Finite Group Databasebeta

← #5827⋊C9 #13147288

Subgroups — #5827⋊C9 #13147288

Order 1728.

343 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 343 conjugacy classes of subgroups — too many to draw. Only its 77 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×C3×#41⋊C9C211yesmaximal, characteristic
2Dic3×Q8⋊C9C212yesmaximal, complemented
3C3×#5827C311yesmaximal, characteristic
4#520⋊C9—31nomaximal
5C3×#41⋊C9C412yescomplemented
6C2×C3×Q8⋊C9C2211yescharacteristic
7C9×C4⋊Dic3—41nomaximal
8C2×C32×#41C2×C311yescharacteristic
9C3×Q8×Dic3C2×C312yes—
10C3×D4×Dic3—32no—
11C3×#926—31no—
12C2×C3×C4⋊Dic3—31no—
13C2×#41⋊C9—21no—
14C2×#41⋊C9D311yescharacteristic
15C4×Q8⋊C9—32no—
16C2×C3×C4×C9—41no—
17C2×C9×Dic3—42no—
18C3×Q8⋊C9C2×C411yescharacteristic, complemented
19C3×#520—31no—
20#5827C911yescharacteristic, complemented
21C32×#41—32no—
22C32×#41C3×C412yes—
23C2×C32×Q8C22×C311yescharacteristic
24C2×C32×D4—31no—
25C22×C32×C4—31no—
26C22×C3×Dic3—32no—
27C3×#217—32no—
28C3×C4⋊Dic3—31no—
29C3×C4⋊Dic3—32no—
30C3×C4⋊Dic3A411yescharacteristic
31C3×C4×Dic3—32no—
32#41⋊C9—22no—
33#41⋊C9Dic312yescomplemented
34C2×Q8⋊C9—21no—
35C2×Q8⋊C9C2×D311yescharacteristic
36C9×C4⋊C4—121no—
37C22×C3×C9—41no—
38C3×C4×C9—42no—
39C9×Dic3—82no—
40C2×C3×#41—21no—
41C2×C3×#41C2×C911yescharacteristic
42C2×C3×#41C3×D311yescharacteristic
43C3×C4×Q8—32no—
44C3×C4×D4—92no—
45C3×#117—91no—
46C2×C3×C4⋊C4—91no—
47Q8×Dic3C2×C912yes—
48D4×Dic3—32no—
49#926—31no—
50C2×C4⋊Dic3—31no—
51C23×C32—31no—
52C32×Q8—31no—
53C32×Q8C2×C3×C411yescharacteristic
54C32×D4—32no—
55C32×D4—32no—
56C2×C32×C4—31no—
57C2×C32×C4—32no—
58C2×C32×C4—32no—
59C2×C32×C4C2×A411yescharacteristic
60C2×C3×Dic3—32no—
61C2×C3×Dic3—122no—
62C2×C3×Dic3C2×A412yes—
63C2×C4×C9—41no—
64C2×C4×C9—81no—
65C2×C4×C9—122no—
66Q8⋊C9—21no—
67Q8⋊C9C2×Dic311yescharacteristic, complemented
68#520—31noSylow
69C2×C3×C9—41no—
70C2×C3×C9—41no—
71C2×C3×C9—41no—
72C3×#41—22no—
73C3×#41—32no—
74C3×#41—32no—
75C3×#41—62no—
76C3×#41C4×C912yescomplemented
77C3×#41C3×Dic312yes—
78C2×C3×Q8—21no—
79C2×C3×Q8C22×C911yescharacteristic
80C2×C3×Q8C2×C3×D311yescharacteristic
81C2×C3×D4—31no—
82C2×C3×D4—31no—
83C2×C3×D4—61no—
84C22×C3×C4—31no—
85C22×C3×C4—31no—
86C22×C3×C4—61no—
87C22×C3×C4—92no—
88C22×Dic3—32no—
89C3×C4⋊C4—31no—
90C3×C4⋊C4—92no—
91C3×C4⋊C4—91no—
92C3×C22⋊C4—92no—
93C3×C42—92no—
94#217—32no—
95C4⋊Dic3—31no—
96C4⋊Dic3—32no—
97C4⋊Dic3C22⋊C911yescharacteristic
98C4×Dic3—32no—
99C22×C32—32no—
100C22×C32—61no—
101C22×C32—61no—
102C22×C32C22×A411yescharacteristic
103C32×C4—31no—
104C32×C4—31no—
105C32×C4C4×A412yes—
106C3×Dic3—22no—
107C3×Dic3—62no—
108C22×C9—41no—
109C22×C9—81no—
110C4×C9—42no—
111C4×C9—82no—
112C4×C9—242no—
113C2×#41D3×C911yescharacteristic
114C4×Q8—32no—
115C4×D4—92no—
116#117—91no—
117C2×C4⋊C4—91no—
118C3×C9—41noSylow
119C23×C3—31no—
120C23×C3—31no—
121C23×C3—61no—
122C3×Q8—21no—
123C3×Q8—31no—
124C3×Q8—31no—
125C3×Q8—61no—
