Finite Group Databasebeta

← C2×#11722724 #13166581

Subgroups — C2×#11722724 #13166581

Order 1728.

616 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 138 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⋊#622C211yesmaximal, characteristic
2C22×C3×#622C211yesmaximal, characteristic
3C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic
4#11722724C214yesmaximal, direct product factor
5C2×#63777—31nomaximal
6C22×C3⋊(C3⋊Dic3)C411yescharacteristic
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C22×C32×Dic3C411yescharacteristic
9C2×C3⋊#622C414yescomplemented
10C2×C3⋊#622C2212yes—
11C2×C3×#622C414yescomplemented
12C2×C3×#622C2212yes—
13Dic3×C3⋊Dic3C412yes—
14Dic3×C3⋊Dic3C2212yes—
15C22×#622—31no—
16C22×#622D311yescharacteristic
17C2×C4×C3⋊Dic3—31no—
18C2×Dic32—62no—
19#63777—34no—
20C23×C33C2×C411yescharacteristic
21C2×C3⋊(C3⋊Dic3)—22no—
22C2×C3⋊(C3⋊Dic3)C2×C412yes—
23C2×C3×C3⋊Dic3C2×C411yescharacteristic
24C2×C3×C3⋊Dic3C2×C412yes—
25C2×C3×C3⋊Dic3C2×C412yes—
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C2×C32×Dic3—22no—
28C2×C32×Dic3C2×C412yes—
29C3⋊#622C2×C414yescomplemented
30C3×#622C2×C414yescomplemented
31C2×#815—91nomaximal
32C22×C32×C4—31no—
33C22×C3⋊Dic3—31no—
34C22×C3⋊Dic3—62no—
35C22×C3⋊Dic3—122no—
36C22×C3⋊Dic3C2×D311yescharacteristic
37C22×C3×Dic3—22no—
38C22×C3×Dic3—62no—
39C2×#622—34no—
40C2×#622—32no—
41C2×#622Dic314yescomplemented
42C2×#622C2×D312yes—
43C4×C3⋊Dic3—32no—
44C4×C3⋊Dic3—32no—
45Dic32—64no—
46Dic32—64no—
47C22×C33C4212yes—
48C22×C33M1612yes—
49C22×C33M1612yes—
50C22×C33C22×C411yescharacteristic
51C3⋊(C3⋊Dic3)—22no—
52C3×C3⋊Dic3C4212yes—
53C3×C3⋊Dic3C22×C411yescharacteristic
54C3×C3⋊Dic3C22×C411yescharacteristic
55C32×Dic3—22no—
56C22×C3×C8—91no—
57C2×C4×Dic3—91no—
58C2×C4×Dic3—182no—
59C22×C3⋊C8—91no—
60#815—94no—
61C23×C32—22no—
62C23×C32—42no—
63C23×C32C4×D311yescharacteristic
64C2×C32×C4—32no—
65C2×C32×C4—62no—
66C2×C3⋊Dic3—32no—
67C2×C3⋊Dic3—64no—
68C2×C3⋊Dic3—62no—
69C2×C3⋊Dic3—68no—
70C2×C3⋊Dic3—128no—
71C2×C3⋊Dic3—124no—
72C2×C3⋊Dic3C4×D311yescharacteristic
73C2×C3⋊Dic3C4×D312yes—
74C2×C3⋊Dic3C2×Dic312yes—
75C2×C3⋊Dic3C22×D311yescharacteristic
76C2×C3×Dic3—28no—
77C2×C3×Dic3—24no—
78C2×C3×Dic3—62no—
79C2×C3×Dic3—62no—
80C2×C3×Dic3—64no—
81C2×C3×Dic3—64no—
82#622—34no—
83#622C2×Dic314yescomplemented
84C2×#97—271noSylow
85C2×C33#9714yescomplemented
86C2×C33C2×C4211yescharacteristic
87C2×C33C2×M1611yescharacteristic
88C2×C33C2×M1611yescharacteristic
89C22×C3×C4—62no—
90C22×C3×C4—91no—
91C22×Dic3—62no—
92C22×Dic3—91no—
93C22×Dic3—182no—
94C22×Dic3—362no—
95C22×Dic3#15711yescharacteristic
96C2×C3×C8—92no—
97C2×C3×C8—94no—
98C4×Dic3—92no—
99C4×Dic3—92no—
100C4×Dic3—184no—
101C4×Dic3—184no—
102C2×C3⋊C8—92no—
103C2×C3⋊C8—94no—
104C22×C32—22no—
105C22×C32—24no—
106C22×C32—24no—
107C22×C32—24no—
108C22×C32—44no—
109C22×C32—44no—
110C22×C32—44no—
111C22×C32—42no—
112C22×C32ΓL(1,𝔽25)12yes—
113C22×C32ΓL(1,𝔽25)12yes—
114C22×C32C4×Dic312yes—
115C22×C32C2×C4×D311yescharacteristic
116C32×C4—62no—
117C3⋊Dic3—62no—
118C3⋊Dic3—68no—
119C3⋊Dic3—128no—
120C3⋊Dic3C4×Dic312yes—
121C3⋊Dic3C2×C4×D311yescharacteristic
