Finite Group Databasebeta

← D4⋊(C3⋊#622) #13157230

Subgroups — D4⋊(C3⋊#622) #13157230

Order 1728.

460 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C3×D4×C3⋊Dic3C211yesmaximal, characteristic
2C4⋊(C3⋊#622)C211yesmaximal, characteristic, complemented
3C4×C3⋊#622C211yesmaximal, characteristic, complemented
4D4⋊#622—31nomaximal
5C2×C33×D4C411yescharacteristic
6C22×C3×C3⋊Dic3—21no—
7C2×C3⋊#622—21no—
8C2×C3⋊#622—21no—
9C3×#3757—21no—
10C3×C4⋊(C3⋊Dic3)C411yescharacteristic
11C3×C4×C3⋊Dic3C2211yescharacteristic
12D4×C3⋊Dic3D311yescharacteristic
13C3×D4×Dic3—62no—
14C4⋊#622—31no—
15C4×#622—31no—
16C23×C33—21no—
17C33×D4—21no—
18C33×D4C812yescomplemented
19C2×C33×C4C2×C411yescharacteristic
20C2×C3×C3⋊Dic3—21no—
21C2×C3×C3⋊Dic3—41no—
22C2×C3×C3⋊Dic3—41no—
23C2×C3×C3⋊Dic3D411yescharacteristic
24C2×C3×C3⋊Dic3D411yescharacteristic
25C3⋊#622—22no—
26C3⋊#622—41no—
27D4⋊(C3⋊C8)—91nomaximal
28C2×C32×D4—22no—
29C2×C32×D4—42no—
30C2×C32×D4Dic311yescharacteristic
31C22×C3⋊Dic3—21no—
32C22×C3×Dic3—64no—
33C2×#622—61no—
34C2×#622—61no—
35#3757—21no—
36C4⋊(C3⋊Dic3)Dic311yescharacteristic
37C4×C3⋊Dic3C2×D311yescharacteristic
38C3×#217—64no—
39C3×C4⋊Dic3—62no—
40C3×C4×Dic3—62no—
41C22×C33—22no—
42C22×C33—41no—
43C22×C33—41no—
44C22×C33C22⋊C411yescharacteristic
45C33×C4C2×C811yescharacteristic
46C33×C4M1611yescharacteristic
47C3×C3⋊Dic3—21no—
48C3×C3⋊Dic3—41no—
49C3×C3⋊Dic3D811yescharacteristic
50C3×C3⋊Dic3SD1611yescharacteristic
51C3×C4×D4—91no—
52D4×Dic3—62no—
53C4⋊(C3⋊C8)—91no—
54C4×C3⋊C8—91no—
55C23×C32—21no—
56C23×C32—24no—
57C23×C32—44no—
58C32×D4—21no—
59C32×D4—24no—
60C32×D4—24no—
61C32×D4—44no—
62C32×D4—44no—
63C32×D4C3⋊C812yescomplemented
64C2×C32×C4—22no—
65C2×C32×C4—42no—
66C2×C32×C4C2×Dic311yescharacteristic
67C2×C3⋊Dic3—21no—
68C2×C3⋊Dic3—41no—
69C2×C3⋊Dic3—41no—
70C2×C3⋊Dic3#6711yescharacteristic
71C2×C3⋊Dic3#6711yescharacteristic
72C2×C3×Dic3—62no—
73C2×C3×Dic3—62no—
74C2×C3×Dic3—64no—
75C2×C3×Dic3—124no—
76C2×C3×Dic3—124no—
77#622—62no—
78#622—121no—
79D4⋊C8—271noSylow
80C2×C33—42no—
81C2×C33C22⋊C811yescharacteristic
82C2×C33#10211yescharacteristic
83C2×C33#10411yescharacteristic
84C2×C3×D4—22no—
85C2×C3×D4—42no—
86C2×C3×D4#15711yescharacteristic
87C22×C3×C4—181no—
88C22×Dic3—64no—
89C3×C4⋊C4—91no—
90C3×C22⋊C4—181no—
91C3×C42—91no—
92#217—64no—
93C4⋊Dic3—62no—
94C4×Dic3—62no—
95C2×C3⋊C8—181no—
96C2×C3⋊C8—181no—
97C22×C32—22no—
98C22×C32—28no—
99C22×C32—22no—
100C22×C32—48no—
101C22×C32—42no—
102C22×C32—41no—
103C22×C32—41no—
104C22×C32—44no—
105C22×C32—44no—
106C22×C32—84no—
107C22×C32—84no—
108C22×C32#21711yescharacteristic
109C32×C4—22no—
110C32×C4—22no—
111C32×C4—42no—
112C32×C4—42no—
113C32×C4C2×C3⋊C811yescharacteristic
114C32×C4#20811yescharacteristic
115C3⋊Dic3—21no—
116C3⋊Dic3—41no—
117C3⋊Dic3#21311yescharacteristic
118C3⋊Dic3#21411yescharacteristic
119C3×Dic3—62no—
120C3×Dic3—62no—
121C3×Dic3—122no—
