Finite Group Databasebeta

← C2×#11722796 #13167577

Subgroups — C2×#11722796 #13167577

Order 1728.

552 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 119 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
2C2×#88815C212yesmaximal
3#11722796C214yesmaximal, direct product factor
4C2×#63742—31nomaximal
5C22×C3×C3⋊Dic3—22no—
6C22×C3×C3⋊Dic3C2211yescharacteristic
7C2×C3×#622C414yescomplemented
8C2×C3×#622C2212yes—
9#88815—22no—
10#88815C2214yes—
11C22×#622D311yescharacteristic
12C2×#3724—32no—
13C2×Dic3⋊Dic3—62no—
14#63742—34no—
15C23×C33D411yescharacteristic
16C2×C3×C3⋊Dic3—24no—
17C2×C3×C3⋊Dic3—44no—
18C2×C3×C3⋊Dic3C2×C412yes—
19C2×C3×C3⋊Dic3D411yescharacteristic
20C2×C3×C3⋊Dic3Q812yes—
21C2×C3×C3⋊Dic3C2311yescharacteristic
22C3×#622C2×C414yescomplemented
23C2×#817—91nomaximal
24C22×C3⋊Dic3—22no—
25C22×C3⋊Dic3C2×D311yescharacteristic
26C22×C3×Dic3—62no—
27C22×C3×Dic3—62no—
28C22×C3×Dic3—62no—
29C22×C3×Dic3—122no—
30C2×#622Dic314yescomplemented
31C2×#622C2×D312yes—
32#3724—34no—
33#3724—62no—
34Dic3⋊Dic3—64no—
35Dic3⋊Dic3—64no—
36C22×C33C4⋊C412yes—
37C22×C33SD1614yes—
38C22×C33C2×D411yescharacteristic
39C3×C3⋊Dic3—44no—
40C3×C3⋊Dic3C4⋊C412yes—
41C3×C3⋊Dic3C22×C411yescharacteristic
42C3×C3⋊Dic3C2×Q811yescharacteristic
43C22×C3×C8—91no—
44C2×C4⋊Dic3—92no—
45C2×Dic3⋊C4—182no—
46#817—94no—
47C23×C32—22no—
48C23×C32—42no—
49C23×C32D1211yescharacteristic
50C2×C3⋊Dic3—24no—
51C2×C3⋊Dic3—44no—
52C2×C3⋊Dic3Dic612yes—
53C2×C3⋊Dic3D1211yescharacteristic
54C2×C3⋊Dic3C2×Dic312yes—
55C2×C3⋊Dic3C22×D311yescharacteristic
56C2×C3×Dic3—62no—
57C2×C3×Dic3—62no—
58C2×C3×Dic3—64no—
59C2×C3×Dic3—64no—
60C2×C3×Dic3—64no—
61C2×C3×Dic3—64no—
62C2×C3×Dic3—124no—
63C2×C3×Dic3—124no—
64C2×C3×Dic3—128no—
65C2×C3×Dic3—124no—
66#622C2×Dic314yescomplemented
67C2×#106—271noSylow
68C2×C33#10614yescomplemented
69C2×C33C2×C4⋊C411yescharacteristic
70C2×C33C2×SD1612yes—
71C22×C3×C4—91no—
72C22×C3×C4—182no—
73C22×Dic3—62no—
74C22×Dic3—62no—
75C22×Dic3—62no—
76C22×Dic3—122no—
77C2×C3×C8—92no—
78C2×C3×C8—94no—
79C4⋊Dic3—94no—
80C4⋊Dic3—182no—
81Dic3⋊C4—184no—
82Dic3⋊C4—184no—
83C22×C32—22no—
84C22×C32—24no—
85C22×C32—24no—
86C22×C32—24no—
87C22×C32—44no—
88C22×C32—44no—
89C22×C32—44no—
90C22×C32—42no—
91C22×C32#20414yes—
92C22×C32C4⋊Dic312yes—
93C22×C32C2×D1211yescharacteristic
94C3⋊Dic3—44no—
95C3⋊Dic3C4⋊Dic312yes—
96C3⋊Dic3C2×Dic611yescharacteristic
97C3⋊Dic3C22×Dic311yescharacteristic
98C3×Dic3—64no—
99C3×Dic3—62no—
100C3×Dic3—62no—
101C3×Dic3—124no—
102C3×Dic3—128no—
103C3×Dic3—124no—
104C22×C8—91no—
