Finite Group Databasebeta

← #13167594

Subgroups — #13167594

Order 1728.

564 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C22×C3×#622C211yesmaximal, characteristic, complemented
2C2×#88941C211yesmaximal, characteristic, complemented
3#11723717C212yesmaximal, complemented
4#11722797C211yesmaximal, characteristic, complemented
5#11722793C212yesmaximal, complemented
6#10593520—31nomaximal
7C2×C3×#638—22no—
8C22×C3×C3⋊Dic3C2211yescharacteristic
9C2×#6295C2212yes—
10C2×C3×#622—22no—
11C2×C3×#622C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13#88941—22no—
14#88815C2212yes—
15#88814—22no—
16C22×#622D311yescharacteristic, complemented
17C2×#3774—31no—
18#63980—32no—
19#63978—62no—
20#63743—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#5115—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—
43C2×#622—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—21no—
54C22×C33D812yescomplemented
55C22×C33C2×D411yescharacteristic
56C2×C3×C3⋊D3—42no—
57C2×C3×C3⋊D3—42no—
58C2×C3×C3⋊D3—42no—
59C3×C3⋊Dic3—21no—
60C3×C3⋊Dic3—42no—
61C3×C3⋊Dic3C2×D411yescharacteristic
62C3×C3⋊Dic3#4111yescharacteristic
63C22×C3×C8—91no—
64#939—182no—
65#930—92no—
66C2×D24—91no—
67#821—92no—
68#818—91no—
69C23×C32—22no—
70C23×C32—42no—
71C23×C32D1211yescharacteristic
72C22×C3⋊D3—22no—
73C22×C3×D3—62no—
74C22×C3×D3—62no—
75C22×C3×D3—122no—
76#638—44no—
77C2×C3⋊Dic3—22no—
78C2×C3⋊Dic3—21no—
79C2×C3⋊Dic3—21no—
80C2×C3⋊Dic3D1211yescharacteristic
81C2×C3⋊Dic3C22×D311yescharacteristic
82C3×#67—124no—
83C3×#67—128no—
84C3×#67—124no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3—62no—
87C2×C3×Dic3—62no—
88C2×C3×Dic3—62no—
89C2×C3×Dic3—122no—
90C2×C3×Dic3—122no—
91C2×C3×Dic3—122no—
92#626—122no—
93#626—122no—
94#625—62no—
95#625—62no—
96#622—21no—
97#622#6712yescomplemented
98#466—271noSylow
99C2×C33—22no—
100C2×C33#12111yescharacteristic
101C2×C33C2×D811yescharacteristic
102C2×C33#13511yescharacteristic
103C3×C3⋊D3—82no—
104C2×C3×D4—182no—
105C22×C3×C4—91no—
106C2×#67—62no—
107C2×#67—62no—
108C2×#67—122no—
109C2×#67—182no—
110C22×Dic3—62no—
111C2×D12—92no—
112C2×C3×C8—91no—
113C2×C3×C8—91no—
114C2×C3×C8—182no—
115#217—182no—
116(C2×D3)⋊C4—182no—
117(C2×D3)⋊C4—182no—
118C4⋊Dic3—92no—
119Dic3⋊C4—182no—
120D24—182no—
121C22×C32—22no—
122C22×C32—24no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—42no—
126C22×C32—44no—
127C22×C32—42no—
128C22×C32—42no—
129C22×C32—82no—
130C22×C32—82no—
131C22×C32D2412yescomplemented
132C22×C32C2×D1211yescharacteristic
133C2×C3⋊D3—42no—
134C2×C3⋊D3—42no—
135C2×C3⋊D3—42no—
136C2×C3×D3—122no—
137C2×C3×D3—122no—
138C2×C3×D3—122no—
139C2×C3×D3—124no—
140C2×C3×D3—122no—
141C2×C3×D3—122no—
142C2×C3×D3—122no—
143C2×C3×D3—242no—
144C2×C3×D3—242no—
145C3⋊Dic3—21no—
146C3⋊Dic3—42no—
147C3⋊Dic3#23511yescharacteristic
148C3⋊Dic3C2×#6711yescharacteristic
149C3×Dic3—62no—
150C3×Dic3—62no—
151C3×Dic3—122no—
152C3×Dic3—122no—
153C3×Dic3—124no—
154C3×Dic3—122no—
155C2×D8—271no—
156C22×C8—91no—
157#121—272no—
158#107—271no—
159#102—272no—
160C33#46611yescharacteristic, complemented, Sylow
161C23×C3—22no—
162C23×C3—42no—
