Finite Group Databasebeta

← #13146549

Subgroups — #13146549

Order 1728.

597 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 57 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×#88655C211yesmaximal, characteristic
2C2×#88627C211yesmaximal, characteristic, complemented
3#11721097C211yesmaximal, characteristic, complemented
4#11720705C214yesmaximal, complemented
5C22×((C3⋊D3)⋊C3)⋊C2—21no—
6C2×C4×#272C2211yescharacteristic
7C2×C4⋊((C3⋊D3)⋊C3)—21no—
8C2×#6198C2212yes—
9#88655—22no—
10#88655C414yescomplemented
11C2×#6188C2212yescomplemented
12#88627—22no—
13C4×(C3⋊Dic3)⋊C3—21no—
14#88451—21no—
15#88449C2212yescomplemented
16C2×D3×D12—61no—
17#63855—61no—
18C22×#272D411yescharacteristic
19C22×(C3⋊D3)⋊C3—22no—
20C2×((C3⋊D3)⋊C3)⋊C2—24no—
21C2×((C3⋊D3)⋊C3)⋊C2—44no—
22C2×C4×H(3,𝔽3)D411yescharacteristic
23C2×#1084C2311yescharacteristic
24C4×#272C2×C412yes—
25C4×#272D412yes—
26C2×(C3⋊Dic3)⋊C3—22no—
27C4⋊((C3⋊D3)⋊C3)—24no—
28#6198—22no—
29#6198C2×C414yes—
30#6188—22no—
31#5131—91nomaximal
32C22×D32—62no—
33C2×C4⋊(C3⋊D3)—21no—
34C2×C3×D12—61no—
35C2×C3×C4×D3—61no—
36C2×C3×C4×D3—61no—
37C2×#626—62no—
38D3×D12—64no—
39D3×D12—64no—
40C4×C3⋊Dic3—21no—
41C3×C4×Dic3—61no—
42Dic3⋊Dic3—62no—
43(C2×D3)⋊Dic3—62no—
44C22×H(3,𝔽3)C2×D411yescharacteristic
45C2×#272—22no—
46C2×#272C22⋊C412yes—
47C2×(C3⋊D3)⋊C3—24no—
48C2×(C3⋊D3)⋊C3—44no—
49((C3⋊D3)⋊C3)⋊C2—48no—
50C4×H(3,𝔽3)C22⋊C412yes—
51#1084C22×C411yescharacteristic
52#1084C2×D411yescharacteristic
53(C3⋊Dic3)⋊C3—42no—
54C2×D3×D4—181no—
55C22×D12—91no—
56C2×C3×M16—91no—
57#926—91no—
58#872—181no—
59#821—94no—
60C22×C3⋊D3—22no—
61C22×C3×D3—61no—
62C22×C3×D3—61no—
63C22×C3×D3—62no—
64C2×D32—68no—
65C2×D32—128no—
66C2×C32×C4—21no—
67C2×C32×C4—21no—
68C2×C3⋊Dic3—22no—
69C4⋊(C3⋊D3)—24no—
70C2×C3×Dic3—61no—
71C2×C3×Dic3—61no—
72C2×C3×Dic3—62no—
73C3×D12—64no—
74C3×C4×D3—62no—
75C3×C4×D3—62no—
76C3×C4×D3—62no—
77C3×C4×D3—62no—
78#626—64no—
79#626—64no—
80#418—271noSylow
81C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
82C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
83#272—22no—
84(C3⋊D3)⋊C3—44no—
85C23×D3—181no—
86C23×D3—182no—
87C2×C3×D4—181no—
88C22×C3×C4—91no—
89C2×#67—182no—
90D3×D4—184no—
91D3×D4—184no—
92C2×D12—61no—
93C2×D12—94no—
94C2×D12—92no—
95C2×D12—181no—
96C2×D12—182no—
97C2×C4×D3—61no—
98C2×C4×D3—61no—
99C3×M16—182no—
100C2×C3×C8—92no—
101C3×C42—181no—
102#217—181no—
103(C2×D3)⋊C4—182no—
104C4⋊Dic3—92no—
105Dic3⋊C4—182no—
106C4×Dic3—61no—
107C4×Dic3—181no—
108C22×C32—21no—
109C22×C32—21no—
110C2×C3⋊D3—24no—
111C2×C3⋊D3—44no—
112C2×C3×D3—62no—
113C2×C3×D3—62no—
114C2×C3×D3—64no—
115C2×C3×D3—64no—
116C2×C3×D3—122no—
117C2×C3×D3—124no—
118D32—1216no—
119C32×C4—22no—
120C32×C4—22no—
121C3⋊Dic3—42no—
122C3×Dic3—61no—
123C3×Dic3—61no—
