Finite Group Databasebeta

← C2×#11721078 #13150653

Subgroups — C2×#11721078 #13150653

Order 1728.

1151 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1151 conjugacy classes of subgroups — too many to draw. Only its 251 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×#6196C211yesmaximal, characteristic
2C2×C4×#1084C211yesmaximal, characteristic
3C2×#3750⋊C3C212yesmaximal
4C2×#88449C211yesmaximal, characteristic
5C2×#88447C212yesmaximal
6#11721078C218yesmaximal, direct product factor
7C2×#63837—32nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×#1084C2211yescharacteristic
10C22×#1084C2211yescharacteristic
11C22×(C3⋊Dic3)⋊C3C2212yes—
12C22×(C3⋊Dic3)⋊C3C2212yes—
13C2×#6196C418yescomplemented
14C2×#6196C2214yes—
15C4×#1084C2214yes—
16#3750⋊C3C2218yes—
17#88449C2214yes—
18#88447C2218yes—
19C22×#627—32no—
20C2×#3750D312yes—
21C2×C3×Dic3⋊C4—32no—
22C2×C3×C4×Dic3—32no—
23C2×C3×C4×Dic3—61no—
24C2×Dic3⋊Dic3—32no—
25C2×Dic32—32no—
26C2×Dic32—32no—
27#63837—316no—
28C23×H(3,𝔽3)C2311yescharacteristic
29C2×C4×H(3,𝔽3)—22no—
30C2×C4×H(3,𝔽3)C2312yes—
31C2×#1084C2×C414yes—
32C2×#1084Q814yescomplemented
33C2×#1084C2312yes—
34C2×#1084C2312yes—
35C2×(C3⋊Dic3)⋊C3—24no—
36C2×(C3⋊Dic3)⋊C3C2×C418yes—
37C2×(C3⋊Dic3)⋊C3C2314yes—
38C2×(C3⋊Dic3)⋊C3C2314yes—
39#6196C2×C4116yescomplemented
40C2×#887—92no—
41C2×C4×Dic6—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D312yes—
44C22×C3⋊Dic3C2×D312yes—
45C22×C3⋊Dic3C2×D312yes—
46C22×C3×Dic3—32no—
47C22×C3×Dic3—32no—
48C22×C3×Dic3—32no—
49C22×C3×Dic3—32no—
50C22×C3×Dic3—61no—
51C22×C3×Dic3—61no—
52C2×#627—316no—
53C2×#627—38no—
54#3750C2×D318yes—
55C3×Dic3⋊C4—38no—
56C3×C4×Dic3—38no—
57C3×C4×Dic3—64no—
58Dic3⋊Dic3—38no—
59Dic32—38no—
60Dic32—38no—
61C22×H(3,𝔽3)C22×C412yes—
62C22×H(3,𝔽3)C2×Q812yes—
63C22×H(3,𝔽3)#4112yes—
64C22×H(3,𝔽3)C2411yescharacteristic
65C4×H(3,𝔽3)—22no—
66#1084C22×C414yes—
67#1084C2×Q814yescomplemented
68(C3⋊Dic3)⋊C3—24no—
69(C3⋊Dic3)⋊C3C22×C418yes—
70C22×Dic6—91no—
71C22×Dic6—92no—
72C2×C3×C4⋊C4—92no—
73C2×C3×C42—91no—
74C2×C4⋊Dic3—91no—
75C2×Dic3⋊C4—32no—
76C2×Dic3⋊C4—92no—
77C2×Dic3⋊C4—92no—
78C2×C4×Dic3—32no—
79C2×C4×Dic3—61no—
80C2×C4×Dic3—92no—
81C2×C4×Dic3—92no—
82C2×C4×Dic3—92no—
83#887—916no—
84C4×Dic6—98no—
85C23×C32—21no—
86C23×C32C22×D312yes—
87C2×C32×C4—24no—
88C2×C32×C4—24no—
89C2×C32×C4—22no—
90C2×C32×C4C22×D314yes—
91C2×C3⋊Dic3—24no—
92C2×C3⋊Dic3C4×D318yes—
93C2×C3⋊Dic3C22×D314yes—
94C2×C3⋊Dic3C22×D314yes—
95C2×C3×Dic3—38no—
96C2×C3×Dic3—38no—
97C2×C3×Dic3—38no—
98C2×C3×Dic3—34no—
99C2×C3×Dic3—34no—
100C2×C3×Dic3—34no—
101C2×C3×Dic3—34no—
102C2×C3×Dic3—64no—
103C2×C3×Dic3—64no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C2×C3×Dic3—64no—
107#627—332no—
108C2×C4×Q8—271noSylow
109C2×H(3,𝔽3)C4×Q814yescomplemented
110C2×H(3,𝔽3)C23×C411yescharacteristic
111C2×H(3,𝔽3)C22×Q811yescharacteristic
112C2×H(3,𝔽3)C2×#4111yescharacteristic
113C22×C3×C4—32no—
114C22×C3×C4—61no—
115C22×C3×C4—92no—
116C22×C3×C4—91no—
117C22×C3×C4—91no—
118C22×C3×C4—92no—
119C22×C3×C4D3211yescharacteristic
120C22×Dic3—32no—
121C22×Dic3—32no—
122C22×Dic3—32no—
123C22×Dic3—32no—
124C22×Dic3—61no—
125C22×Dic3—61no—
126C22×Dic3—92no—
127C22×Dic3—92no—
128C2×Dic6—94no—
129C2×Dic6—98no—
130C2×Dic6—98no—
131C2×Dic6—916no—
132C3×C4⋊C4—98no—
133C3×C42—94no—
134C4⋊Dic3—94no—
135Dic3⋊C4—38no—
136Dic3⋊C4—98no—
137Dic3⋊C4—98no—
138C4×Dic3—38no—
139C4×Dic3—64no—
140C4×Dic3—98no—
141C4×Dic3—98no—
142C4×Dic3—98no—
143C22×C32—21no—
144C22×C32—22no—
145C22×C32—22no—
146C22×C32—22no—
147C22×C32C2×C4×D314yes—
148C22×C32#23714yes—
