Finite Group Databasebeta

← #13167496

Subgroups — #13167496

Order 1728.

524 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 59 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×#3787C211yesmaximal, characteristic, complemented
2C2×#88941C211yesmaximal, characteristic, complemented
3#11723695C212yesmaximal, complemented
4#11722797C211yesmaximal, characteristic, complemented
5#11722793C212yesmaximal, complemented
6#10593423—31nomaximal
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×#6295—22no—
9C2×#6295C2212yes—
10C2×C3×#622C2211yescharacteristic
11C2×C3×#622C2211yescharacteristic
12C3×#3787—22no—
13#88941—22no—
14C3×#3752—22no—
15#88815C2212yes—
16C2×#3787D311yescharacteristic, complemented
17C2×#3774—31no—
18#63912—32no—
19#63906—62no—
20#63743—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#5115—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#3787—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—21no—
55C2×C3×C3⋊D3—42no—
56C2×C3×C3⋊D3—42no—
57C2×C3×C3⋊D3—42no—
58C2×C3×C3⋊D3D812yescomplemented
59C33×C4—21no—
60C3×C3⋊Dic3—42no—
61C3×C3⋊Dic3C2×D411yescharacteristic
62C3×C3⋊Dic3#4111yescharacteristic
63C22×C3×C8—91no—
64#930—92no—
65C2×D24—91no—
66#884—182no—
67#821—92no—
68#818—91no—
69C22×C3⋊D3—22no—
70C22×C3⋊D3D1211yescharacteristic
71C22×C3×D3—62no—
72C22×C3×D3—62no—
73C22×C3×D3—62no—
74C22×C3×D3—122no—
75C2×C32×C4—22no—
76C2×C32×C4—42no—
77C2×C32×C4D1211yescharacteristic
78C2×C3⋊Dic3—22no—
79C2×C3⋊Dic3C22×D311yescharacteristic
80C4×C3⋊D3—21no—
81C4×C3⋊D3—21no—
82C2×C3×Dic3—62no—
83C2×C3×Dic3—62no—
84C2×C3×Dic3—62no—
85C2×C3×Dic3—122no—
86C3×C4×D3—122no—
87C3×C4×D3—122no—
88#626—122no—
89#626—122no—
90#626—124no—
91#626—124no—
92#625—62no—
93#625—62no—
94#625—124no—
95#622—21no—
96#622#6712yescomplemented
97#466—271noSylow
98C2×C33#12111yescharacteristic
99C2×C33C2×D811yescharacteristic
100C2×C33#13511yescharacteristic
101C3×C3⋊D3—22no—
102C3×C3⋊D3—82no—
103C22×C3×C4—91no—
104C2×#67—182no—
105C2×#67—182no—
106C2×#67—182no—
107C2×D12—92no—
108C2×D12—182no—
109C2×C4×D3—62no—
110C2×C3×C8—91no—
111C2×C3×C8—91no—
112C2×C3×C8—182no—
113C3×C22⋊C4—182no—
114(C2×D3)⋊C4—62no—
115(C2×D3)⋊C4—62no—
116(C2×D3)⋊C4—122no—
117C4⋊Dic3—92no—
118Dic3⋊C4—182no—
119D24—182no—
120C22×C32—22no—
121C22×C32—42no—
122C22×C32C2×D1211yescharacteristic
123C2×C3⋊D3—21no—
124C2×C3⋊D3—21no—
125C2×C3⋊D3—42no—
126C2×C3⋊D3—42no—
127C2×C3⋊D3—42no—
128C2×C3⋊D3D2412yescomplemented
129C2×C3×D3—64no—
130C2×C3×D3—122no—
131C2×C3×D3—122no—
132C2×C3×D3—122no—
133C2×C3×D3—122no—
134C2×C3×D3—122no—
135C2×C3×D3—124no—
136C2×C3×D3—122no—
137C2×C3×D3—122no—
138C2×C3×D3—122no—
139C2×C3×D3—242no—
140C2×C3×D3—242no—
141C32×C4—21no—
142C32×C4—42no—
143C32×C4—82no—
144C3⋊Dic3—42no—
145C3⋊Dic3#23511yescharacteristic
146C3⋊Dic3C2×#6711yescharacteristic
147C3×Dic3—62no—
148C3×Dic3—62no—
149C3×Dic3—122no—
150C3×Dic3—122no—
151C3×Dic3—242no—
152C2×D8—271no—
153C22×C8—91no—
154#121—272no—
155#107—271no—
156#102—272no—
