Finite Group Databasebeta

← C2×#11721353 #13152188

Subgroups — C2×#11721353 #13152188

Order 1728.

588 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 56 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
1#11721353C212yesmaximal, direct product factor
2C2×#88455C211yesmaximal, characteristic, complemented
3C2×C32×#127C311yesmaximal, characteristic
4C2×#127⋊C9C313yesmaximal, complemented
5C22×#1094—41nomaximal
6#88455C2211yescharacteristic, complemented
7C22×C32×D4—31no—
8C22×C32×D4—31no—
9C32×#127—32no—
10C32×#127C2×C312yes—
11C2×C32×C42C2×C311yescharacteristic
12#127⋊C9C2×C316yescomplemented
13C2×C42⋊C9C2×C313yescomplemented
14C2×#1094—41no—
15C2×#1094—42no—
16C2×C3×#127—31no—
17C2×C3×#127C3211yescharacteristic
18C24×C32—31no—
19C24×C32A411yescharacteristic
20C2×C32×D4—32no—
21C2×C32×D4—32no—
22C2×C32×D4—32no—
23C2×C32×D4—32no—
24C2×C32×D4—62no—
25C2×C32×D4—62no—
26C2×C32×D4—62no—
27C2×C32×D4—62no—
28C22×C32×C4—31no—
29C22×C22⋊C9—43no—
30C32×C42—31no—
31C32×C42C22×C311yescharacteristic
32C42⋊C9C22×C313yescomplemented
33C22×3−1+2—161no—
34#1094—41no—
35C22×C3×D4—31no—
36C22×C3×D4—33no—
37C22×C3×D4—33no—
38C22×C3×D4—31no—
39C3×#127—32no—
40C3×#127—36no—
41C3×#127—32no—
42C3×#127C2×C3212yes—
43C2×C3×C42—31no—
44C2×C3×C42C2×C3211yescharacteristic
45C23×C32—32no—
46C23×C32—61no—
47C23×C32—61no—
48C23×C32—61no—
49C23×C32—62no—
50C23×C32—61no—
51C23×C32—62no—
52C23×C32C2×A411yescharacteristic
53C23×C32C2×A412yes—
54C32×D4—64no—
55C32×D4—64no—
56C32×D4—64no—
57C32×D4—64no—
58C2×C32×C4—32no—
59C2×C32×C4—32no—
60C2×C32×C4—31no—
61C2×C32×C4—31no—
62C2×C22⋊C9—43no—
63C2×C22⋊C9—46no—
64C2×#1273−1+211yescharacteristic, complemented, Sylow
65C2×3−1+2—162no—
66C2×3−1+2—161no—
67C24×C3—33no—
68C24×C3—31no—
69C24×C3—31no—
70C24×C3C3×A411yescharacteristic
71C2×C3×D4—32no—
72C2×C3×D4—36no—
73C2×C3×D4—36no—
74C2×C3×D4—32no—
75C2×C3×D4—36no—
76C2×C3×D4—36no—
77C2×C3×D4—32no—
78C2×C3×D4—32no—
79C2×C3×D4—62no—
80C2×C3×D4—62no—
81C2×C3×D4—62no—
82C2×C3×D4—62no—
83C2×C3×D4—66no—
84C2×C3×D4—66no—
85C2×C3×D4—66no—
86C2×C3×D4—66no—
87C22×C3×C4—33no—
88C22×C3×C4—31no—
89C3×C42—31no—
90C3×C42—33no—
91C3×C42—31no—
92C3×C42C22×C3211yescharacteristic
93C22×C32—31no—
94C22×C32—31no—
95C22×C32—41no—
96C22×C32—62no—
97C22×C32—62no—
98C22×C32—64no—
99C22×C32—121no—
100C22×C32—122no—
101C22×C32—121no—
102C22×C32—121no—
103C22×C32C22×A411yescharacteristic
104C32×C4—32no—
105C32×C4—32no—
106C22×C9—163no—
107C22⋊C9—43no—
108C22×D4—31no—
109C22×D4—31no—
110#127—32no—
111#127C2×3−1+212yescomplemented
112C2×C42C2×3−1+211yescharacteristic, complemented
1133−1+2—161noSylow
114C23×C3—36no—
115C23×C3—32no—
116C23×C3—31no—
117C23×C3—32no—
118C23×C3—61no—
119C23×C3—61no—
120C23×C3—63no—
121C23×C3—63no—
122C23×C3—61no—
123C23×C3—62no—
124C23×C3—63no—
125C23×C3—66no—
