Finite Group Databasebeta

← #13167497

Subgroups — #13167497

Order 1728.

516 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 55 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×#3788C211yesmaximal, characteristic, complemented
2C2×#88941C211yesmaximal, characteristic, complemented
3#11723695C212yesmaximal, complemented
4#11722796C211yesmaximal, characteristic, complemented
5#11722793C212yesmaximal, complemented
6#10593429—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×#6295—22no—
9C2×#6295C2212yes—
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C3×#3788—22no—
13#88941—22no—
14C3×#3752—22no—
15#88815C2212yes—
16C2×#3788D311yescharacteristic, complemented
17C2×#3774—31no—
18#63912—32no—
19#63906—62no—
20#63742—31no—
21#63739—32no—
22C22×C3×C3⋊D3—22no—
23C22×C3×C3⋊D3D411yescharacteristic
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3—22no—
26C2×C3×C3⋊Dic3C2311yescharacteristic
27C3×C4×C3⋊D3—21no—
28C3×C4×C3⋊D3—21no—
29#6295—22no—
30#6295—22no—
31#6295—44no—
32C3×#622—21no—
33C3×#622D412yescomplemented
34#5135—91nomaximal
35C2×C4×C3⋊D3C2×D311yescharacteristic
36C2×C3×C4×D3—62no—
37C2×#626—62no—
38C2×#626—62no—
39C2×#626—62no—
40C2×#625—32no—
41C2×#625—62no—
42C2×#622C2×D311yescharacteristic
43C2×#622C2×D311yescharacteristic
44#3788—22no—
45#3774—62no—
46#3752—22no—
47C3×(C2×D3)⋊C4—62no—
48C3×(C2×D3)⋊C4—62no—
49C3×(C2×D3)⋊C4—122no—
50#3724—32no—
51Dic3⋊Dic3—62no—
52C22×C33C2×D411yescharacteristic
53C2×C3×C3⋊D3—21no—
54C2×C3×C3⋊D3—22no—
55C2×C3×C3⋊D3—42no—
56C2×C3×C3⋊D3—42no—
57C2×C3×C3⋊D3—42no—
58C33×C4—21no—
59C3×C3⋊Dic3—42no—
60C3×C3⋊Dic3C2×D411yescharacteristic
61C3×C3⋊Dic3#4111yescharacteristic
62C2×C3×M16—91no—
63#930—92no—
64C2×D24—91no—
65#884—182no—
66#821—92no—
67#817—91no—
68C22×C3⋊D3—22no—
69C22×C3⋊D3D1211yescharacteristic
70C22×C3×D3—62no—
71C22×C3×D3—62no—
72C22×C3×D3—62no—
73C22×C3×D3—122no—
74C2×C32×C4—22no—
75C2×C32×C4—42no—
76C2×C32×C4D1211yescharacteristic
77C2×C3⋊Dic3—22no—
78C2×C3⋊Dic3C22×D311yescharacteristic
79C4×C3⋊D3—21no—
80C4×C3⋊D3—21no—
81C2×C3×Dic3—62no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—122no—
85C3×C4×D3—122no—
86C3×C4×D3—122no—
87#626—122no—
88#626—122no—
89#626—124no—
90#626—124no—
91#625—62no—
92#625—62no—
93#625—124no—
94#622—21no—
95#622#6712yescomplemented
96#469—271noSylow
97C2×C33#12111yescharacteristic
98C2×C33AGL(1,ℤ/8)12yes—
99C3×C3⋊D3—41no—
100C3×C3⋊D3—82no—
101C22×C3×C4—91no—
102C2×#67—182no—
103C2×#67—182no—
104C2×#67—182no—
105C2×D12—92no—
106C2×D12—182no—
107C2×C4×D3—62no—
108C3×M16—182no—
109C2×C3×C8—91no—
110C2×C3×C8—91no—
111C3×C22⋊C4—182no—
112(C2×D3)⋊C4—62no—
113(C2×D3)⋊C4—62no—
114(C2×D3)⋊C4—122no—
115C4⋊Dic3—92no—
116Dic3⋊C4—182no—
117D24—182no—
118C22×C32—22no—
119C22×C32—42no—
120C22×C32C2×D1211yescharacteristic
121C2×C3⋊D3—22no—
122C2×C3⋊D3—21no—
123C2×C3⋊D3—42no—
124C2×C3⋊D3—42no—
125C2×C3⋊D3—42no—
126C2×C3×D3—64no—
127C2×C3×D3—122no—
128C2×C3×D3—122no—
129C2×C3×D3—122no—
130C2×C3×D3—122no—
131C2×C3×D3—122no—
132C2×C3×D3—124no—
133C2×C3×D3—122no—
134C2×C3×D3—122no—
135C2×C3×D3—122no—
