Finite Group Databasebeta

← C2×#11722717 #13166395

Subgroups — C2×#11722717 #13166395

Order 1728.

408 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 82 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×#88989C211yesmaximal, characteristic
2C2×C3×#3788C211yesmaximal, characteristic
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic
4#11722717C214yesmaximal, direct product factor
5C2×#63770—31nomaximal
6C2×C3×C4×C3⋊D3C2211yescharacteristic
7C2×C3⋊#622—21no—
8#88989—21no—
9#88989C2212yes—
10C2×C3×#622—21no—
11C3×#3788—21no—
12C3×#3788C2212yes—
13C2×C3⋊(C3⋊(C3⋊C8))C411yescharacteristic
14C2×C32×C3⋊C8C411yescharacteristic
15C3⋊D3×C3⋊C8C412yes—
16C3⋊D3×C3⋊C8C2212yes—
17C2×#3788—31no—
18C2×#3788D311yescharacteristic
19C2×C8×C3⋊D3—31no—
20C2×D3×C3⋊C8—62no—
21#63770—34no—
22C22×C3×C3⋊D3Q811yescharacteristic
23C2×C33×C4C2×C411yescharacteristic
24C2×C3×C3⋊Dic3D411yescharacteristic
25C3×C4×C3⋊D3C2×C411yescharacteristic
26C3×C4×C3⋊D3D411yescharacteristic
27C3×C4×C3⋊D3Q811yescharacteristic
28C3×C4×C3⋊D3C2311yescharacteristic
29C3⋊#622—22no—
30C3×#622—22no—
31C3⋊(C3⋊(C3⋊C8))C2×C412yes—
32C32×C3⋊C8C2×C412yes—
33C2×#822—91nomaximal
34C2×C4×C3⋊D3C2×D311yescharacteristic
35C2×C3×C4×D3—62no—
36C2×#622—21no—
37C2×#622—61no—
38#3788—21no—
39#3788—32no—
40#3788—61no—
41#3788C2×D312yes—
42C2×C32×C8—31no—
43C2×C3⋊(C3⋊C8)—31no—
44C2×C3⋊(C3⋊C8)—62no—
45C2×C3⋊(C3⋊C8)—122no—
46C8×C3⋊D3—32no—
47C8×C3⋊D3—32no—
48C2×C3×C3⋊C8—22no—
49D3×C3⋊C8—64no—
50D3×C3⋊C8—64no—
51C22×C33C4⋊C411yescharacteristic
52C2×C3×C3⋊D3—22no—
53C2×C3×C3⋊D3C4⋊C411yescharacteristic
54C2×C3×C3⋊D3C2×Q811yescharacteristic
55C33×C4C4⋊C411yescharacteristic
56C33×C4C22×C411yescharacteristic
57C3×C3⋊Dic3C4⋊C411yescharacteristic
58C3×C3⋊Dic3C2×D411yescharacteristic
59C2×C3×M16—91no—
60C2×#208—91no—
61C22×C3⋊C8—91no—
62C2×D3×C8—182no—
63#822—94no—
64C22×C3⋊D3Dic611yescharacteristic
65C22×C3×D3—62no—
66C2×C32×C4—22no—
67C2×C32×C4—42no—
68C2×C32×C4C4×D311yescharacteristic
69C2×C3⋊Dic3#6711yescharacteristic
70C4×C3⋊D3Dic611yescharacteristic
71C4×C3⋊D3C4×D311yescharacteristic
72C4×C3⋊D3#6711yescharacteristic
73C4×C3⋊D3C22×D311yescharacteristic
74C2×C3×Dic3—62no—
75C3×C4×D3—62no—
76C3×C4×D3—62no—
77C3×C4×D3—62no—
78C3×C4×D3—62no—
79#622—22no—
80#622—62no—
81C32×C8—32no—
82C3⋊(C3⋊C8)—32no—
83C3⋊(C3⋊C8)—64no—
84C3⋊(C3⋊C8)—124no—
85C3×C3⋊C8—24no—
86C2×#108—271noSylow
87C2×C33#10812yescomplemented
88C2×C33C2×C4⋊C411yescharacteristic
89C3×C3⋊D3—21no—
90C3×C3⋊D3—21no—
91C22×C3×C4—91no—
92C2×C4×D3—62no—
93C3×M16—92no—
94C3×M16—181no—
95C2×C3×C8—62no—
96C2×C3×C8—181no—
97#208—92no—
98#208—181no—
99C2×C3⋊C8—91no—
100C2×C3⋊C8—92no—
101C2×C3⋊C8—92no—
102C2×C3⋊C8—182no—
103C2×C3⋊C8—181no—
104C2×C3⋊C8—362no—
105C2×C3⋊C8#15711yescharacteristic
106D3×C8—184no—
107D3×C8—184no—
108C22×C32—22no—
109C22×C32—42no—
110C22×C32Dic3⋊C411yescharacteristic
111C2×C3⋊D3—22no—
112C2×C3⋊D3Dic3⋊C411yescharacteristic
