Finite Group Databasebeta

← #13155105

Subgroups — #13155105

Order 1728.

324 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 45 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×#88810C211yesmaximal, characteristic
2C2×C3×#3708C211yesmaximal, characteristic, complemented
3#11722080C211yesmaximal, characteristic, complemented
4#10590061—31nomaximal
5C2×C3×C4⋊(C3⋊D3)—21no—
6C3×C4⋊(C3⋊Dic3)—21no—
7C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
8#88810—21no—
9#88810C412yescomplemented
10C3×#3708—21no—
11C2×#3714—61no—
12C2×#3713—31no—
13C2×#3708D311yescharacteristic, complemented
14#63576—31no—
15#63575—61no—
16C22×C3×C3⋊D3—41no—
17C2×C33×C4D411yescharacteristic
18C2×C3×C3⋊Dic3—41no—
19C3×C4⋊(C3⋊D3)—22no—
20C3×C4⋊(C3⋊D3)—41no—
21C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
22C3×C3⋊(C3⋊C8)D411yescharacteristic
23#4509—91nomaximal
24C2×C4⋊(C3⋊D3)—21no—
25C2×C3×D12—61no—
26C2×C3×D12—61no—
27C2×C3×D12—121no—
28C4⋊(C3⋊Dic3)—21no—
29C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
30C3×C4⋊Dic3—61no—
31C3×C4⋊Dic3—61no—
32C3×C4⋊Dic3—121no—
33C2×C3×C3⋊C8—61no—
34C2×C3×C3⋊C8—61no—
35#3714—62no—
36#3714—62no—
37#3713—32no—
38#3713—61no—
39#3708—21no—
40C22×C33D811yescharacteristic
41C2×C3×C3⋊D3—42no—
42C2×C3×C3⋊D3—81no—
43C2×C3×C3⋊D3—81no—
44C33×C4C22⋊C411yescharacteristic
45C33×C4SD1611yescharacteristic
46C3×C3⋊Dic3—81no—
47C2×#213—181no—
48C2×D24—91no—
49C2×C3×C16—91no—
50#818—91no—
51#807—181no—
52C22×C3⋊D3—41no—
53C22×C3×D3—121no—
54C22×C3×D3—122no—
55C22×C3×D3—121no—
56C2×C32×C4—21no—
57C2×C32×C4—21no—
58C2×C32×C4—41no—
59C2×C32×C4—41no—
60C2×C32×C4D1211yescharacteristic
61C2×C3⋊Dic3—41no—
62C4⋊(C3⋊D3)—22no—
63C4⋊(C3⋊D3)—41no—
64C2×C3×Dic3—121no—
65C2×C3×Dic3—122no—
66C2×C3×Dic3—121no—
67C3×D12—62no—
68C3×D12—62no—
69C3×D12—122no—
70C3×D12—121no—
71C3×D12—122no—
72C3×D12—121no—
73C3⋊(C3⋊C8)C4×D311yescharacteristic
74C3⋊(C3⋊C8)#6711yescharacteristic
75C3×C3⋊C8—61no—
76C3×C3⋊C8—61no—
77C3×C3⋊C8—61no—
78C3×C3⋊C8—61no—
79#357—271noSylow
80C2×C33#10211yescharacteristic
81C2×C33D1611yescharacteristic
82C2×C33SD3211yescharacteristic
83C3×C3⋊D3—82no—
84C2×C3×D4—181no—
85C2×D12—61no—
86C2×D12—61no—
87C2×D12—121no—
88C2×C3×C8—91no—
89C3×C4⋊C4—181no—
90#213—182no—
91#213—182no—
92C4⋊Dic3—61no—
93C4⋊Dic3—61no—
94C4⋊Dic3—121no—
95C2×C3⋊C8—61no—
96C2×C3⋊C8—61no—
97D24—92no—
98D24—181no—
99C3×C16—181no—
100C22×C32—21no—
101C22×C32—21no—
102C22×C32—41no—
103C22×C32—41no—
104C22×C32D2411yescharacteristic
105C2×C3⋊D3—42no—
106C2×C3⋊D3—81no—
107C2×C3⋊D3—81no—
108C2×C3×D3—122no—
109C2×C3×D3—124no—
110C2×C3×D3—122no—
111C2×C3×D3—241no—
112C2×C3×D3—241no—
113C2×C3×D3—242no—
114C2×C3×D3—242no—
115C2×C3×D3—241no—
116C2×C3×D3—241no—
117C32×C4—21no—
118C32×C4—21no—
119C32×C4—21no—
120C32×C4—21no—
121C32×C4—41no—
122C32×C4—41no—
123C32×C4—41no—
124C32×C4—41no—
125C32×C4#20411yescharacteristic
126C32×C4(C2×D3)⋊C411yescharacteristic
127C3⋊Dic3—81no—
128C3×Dic3—241no—
129C3×Dic3—242no—
130C3×Dic3—241no—
131C2×D8—271no—
132C2×C16—91no—