149C22×C32D3×Q814yes—
150C22×C32C23×D312yes—
151C32×C4—24no—
152C32×C4—24no—
153C3⋊Dic3—24no—
154C3⋊Dic3C2×C4×D318yes—
155C3×Dic3—38no—
156C3×Dic3—38no—
157C3×Dic3—38no—
158C3×Dic3—64no—
159C3×Dic3—64no—
160C3×Dic3—64no—
161C22×Q8—271no—
162C4×Q8—278no—
163C2×C4⋊C4—92no—
164C2×C4⋊C4—271no—
165C2×C42—91no—
166C2×C42—272no—
167H(3,𝔽3)C2×C4×Q811yescharacteristic, complemented, Sylow
168C23×C3—32no—
169C23×C3—61no—
170C23×C3C2×D3211yescharacteristic
171C2×C3×C4—22no—
172C2×C3×C4—34no—
173C2×C3×C4—64no—
174C2×C3×C4—64no—
175C2×C3×C4—62no—
176C2×C3×C4—94no—
177C2×C3×C4—98no—
178C2×C3×C4—92no—
179C2×C3×C4—92no—
180C2×C3×C4—94no—
181C2×C3×C4—94no—
182C2×C3×C4—94no—
183C2×C3×C4—184no—
184C2×C3×C4C2×D3212yes—
185C2×Dic3—34no—
186C2×Dic3—34no—
187C2×Dic3—38no—
188C2×Dic3—38no—
189C2×Dic3—34no—
190C2×Dic3—38no—
191C2×Dic3—34no—
192C2×Dic3—62no—
193C2×Dic3—62no—
194C2×Dic3—64no—
195C2×Dic3—64no—
196C2×Dic3—64no—
197C2×Dic3—98no—
198C2×Dic3—94no—
199C2×Dic3—94no—
200C2×Dic3—184no—
201Dic6—916no—
202Dic6—932no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—24no—
207C2×C32#88718yescomplemented
208C2×C32C22×C4×D312yes—
209C2×C32C2×#23712yes—
210C2×C32C2×D3×Q812yes—
211C2×Q8—274no—
212C2×Q8—278no—
213C22×C4—91no—
214C22×C4—91no—
215C22×C4—92no—
216C22×C4—92no—
217C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
218C4⋊C4—98no—
219C4⋊C4—274no—
220C42—94no—
221C42—278no—
222C22×C3—32no—
223C22×C3—34no—
224C22×C3—34no—
225C22×C3—34no—
226C22×C3—61no—
227C22×C3—62no—
228C22×C3—62no—
229C22×C3—62no—
230C22×C3#379712yes—
231C22×C3C4×D3212yes—
232C22×C3#380512yes—
233C22×C3C22×D3211yescharacteristic
234C3×C4—22no—
235C3×C4—64no—
236C3×C4—64no—
237C3×C4—98no—
238C3×C4—94no—
239C3×C4—94no—
240C3×C4—184no—
241Dic3—38no—
242Dic3—38no—
243Dic3—38no—
244Dic3—64no—
245Dic3—64no—
246Dic3—64no—
247Dic3—98no—
248Dic3—184no—
249C32—21no—
250C32C2×#88712yescomplemented
251C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
252Q8—2716no—
253C2×C4—22no—
254C2×C4—92no—
255C2×C4—92no—
256C2×C4—94no—
257C2×C4—94no—
258C2×C4—94no—
259C2×C4—98no—
260C2×C4—94no—
261C2×C4—184no—
262C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
263C2×C3—32no—
264C2×C3—38no—
265C2×C3—32no—
266C2×C3—32no—
267C2×C3—61no—
268C2×C3—64no—
269C2×C3—61no—
270C2×C3—61no—
271C2×C3#6383814yes—
272C2×C3C2×#379711yescharacteristic
273C2×C3C2×C4×D3211yescharacteristic
274C2×C3C2×#380511yescharacteristic
275C22#8865112yes—
276C22C4×((C3⋊D3)⋊C3)⋊C212yes—
277C22#8867212yes—
278C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
279C4—22no—
280C4—94no—
281C4—94no—
282C4—98no—
283C4—184no—
284C3—32no—
285C3—61no—
286C3C2×#6383811yescharacteristic
287C2#1172107814yesdirect product factor
288C2C2×#8865111yescharacteristic
289C2C2×C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
290C2C2×#8867211yescharacteristic
subgroupG/H
centerC23C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C23×C4
FrattiniC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
socleC23×C3C2×D32
prime radicalC23×H(3,𝔽3)C23
cosocleC22×H(3,𝔽3)C24
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C4×Q8
LCSG ▷ 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
6. #11721078C2direct (N × H)
13. C2×#6196C4semidirect
32. C2×#1084Q8semidirect
39. #6196C2×C4semidirect
67. #1084C2×Q8semidirect
109. C2×H(3,𝔽3)C4×Q8semidirect
167. H(3,𝔽3)C2×C4×Q8semidirect
207. C2×C32#887semidirect
250. C32C2×#887semidirect
287. C2#11721078direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#61962
C2×C4×#10842
C2×#3750⋊C32
C2×#884492
C2×#884472
#117210782
C2×#638373