Finite Group Databasebeta

← C2×#11722721 #13166409

Subgroups — C2×#11722721 #13166409

Order 1728.

440 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 74 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×#88785C211yesmaximal, characteristic
2C2×C3⋊#3708C211yesmaximal, characteristic, complemented
3C2×C3×#3708C211yesmaximal, characteristic, complemented
4#11722721C214yesmaximal, direct product factor
5C2×#63765—31nomaximal
6C2×C4×C3⋊(C3⋊D3)C411yescharacteristic
7C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
8C2×C32×C3⋊C8C411yescharacteristic
9#88785C412yes—
10#88785C2212yes—
11C3⋊#3708C2212yescomplemented
12C3×#3708C2212yescomplemented
13C2×C8×C3⋊D3—31no—
14C2×#3710—62no—
15C2×#3708—31no—
16C2×#3708D311yescharacteristic, complemented
17#63765—34no—
18C22×C3⋊(C3⋊D3)C811yescharacteristic
19C2×C33×C4C2×C411yescharacteristic
20C2×C3⋊(C3⋊Dic3)C811yescharacteristic
21C4×C3⋊(C3⋊D3)C812yes—
22C4×C3⋊(C3⋊D3)C2×C412yes—
23C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
24C3×C3⋊(C3⋊C8)C2311yescharacteristic
25C32×C3⋊C8C2×C412yes—
26C2×#798—91nomaximal
27C2×C4×C3⋊D3—31no—
28C2×C4×C3⋊D3—62no—
29C2×C4×C3⋊D3—122no—
30C2×C32×C8—31no—
31C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
32C8×C3⋊D3—32no—
33C8×C3⋊D3—32no—
34C2×C3×C3⋊C8—22no—
35C2×C3×C3⋊C8—62no—
36#3710—64no—
37#3710—64no—
38#3708—32no—
39#3708C2×D312yescomplemented
40C22×C33C2×C811yescharacteristic
41C2×C3⋊(C3⋊D3)—22no—
42C2×C3⋊(C3⋊D3)C2×C812yes—
43C33×C4C2×C811yescharacteristic
44C33×C4C22×C411yescharacteristic
45C3⋊(C3⋊Dic3)C2×C812yes—
46C2×D3×C8—91no—
47C2×D3×C8—182no—
48C2×C3×C16—91no—
49C2×C3⋊C16—91no—
50#798—94no—
51C22×C3⋊D3—31no—
52C22×C3⋊D3—62no—
53C22×C3⋊D3—122no—
54C2×C32×C4—22no—
55C2×C32×C4—42no—
56C2×C32×C4C4×D311yescharacteristic
57C2×C3⋊Dic3—31no—
58C2×C3⋊Dic3—62no—
59C2×C3⋊Dic3—122no—
60C4×C3⋊D3—32no—
61C4×C3⋊D3—32no—
62C4×C3⋊D3—64no—
63C4×C3⋊D3—64no—
64C4×C3⋊D3—124no—
65C4×C3⋊D3—124no—
66C32×C8—32no—
67C3⋊(C3⋊C8)C4×D311yescharacteristic
68C3⋊(C3⋊C8)C22×D311yescharacteristic
69C3×C3⋊C8—24no—
70C3×C3⋊C8—62no—
71C3×C3⋊C8—62no—
72C2×M32—271noSylow
73C2×C33M3212yescomplemented
74C2×C33C22×C811yescharacteristic
75C3⋊(C3⋊D3)—22no—
76C2×C4×D3—91no—
77C2×C4×D3—182no—
78C2×C4×D3—362no—
79C2×C3×C8—62no—
80C2×C3×C8—91no—
81C2×C3⋊C8—62no—
82C2×C3⋊C8#15711yescharacteristic
83D3×C8—92no—
84D3×C8—92no—
85D3×C8—184no—
86D3×C8—184no—
87C3×C16—92no—
88C3⋊C16—92no—
89C22×C32—22no—
90C22×C32—42no—
91C22×C32D3×C811yescharacteristic
92C2×C3⋊D3—32no—
93C2×C3⋊D3—64no—
94C2×C3⋊D3—62no—
95C2×C3⋊D3—68no—
96C2×C3⋊D3—128no—
97C2×C3⋊D3—124no—
98C32×C4—22no—
99C32×C4—22no—
100C32×C4—42no—
101C32×C4—42no—
