Finite Group Databasebeta

← C2×(((C3⋊D3)⋊C3)⋊C2)⋊C4 #11722313

Subgroups — C2×(((C3⋊D3)⋊C3)⋊C2)⋊C4 #11722313

Order 864.

386 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 48 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⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic
2C22×#1088C211yesmaximal, characteristic, complemented
3C2×#6199C211yesmaximal, characteristic, complemented
4(((C3⋊D3)⋊C3)⋊C2)⋊C4C214yesmaximal, direct product factor
5C22×#272C2211yescharacteristic
6C22×(C3⋊D3)⋊C3—21no—
7C2×((C3⋊D3)⋊C3)⋊C2—21no—
8C2×((C3⋊D3)⋊C3)⋊C2—22no—
9C2×((C3⋊D3)⋊C3)⋊C2C414yescomplemented
10C2×((C3⋊D3)⋊C3)⋊C2C2212yes—
11C2×#1088—22no—
12C2×#1088C2212yescomplemented
13C2×(C3⋊Dic3)⋊C3—21no—
14#6199—21no—
15#6199C2212yescomplemented
16C22×D32—61no—
17C2×D3×Dic3—61no—
18C22×H(3,𝔽3)D411yescharacteristic
19C2×#272C2×C412yes—
20C2×#272D411yescharacteristic
21C2×#272D412yes—
22C2×#272C2311yescharacteristic
23C2×(C3⋊D3)⋊C3—24no—
24C2×(C3⋊D3)⋊C3—22no—
25((C3⋊D3)⋊C3)⋊C2—24no—
26((C3⋊D3)⋊C3)⋊C2—22no—
27((C3⋊D3)⋊C3)⋊C2C2×C414yescomplemented
28#1088—22no—
29(C3⋊Dic3)⋊C3—22no—
30C2×(C2×D3)⋊C4—91nomaximal
31C22×C3⋊D3—21no—
32C22×C3×D3—61no—
33C22×C3×D3—61no—
34C22×C3×D3—61no—
35C2×D32—64no—
36C2×D32—62no—
37C2×D32—62no—
38C2×D32—64no—
39C2×C3⋊Dic3—21no—
40C2×C3×Dic3—61no—
41D3×Dic3—62no—
42D3×Dic3—62no—
43C2×H(3,𝔽3)C22⋊C412yescomplemented
44C2×H(3,𝔽3)C2×D411yescharacteristic
45#272C22⋊C412yescomplemented
46#272C22×C411yescharacteristic
47#272C2×D411yescharacteristic
48(C3⋊D3)⋊C3—24no—
49C23×D3—91no—
50C23×D3—181no—
51C22×C3×C4—91no—
52C22×Dic3—91no—
53C2×C4×D3—181no—
54(C2×D3)⋊C4—94no—
55C22×C32—21no—
56C22×C32—21no—
57C2×C3⋊D3—22no—
58C2×C3⋊D3—24no—
59C2×C3×D3—62no—
60C2×C3×D3—62no—
61C2×C3×D3—61no—
62C2×C3×D3—61no—
63C2×C3×D3—62no—
64C2×C3×D3—62no—
65C2×C3×D3—61no—
66C2×C3×D3—61no—
67C2×C3×D3—62no—
68C2×C3×D3—64no—
69D32—64no—
70D32—68no—
71D32—64no—
72C3⋊Dic3—22no—
73C3×Dic3—62no—
74C2×C22⋊C4—271noSylow
75H(3,𝔽3)C2×C22⋊C411yescharacteristic, complemented, Sylow
76C23×C3—91no—
77C23×C3—181no—
78C22×D3—61no—
79C22×D3—61no—
80C22×D3—61no—
81C22×D3—92no—
82C22×D3—94no—
83C22×D3—181no—
84C22×D3—182no—
85C22×D3—184no—
86C22×D3—181no—
87C22×D3—182no—
88C22×D3—184no—
89C22×D3—182no—
90C2×C3×C4—92no—
91C2×C3×C4—181no—
92C2×C3×C4—182no—
93C2×Dic3—61no—
94C2×Dic3—92no—
95C2×Dic3—181no—
96C2×Dic3—181no—
97C4×D3—182no—
98C4×D3—182no—
99C2×C32—21no—