126C23×C3—63no—
127C23×C3—66no—
128C23×C3—61no—
129C23×C3—62no—
130C23×C3C2×C3×A411yescharacteristic
131C23×C3C2×C3×A412yes—
132C3×D4—64no—
133C3×D4—64no—
134C3×D4—64no—
135C3×D4—612no—
136C3×D4—612no—
137C3×D4—612no—
138C3×D4—612no—
139C3×D4—64no—
140C2×C3×C4—33no—
141C2×C3×C4—33no—
142C2×C3×C4—32no—
143C2×C3×C4—36no—
144C2×C3×C4—36no—
145C2×C3×C4—32no—
146C2×C3×C4—31no—
147C2×C3×C4—31no—
148C2×C32—31no—
149C2×C32—31no—
150C2×C32—42no—
151C2×C32—122no—
152C2×C32#127⋊C311yescharacteristic
153C2×C9—166no—
154C2×C9—163no—
155C24—31no—
156C24#109411yescharacteristic
157C2×D4—32no—
158C2×D4—32no—
159C2×D4—32no—
160C2×D4—32no—
161C2×D4—62no—
162C2×D4—62no—
163C2×D4—62no—
164C2×D4—62no—
165C22×C4—31no—
166C42—31no—
167C42C22×3−1+211yescharacteristic, complemented
168C22×C3—31no—
169C22×C3—33no—
170C22×C3—33no—
171C22×C3—31no—
172C22×C3—31no—
173C22×C3—41no—
174C22×C3—62no—
175C22×C3—66no—
176C22×C3—66no—
177C22×C3—612no—
178C22×C3—62no—
179C22×C3—64no—
180C22×C3—121no—
181C22×C3—121no—
182C22×C3—123no—
183C22×C3—126no—
184C22×C3—123no—
185C22×C3—122no—
186C22×C3—121no—
187C22×C3—123no—
188C22×C3—121no—
189C22×C3C22×C3×A411yescharacteristic
190C3×C4—36no—
191C3×C4—36no—
192C3×C4—32no—
193C3×C4—32no—
194C32C2×#127⋊C311yescharacteristic
195C9—163no—
196C23—32no—
197C23—61no—
198C23—61no—
199C23—61no—
200C23—62no—
201C23—61no—
202C23—62no—
203C23C2×#109411yescharacteristic
204C23C2×#109412yes—
205D4—64no—
206D4—64no—
207D4—64no—
208D4—64no—
209C2×C4—32no—
210C2×C4—32no—
211C2×C4—31no—
212C2×C4—31no—
213C2×C3—33no—
214C2×C3—33no—
215C2×C3—31no—
216C2×C3—31no—
217C2×C3—31no—
218C2×C3—42no—
219C2×C3—122no—
220C2×C3—126no—
221C2×C3—122no—
222C2×C3C3×#127⋊C311yescharacteristic
223C22—31no—
224C22—31no—
225C22—41no—
226C22—62no—
227C22—62no—
228C22—64no—
229C22—121no—
230C22—122no—
231C22—121no—
232C22—121no—
233C22C22×#109411yescharacteristic
234C4—32no—
235C4—32no—
236C3—31no—
237C3C2×C3×#127⋊C311yescharacteristic
238C2—31no—
239C2—31no—
240C2—42no—
241C2—122no—
242C2#1172135311yescharacteristic, direct product factor
subgroupG/H
centerC2×C3C3×#127⋊C3
commutator, cosocleC3×C42C22×C32
FrattiniC22×C3C22×C3×A4
Fitting, prime radicalC2×C32×#127C3
socleC23×C3C2×C3×A4
hypercenterC2×C32#127⋊C3
hypocenterC42C22×3−1+2
LCSG ▷ C3×C42 ▷ C42
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. #11721353C2direct (N × H)
2. C2×#88455C2semidirect
4. C2×#127⋊C9C3semidirect
6. #88455C22semidirect
12. #127⋊C9C2×C3semidirect
13. C2×C42⋊C9C2×C3semidirect
32. C42⋊C9C22×C3semidirect
64. C2×#1273−1+2semidirect
111. #127C2×3−1+2semidirect
112. C2×C42C2×3−1+2semidirect
167. C42C22×3−1+2semidirect
242. C2#11721353direct (N × H)

Maximal subgroups

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

subgroupindex
#117213532
C2×#884552
C2×C32×#1273
C2×#127⋊C93
C22×#10944