Finite Group Databasebeta

← C2×#11722797 #13167578

Subgroups — C2×#11722797 #13167578

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#11722797C214yesmaximal, direct product factor
4C2×#63743—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#63743—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×#818—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×C33D812yes—
38C22×C33Q1612yes—
39C22×C33C2×D411yescharacteristic
40C3×C3⋊Dic3—44no—
41C3×C3⋊Dic3C4⋊C412yes—
42C3×C3⋊Dic3C22×C411yescharacteristic
43C3×C3⋊Dic3C2×Q811yescharacteristic
44C22×C3×C8—91no—
45C2×C4⋊Dic3—92no—
46C2×Dic3⋊C4—182no—
47#818—94no—
48C23×C32—22no—
49C23×C32—42no—
50C23×C32D1211yescharacteristic
51C2×C3⋊Dic3—24no—
52C2×C3⋊Dic3—44no—
53C2×C3⋊Dic3Dic612yes—
54C2×C3⋊Dic3D1211yescharacteristic
55C2×C3⋊Dic3C2×Dic312yes—
56C2×C3⋊Dic3C22×D311yescharacteristic
57C2×C3×Dic3—62no—
58C2×C3×Dic3—62no—
59C2×C3×Dic3—64no—
60C2×C3×Dic3—64no—
61C2×C3×Dic3—64no—
62C2×C3×Dic3—64no—
63C2×C3×Dic3—124no—
64C2×C3×Dic3—124no—
65C2×C3×Dic3—128no—
66C2×C3×Dic3—124no—
67#622C2×Dic314yescomplemented
68C2×#107—271noSylow
69C2×C33#10714yescomplemented
70C2×C33C2×C4⋊C411yescharacteristic
71C2×C33C2×D811yescharacteristic
72C2×C33C2×Q1611yescharacteristic
73C22×C3×C4—91no—
74C22×C3×C4—182no—
75C22×Dic3—62no—
76C22×Dic3—62no—
77C22×Dic3—62no—
78C22×Dic3—122no—
79C2×C3×C8—92no—
80C2×C3×C8—94no—
81C4⋊Dic3—94no—
82C4⋊Dic3—182no—
83Dic3⋊C4—184no—
84Dic3⋊C4—184no—
85C22×C32—22no—
86C22×C32—24no—
87C22×C32—24no—
88C22×C32—24no—
89C22×C32—44no—
90C22×C32—44no—
91C22×C32—44no—
92C22×C32—42no—
93C22×C32D2412yes—
94C22×C32Dic1212yes—
95C22×C32C4⋊Dic312yes—
96C22×C32C2×D1211yescharacteristic
97C3⋊Dic3—44no—
98C3⋊Dic3C4⋊Dic312yes—
99C3⋊Dic3C2×Dic611yescharacteristic
100C3⋊Dic3C22×Dic311yescharacteristic
101C3×Dic3—64no—
102C3×Dic3—62no—
103C3×Dic3—62no—
104C3×Dic3—124no—
105C3×Dic3—128no—
106C3×Dic3—124no—
107C22×C8—91no—
108C2×C4⋊C4—272no—
109#107—274no—
110C33C2×#10711yescharacteristic, complemented, Sylow
111C23×C3—22no—
112C23×C3—42no—
113C23×C3GO+(4,𝔽2)11yescharacteristic
114C2×C3×C4—91no—
115C2×C3×C4—91no—
116C2×C3×C4—92no—
117C2×C3×C4—92no—
118C2×C3×C4—184no—
119C2×C3×C4—364no—
120C2×Dic3—64no—
121C2×Dic3—62no—
122C2×Dic3—64no—
123C2×Dic3—62no—
124C2×Dic3—64no—
125C2×Dic3—64no—
126C2×Dic3—124no—
127C2×Dic3—124no—
128C2×Dic3—128no—
129C2×Dic3—124no—
130C3×C8—94no—
131C2×C32—22no—
132C2×C32—22no—
133C2×C32—22no—
134C2×C32—28no—
135C2×C32—48no—
136C2×C32—42no—
137C2×C32—42no—
138C2×C32—42no—
139C2×C32#81814yescomplemented
140C2×C32C2×D2411yescharacteristic
141C2×C32C2×Dic1211yescharacteristic
142C2×C32C2×C4⋊Dic311yescharacteristic
143C22×C4—91no—
144C22×C4—182no—
145C2×C8—92no—
146C2×C8—94no—
147C4⋊C4—274no—
148C4⋊C4—542no—
149C22×C3—22no—
150C22×C3—24no—
151C22×C3—24no—
152C22×C3—24no—
153C22×C3—42no—
154C22×C3—44no—
155C22×C3—44no—
156C22×C3—44no—
157C22×C3#377312yes—
158C22×C3#377412yes—
159C22×C3#377612yes—
160C22×C3C2×GO+(4,𝔽2)11yescharacteristic
161C3×C4—91no—
162C3×C4—91no—
163C3×C4—92no—
164C3×C4—364no—
165Dic3—64no—
166Dic3—62no—
167Dic3—62no—
168Dic3—124no—
169Dic3—128no—
170Dic3—124no—
171C32—22no—
172C32—42no—
173C32C2×#81811yescharacteristic, complemented
174C23#157⋊D311yescharacteristic
175C2×C4—91no—
176C2×C4—91no—
177C2×C4—92no—
178C2×C4—92no—
179C2×C4—184no—
180C2×C4—364no—
181C8—94no—
182C2×C3—28no—
183C2×C3—22no—
184C2×C3—22no—
185C2×C3—22no—
186C2×C3—48no—
187C2×C3—42no—
188C2×C3—42no—
189C2×C3—42no—
190C2×C3#6374314yescomplemented
191C2×C3C2×#377311yescharacteristic
192C2×C3C2×#377411yescharacteristic
193C2×C3C2×#377611yescharacteristic
194C22#8894012yes—
195C22#8894112yes—
196C22#8894312yes—
197C22C2×#157⋊D311yescharacteristic
198C4—91no—
199C4—91no—
200C4—92no—
201C4—364no—
202C3—22no—
203C3—42no—
204C3C2×#6374311yescharacteristic, complemented
205C2#1172279714yesdirect product factor
206C2C2×#8894011yescharacteristic
207C2C2×#8894111yescharacteristic
208C2C2×#8894311yescharacteristic
subgroupG/H
center, hypercenterC23#157⋊D3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×#157⋊D3
Fitting, socle, prime radicalC23×C33D4
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C2×#107
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
3. #11722797C2direct (N × H)
7. C2×C3×#622C4semidirect
22. C3×#622C2×C4semidirect
30. C2×#622Dic3semidirect
67. #622C2×Dic3semidirect
69. C2×C33#107semidirect
110. C33C2×#107semidirect
139. C2×C32#818semidirect
173. C32C2×#818semidirect
190. C2×C3#63743semidirect
204. C3C2×#63743semidirect
205. C2#11722797direct (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
#117227972
C2×#637433
C2×#8189