Finite Group Databasebeta

← C2×#11721098 #13150620

Subgroups — C2×#11721098 #13150620

Order 1728.

899 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 899 conjugacy classes of subgroups — too many to draw. Only its 187 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×#1084C211yesmaximal, characteristic
2C2×#3750⋊C3C212yesmaximal
3C2×#88449C211yesmaximal, characteristic
4C2×#88449C211yesmaximal, characteristic
5C2×#88448C212yesmaximal
6#11721098C218yesmaximal, direct product factor
7C2×#63867—32nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C4×#1084C2214yes—
14#3750⋊C3C2218yes—
15#88449C2214yes—
16#88449C2214yes—
17#88448C2218yes—
18C2×#3750D312yes—
19C2×C3×Dic3⋊C4—32no—
20C2×C3×C4×Dic3—32no—
21C2×C3×C4×Dic3—61no—
22C2×#3724—32no—
23C2×Dic3⋊Dic3—32no—
24C2×Dic3⋊Dic3—32no—
25C2×Dic3⋊Dic3—32no—
26#63867—316no—
27C23×H(3,𝔽3)C2311yescharacteristic
28C2×C4×H(3,𝔽3)—22no—
29C2×C4×H(3,𝔽3)C2312yes—
30C2×#1084—22no—
31C2×#1084Q814yes—
32C2×#1084C2312yes—
33C2×#1084C2312yes—
34C2×(C3⋊Dic3)⋊C3—24no—
35C2×(C3⋊Dic3)⋊C3—24no—
36C2×(C3⋊Dic3)⋊C3C2314yes—
37C2×(C3⋊Dic3)⋊C3C2314yes—
38C2×#889—92no—
39C2×#870—91no—
40C22×C32×C4—21no—
41C22×C32×C4C2×D312yes—
42C22×C3⋊Dic3C2×D312yes—
43C22×C3⋊Dic3C2×D312yes—
44C22×C3×Dic3—32no—
45C22×C3×Dic3—32no—
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—61no—
49C22×C3×Dic3—61no—
50#3750C2×D318yes—
51C3×Dic3⋊C4—38no—
52C3×C4×Dic3—38no—
53C3×C4×Dic3—64no—
54#3724—38no—
55Dic3⋊Dic3—38no—
56Dic3⋊Dic3—38no—
57Dic3⋊Dic3—38no—
58C22×H(3,𝔽3)C2×Q812yes—
59C22×H(3,𝔽3)#4112yes—
60C22×H(3,𝔽3)#4112yes—
61C22×H(3,𝔽3)C2411yescharacteristic
62C4×H(3,𝔽3)—22no—
63#1084—22no—
64#1084C2×Q814yes—
65(C3⋊Dic3)⋊C3—24no—
66(C3⋊Dic3)⋊C3—24no—
67C2×C3×C4⋊C4—92no—
68C2×C3×C42—91no—
69C2×C4⋊Dic3—91no—
70C2×C4⋊Dic3—91no—
71C2×C4⋊Dic3—92no—
72C2×Dic3⋊C4—32no—
73C2×Dic3⋊C4—92no—
74C2×Dic3⋊C4—92no—
75C2×Dic3⋊C4—92no—
76C2×Dic3⋊C4—92no—
77C2×Dic3⋊C4—92no—
78C2×C4×Dic3—32no—
79C2×C4×Dic3—61no—
80#889—916no—
81#870—98no—
82C23×C32—21no—
83C23×C32C22×D312yes—
84C2×C32×C4—24no—
85C2×C32×C4—24no—
86C2×C32×C4—22no—
87C2×C32×C4C22×D314yes—
88C2×C3⋊Dic3—24no—
89C2×C3⋊Dic3—24no—
90C2×C3⋊Dic3C22×D314yes—
91C2×C3⋊Dic3C22×D314yes—
92C2×C3×Dic3—38no—
93C2×C3×Dic3—34no—
94C2×C3×Dic3—34no—
95C2×C3×Dic3—34no—
96C2×C3×Dic3—34no—
97C2×C3×Dic3—64no—
98C2×C3×Dic3—62no—
99C2×C3×Dic3—62no—
100C2×C3×Dic3—64no—
101C2×C3×Dic3—64no—
102C2×C3×Dic3—64no—
103C2×C3×Dic3—64no—
104C2×#125—271noSylow
105C2×H(3,𝔽3)#12514yescomplemented
106C2×H(3,𝔽3)C22×Q811yescharacteristic
107C2×H(3,𝔽3)C2×#4111yescharacteristic
108C2×H(3,𝔽3)C2×#4111yescharacteristic
109C22×C3×C4—32no—
110C22×C3×C4—61no—
111C22×C3×C4—91no—
112C22×C3×C4—92no—
113C22×C3×C4—92no—
114C22×C3×C4—91no—
115C22×C3×C4D3211yescharacteristic
116C22×Dic3—32no—
117C22×Dic3—32no—
118C22×Dic3—32no—
119C22×Dic3—32no—
120C22×Dic3—61no—
121C22×Dic3—61no—
122C22×Dic3—92no—
123C22×Dic3—92no—
124C3×C4⋊C4—98no—
125C3×C42—94no—
126C4⋊Dic3—94no—
127C4⋊Dic3—94no—
128C4⋊Dic3—98no—
129Dic3⋊C4—38no—
130Dic3⋊C4—98no—
131Dic3⋊C4—98no—
132Dic3⋊C4—98no—
133Dic3⋊C4—98no—
134Dic3⋊C4—98no—
135C4×Dic3—38no—
136C4×Dic3—64no—
137C22×C32—21no—
138C22×C32—22no—
139C22×C32—22no—
