Finite Group Databasebeta

← C2×#11723112 #13173441

Subgroups — C2×#11723112 #13173441

Order 1728.

1144 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic
2C22×C3⋊#622C212yesmaximal, complemented
3#11723112C214yesmaximal, direct product factor
4C2×C22⋊#622—31nomaximal
5C24×C33C411yescharacteristic
6C22×C3×C3⋊Dic3—24no—
7C22×C3×C3⋊Dic3C411yescharacteristic
8C22×C3×C3⋊Dic3C412yes—
9C22×C3×C3⋊Dic3C2212yes—
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C2×C3⋊#622—24no—
12C2×C3⋊#622C2214yescomplemented
13C23×C3⋊Dic3D311yescharacteristic
14C23×C3×Dic3—62no—
15C22×#622—32no—
16C22⋊#622—34no—
17C23×C33—22no—
18C23×C33—22no—
19C23×C33C814yescomplemented
20C23×C33C2×C412yes—
21C23×C33C2×C411yescharacteristic
22C2×C3×C3⋊Dic3—22no—
23C2×C3×C3⋊Dic3—24no—
24C2×C3×C3⋊Dic3—24no—
25C2×C3×C3⋊Dic3C2×C412yes—
26C2×C3×C3⋊Dic3C2×C411yescharacteristic
27C2×C3×C3⋊Dic3D414yes—
28C2×C3×C3⋊Dic3C2311yescharacteristic
29C3⋊#622—24no—
30C2×#830—91nomaximal
31C24×C32—22no—
32C24×C32—42no—
33C24×C32Dic311yescharacteristic
34C22×C3⋊Dic3—24no—
35C22×C3⋊Dic3Dic311yescharacteristic
36C22×C3⋊Dic3Dic312yes—
37C22×C3⋊Dic3C2×D312yes—
38C22×C3⋊Dic3C2×D311yescharacteristic
39C22×C3×Dic3—64no—
40C22×C3×Dic3—62no—
41C22×C3×Dic3—62no—
42C22×C3×Dic3—64no—
43C22×C3×Dic3—616no—
44C2×#622—34no—
45C2×#622—64no—
46C22×C33—28no—
47C22×C33—22no—
48C22×C33—22no—
49C22×C33C22⋊C412yes—
50C22×C33C2×C812yes—
51C22×C33C2×C814yescomplemented
52C22×C33M1612yes—
53C22×C33C22×C411yescharacteristic
54C3×C3⋊Dic3—22no—
55C3×C3⋊Dic3C22⋊C412yes—
56C3×C3⋊Dic3C2×D412yes—
57C23×C3×C4—91no—
58C23×Dic3—62no—
59C22×C3⋊C8—92no—
60#830—94no—
61C23×C32—22no—
62C23×C32—22no—
63C23×C32—28no—
64C23×C32—28no—
65C23×C32—28no—
66C23×C32—24no—
67C23×C32—22no—
68C23×C32—48no—
69C23×C32—48no—
70C23×C32—48no—
71C23×C32—44no—
72C23×C32—42no—
73C23×C32C3⋊C814yescomplemented
74C23×C32C2×Dic312yes—
75C23×C32C2×Dic311yescharacteristic
76C2×C3⋊Dic3—22no—
77C2×C3⋊Dic3—24no—
78C2×C3⋊Dic3—24no—
79C2×C3⋊Dic3C2×Dic312yes—
80C2×C3⋊Dic3C2×Dic311yescharacteristic
81C2×C3⋊Dic3#6714yes—
82C2×C3⋊Dic3C22×D311yescharacteristic
83C2×C3×Dic3—62no—
84C2×C3×Dic3—64no—
85C2×C3×Dic3—68no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—616no—
88C2×C3×Dic3—616no—
89C2×C3×Dic3—68no—
90#622—64no—
91C2×C22⋊C8—271noSylow
92C2×C33—24no—
93C2×C33C22⋊C814yescomplemented
94C2×C33C2×C22⋊C411yescharacteristic
95C2×C33C22×C811yescharacteristic
96C2×C33C2×M1611yescharacteristic
97C24×C3—22no—
98C24×C3—42no—
99C24×C3#15711yescharacteristic
100C22×C3×C4—91no—
101C22×C3×C4—92no—
102C22×C3×C4—92no—
103C22×C3×C4—91no—
104C22×C3×C4—184no—
105C22×Dic3—64no—
106C22×Dic3—62no—
107C22×Dic3—62no—
108C22×Dic3—64no—
109C22×Dic3—616no—
110C2×C3⋊C8—94no—
111C2×C3⋊C8—184no—
112C22×C32—28no—
113C22×C32—22no—
114C22×C32—22no—
115C22×C32—28no—
116C22×C32—28no—
117C22×C32—232no—
118C22×C32—22no—
119C22×C32—24no—
120C22×C32—28no—
121C22×C32—24no—
122C22×C32—24no—
123C22×C32—48no—
124C22×C32—432no—
125C22×C32—48no—
126C22×C32—44no—
127C22×C32—44no—
128C22×C32—48no—
129C22×C32—44no—
130C22×C32—42no—
131C22×C32C2×C3⋊C812yes—
132C22×C32C2×C3⋊C814yescomplemented
133C22×C32#20812yes—
134C22×C32#21712yes—
135C22×C32C22×Dic311yescharacteristic
136C3⋊Dic3—22no—
137C3⋊Dic3#21712yes—
138C3⋊Dic3C2×#6712yes—
139C3×Dic3—64no—
140C3×Dic3—64no—
141C3×Dic3—68no—
142C23×C4—91no—
143C22×C8—272no—
144C22⋊C8—274no—
145C33C2×C22⋊C811yescharacteristic, complemented, Sylow
146C23×C3—28no—
147C23×C3—28no—
148C23×C3—22no—