157C33#46611yescharacteristic, complemented, Sylow
158C23×C3—91no—
159C23×C3—182no—
160C22×D3—62no—
161C22×D3—62no—
162C22×D3—62no—
163C22×D3—122no—
164C2×C3×C4—22no—
165C2×C3×C4—42no—
166C2×C3×C4—91no—
167C2×C3×C4—181no—
168C2×C3×C4—181no—
169C2×C3×C4—182no—
170C2×C3×C4GO+(4,𝔽2)11yescharacteristic
171#67—362no—
172#67—362no—
173#67—364no—
174#67—364no—
175C2×Dic3—62no—
176C2×Dic3—62no—
177C2×Dic3—62no—
178C2×Dic3—122no—
179D12—182no—
180D12—182no—
181D12—364no—
182C4×D3—122no—
183C4×D3—122no—
184C3×C8—92no—
185C3×C8—181no—
186C2×C32—22no—
187C2×C32—22no—
188C2×C32—22no—
189C2×C32—42no—
190C2×C32—42no—
191C2×C32—42no—
192C2×C32C2×D2411yescharacteristic
193C2×C32#90411yescharacteristic
194C2×C32#93011yescharacteristic
195C3⋊D3—22no—
196C3⋊D3—82no—
197C3×D3—124no—
198C3×D3—242no—
199C3×D3—244no—
200C3×D3—242no—
201C2×D4—272no—
202C2×D4—542no—
203C22×C4—91no—
204D8—542no—
205C2×C8—91no—
206C2×C8—91no—
207C2×C8—182no—
208C4⋊C4—272no—
209C22⋊C4—182no—
210C22×C3—22no—
211C22×C3—42no—
212C22×C3—92no—
213C22×C3—181no—
214C22×C3—181no—
215C22×C3—362no—
216C22×C3—362no—
217C22×C3—362no—
218C22×C3C2×GO+(4,𝔽2)11yescharacteristic
219C2×D3—64no—
220C2×D3—122no—
221C2×D3—122no—
222C2×D3—122no—
223C2×D3—122no—
224C2×D3—122no—
225C2×D3—124no—
226C2×D3—122no—
227C2×D3—122no—
228C2×D3—122no—
229C2×D3—242no—
230C2×D3—242no—
231C3×C4—21no—
232C3×C4—42no—
233C3×C4—82no—
234C3×C4—91no—
235C3×C4—91no—
236C3×C4—362no—
237Dic3—62no—
238Dic3—62no—
239Dic3—122no—
240Dic3—122no—
241Dic3—242no—
242C32—22no—
243C32—42no—
244C32#511511yescharacteristic, complemented
245C23—91no—
246C23—182no—
247D4—542no—
248D4—542no—
249D4—1084no—
250C2×C4—91no—
251C2×C4—181no—
252C2×C4—181no—
253C2×C4—182no—
254C2×C4#157⋊D311yescharacteristic
255C8—92no—
256C8—181no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—22no—
260C2×C3—42no—
261C2×C3—42no—
262C2×C3—42no—
263C2×C3—182no—
264C2×C3—722no—
265C2×C3#6422311yescharacteristic
266C2×C3#6422511yescharacteristic
267C2×C3#6424211yescharacteristic
268D3—124no—
269D3—242no—
270D3—244no—
271D3—242no—
272C22—92no—
273C22—181no—
274C22—181no—
275C22—362no—
276C22—362no—
277C22—362no—
278C22C2×#157⋊D311yescharacteristic
279C4—21no—
280C4—91no—
281C4—91no—
282C4—362no—
283C3—22no—
284C3—42no—
285C3#1059342311yescharacteristic, complemented
286C2—182no—
287C2—722no—
288C2#1172433611yescharacteristic
289C2#1172433711yescharacteristic
290C2#1172435511yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C4#157⋊D3
hypocenterC33#466
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×#3787C2semidirect
2. C2×#88941C2semidirect
3. #11723695C2semidirect
4. #11722797C2semidirect
5. #11722793C2semidirect
16. C2×#3787D3semidirect
33. C3×#622D4semidirect
58. C2×C3×C3⋊D3D8semidirect
96. #622#67semidirect
128. C2×C3⋊D3D24semidirect
157. C33#466semidirect
244. C32#5115semidirect
285. C3#10593423semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37872
C2×#889412
#117236952
#117227972
#117227932
#105934233
#51159