Q8semidirect
140. C2×C4⋊C4#272semidirect
142. H(3,𝔽3)C2×C22⋊Q8semidirect
173. C2×C32#897semidirect
188. C4⋊C4C2×#272semidirect
214. C32C2×#897semidirect
255. C2#11721765direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×#2722
C22×#62292
C2×C4⋊C4×H(3,𝔽3)2
C2×#885422
C2×C4⋊#10842
#117217652
C2×C3×#8973