149C23×C3—22no—
150C23×C3—28no—
151C23×C3—24no—
152C23×C3—22no—
153C23×C3—48no—
154C23×C3—48no—
155C23×C3—48no—
156C23×C3—44no—
157C23×C3—42no—
158C23×C3#62214yescomplemented
159C23×C3C2×#15712yes—
160C23×C3C2×#15711yescharacteristic
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4—94no—
164C2×C3×C4—92no—
165C2×C3×C4—184no—
166C2×C3×C4—184no—
167C2×C3×C4—182no—
168C2×Dic3—62no—
169C2×Dic3—64no—
170C2×Dic3—68no—
171C2×Dic3—62no—
172C2×Dic3—616no—
173C2×Dic3—616no—
174C2×Dic3—68no—
175C3⋊C8—184no—
176C2×C32—24no—
177C2×C32—216no—
178C2×C32—22no—
179C2×C32—22no—
180C2×C32—22no—
181C2×C32—28no—
182C2×C32—416no—
183C2×C32—48no—
184C2×C32—42no—
185C2×C32—42no—
186C2×C32—42no—
187C2×C32#83014yescomplemented
188C2×C32C22×C3⋊C811yescharacteristic
189C2×C32C2×#20811yescharacteristic
190C2×C32C2×#21711yescharacteristic
191C24C3⋊#15711yescharacteristic
192C22×C4—91no—
193C22×C4—92no—
194C22×C4—92no—
195C22×C4—91no—
196C22×C4—184no—
197C2×C8—274no—
198C2×C8—544no—
199C22×C3—28no—
200C22×C3—232no—
201C22×C3—28no—
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—28no—
205C22×C3—22no—
206C22×C3—24no—
207C22×C3—28no—
208C22×C3—24no—
209C22×C3—24no—
210C22×C3—432no—
211C22×C3—48no—
212C22×C3—48no—
213C22×C3—42no—
214C22×C3—44no—
215C22×C3—48no—
216C22×C3—44no—
217C22×C3—44no—
218C22×C3C2×#62212yes—
219C22×C3C2×#62214yescomplemented
220C22×C3#379212yes—
221C22×C3C22⋊#15712yes—
222C22×C3C22×#15711yescharacteristic
223C3×C4—92no—
224C3×C4—92no—
225C3×C4—182no—
226Dic3—64no—
227Dic3—64no—
228Dic3—68no—
229C32—22no—
230C32—42no—
231C32C2×#83011yescharacteristic, complemented
232C23—22no—
233C23—22no—
234C23C3⋊#62214yescomplemented
235C23C2×C3⋊#15712yes—
236C23C2×C3⋊#15711yescharacteristic
237C2×C4—91no—
238C2×C4—91no—
239C2×C4—94no—
240C2×C4—92no—
241C2×C4—184no—
242C2×C4—184no—
243C2×C4—182no—
244C8—544no—
245C2×C3—216no—
246C2×C3—24no—
247C2×C3—28no—
248C2×C3—22no—
249C2×C3—22no—
250C2×C3—22no—
251C2×C3—416no—
252C2×C3—48no—
253C2×C3—42no—
254C2×C3—42no—
255C2×C3—42no—
256C2×C3C22⋊#62214yescomplemented
257C2×C3C22×#62211yescharacteristic
258C2×C3C2×#379211yescharacteristic
259C2×C3C2×C22⋊#15711yescharacteristic
260C22—28no—
261C22—22no—
262C22—22no—
263C22C2×C3⋊#62212yes—
264C22C2×C3⋊#62214yescomplemented
265C22#8899312yes—
266C22#8899412yes—
267C22C22×C3⋊#15711yescharacteristic
268C4—92no—
269C4—92no—
270C4—182no—
271C3—22no—
272C3—42no—
273C3C2×C22⋊#62211yescharacteristic, complemented
274C2—24no—
275C2#1172311214yesdirect product factor
276C2C22×C3⋊#62211yescharacteristic
277C2C2×#8899311yescharacteristic
278C2C2×#8899411yescharacteristic
subgroupG/H
centerC23C2×C3⋊#157
commutatorC2×C33C22×C8
FrattiniC22C22×C3⋊#157
Fitting, prime radicalC24×C33C4
socleC23×C33C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypercenterC24C3⋊#157
hypocenterC33C2×C22⋊C8
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C22×C3⋊#622C2semidirect
3. #11723112C2direct (N × H)
12. C2×C3⋊#622C22semidirect
19. C23×C33C8semidirect
51. C22×C33C2×C8semidirect
73. C23×C32C3⋊C8semidirect
93. C2×C33C22⋊C8semidirect
132. C22×C32C2×C3⋊C8semidirect
145. C33C2×C22⋊C8semidirect
158. C23×C3#622semidirect
187. C2×C32#830semidirect
219. C22×C3C2×#622semidirect
231. C32C2×#830semidirect
234. C23C3⋊#622semidirect
256. C2×C3C22⋊#622semidirect
264. C22C2×C3⋊#622semidirect
273. C3C2×C22⋊#622semidirect
275. C2#11723112direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C22×C3⋊#6222
#117231122
C2×C22⋊#6223
C2×#8309