100C2×C32—21no—
101C2×C32—22no—
102C2×C32—22no—
103C3⋊D3—24no—
104C3×D3—62no—
105C3×D3—62no—
106C3×D3—61no—
107C3×D3—61no—
108C3×D3—61no—
109C3×D3—61no—
110C3×D3—64no—
111C24—271no—
112C22×C4—91no—
113C22×C4—271no—
114C22⋊C4—274no—
115C22×C3—61no—
116C22×C3—61no—
117C22×C3—91no—
118C22×C3—91no—
119C22×C3—92no—
120C22×C3—92no—
121C22×C3—184no—
122C22×C3—182no—
123C22×C3GO+(4,𝔽2)11yescharacteristic
124C2×D3—61no—
125C2×D3—61no—
126C2×D3—62no—
127C2×D3—62no—
128C2×D3—61no—
129C2×D3—61no—
130C2×D3—62no—
131C2×D3—62no—
132C2×D3—62no—
133C2×D3—64no—
134C2×D3—94no—
135C2×D3—184no—
136C2×D3—182no—
137C2×D3—184no—
138C2×D3—182no—
139C2×D3—184no—
140C2×D3—184no—
141C2×D3—188no—
142C3×C4—182no—
143C3×C4—182no—
144Dic3—62no—
145Dic3—182no—
146C32—21no—
147C32—21no—
148C23—91no—
149C23—181no—
150C23—272no—
151C23—274no—
152C23—542no—
153C23—541no—
154C2×C4—92no—
155C2×C4—182no—
156C2×C4—181no—
157C2×C4—272no—
158C2×C4—541no—
159C2×C3—62no—
160C2×C3—62no—
161C2×C3—61no—
162C2×C3—61no—
163C2×C3—91no—
164C2×C3—91no—
165C2×C3—92no—
166C2×C3—184no—
167C2×C3D32⋊C412yes—
168C2×C3C2×GO+(4,𝔽2)11yescharacteristic
169D3—61no—
170D3—61no—
171D3—62no—
172D3—61no—
173D3—61no—
174D3—62no—
175D3—64no—
176D3—184no—
177C22—92no—
178C22—92no—
179C22—91no—
180C22—91no—
181C22—184no—
182C22—182no—
183C22—274no—
184C22—542no—
185C22—544no—
186C22#1088⋊C211yescharacteristic
187C4—182no—
188C4—182no—
189C3—61no—
190C3—61no—
191C3C2×D32⋊C411yescharacteristic
192C2—91no—
193C2—91no—
194C2—92no—
195C2—184no—
196C2(((C3⋊D3)⋊C3)⋊C2)⋊C412yesdirect product factor
197C2C2×#1088⋊C211yescharacteristic
subgroupG/H
center, hypercenterC22#1088⋊C2
commutator#272C22×C4
FrattiniC2×C3C2×GO+(4,𝔽2)
FittingC22×H(3,𝔽3)D4
socleC22×C3GO+(4,𝔽2)
prime radicalC22×((C3⋊D3)⋊C3)⋊C2C2
cosocleC2×#272C23
hypocenterH(3,𝔽3)C2×C22⋊C4
LCSG ▷ #272 ▷ H(3,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
2. C22×#1088C2semidirect
3. C2×#6199C2semidirect
4. (((C3⋊D3)⋊C3)⋊C2)⋊C4C2direct (N × H)
9. C2×((C3⋊D3)⋊C3)⋊C2C4semidirect
12. C2×#1088C22semidirect
15. #6199C22semidirect
27. ((C3⋊D3)⋊C3)⋊C2C2×C4semidirect
43. C2×H(3,𝔽3)C22⋊C4semidirect
45. #272C22⋊C4semidirect
75. H(3,𝔽3)C2×C22⋊C4semidirect
196. C2(((C3⋊D3)⋊C3)⋊C2)⋊C4direct (N × H)

Maximal subgroups

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

subgroupindex
C22×((C3⋊D3)⋊C3)⋊C22
C22×#10882
C2×#61992
(((C3⋊D3)⋊C3)⋊C2)⋊C42
C2×(C2×D3)⋊C49