102C32×C4D3×C811yescharacteristic
103C32×C4C2×C4×D311yescharacteristic
104C3⋊Dic3—32no—
105C3⋊Dic3—64no—
106C3⋊Dic3—124no—
107C22×C8—271no—
108M32—274no—
109C2×C16—91no—
110C2×C16—271no—
111C33C2×M3211yescharacteristic, complemented, Sylow
112C22×D3—91no—
113C22×D3—182no—
114C22×D3—362no—
115C2×C3×C4—22no—
116C2×C3×C4—42no—
117C2×C3×C4C2×#15711yescharacteristic
118C2×Dic3—91no—
119C2×Dic3—182no—
120C2×Dic3—362no—
121C4×D3—92no—
122C4×D3—92no—
123C4×D3—184no—
124C4×D3—184no—
125C4×D3—364no—
126C4×D3—364no—
127C3×C8—64no—
128C3×C8—91no—
129C3×C8—91no—
130C3⋊C8—62no—
131C3⋊C8—62no—
132C3⋊C8C2×#15712yes—
133C2×C32—24no—
134C2×C32—22no—
135C2×C32—44no—
136C2×C32—42no—
137C2×C32#79812yescomplemented
138C2×C32C2×D3×C811yescharacteristic
139C3⋊D3—62no—
140C3⋊D3—68no—
141C3⋊D3—128no—
142C22×C4—271no—
143C2×C8—31no—
144C2×C8—91no—
145C2×C8—272no—
146C2×C8—272no—
147C16—92no—
148C16—272no—
149C22×C3—22no—
150C22×C3—42no—
151C22×C3#378711yescharacteristic
152C2×D3—92no—
153C2×D3—184no—
154C2×D3—188no—
155C2×D3—182no—
156C2×D3—364no—
157C2×D3—368no—
158C3×C4—22no—
159C3×C4—22no—
160C3×C4—42no—
161C3×C4—42no—
162C3×C4#378711yescharacteristic
163C3×C4C22×#15711yescharacteristic
164Dic3—92no—
165Dic3—184no—
166Dic3—364no—
167C32—22no—
168C32—42no—
169C32C2×#79811yescharacteristic, complemented
170C23—271no—
171C2×C4—271no—
172C2×C4—272no—
173C2×C4—272no—
174C2×C4D3×#15711yescharacteristic
175C8—32no—
176C8—91no—
177C8—91no—
178C2×C3—24no—
179C2×C3—22no—
180C2×C3—44no—
181C2×C3—42no—
182C2×C3#6376512yescomplemented
183C2×C3C2×#378711yescharacteristic
184D3—188no—
185D3—182no—
186D3—368no—
187C22—272no—
188C22—542no—
189C22#8892411yescharacteristic
190C4—272no—
191C4#8892411yescharacteristic
192C4C2×D3×#15711yescharacteristic
193C3—22no—
194C3—42no—
195C3C2×#6376511yescharacteristic, complemented
196C2—542no—
197C2#1172272112yesdirect product factor
198C2C2×#8892411yescharacteristic
subgroupG/H
center, hypercenterC2×C4D3×#157
commutatorC2×C33C22×C8
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C2×C8
prime radicalC22×C3⋊(C3⋊D3)C8
cosocleC3×C3⋊(C3⋊C8)C23
hypocenterC33C2×M32
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3⋊#3708C2semidirect
3. C2×C3×#3708C2semidirect
4. #11722721C2direct (N × H)
11. C3⋊#3708C22semidirect
12. C3×#3708C22semidirect
16. C2×#3708D3semidirect
39. #3708C2×D3semidirect
73. C2×C33M32semidirect
111. C33C2×M32semidirect
137. C2×C32#798semidirect
169. C32C2×#798semidirect
182. C2×C3#63765semidirect
195. C3C2×#63765semidirect
197. C2#11722721direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#887852
C2×C3⋊#37082
C2×C3×#37082
#117227212
C2×#637653
C2×#7989