124C3×Dic3—61no—
125C3×Dic3—61no—
126C3×Dic3—122no—
127C22×D4—271no—
128C2×M16—91no—
129#117—271no—
130#102—274no—
131H(3,𝔽3)#41811yescharacteristic, complemented, Sylow
132C23×C3—91no—
133C23×C3—182no—
134C22×D3—61no—
135C22×D3—61no—
136C22×D3—62no—
137C22×D3—182no—
138C22×D3—184no—
139C22×D3—188no—
140C22×D3—368no—
141C22×D3—364no—
142C3×D4—184no—
143C2×C3×C4—61no—
144C2×C3×C4—61no—
145C2×C3×C4—91no—
146C2×C3×C4—92no—
147C2×C3×C4—92no—
148C2×C3×C4—182no—
149C2×C3×C4GO+(4,𝔽2)11yescharacteristic
150#67—184no—
151#67—184no—
152C2×Dic3—61no—
153C2×Dic3—61no—
154C2×Dic3—62no—
155C2×Dic3—182no—
156D12—64no—
157D12—94no—
158D12—184no—
159D12—182no—
160C4×D3—62no—
161C4×D3—62no—
162C4×D3—62no—
163C4×D3—62no—
164C3×C8—182no—
165C2×C32—22no—
166C2×C32—22no—
167C2×C32—21no—
168C2×C32—21no—
169C3⋊D3—44no—
170C3×D3—64no—
171C3×D3—122no—
172C3×D3—124no—
173C24—541no—
174C2×D4—181no—
175C2×D4—274no—
176C2×D4—272no—
177C2×D4—542no—
178C22×C4—91no—
179M16—182no—
180C2×C8—92no—
181C4⋊C4—272no—
182C22⋊C4—541no—
183C42—181no—
184C22×C3—61no—
185C22×C3—61no—
186C22×C3—92no—
187C22×C3—184no—
188C22×C3—182no—
189C22×C3—364no—
190C22×C3C2×GO+(4,𝔽2)11yescharacteristic
191C2×D3—62no—
192C2×D3—62no—
193C2×D3—64no—
194C2×D3—64no—
195C2×D3—122no—
196C2×D3—124no—
197C2×D3—184no—
198C2×D3—364no—
199C2×D3—3616no—
200C2×D3—368no—
201C3×C4—62no—
202C3×C4—62no—
203C3×C4—91no—
204C3×C4—91no—
205C3×C4—362no—
206C3×C4D32⋊C412yes—
207Dic3—61no—
208Dic3—61no—
209Dic3—61no—
210Dic3—61no—
211Dic3—122no—
212Dic3—362no—
213C32—21no—
214C32—21no—
215C23—91no—
216C23—182no—
217C23—544no—
218C23—1084no—
219D4—184no—
220D4—274no—
221D4—542no—
222C2×C4—91no—
223C2×C4—92no—
224C2×C4—92no—
225C2×C4—182no—
226C2×C4#1088⋊C211yescharacteristic
227C8—182no—
228C2×C3—62no—
229C2×C3—62no—
230C2×C3—61no—
231C2×C3—61no—
232C2×C3—182no—
233C2×C3—364no—
234C2×C3#6422412yes—
235C2×C3C2×D32⋊C411yescharacteristic
236D3—64no—
237D3—122no—
238D3—124no—
239D3—364no—
240C22—92no—
241C22—182no—
242C22—184no—
243C22—364no—
244C22—1088no—
245C22C2×#1088⋊C211yescharacteristic
246C4—91no—
247C4—91no—
248C4—362no—
249C4(((C3⋊D3)⋊C3)⋊C2)⋊C412yes—
250C3—61no—
251C3—61no—
252C3#1059336311yescharacteristic
253C2—182no—
254C2—364no—
255C2#1172229712yes—
256C2C2×(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator#1084C22×C4
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radicalC2×#88655C2
cosocleC2×#1084C23
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#418
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88627C2semidirect
3. #11721097C2semidirect
4. #11720705C2semidirect
10. #88655C4semidirect
11. C2×#6188C22semidirect
15. #88449C22semidirect
131. H(3,𝔽3)#418semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
C2×#886272
#117210972
#117207052
#51319