140C22×C32—22no—
141C22×C32#23514yes—
142C22×C32#23714yes—
143C22×C32D3×Q814yes—
144C22×C32C23×D312yes—
145C32×C4—24no—
146C32×C4—24no—
147C3⋊Dic3—24no—
148C3⋊Dic3—24no—
149C3×Dic3—38no—
150C3×Dic3—64no—
151C3×Dic3—64no—
152C3×Dic3—64no—
153C3×Dic3—64no—
154C3×Dic3—64no—
155#125—278no—
156C2×C4⋊C4—92no—
157C2×C4⋊C4—271no—
158C2×C4⋊C4—271no—
159C2×C4⋊C4—272no—
160C2×C42—91no—
161H(3,𝔽3)C2×#12511yescharacteristic, complemented, Sylow
162C23×C3—32no—
163C23×C3—61no—
164C23×C3C2×D3211yescharacteristic
165C2×C3×C4—22no—
166C2×C3×C4—34no—
167C2×C3×C4—64no—
168C2×C3×C4—64no—
169C2×C3×C4—62no—
170C2×C3×C4—92no—
171C2×C3×C4—94no—
172C2×C3×C4—94no—
173C2×C3×C4—94no—
174C2×C3×C4—92no—
175C2×C3×C4—184no—
176C2×C3×C4—184no—
177C2×C3×C4—182no—
178C2×C3×C4C2×D3212yes—
179C2×Dic3—34no—
180C2×Dic3—38no—
181C2×Dic3—34no—
182C2×Dic3—34no—
183C2×Dic3—34no—
184C2×Dic3—62no—
185C2×Dic3—62no—
186C2×Dic3—64no—
187C2×Dic3—64no—
188C2×Dic3—64no—
189C2×Dic3—64no—
190C2×Dic3—64no—
191C2×Dic3—94no—
192C2×Dic3—94no—
193C2×Dic3—184no—
194C2×Dic3—184no—
195C2×C32—21no—
196C2×C32—21no—
197C2×C32—21no—
198C2×C32—24no—
199C2×C32#88918yescomplemented
200C2×C32C2×#23512yes—
201C2×C32C2×#23712yes—
202C2×C32C2×D3×Q812yes—
203C22×C4—91no—
204C22×C4—91no—
205C22×C4—92no—
206C22×C4—92no—
207C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
208C4⋊C4—98no—
209C4⋊C4—274no—
210C4⋊C4—274no—
211C4⋊C4—278no—
212C42—94no—
213C22×C3—32no—
214C22×C3—34no—
215C22×C3—34no—
216C22×C3—34no—
217C22×C3—61no—
218C22×C3—62no—
219C22×C3—62no—
220C22×C3—62no—
221C22×C3#379712yes—
222C22×C3#379812yes—
223C22×C3#380512yes—
224C22×C3C22×D3211yescharacteristic
225C3×C4—22no—
226C3×C4—64no—
227C3×C4—64no—
228C3×C4—94no—
229C3×C4—184no—
230C3×C4—184no—
231C3×C4—182no—
232Dic3—38no—
233Dic3—64no—
234Dic3—64no—
235Dic3—64no—
236Dic3—64no—
237Dic3—64no—
238Dic3—184no—
239Dic3—184no—
240C32—21no—
241C32C2×#88912yescomplemented
242C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
243C2×C4—22no—
244C2×C4—92no—
245C2×C4—94no—
246C2×C4—92no—
247C2×C4—94no—
248C2×C4—94no—
249C2×C4—182no—
250C2×C4—184no—
251C2×C4—184no—
252C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
253C2×C3—32no—
254C2×C3—38no—
255C2×C3—32no—
256C2×C3—32no—
257C2×C3—61no—
258C2×C3—64no—
259C2×C3—61no—
260C2×C3—61no—
261C2×C3#6387014yes—
262C2×C3C2×#379711yescharacteristic
263C2×C3C2×#379811yescharacteristic
264C2×C3C2×#380511yescharacteristic
265C22#8865012yes—
266C22#8865112yes—
267C22#8867212yes—
268C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
269C4—22no—
270C4—94no—
271C4—184no—
272C4—182no—
273C4—184no—
274C3—32no—
275C3—61no—
276C3C2×#6387011yescharacteristic
277C2#1172109814yesdirect product factor
278C2C2×#8865011yescharacteristic
279C2C2×#8865111yescharacteristic
280C2C2×#8867211yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#125
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #11721098C2direct (N × H)
105. C2×H(3,𝔽3)#125semidirect
161. H(3,𝔽3)C2×#125semidirect
199. C2×C32#889semidirect
241. C32C2×#889semidirect
277. C2#11721098direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×#10842
C2×#3750⋊C32
C2×#884492
C2×#884482
#117210982
C2×#638673