Finite Group Databasebeta

← C2×#11723103 #13173393

Subgroups — C2×#11723103 #13173393

Order 1728.

984 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 141 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×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C22×C3⋊#622C212yesmaximal, complemented
3#11723103C214yesmaximal, direct product factor
4C2×#63778—31nomaximal
5C23×C3×C3⋊D3C411yescharacteristic
6C22×C33×C4C411yescharacteristic
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C2×C3×C4×C3⋊D3—24no—
9C2×C3×C4×C3⋊D3C412yes—
10C2×C3×C4×C3⋊D3C2212yes—
11C2×C3⋊#622—24no—
12C2×C3⋊#622C2214yescomplemented
13C22×C4×C3⋊D3D311yescharacteristic
14C22×C3×C4×D3—62no—
15C22×#622—32no—
16#63778—34no—
17C23×C33C2×C411yescharacteristic
18C22×C3×C3⋊D3—22no—
19C22×C3×C3⋊D3—22no—
20C22×C3×C3⋊D3C814yescomplemented
21C22×C3×C3⋊D3C2×C412yes—
22C2×C33×C4—22no—
23C2×C33×C4C2×C412yes—
24C2×C3×C3⋊Dic3C2×C411yescharacteristic
25C2×C3×C3⋊Dic3D414yes—
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3×C4×C3⋊D3—24no—
28C3×C4×C3⋊D3—24no—
29C3⋊#622—24no—
30C2×#830—91nomaximal
31C23×C3⋊D3Dic311yescharacteristic
32C23×C3×D3—62no—
33C22×C32×C4—22no—
34C22×C32×C4—42no—
35C22×C32×C4Dic311yescharacteristic
36C22×C3⋊Dic3C2×D311yescharacteristic
37C2×C4×C3⋊D3—24no—
38C2×C4×C3⋊D3Dic312yes—
39C2×C4×C3⋊D3C2×D312yes—
40C22×C3×Dic3—62no—
41C2×C3×C4×D3—64no—
42C2×C3×C4×D3—64no—
43C2×C3×C4×D3—616no—
44C2×#622—34no—
45C2×#622—64no—
46C22×C33C22⋊C412yes—
47C22×C33C2×C812yes—
48C22×C33M1612yes—
49C22×C33C22×C411yescharacteristic
50C2×C3×C3⋊D3—28no—
51C2×C3×C3⋊D3—22no—
52C2×C3×C3⋊D3—22no—
53C2×C3×C3⋊D3C2×C814yescomplemented
54C33×C4—22no—
55C3×C3⋊Dic3C22⋊C412yes—
56C3×C3⋊Dic3C2×D412yes—
57C23×C3×C4—91no—
58C22×C4×D3—62no—
59C22×C3⋊C8—92no—
60#830—94no—
61C23×C32—22no—
62C23×C32—42no—
63C23×C32C2×Dic311yescharacteristic
64C22×C3⋊D3—22no—
65C22×C3⋊D3—22no—
66C22×C3⋊D3C3⋊C814yescomplemented
67C22×C3⋊D3C2×Dic312yes—
68C22×C3×D3—68no—
69C22×C3×D3—64no—
70C22×C3×D3—68no—
71C22×C3×D3—68no—
72C2×C32×C4—22no—
73C2×C32×C4—28no—
74C2×C32×C4—24no—
75C2×C32×C4—48no—
76C2×C32×C4—44no—
77C2×C32×C4C2×Dic312yes—
78C2×C3⋊Dic3C2×Dic311yescharacteristic
79C2×C3⋊Dic3#6714yes—
80C2×C3⋊Dic3C22×D311yescharacteristic
81C4×C3⋊D3—24no—
82C4×C3⋊D3—24no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—68no—
85C2×C3×Dic3—62no—
86C3×C4×D3—616no—
87C3×C4×D3—616no—
88#622—64no—
89C2×C22⋊C8—271noSylow
90C2×C33C22⋊C814yescomplemented
91C2×C33C2×C22⋊C411yescharacteristic
92C2×C33C22×C811yescharacteristic
93C2×C33C2×M1611yescharacteristic
94C3×C3⋊D3—24no—
95C24×C3—91no—
96C23×D3—62no—
97C22×C3×C4—22no—
98C22×C3×C4—42no—
99C22×C3×C4—91no—
100C22×C3×C4—92no—
101C22×C3×C4—92no—
102C22×C3×C4—184no—
103C22×C3×C4#15711yescharacteristic
104C22×Dic3—62no—
105C2×C4×D3—64no—
106C2×C4×D3—64no—
107C2×C4×D3—616no—
108C2×C3⋊C8—94no—
109C2×C3⋊C8—184no—
110C22×C32—22no—
111C22×C32—24no—
112C22×C32—24no—
113C22×C32—24no—
114C22×C32—44no—
115C22×C32—44no—
116C22×C32—44no—
117C22×C32—42no—
118C22×C32C2×C3⋊C812yes—
119C22×C32#20812yes—
120C22×C32#21712yes—
121C22×C32C22×Dic311yescharacteristic
122C2×C3⋊D3—28no—
123C2×C3⋊D3—22no—
124C2×C3⋊D3—22no—
125C2×C3⋊D3C2×C3⋊C814yescomplemented
126C2×C3×D3—68no—
127C2×C3×D3—68no—
128C2×C3×D3—68no—
129C2×C3×D3—632no—
130C32×C4—22no—
131C32×C4—28no—
132C32×C4—48no—
133C3⋊Dic3#21712yes—
134C3⋊Dic3C2×#6712yes—