163C23×C3—182no—
164C23×C3GO+(4,𝔽2)11yescharacteristic
165C22×D3—62no—
166C22×D3—62no—
167C22×D3—122no—
168C3×D4—364no—
169C2×C3×C4—91no—
170C2×C3×C4—91no—
171C2×C3×C4—181no—
172C2×C3×C4—181no—
173C2×C3×C4—182no—
174#67—124no—
175#67—128no—
176#67—124no—
177#67—362no—
178#67—362no—
179C2×Dic3—62no—
180C2×Dic3—62no—
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—122no—
184C2×Dic3—122no—
185C2×Dic3—122no—
186D12—182no—
187D12—182no—
188C3×C8—92no—
189C3×C8—181no—
190C2×C32—22no—
191C2×C32—22no—
192C2×C32—22no—
193C2×C32—22no—
194C2×C32—42no—
195C2×C32—42no—
196C2×C32—42no—
197C2×C32—44no—
198C2×C32—84no—
199C2×C32C2×D2411yescharacteristic
200C2×C32#90411yescharacteristic
201C2×C32#93011yescharacteristic
202C3⋊D3—82no—
203C3×D3—242no—
204C3×D3—244no—
205C3×D3—242no—
206C2×D4—182no—
207C2×D4—272no—
208C22×C4—91no—
209D8—542no—
210C2×C8—91no—
211C2×C8—91no—
212C2×C8—182no—
213C4⋊C4—272no—
214C22⋊C4—542no—
215C22×C3—21no—
216C22×C3—21no—
217C22×C3—22no—
218C22×C3—24no—
219C22×C3—42no—
220C22×C3—42no—
221C22×C3—42no—
222C22×C3—44no—
223C22×C3—82no—
224C22×C3—82no—
225C22×C3—362no—
226C22×C3—362no—
227C22×C3—362no—
228C22×C3#377412yescomplemented
229C22×C3C2×GO+(4,𝔽2)11yescharacteristic
230C2×D3—122no—
231C2×D3—122no—
232C2×D3—122no—
233C2×D3—124no—
234C2×D3—122no—
235C2×D3—122no—
236C2×D3—122no—
237C2×D3—242no—
238C2×D3—242no—
239C3×C4—91no—
240C3×C4—91no—
241C3×C4—181no—
242C3×C4—362no—
243Dic3—62no—
244Dic3—62no—
245Dic3—122no—
246Dic3—122no—
247Dic3—124no—
248Dic3—122no—
249C32—22no—
250C32—42no—
251C32#511511yescharacteristic, complemented
252C23—182no—
253C23#157⋊D311yescharacteristic
254D4—364no—
255D4—542no—
256D4—542no—
257C2×C4—91no—
258C2×C4—91no—
259C2×C4—181no—
260C2×C4—181no—
261C2×C4—182no—
262C8—92no—
263C8—181no—
264C2×C3—22no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3—22no—
268C2×C3—44no—
269C2×C3—42no—
270C2×C3—42no—
271C2×C3—42no—
272C2×C3—84no—
273C2×C3—722no—
274C2×C3C2×#377411yescharacteristic
275C2×C3#6423711yescharacteristic
276C2×C3#6424211yescharacteristic
277D3—242no—
278D3—244no—
279D3—242no—
280C22—21no—
281C22—21no—
282C22—362no—
283C22—362no—
284C22—362no—
285C22#8894112yescomplemented
286C22C2×#157⋊D311yescharacteristic
287C4—91no—
288C4—91no—
289C4—181no—
290C4—362no—
291C3—22no—
292C3—42no—
293C3#1059352011yescharacteristic, complemented
294C2—22no—
295C2—722no—
296C2C2×#8894111yescharacteristic
297C2#1172435111yescharacteristic
298C2#1172435511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC23×C33D4
socleC22×C33C2×D4
hypercenterC23#157⋊D3
hypocenterC33#466
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. C22×C3×#622C2semidirect
2. C2×#88941C2semidirect
3. #11723717C2semidirect
4. #11722797C2semidirect
5. #11722793C2semidirect
16. C22×#622D3semidirect
33. C3×#622D4semidirect
54. C22×C33D8semidirect
97. #622#67semidirect
131. C22×C32D24semidirect
160. C33#466semidirect
228. C22×C3#3774semidirect
251. C32#5115semidirect
285. C22#88941semidirect
293. C3#10593520semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6222
C2×#889412
#117237172
#117227972
#117227932
#105935203
#51159