136C2×C3×D3—242no—
137C2×C3×D3—242no—
138C32×C4—21no—
139C32×C4—42no—
140C32×C4—82no—
141C3⋊Dic3—42no—
142C3⋊Dic3#23511yescharacteristic
143C3⋊Dic3C2×#6711yescharacteristic
144C3×Dic3—62no—
145C3×Dic3—62no—
146C3×Dic3—122no—
147C3×Dic3—122no—
148C3×Dic3—242no—
149C2×D8—271no—
150C2×M16—91no—
151#121—272no—
152#106—271no—
153#102—272no—
154C33#46911yescharacteristic, complemented, Sylow
155C23×C3—91no—
156C23×C3—182no—
157C22×D3—62no—
158C22×D3—62no—
159C22×D3—62no—
160C22×D3—122no—
161C2×C3×C4—22no—
162C2×C3×C4—42no—
163C2×C3×C4—91no—
164C2×C3×C4—181no—
165C2×C3×C4—181no—
166C2×C3×C4—182no—
167C2×C3×C4GO+(4,𝔽2)11yescharacteristic
168#67—362no—
169#67—362no—
170#67—364no—
171#67—364no—
172C2×Dic3—62no—
173C2×Dic3—62no—
174C2×Dic3—62no—
175C2×Dic3—122no—
176D12—182no—
177D12—182no—
178D12—364no—
179C4×D3—122no—
180C4×D3—122no—
181C3×C8—92no—
182C3×C8—181no—
183C2×C32—22no—
184C2×C32—22no—
185C2×C32—22no—
186C2×C32—42no—
187C2×C32—42no—
188C2×C32—42no—
189C2×C32#90812yes—
190C2×C32#93011yescharacteristic
191C3⋊D3—41no—
192C3⋊D3—82no—
193C3×D3—124no—
194C3×D3—242no—
195C3×D3—244no—
196C3×D3—242no—
197C2×D4—272no—
198C2×D4—542no—
199C22×C4—91no—
200D8—542no—
201M16—182no—
202C2×C8—91no—
203C2×C8—91no—
204C4⋊C4—272no—
205C22⋊C4—182no—
206C22×C3—22no—
207C22×C3—42no—
208C22×C3—182no—
209C22×C3—181no—
210C22×C3—362no—
211C22×C3—362no—
212C22×C3—362no—
213C22×C3C2×GO+(4,𝔽2)11yescharacteristic
214C2×D3—64no—
215C2×D3—122no—
216C2×D3—122no—
217C2×D3—122no—
218C2×D3—122no—
219C2×D3—122no—
220C2×D3—124no—
221C2×D3—122no—
222C2×D3—122no—
223C2×D3—122no—
224C2×D3—242no—
225C2×D3—242no—
226C3×C4—21no—
227C3×C4—42no—
228C3×C4—82no—
229C3×C4—91no—
230C3×C4—91no—
231C3×C4—362no—
232Dic3—62no—
233Dic3—62no—
234Dic3—122no—
235Dic3—122no—
236Dic3—242no—
237C32—22no—
238C32—42no—
239C32#513511yescharacteristic, complemented
240C23—91no—
241C23—182no—
242D4—542no—
243D4—542no—
244D4—1084no—
245C2×C4—91no—
246C2×C4—181no—
247C2×C4—181no—
248C2×C4—182no—
249C2×C4#157⋊D311yescharacteristic
250C8—92no—
251C8—181no—
252C2×C3—22no—
253C2×C3—22no—
254C2×C3—22no—
255C2×C3—42no—
256C2×C3—42no—
257C2×C3—42no—
258C2×C3—361no—
259C2×C3—722no—
260C2×C3#6422412yes—
261C2×C3#6424211yescharacteristic
262D3—124no—
263D3—242no—
264D3—244no—
265D3—242no—
266C22—182no—
267C22—181no—
268C22—362no—
269C22—362no—
270C22—362no—
271C22C2×#157⋊D311yescharacteristic
272C4—21no—
273C4—91no—
274C4—91no—
275C4—362no—
276C3—22no—
277C3—42no—
278C3#1059342911yescharacteristic, complemented
279C2—361no—
280C2—722no—
281C2#1172434012yes—
282C2#1172435511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#469
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3788C2semidirect
2. C2×#88941C2semidirect
3. #11723695C2semidirect
4. #11722796C2semidirect
5. #11722793C2semidirect
16. C2×#3788D3semidirect
33. C3×#622D4semidirect
95. #622#67semidirect
154. C33#469semidirect
239. C32#5135semidirect
278. C3#10593429semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37882
C2×#889412
#117236952
#117227962
#117227932
#105934293
#51359