Finite Group Databasebeta

← #13167595

Subgroups — #13167595

Order 1728.

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

normal subgroups only

This group has 556 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×#3792C211yesmaximal, characteristic, complemented
2C2×#88941C211yesmaximal, characteristic, complemented
3#11723717C212yesmaximal, complemented
4#11722796C211yesmaximal, characteristic, complemented
5#11722793C212yesmaximal, complemented
6#10593522—31nomaximal
7C2×C3×#638—22no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×#6295C2212yes—
10C3×#3792—22no—
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13#88941—22no—
14#88815C2212yes—
15#88814—22no—
16C2×#3792D311yescharacteristic, complemented
17C2×#3774—31no—
18#63980—32no—
19#63978—62no—
20#63742—31no—
21#63739—32no—
22C23×C33D411yescharacteristic
23C22×C3×C3⋊D3—22no—
24C3×#638—44no—
25C2×C3×C3⋊Dic3—21no—
26C2×C3×C3⋊Dic3—21no—
27C2×C3×C3⋊Dic3—22no—
28C2×C3×C3⋊Dic3D411yescharacteristic
29C2×C3×C3⋊Dic3C2311yescharacteristic
30#6295—22no—
31#6295—22no—
32C3×#622—21no—
33C3×#622D412yescomplemented
34#5135—91nomaximal
35C2×#638—22no—
36C22×C3⋊Dic3C2×D311yescharacteristic
37C2×C3×#67—62no—
38C2×C3×#67—62no—
39C2×C3×#67—122no—
40C22×C3×Dic3—62no—
41C2×#626—62no—
42C2×#625—32no—
43#3792—22no—
44C2×#622C2×D311yescharacteristic
45C2×#622C2×D311yescharacteristic
46#3774—62no—
47#3724—32no—
48Dic3⋊Dic3—62no—
49#3722—62no—
50(C2×D3)⋊Dic3—62no—
51(C2×D3)⋊Dic3—62no—
52C22×C33—21no—
53C22×C33—22no—
54C22×C33C2×D411yescharacteristic
55C2×C3×C3⋊D3—42no—
56C2×C3×C3⋊D3—42no—
57C2×C3×C3⋊D3—42no—
58C3×C3⋊Dic3—21no—
59C3×C3⋊Dic3—42no—
60C3×C3⋊Dic3C2×D411yescharacteristic
61C3×C3⋊Dic3#4111yescharacteristic
62C2×C3×M16—91no—
63#939—182no—
64#930—92no—
65C2×D24—91no—
66#821—92no—
67#817—91no—
68C23×C32—22no—
69C23×C32—42no—
70C23×C32D1211yescharacteristic
71C22×C3⋊D3—22no—
72C22×C3×D3—62no—
73C22×C3×D3—62no—
74C22×C3×D3—122no—
75#638—44no—
76C2×C3⋊Dic3—22no—
77C2×C3⋊Dic3—21no—
78C2×C3⋊Dic3—21no—
79C2×C3⋊Dic3D1211yescharacteristic
80C2×C3⋊Dic3C22×D311yescharacteristic
81C3×#67—124no—
82C3×#67—124no—
83C3×#67—128no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—122no—
89C2×C3×Dic3—122no—
90C2×C3×Dic3—122no—
91#626—122no—
92#626—122no—
93#625—62no—
94#625—62no—
95#622—21no—
96#622#6712yescomplemented
97#469—271noSylow
98C2×C33—41no—
99C2×C33#12111yescharacteristic
100C2×C33AGL(1,ℤ/8)12yes—
101C3×C3⋊D3—82no—
102C2×C3×D4—182no—
103C22×C3×C4—91no—
104C2×#67—62no—
105C2×#67—62no—
106C2×#67—122no—
107C2×#67—182no—
108C22×Dic3—62no—
109C2×D12—92no—
110C3×M16—182no—
111C2×C3×C8—91no—
112C2×C3×C8—91no—
113#217—182no—
114(C2×D3)⋊C4—182no—
115(C2×D3)⋊C4—182no—
116C4⋊Dic3—92no—
117Dic3⋊C4—182no—
118D24—182no—
119C22×C32—24no—
120C22×C32—22no—
121C22×C32—22no—
122C22×C32—21no—
123C22×C32—42no—
124C22×C32—42no—
125C22×C32—42no—
126C22×C32—44no—
127C22×C32—82no—
128C22×C32—82no—
129C22×C32C2×D1211yescharacteristic
130C2×C3⋊D3—42no—
131C2×C3⋊D3—42no—
132C2×C3⋊D3—42no—
133C2×C3×D3—122no—
134C2×C3×D3—122no—
135C2×C3×D3—122no—
136C2×C3×D3—124no—
137C2×C3×D3—122no—
138C2×C3×D3—122no—
139C2×C3×D3—122no—
140C2×C3×D3—242no—
141C2×C3×D3—242no—