122C3⋊Dic3C22×Dic311yescharacteristic
123C3×Dic3—28no—
124C3×Dic3—64no—
125C3×Dic3—62no—
126C3×Dic3—62no—
127C22×C8—91no—
128C22×C8—271no—
129C2×C42—271no—
130#97—274no—
131C33C2×#9711yescharacteristic, complemented, Sylow
132C23×C3—22no—
133C23×C3—42no—
134C23×C3C2×#15711yescharacteristic
135C2×C3×C4—64no—
136C2×C3×C4—68no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4—92no—
140C2×C3×C4—92no—
141C2×Dic3—22no—
142C2×Dic3—62no—
143C2×Dic3—64no—
144C2×Dic3—64no—
145C2×Dic3—62no—
146C2×Dic3—92no—
147C2×Dic3—184no—
148C2×Dic3—188no—
149C2×Dic3—182no—
150C2×Dic3—364no—
151C2×Dic3—368no—
152C2×Dic3C2×#15712yes—
153C3×C8—94no—
154C3⋊C8—94no—
155C2×C32—22no—
156C2×C32—22no—
157C2×C32—28no—
158C2×C32—22no—
159C2×C32—48no—
160C2×C32—42no—
161C2×C32—42no—
162C2×C32—42no—
163C2×C32#81514yescomplemented
164C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
165C2×C32C2×ΓL(1,𝔽25)11yescharacteristic
166C2×C32C2×C4×Dic311yescharacteristic
167C22×C4—31no—
168C22×C4—91no—
169C22×C4—271no—
170C2×C8—92no—
171C2×C8—94no—
172C2×C8—272no—
173C2×C8—274no—
174C42—272no—
175C42—272no—
176C22×C3—22no—
177C22×C3—24no—
178C22×C3—24no—
179C22×C3—24no—
180C22×C3—42no—
181C22×C3—44no—
182C22×C3—44no—
183C22×C3—44no—
184C22×C3#378812yes—
185C22×C3C4×#15712yes—
186C22×C3#379212yes—
187C22×C3C22×#15711yescharacteristic
188C3×C4—68no—
189C3×C4—91no—
190C3×C4—91no—
191C3×C4—92no—
192Dic3—22no—
193Dic3—64no—
194Dic3—62no—
195Dic3—62no—
196Dic3—188no—
197Dic3—182no—
198Dic3—368no—
199C32—22no—
200C32—42no—
201C32C2×#81511yescharacteristic, complemented
202C23D3×#15711yescharacteristic
203C2×C4—32no—
204C2×C4—62no—
205C2×C4—91no—
206C2×C4—91no—
207C2×C4—92no—
208C2×C4—92no—
209C2×C4—272no—
210C2×C4—542no—
211C8—94no—
212C8—274no—
213C2×C3—28no—
214C2×C3—22no—
215C2×C3—22no—
216C2×C3—22no—
217C2×C3—48no—
218C2×C3—42no—
219C2×C3—42no—
220C2×C3—42no—
221C2×C3#6377714yescomplemented
222C2×C3C2×#378811yescharacteristic
223C2×C3C2×C4×#15711yescharacteristic
224C2×C3C2×#379211yescharacteristic
225C22Dic3×#15712yes—
226C22#8892512yes—
227C22#8892612yes—
228C22C2×D3×#15711yescharacteristic
229C4—62no—
230C4—91no—
231C4—91no—
232C4—92no—
233C4—542no—
234C3—22no—
235C3—42no—
236C3C2×#6377711yescharacteristic, complemented
237C2#1172272414yesdirect product factor
238C2C2×Dic3×#15711yescharacteristic
239C2C2×#8892511yescharacteristic
240C2C2×#8892611yescharacteristic
subgroupG/H
center, hypercenterC23D3×#157
commutatorC2×C33C2×C42
FrattiniC22C2×D3×#157
Fitting, socle, prime radicalC23×C33C2×C4
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C2×#97
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722724C2direct (N × H)
9. C2×C3⋊#622C4semidirect
11. C2×C3×#622C4semidirect
29. C3⋊#622C2×C4semidirect
30. C3×#622C2×C4semidirect
41. C2×#622Dic3semidirect
83. #622C2×Dic3semidirect
85. C2×C33#97semidirect
131. C33C2×#97semidirect
163. C2×C32#815semidirect
201. C32C2×#815semidirect
221. C2×C3#63777semidirect
236. C3C2×#63777semidirect
237. C2#11722724direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊#6222
C22×C3×#6222
C2×Dic3×C3⋊Dic32
#117227242
C2×#637773
C2×#8159