113C2×C3⋊D3C2×Dic611yescharacteristic
114C2×C3×D3—62no—
115C2×C3×D3—62no—
116C2×C3×D3—68no—
117C32×C4—22no—
118C32×C4—22no—
119C32×C4—42no—
120C32×C4—42no—
121C32×C4Dic3⋊C411yescharacteristic
122C32×C4C2×C4×D311yescharacteristic
123C3⋊Dic3Dic3⋊C411yescharacteristic
124C3⋊Dic3C2×#6711yescharacteristic
125C3×Dic3—62no—
126C3×Dic3—62no—
127C2×M16—91no—
128C2×M16—271no—
129C22×C8—271no—
130#108—274no—
131C33C2×#10811yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C22×D3—62no—
134C2×C3×C4—22no—
135C2×C3×C4—42no—
136C2×C3×C4—91no—
137C2×C3×C4—91no—
138C2×C3×C4—91no—
139C2×C3×C4—91no—
140C2×C3×C4—91no—
141C2×C3×C4C2×#15711yescharacteristic
142C2×Dic3—62no—
143C4×D3—62no—
144C4×D3—62no—
145C4×D3—62no—
146C4×D3—62no—
147C3×C8—64no—
148C3×C8—182no—
149C3⋊C8—92no—
150C3⋊C8—184no—
151C3⋊C8—182no—
152C3⋊C8—364no—
153C3⋊C8C2×#15712yes—
154C2×C32—24no—
155C2×C32—22no—
156C2×C32—44no—
157C2×C32—42no—
158C2×C32#82212yescomplemented
159C2×C32C2×Dic3⋊C411yescharacteristic
160C3⋊D3—21no—
161C3⋊D3—21no—
162C3×D3—64no—
163C3×D3—64no—
164C22×C4—91no—
165M16—92no—
166M16—181no—
167M16—272no—
168M16—541no—
169C2×C8—31no—
170C2×C8—181no—
171C2×C8—272no—
172C2×C8—272no—
173C2×C8—271no—
174C2×C8—541no—
175C22×C3—22no—
176C22×C3—42no—
177C22×C3—91no—
178C22×C3—91no—
179C22×C3—182no—
180C22×C3C4⋊#15711yescharacteristic
181C2×D3—62no—
182C2×D3—62no—
183C2×D3—68no—
184C3×C4—22no—
185C3×C4—22no—
186C3×C4—42no—
187C3×C4—42no—
188C3×C4—91no—
189C3×C4—91no—
190C3×C4C4⋊#15711yescharacteristic
191C3×C4C22×#15711yescharacteristic
192Dic3—62no—
193Dic3—62no—
194C32—22no—
195C32—42no—
196C32C2×#82211yescharacteristic, complemented
197C23—91no—
198C2×C4—91no—
199C2×C4—91no—
200C2×C4—91no—
201C2×C4—91no—
202C2×C4—91no—
203C2×C4D3×#15711yescharacteristic
204C8—32no—
205C8—182no—
206C8—272no—
207C8—542no—
208C2×C3—24no—
209C2×C3—22no—
210C2×C3—44no—
211C2×C3—42no—
212C2×C3—181no—
213C2×C3—181no—
214C2×C3#6377012yescomplemented
215C2×C3C2×C4⋊#15711yescharacteristic
216D3—64no—
217D3—64no—
218C22—91no—
219C22—91no—
220C22—182no—
221C22Dic3⋊#15711yescharacteristic
222C4—91no—
223C4—91no—
224C4Dic3⋊#15711yescharacteristic
225C4C2×D3×#15711yescharacteristic
226C3—22no—
227C3—42no—
228C3C2×#6377011yescharacteristic, complemented
229C2—181no—
230C2—181no—
231C2#1172271712yesdirect product factor
232C2C2×Dic3⋊#15711yescharacteristic
subgroupG/H
centerC22Dic3⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C4⋊C4
prime radicalC22×C3×C3⋊D3Q8
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33C2×#108
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722717C2direct (N × H)
87. C2×C33#108semidirect
131. C33C2×#108semidirect
158. C2×C32#822semidirect
196. C32C2×#822semidirect
214. C2×C3#63770semidirect
228. C3C2×#63770semidirect
231. C2#11722717direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#889892
C2×C3×#37882
C2×C3⋊D3×C3⋊C82
#117227172
C2×#637703
C2×#8229