135C3×Dic3—64no—
136C3×Dic3—64no—
137C23×C4—91no—
138C22×C8—272no—
139C22⋊C8—274no—
140C33C2×C22⋊C811yescharacteristic, complemented, Sylow
141C23×C3—22no—
142C23×C3—42no—
143C23×C3—92no—
144C23×C3—94no—
145C23×C3—182no—
146C23×C3—182no—
147C23×C3C2×#15711yescharacteristic
148C22×D3—68no—
149C22×D3—64no—
150C22×D3—68no—
151C22×D3—68no—
152C2×C3×C4—28no—
153C2×C3×C4—22no—
154C2×C3×C4—24no—
155C2×C3×C4—48no—
156C2×C3×C4—44no—
157C2×C3×C4—91no—
158C2×C3×C4—91no—
159C2×C3×C4—94no—
160C2×C3×C4—184no—
161C2×C3×C4—184no—
162C2×C3×C4C2×#15712yes—
163C2×Dic3—62no—
164C2×Dic3—68no—
165C2×Dic3—62no—
166C4×D3—616no—
167C4×D3—616no—
168C3⋊C8—184no—
169C2×C32—22no—
170C2×C32—22no—
171C2×C32—28no—
172C2×C32—22no—
173C2×C32—48no—
174C2×C32—42no—
175C2×C32—42no—
176C2×C32—42no—
177C2×C32#83014yescomplemented
178C2×C32C22×C3⋊C811yescharacteristic
179C2×C32C2×#20811yescharacteristic
180C2×C32C2×#21711yescharacteristic
181C3⋊D3—24no—
182C3×D3—616no—
183C24—91no—
184C22×C4—91no—
185C22×C4—92no—
186C22×C4—92no—
187C22×C4—184no—
188C22×C4C3⋊#15711yescharacteristic
189C2×C8—274no—
190C2×C8—544no—
191C22×C3—22no—
192C22×C3—24no—
193C22×C3—24no—
194C22×C3—24no—
195C22×C3—42no—
196C22×C3—44no—
197C22×C3—44no—
198C22×C3—44no—
199C22×C3—94no—
200C22×C3—182no—
201C22×C3—188no—
202C22×C3—182no—
203C22×C3#378712yes—
204C22×C3#378812yes—
205C22×C3C22⋊#15712yes—
206C22×C3C22×#15711yescharacteristic
207C2×D3—68no—
208C2×D3—68no—
209C2×D3—632no—
210C2×D3—68no—
211C3×C4—28no—
212C3×C4—22no—
213C3×C4—48no—
214C3×C4—92no—
215C3×C4—92no—
216Dic3—64no—
217Dic3—64no—
218C32—22no—
219C32—42no—
220C32C2×#83011yescharacteristic, complemented
221C23—92no—
222C23—94no—
223C23—182no—
224C23—182no—
225C23C2×C3⋊#15711yescharacteristic
226C2×C4—22no—
227C2×C4—91no—
228C2×C4—91no—
229C2×C4—94no—
230C2×C4—184no—
231C2×C4—184no—
232C2×C4C2×C3⋊#15712yes—
233C8—544no—
234C2×C3—28no—
235C2×C3—22no—
236C2×C3—22no—
237C2×C3—22no—
238C2×C3—48no—
239C2×C3—42no—
240C2×C3—42no—
241C2×C3—42no—
242C2×C3—184no—
243C2×C3#6377814yescomplemented
244C2×C3C2×#378711yescharacteristic
245C2×C3C2×#378811yescharacteristic
246C2×C3C2×C22⋊#15711yescharacteristic
247D3—616no—
248C22—94no—
249C22—188no—
250C22—182no—
251C22—182no—
252C22#8898812yes—
253C22#8898912yes—
254C22#8899412yes—
255C22C22×C3⋊#15711yescharacteristic
256C4—22no—
257C4—92no—
258C4—92no—
259C3—22no—
260C3—42no—
261C3C2×#6377811yescharacteristic, complemented
262C2—184no—
263C2#1172310314yesdirect product factor
264C2C2×#8898811yescharacteristic
265C2C2×#8898911yescharacteristic
266C2C2×#8899411yescharacteristic
subgroupG/H
centerC23C2×C3⋊#157
commutatorC2×C33C22×C8
FrattiniC22C22×C3⋊#157
FittingC22×C33×C4C4
socleC23×C33C2×C4
prime radicalC23×C3×C3⋊D3C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC22×C4C3⋊#157
hypocenterC33C2×C22⋊C8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#622C2semidirect
3. #11723103C2direct (N × H)
12. C2×C3⋊#622C22semidirect
20. C22×C3×C3⋊D3C8semidirect
53. C2×C3×C3⋊D3C2×C8semidirect
66. C22×C3⋊D3C3⋊C8semidirect
90. C2×C33C22⋊C8semidirect
125. C2×C3⋊D3C2×C3⋊C8semidirect
140. C33C2×C22⋊C8semidirect
177. C2×C32#830semidirect
220. C32C2×#830semidirect
243. C2×C3#63778semidirect
261. C3C2×#63778semidirect
263. C2#11723103direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C22×C3⋊#6222
#117231032
C2×#637783
C2×#8309