133#107—271no—
134C33#35711yescharacteristic, complemented, Sylow
135C23×C3—361no—
136C22×D3—121no—
137C22×D3—122no—
138C22×D3—121no—
139C3×D4—182no—
140C3×D4—361no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—41no—
144C2×C3×C4—41no—
145C2×C3×C4—361no—
146C2×C3×C4GO+(4,𝔽2)11yescharacteristic
147C2×Dic3—121no—
148C2×Dic3—122no—
149C2×Dic3—121no—
150D12—62no—
151D12—62no—
152D12—122no—
153D12—121no—
154D12—122no—
155D12—121no—
156C3×C8—91no—
157C3×C8—91no—
158C3⋊C8—61no—
159C3⋊C8—61no—
160C3⋊C8—61no—
161C3⋊C8—61no—
162C2×C32—21no—
163C2×C32—21no—
164C2×C32—21no—
165C2×C32—21no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32—41no—
169C2×C32—41no—
170C2×C32—41no—
171C2×C32—41no—
172C2×C32—41no—
173C2×C32—41no—
174C2×C32D4811yescharacteristic
175C2×C32#80011yescharacteristic
176C2×C32#82111yescharacteristic
177C3⋊D3—82no—
178C3×D3—242no—
179C3×D3—244no—
180C3×D3—242no—
181C2×D4—181no—
182D8—272no—
183D8—541no—
184C2×C8—91no—
185C4⋊C4—181no—
186C16—181no—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—362no—
192C22×C3—721no—
193C22×C3—721no—
194C22×C3#377411yescharacteristic
195C2×D3—122no—
196C2×D3—124no—
197C2×D3—122no—
198C2×D3—241no—
199C2×D3—241no—
200C2×D3—242no—
201C2×D3—242no—
202C2×D3—241no—
203C2×D3—241no—
204C3×C4—21no—
205C3×C4—21no—
206C3×C4—21no—
207C3×C4—21no—
208C3×C4—41no—
209C3×C4—41no—
210C3×C4—41no—
211C3×C4—41no—
212C3×C4—721no—
213C3×C4D32⋊C411yescharacteristic
214C3×C4#377511yescharacteristic
215Dic3—241no—
216Dic3—242no—
217Dic3—241no—
218C32—21no—
219C32—21no—
220C32—41no—
221C32—41no—
222C32#450911yescharacteristic, complemented
223C23—361no—
224D4—182no—
225D4—361no—
226C2×C4—361no—
227C2×C4#157⋊D311yescharacteristic
228C8—91no—
229C8—91no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—21no—
234C2×C3—21no—
235C2×C3—21no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—41no—
239C2×C3—41no—
240C2×C3—41no—
241C2×C3—41no—
242C2×C3—722no—
243C2×C3#6373411yescharacteristic
244C2×C3#6373511yescharacteristic
245C2×C3#6373911yescharacteristic
246D3—242no—
247D3—244no—
248D3—242no—
249C22—362no—
250C22—721no—
251C22—721no—
252C22#8894111yescharacteristic
253C4—721no—
254C4#8893911yescharacteristic
255C4#8894211yescharacteristic
256C3—21no—
257C3—21no—
258C3—41no—
259C3—41no—
260C3#1059006111yescharacteristic, complemented
261C2—722no—
262C2#1172278811yescharacteristic
263C2#1172278911yescharacteristic
264C2#1172279311yescharacteristic
subgroupG/H
centerC22#88941
commutatorC3×C3⋊(C3⋊C8)C2×C4
Frattini, hypercenterC2×C4#157⋊D3
FittingC2×C33×C4D4
socleC22×C33D8
prime radicalC2×#88810C2
cosocleC2×C3×C3⋊(C3⋊C8)C22
hypocenterC33#357
LCSG ▷ C3×C3⋊(C3⋊C8) ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3708C2semidirect
3. #11722080C2semidirect
9. #88810C4semidirect
13. C2×#3708D3semidirect
134. C33#357semidirect
222. C32#4509semidirect
260. C3#10590061semidirect

Maximal subgroups

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

subgroupindex
C2×#888102
C2×C3×#37082
#117220802
#105900613
#45099