105C2×C4⋊C4—272no—
106#106—274no—
107C33C2×#10611yescharacteristic, complemented, Sylow
108C23×C3—22no—
109C23×C3—42no—
110C23×C3GO+(4,𝔽2)11yescharacteristic
111C2×C3×C4—91no—
112C2×C3×C4—91no—
113C2×C3×C4—92no—
114C2×C3×C4—92no—
115C2×C3×C4—184no—
116C2×C3×C4—364no—
117C2×Dic3—64no—
118C2×Dic3—62no—
119C2×Dic3—64no—
120C2×Dic3—62no—
121C2×Dic3—64no—
122C2×Dic3—64no—
123C2×Dic3—124no—
124C2×Dic3—124no—
125C2×Dic3—128no—
126C2×Dic3—124no—
127C3×C8—94no—
128C2×C32—22no—
129C2×C32—22no—
130C2×C32—22no—
131C2×C32—28no—
132C2×C32—48no—
133C2×C32—42no—
134C2×C32—42no—
135C2×C32—42no—
136C2×C32#81714yescomplemented
137C2×C32C2×#20412yes—
138C2×C32C2×C4⋊Dic311yescharacteristic
139C22×C4—91no—
140C22×C4—182no—
141C2×C8—92no—
142C2×C8—94no—
143C4⋊C4—274no—
144C4⋊C4—542no—
145C22×C3—22no—
146C22×C3—24no—
147C22×C3—24no—
148C22×C3—24no—
149C22×C3—42no—
150C22×C3—44no—
151C22×C3—44no—
152C22×C3—44no—
153C22×C3#377312yes—
154C22×C3#377514yes—
155C22×C3C2×GO+(4,𝔽2)11yescharacteristic
156C3×C4—91no—
157C3×C4—91no—
158C3×C4—92no—
159C3×C4—364no—
160Dic3—64no—
161Dic3—62no—
162Dic3—62no—
163Dic3—124no—
164Dic3—128no—
165Dic3—124no—
166C32—22no—
167C32—42no—
168C32C2×#81711yescharacteristic, complemented
169C23#157⋊D311yescharacteristic
170C2×C4—91no—
171C2×C4—91no—
172C2×C4—92no—
173C2×C4—92no—
174C2×C4—184no—
175C2×C4—364no—
176C8—94no—
177C2×C3—28no—
178C2×C3—22no—
179C2×C3—22no—
180C2×C3—22no—
181C2×C3—48no—
182C2×C3—42no—
183C2×C3—42no—
184C2×C3—42no—
185C2×C3#6374214yescomplemented
186C2×C3C2×#377311yescharacteristic
187C2×C3C2×#377512yes—
188C22#8894012yes—
189C22#8894214yes—
190C22C2×#157⋊D311yescharacteristic
191C4—91no—
192C4—91no—
193C4—92no—
194C4—364no—
195C3—22no—
196C3—42no—
197C3C2×#6374211yescharacteristic, complemented
198C2#1172279614yesdirect product factor
199C2C2×#8894011yescharacteristic
200C2C2×#8894212yes—
subgroupG/H
center, hypercenterC23#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×#157⋊D3
Fitting, socle, prime radicalC23×C33D4
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C2×#106
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
3. #11722796C2direct (N × H)
7. C2×C3×#622C4semidirect
22. C3×#622C2×C4semidirect
30. C2×#622Dic3semidirect
66. #622C2×Dic3semidirect
68. C2×C33#106semidirect
107. C33C2×#106semidirect
136. C2×C32#817semidirect
168. C32C2×#817semidirect
185. C2×C3#63742semidirect
197. C3C2×#63742semidirect
198. C2#11722796direct (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
C2×#888152
#117227962
C2×#637423
C2×#8179