126C3×Q8C2×C4×C911yescharacteristic
127C3×Q8C2×C3×Dic311yescharacteristic
128C3×D4—32no—
129C3×D4—32no—
130C3×D4—32no—
131C3×D4—32no—
132C3×D4—62no—
133C3×D4—62no—
134C2×C3×C4—21no—
135C2×C3×C4—31no—
136C2×C3×C4—31no—
137C2×C3×C4—32no—
138C2×C3×C4—32no—
139C2×C3×C4—32no—
140C2×C3×C4—32no—
141C2×C3×C4—32no—
142C2×C3×C4—62no—
143C2×C3×C4—62no—
144C2×C3×C4—61no—
145C2×C3×C4—92no—
146C2×C3×C4—362no—
147C2×C3×C4C2×C22⋊C911yescharacteristic
148C2×C3×C4D3×A411yescharacteristic
149C2×Dic3—32no—
150C2×Dic3—122no—
151C2×Dic3C2×C22⋊C912yes—
152C2×C32—62no—
153C2×C32C2×C4×A411yescharacteristic
154C2×C32#99411yescharacteristic
155C2×C322−1+4⋊C311yescharacteristic
156C2×C9—41no—
157C2×C9—41no—
158C2×C9—41no—
159C2×C9—81no—
160C2×C9—81no—
161C2×C9—81no—
162#41—32no—
163#41C9×Dic312yescomplemented
164C2×Q8C2×D3×C911yescharacteristic
165C2×D4—31no—
166C22×C4—31no—
167C22×C4—92no—
168C4⋊C4—31no—
169C4⋊C4—92no—
170C4⋊C4—91no—
171C22⋊C4—92no—
172C42—92no—
173C22×C3—21no—
174C22×C3—32no—
175C22×C3—32no—
176C22×C3—62no—
177C22×C3—61no—
178C22×C3—61no—
179C22×C3—61no—
180C22×C3—61no—
181C22×C3—121no—
182C22×C3—121no—
183C22×C3C22×C22⋊C911yescharacteristic
184C22×C3C2×D3×A411yescharacteristic
185C3×C4—22no—
186C3×C4—31no—
187C3×C4—31no—
188C3×C4—31no—
189C3×C4—31no—
190C3×C4—61no—
191C3×C4—61no—
192C3×C4—62no—
193C3×C4—182no—
194C3×C4C4×C22⋊C912yes—
195C3×C4Dic3×A412yes—
196Dic3—22no—
197Dic3—62no—
198C32#520⋊C311yescharacteristic
199C9—41no—
200C9—81no—
201C23—31no—
202Q8—31no—
203Q8C2×C9×Dic311yescharacteristic
204D4—32no—
205D4—32no—
206C2×C4—31no—
207C2×C4—32no—
208C2×C4—32no—
209C2×C4—32no—
210C2×C4—92no—
211C2×C4—362no—
212C2×C4D3×C22⋊C911yescharacteristic
213C2×C3—21no—
214C2×C3—21no—
215C2×C3—21no—
216C2×C3—62no—
217C2×C3—62no—
218C2×C3—122no—
219C2×C3C2×C4×C22⋊C911yescharacteristic
220C2×C3#6370011yescharacteristic
221C2×C32−1+4⋊C911yescharacteristic
222C2×C3C3⋊#99411yescharacteristic
223C2×C3#1010⋊C311yescharacteristic
224C2×C3C2×Dic3×A411yescharacteristic
225C22—32no—
226C22—61no—
227C22—61no—
228C22C2×D3×C22⋊C911yescharacteristic
229C4—31no—
230C4—31no—
231C4—62no—
232C4—182no—
233C4Dic3×C22⋊C912yes—
234C3—21no—
235C3#520⋊C911yescharacteristic, complemented
236C3#5827⋊C311yescharacteristic
237C2—62no—
238C2C3⋊#6370011yescharacteristic
239C2#1010⋊C911yescharacteristic
240C2C2×Dic3×C22⋊C911yescharacteristic
subgroupG/H
center, FrattiniC22×C3C2×D3×A4
commutator, hypocenterC3×Q8C2×C4×C9
Fitting, prime radicalC2×C32×#41C2×C3
socleC22×C32C22×A4
cosocleC2×C32×Q8C22×C3
hypercenterC2×C3×C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. Dic3×Q8⋊C9C2semidirect
5. C3×#41⋊C9C4semidirect
18. C3×Q8⋊C9C2×C4semidirect
20. #5827C9semidirect
33. #41⋊C9Dic3semidirect
67. Q8⋊C9C2×Dic3semidirect
76. C3×#41C4×C9semidirect
163. #41C9×Dic3semidirect
235. C3#520⋊C9semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#41⋊C92
Dic3×Q8⋊C92
C3×#58273
#520⋊C93
C9×C4⋊Dic34