Finite Group Databasebeta

← C2×#11722377 #13164470

Subgroups — C2×#11722377 #13164470

Order 1728.

914 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 139 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×C3×C4×C22⋊C9C211yesmaximal, characteristic
2C2×C3⋊((C22⋊C9)⋊C4)C212yesmaximal
3#11722377C214yesmaximal, direct product factor
4C2×#64133—31nomaximal
5C2×(C22⋊C9)⋊Q8—33nomaximal
6C22×C3×C22⋊C9C2211yescharacteristic
7C2×#6226—41nomaximal
8C3×C4×C22⋊C9C2212yes—
9C3⋊((C22⋊C9)⋊C4)C2214yes—
10C23×C32×C4D311yescharacteristic
11C22×#634—31no—
12C2×#3757—32no—
13C2×C4⋊(C3⋊Dic3)—31no—
14#64133—34no—
15#64133—34no—
16C2×#3750—32no—
17C2×C4×C22⋊C9D313yes—
18C2×(C22⋊C9)⋊C4—36no—
19(C22⋊C9)⋊Q8—312no—
20C2×C3×C22⋊C9Q812yescomplemented
21C2×C3×C22⋊C9C2311yescharacteristic
22C2×C3×C4×C9—41no—
23C2×C3⋊Dic9—42no—
24#6226—44no—
25C2×#924—93no—
26C2×#924—91no—
27C24×C32C2×D311yescharacteristic
28C22×C32×C4—31no—
29C22×C32×C4—31no—
30C22×C32×C4—32no—
31C22×C32×C4C2×D312yes—
32C22×C3⋊Dic3—32no—
33C22×C3⋊Dic3—32no—
34C2×#634—34no—
35C2×#634—61no—
36C2×#634—62no—
37C2×#634—61no—
38C22×C22⋊C9C2×D313yes—
39#3757—34no—
40#3757—34no—
41C4⋊(C3⋊Dic3)—34no—
42#3750—38no—
43C2×Dic18—123no—
44C4×C22⋊C9C2×D316yes—
45(C22⋊C9)⋊C4—312no—
46C22×C3×C9—41no—
47C3×C22⋊C9C2×Q811yescharacteristic, complemented
48C3×C4×C9—42no—
49C3⋊Dic9—44no—
50C23×C3×C4D913yes—
51C23×C3×C4C3⋊D311yescharacteristic
52C22×Dic6—93no—
53C22×Dic6—91no—
54C2×#217—96no—
55C2×#217—92no—
56C2×C4⋊Dic3—93no—
57C2×C4⋊Dic3—91no—
58#924—912no—
59#924—94no—
60#924—912no—
61#924—94no—
62C2×Dic3⋊C4—96no—
63C2×Dic3⋊C4—92no—
64C23×C32—31no—
65C23×C32—32no—
66C23×C32—31no—
67C23×C32Dic612yes—
68C23×C32C22×D311yescharacteristic
69C2×C32×C4—31no—
70C2×C32×C4—32no—
71C2×C32×C4—32no—
72C2×C32×C4—32no—
73C2×C32×C4—61no—
74C2×C32×C4S411yescharacteristic
75C2×C3⋊Dic3—34no—
76C2×C3⋊Dic3—34no—
77C2×C3⋊Dic3—62no—
78C2×C3⋊Dic3—64no—
79C2×C3⋊Dic3—62no—
80#634—64no—
81#634—64no—
82C2×C22⋊C9Dic616yescomplemented
83C2×C22⋊C9C22×D313yes—
84C2×C4×C9—43no—
85C2×Dic9—126no—
86Dic18—1212no—
87C2×C22⋊Q8—271noSylow
88C2×C3×C9—42no—
89C2×C3×C9—41no—
90C24×C3C2×D913yes—
91C24×C3C2×C3⋊D311yescharacteristic
92C22×C3×C4—33no—
93C22×C3×C4—31no—
94C22×C3×C4—33no—
95C22×C3×C4—31no—
96C22×C3×C4—36no—
97C22×C3×C4—32no—
98C22×C3×C4C2×D916yes—
99C22×C3×C4C2×C3⋊D312yes—
100C22×Dic3—92no—
101C22×Dic3—96no—
102C22×Dic3—92no—
103C22×Dic3—96no—
104C2×Dic6—94no—
105C2×Dic6—912no—
106C2×Dic6—181no—
107C2×Dic6—183no—
108C2×Dic6—182no—
109C2×Dic6—186no—
110C2×Dic6—183no—
111C2×Dic6—181no—
112#217—912no—
113#217—94no—
114#217—912no—
115#217—94no—
116C4⋊Dic3—94no—
117C4⋊Dic3—912no—
118Dic3⋊C4—924no—
119Dic3⋊C4—98no—
120C22×C32—32no—
121C22×C32—31no—
122C22×C32—32no—
123C22×C32—32no—
124C22×C32—31no—
125C22×C32—31no—
126C22×C32—61no—
127C22×C32C2×Dic611yescharacteristic
128C22×C32C2×S411yescharacteristic
129C32×C4—32no—
130C32×C4C2×S412yes—
131C3⋊Dic3—64no—
132C3⋊Dic3—64no—
133C22×C9—43no—
134C22⋊C9C2×Dic613yescomplemented
135C4×C9—46no—
136Dic9—1212no—
137C22×Q8—271no—
138C23×C4C3⋊D911yescharacteristic
139C22⋊Q8—274no—
140C22⋊Q8—274no—
141C2×C4⋊C4—272no—
142C2×C4⋊C4—271no—
143C2×C22⋊C4—272no—
144C3×C9—41noSylow
145C23×C3—31no—
146C23×C3—33no—
147C23×C3—33no—
148C23×C3—31no—
149C23×C3—36no—
150C23×C3—32no—
151C23×C3Dic1816yescomplemented