122C3×Dic3—124no—
123C4×D4—91no—
124C4⋊C8—271no—
125C4×C8—271no—
126C33D4⋊C811yescharacteristic, complemented, Sylow
127C23×C3—24no—
128C23×C3—21no—
129C23×C3—44no—
130C3×D4—24no—
131C3×D4—21no—
132C3×D4—24no—
133C3×D4—44no—
134C3×D4—44no—
135C3×D4#62212yescomplemented
136C2×C3×C4—22no—
137C2×C3×C4—42no—
138C2×C3×C4—91no—
139C2×C3×C4—91no—
140C2×C3×C4—181no—
141C2×C3×C4—361no—
142C2×C3×C4—361no—
143C2×C3×C4C2×#15711yescharacteristic
144C2×Dic3—62no—
145C2×Dic3—62no—
146C2×Dic3—64no—
147C2×Dic3—124no—
148C2×Dic3—124no—
149C3⋊C8—182no—
150C3⋊C8—361no—
151C2×C32—22no—
152C2×C32—22no—
153C2×C32—22no—
154C2×C32—42no—
155C2×C32—42no—
156C2×C32—42no—
157C2×C32—42no—
158C2×C32—48no—
159C2×C32—88no—
160C2×C32#83011yescharacteristic
161C2×C32#83211yescharacteristic
162C2×C32#83711yescharacteristic
163C2×D4C3⋊#15711yescharacteristic
164C22×C4—181no—
165C2×C8—541no—
166C2×C8—541no—
167C4⋊C4—91no—
168C22⋊C4—181no—
169C42—91no—
170C22×C3—28no—
171C22×C3—22no—
172C22×C3—22no—
173C22×C3—48no—
174C22×C3—44no—
175C22×C3—44no—
176C22×C3—41no—
177C22×C3—41no—
178C22×C3—42no—
179C22×C3—84no—
180C22×C3—84no—
181C22×C3C22⋊#15711yescharacteristic
182C3×C4—22no—
183C3×C4—22no—
184C3×C4—42no—
185C3×C4—42no—
186C3×C4—91no—
187C3×C4—91no—
188C3×C4—181no—
189C3×C4—361no—
190C3×C4C2×#62211yescharacteristic
191C3×C4#379211yescharacteristic
192Dic3—62no—
193Dic3—62no—
194Dic3—122no—
195Dic3—124no—
196C32—22no—
197C32—42no—
198C32D4⋊(C3⋊C8)11yescharacteristic, complemented
199C23—21no—
200D4—21no—
201D4C3⋊#62212yescomplemented
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—181no—
205C2×C4—361no—
206C2×C4—361no—
207C2×C4C2×C3⋊#15711yescharacteristic
208C8—542no—
209C8—1081no—
210C2×C3—22no—
211C2×C3—22no—
212C2×C3—22no—
213C2×C3—48no—
214C2×C3—42no—
215C2×C3—42no—
216C2×C3—42no—
217C2×C3—42no—
218C2×C3—88no—
219C2×C3#6378211yescharacteristic
220C2×C3#6378311yescharacteristic
221C2×C3C22⋊#62211yescharacteristic
222C22—22no—
223C22—41no—
224C22—41no—
225C22#8899411yescharacteristic
226C4—91no—
227C4—91no—
228C4—181no—
229C4—361no—
230C4C2×C3⋊#62211yescharacteristic
231C4#8899311yescharacteristic
232C3—22no—
233C3—42no—
234C3D4⋊#62211yescharacteristic, complemented
235C2—42no—
236C2#1172310711yescharacteristic
237C2#1172310811yescharacteristic
238C2#1172311211yescharacteristic
subgroupG/H
centerC22#88994
commutatorC33×C4C2×C8
FrattiniC2×C4C2×C3⋊#157
Fitting, prime radicalC2×C33×D4C4
socleC22×C33C22⋊C4
cosocleC3×C4×C3⋊Dic3C22
hypercenterC2×D4C3⋊#157
hypocenterC33D4⋊C8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C4⋊(C3⋊#622)C2semidirect
3. C4×C3⋊#622C2semidirect
18. C33×D4C8semidirect
63. C32×D4C3⋊C8semidirect
126. C33D4⋊C8semidirect
135. C3×D4#622semidirect
198. C32D4⋊(C3⋊C8)semidirect
201. D4C3⋊#622semidirect
234. C3D4⋊#622semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C3⋊Dic32
C4⋊(C3⋊#622)2
C4×C3⋊#6222
D4⋊#6223
D4⋊(C3⋊C8)9