142C3⋊Dic3—21no—
143C3⋊Dic3—42no—
144C3⋊Dic3#23511yescharacteristic
145C3⋊Dic3C2×#6711yescharacteristic
146C3×Dic3—62no—
147C3×Dic3—62no—
148C3×Dic3—122no—
149C3×Dic3—122no—
150C3×Dic3—124no—
151C3×Dic3—122no—
152C2×D8—271no—
153C2×M16—91no—
154#121—272no—
155#106—271no—
156#102—272no—
157C33#46911yescharacteristic, complemented, Sylow
158C23×C3—22no—
159C23×C3—42no—
160C23×C3—182no—
161C23×C3GO+(4,𝔽2)11yescharacteristic
162C22×D3—62no—
163C22×D3—62no—
164C22×D3—122no—
165C3×D4—364no—
166C2×C3×C4—91no—
167C2×C3×C4—91no—
168C2×C3×C4—181no—
169C2×C3×C4—181no—
170C2×C3×C4—182no—
171#67—124no—
172#67—124no—
173#67—128no—
174#67—362no—
175#67—362no—
176C2×Dic3—62no—
177C2×Dic3—62no—
178C2×Dic3—62no—
179C2×Dic3—62no—
180C2×Dic3—122no—
181C2×Dic3—122no—
182C2×Dic3—122no—
183D12—182no—
184D12—182no—
185C3×C8—92no—
186C3×C8—181no—
187C2×C32—22no—
188C2×C32—22no—
189C2×C32—22no—
190C2×C32—42no—
191C2×C32—42no—
192C2×C32—42no—
193C2×C32—41no—
194C2×C32—44no—
195C2×C32—84no—
196C2×C32#90812yes—
197C2×C32#93011yescharacteristic
198C3⋊D3—82no—
199C3×D3—242no—
200C3×D3—244no—
201C3×D3—242no—
202C2×D4—182no—
203C2×D4—272no—
204C22×C4—91no—
205D8—542no—
206M16—182no—
207C2×C8—91no—
208C2×C8—91no—
209C4⋊C4—272no—
210C22⋊C4—542no—
211C22×C3—24no—
212C22×C3—22no—
213C22×C3—21no—
214C22×C3—22no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—44no—
218C22×C3—42no—
219C22×C3—82no—
220C22×C3—82no—
221C22×C3—362no—
222C22×C3—362no—
223C22×C3—362no—
224C22×C3C2×GO+(4,𝔽2)11yescharacteristic
225C2×D3—122no—
226C2×D3—122no—
227C2×D3—122no—
228C2×D3—124no—
229C2×D3—122no—
230C2×D3—122no—
231C2×D3—122no—
232C2×D3—242no—
233C2×D3—242no—
234C3×C4—91no—
235C3×C4—91no—
236C3×C4—181no—
237C3×C4—362no—
238Dic3—62no—
239Dic3—62no—
240Dic3—122no—
241Dic3—122no—
242Dic3—124no—
243Dic3—122no—
244C32—22no—
245C32—42no—
246C32#513511yescharacteristic, complemented
247C23—182no—
248C23#157⋊D311yescharacteristic
249D4—364no—
250D4—542no—
251D4—542no—
252C2×C4—91no—
253C2×C4—91no—
254C2×C4—181no—
255C2×C4—181no—
256C2×C4—182no—
257C8—92no—
258C8—181no—
259C2×C3—22no—
260C2×C3—22no—
261C2×C3—22no—
262C2×C3—44no—
263C2×C3—41no—
264C2×C3—42no—
265C2×C3—42no—
266C2×C3—42no—
267C2×C3—84no—
268C2×C3—722no—
269C2×C3#6423612yes—
270C2×C3#6424211yescharacteristic
271D3—242no—
272D3—244no—
273D3—242no—
274C22—22no—
275C22—21no—
276C22—362no—
277C22—362no—
278C22—362no—
279C22C2×#157⋊D311yescharacteristic
280C4—91no—
281C4—91no—
282C4—181no—
283C4—362no—
284C3—22no—
285C3—42no—
286C3#1059352211yescharacteristic, complemented
287C2—41no—
288C2—722no—
289C2#1172435012yes—
290C2#1172435511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC23×C33D4
socleC22×C33C2×D4
hypercenterC23#157⋊D3
hypocenterC33#469
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3792C2semidirect
2. C2×#88941C2semidirect
3. #11723717C2semidirect
4. #11722796C2semidirect
5. #11722793C2semidirect
16. C2×#3792D3semidirect
33. C3×#622D4semidirect
96. #622#67semidirect
157. C33#469semidirect
246. C32#5135semidirect
286. C3#10593522semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37922
C2×#889412
#117237172
#117227962
#117227932
#105935223
#51359