152C23×C3C22×D913yes—
153C23×C3#63412yes—
154C23×C3C22×C3⋊D311yescharacteristic
155C2×C3×C4—31no—
156C2×C3×C4—33no—
157C2×C3×C4—32no—
158C2×C3×C4—32no—
159C2×C3×C4—36no—
160C2×C3×C4—36no—
161C2×C3×C4—36no—
162C2×C3×C4—32no—
163C2×C3×C4—61no—
164C2×C3×C4—63no—
165C2×C3×C4(C22⋊C9)⋊C213yes—
166C2×C3×C4C3⋊S411yescharacteristic
167C2×Dic3—94no—
168C2×Dic3—912no—
169C2×Dic3—94no—
170C2×Dic3—912no—
171C2×Dic3—182no—
172C2×Dic3—186no—
173C2×Dic3—184no—
174C2×Dic3—1812no—
175C2×Dic3—182no—
176C2×Dic3—186no—
177Dic6—184no—
178Dic6—1812no—
179Dic6—184no—
180Dic6—1812no—
181C2×C32—31no—
182C2×C32—31no—
183C2×C32—32no—
184C2×C32A4⋊Q812yes—
185C2×C32C22×S411yescharacteristic
186C2×C9—46no—
187C2×C9—43no—
188C24C2×C3⋊D911yescharacteristic
189C2×Q8—274no—
190C2×Q8—541no—
191C2×Q8—542no—
192C2×Q8—541no—
193C22×C4—31no—
194C22×C4—31no—
195C22×C4—32no—
196C22×C4—272no—
197C22×C4—272no—
198C22×C4C2×C3⋊D912yes—
199C4⋊C4—278no—
200C4⋊C4—274no—
201C22⋊C4—274no—
202C22⋊C4—274no—
203C22×C3—36no—
204C22×C3—33no—
205C22×C3—32no—
206C22×C3—31no—
207C22×C3—32no—
208C22×C3—32no—
209C22×C3—31no—
210C22×C3—31no—
211C22×C3—33no—
212C22×C3—33no—
213C22×C3—36no—
214C22×C3—36no—
215C22×C3—63no—
216C22×C3—61no—
217C22×C3C2×Dic1813yescomplemented
218C22×C3C2×(C22⋊C9)⋊C213yes—
219C22×C3C2×#63411yescharacteristic
220C22×C3C2×C3⋊S411yescharacteristic
221C3×C4—32no—
222C3×C4—36no—
223C3×C4C2×(C22⋊C9)⋊C216yes—
224C3×C4C2×C3⋊S412yes—
225Dic3—1812no—
226Dic3—184no—
227Dic3—184no—
228Dic3—1812no—
229C32C2×A4⋊Q811yescharacteristic
230C9—43no—
231C23—31no—
232C23—31no—
233C23—32no—
234C23#622612yescomplemented
235C23C22×C3⋊D911yescharacteristic
236Q8—544no—
237Q8—544no—
238C2×C4—31no—
239C2×C4—32no—
240C2×C4—32no—
241C2×C4—32no—
242C2×C4—61no—
243C2×C4—274no—
244C2×C4—274no—
245C2×C4—542no—
246C2×C4—544no—
247C2×C4—542no—
248C2×C4C3⋊((C22⋊C9)⋊C2)11yescharacteristic
249C2×C3—33no—
250C2×C3—33no—
251C2×C3—36no—
252C2×C3—31no—
253C2×C3—31no—
254C2×C3—32no—
255C2×C3(C22⋊C9)⋊Q816yescomplemented
256C2×C3C22×(C22⋊C9)⋊C213yes—
257C2×C3#6425912yes—
258C2×C3C22×C3⋊S411yescharacteristic
259C22—32no—
260C22—31no—
261C22—31no—
262C22—31no—
263C22—32no—
264C22—32no—
265C22—61no—
266C22C2×#622611yescharacteristic, complemented
267C22C2×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
268C4—32no—
269C4—544no—
270C4—544no—
271C4C2×C3⋊((C22⋊C9)⋊C2)12yes—
272C3C2×(C22⋊C9)⋊Q813yescomplemented
273C3C2×#6425911yescharacteristic
274C2—31no—
275C2—31no—
276C2—32no—
277C2#1172237712yesdirect product factor
278C2C22×C3⋊((C22⋊C9)⋊C2)11yescharacteristic
subgroupG/H
centerC22C2×C3⋊((C22⋊C9)⋊C2)
commutator, cosocleC2×C3×C22⋊C9C23
FrattiniC2×C3C22×C3⋊S4
FittingC23×C32×C4D3
socle, prime radicalC24×C32C2×D3
hypercenterC2×C4C3⋊((C22⋊C9)⋊C2)
hypocenterC3×C22⋊C9C2×Q8
LCSG ▷ C2×C3×C22⋊C9 ▷ C3×C22⋊C9
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722377C2direct (N × H)
20. C2×C3×C22⋊C9Q8semidirect
47. C3×C22⋊C9C2×Q8semidirect
82. C2×C22⋊C9Dic6semidirect
134. C22⋊C9C2×Dic6semidirect
151. C23×C3Dic18semidirect
217. C22×C3C2×Dic18semidirect
234. C23#6226semidirect
255. C2×C3(C22⋊C9)⋊Q8semidirect
266. C22C2×#6226semidirect
272. C3C2×(C22⋊C9)⋊Q8semidirect
277. C2#11722377direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C92
C2×C3⋊((C22⋊C9)⋊C4)2
#117223772
C2×#641333
C2×(C22⋊C9)⋊Q83
C2×#62264