Finite Group Databasebeta

← #13138876

Subgroups — #13138876

Order 1728.

545 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 71 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×C8×#272C211yesmaximal, characteristic, complemented
2#11721159C212yesmaximal, complemented
3C2×#88432C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#11720315C211yesmaximal, characteristic, complemented
6#10590826—32nomaximal
7C2×C4×#272C2211yescharacteristic
8C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
9C2×#6200—22no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11C2×#6180C2211yescharacteristic
12C8×#272—22no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14#88450—22no—
15#88432—22no—
16C2×C3×D3×C8—32no—
17C2×C3×D3×C8—61no—
18#63909—32no—
19#63908—32no—
20C2×#3714—32no—
21#63648D312yescomplemented
22C3×#821—32no—
23#63575—32no—
24C22×#272D411yescharacteristic
25C22×(C3⋊D3)⋊C3—22no—
26C2×C4×H(3,𝔽3)C2311yescharacteristic
27C2×#1084D411yescharacteristic
28C4×#272—21no—
29C4×#272—21no—
30C2×(C3⋊Dic3)⋊C3—22no—
31C4⋊((C3⋊D3)⋊C3)—22no—
32C4⋊((C3⋊D3)⋊C3)—22no—
33#6200—44no—
34C8×H(3,𝔽3)—21no—
35#6180D412yescomplemented
36#5115—91no—
37#4775—92no—
38C2×C4⋊(C3⋊D3)C2×D312yes—
39C2×C3×D12—32no—
40C2×C3×C4×D3—32no—
41C2×C3×C4×D3—61no—
42C2×#626—62no—
43C2×#625—62no—
44C2×C32×C8—21no—
45C2×C32×C8C2×D312yes—
46C4⋊(C3⋊Dic3)C2×D312yes—
47C3×C4⋊Dic3—32no—
48C2×C3×C3⋊C8—32no—
49C2×C3×C3⋊C8—61no—
50C3×D3×C8—64no—
51C3×D3×C8—64no—
52#3722—62no—
53(C2×D3)⋊Dic3—62no—
54#3714—64no—
55C22×H(3,𝔽3)C2×D411yescharacteristic
56C2×#272—21no—
57C2×#272—21no—
58C2×#272D812yescomplemented
59C2×(C3⋊D3)⋊C3—42no—
60C2×(C3⋊D3)⋊C3—42no—
61C2×(C3⋊D3)⋊C3—42no—
62C4×H(3,𝔽3)C2×D411yescharacteristic
63C4×H(3,𝔽3)#4111yescharacteristic
64#1084—21no—
65(C3⋊Dic3)⋊C3—42no—
66C22×C3×C8—91no—
67#938—92no—
68C2×#213—92no—
69#930—92no—
70C2×D24—91no—
71C2×D3×C8—32no—
72C2×D3×C8—61no—
73#895—92no—
74C3×#102—92no—
75#821—32no—
76#821—92no—
77#818—91no—
78#807—92no—
79C22×C3⋊D3—22no—
80C22×C3×D3—32no—
81C22×C3×D3—61no—
82C22×C3×D3—62no—
83C2×C32×C4—21no—
84C2×C32×C4C22×D312yes—
85C2×C3⋊Dic3—22no—
86C4⋊(C3⋊D3)—22no—
87C4⋊(C3⋊D3)—22no—
88C2×C3×Dic3—32no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—62no—
91C3×D12—62no—
92C3×D12—62no—
93C3×C4×D3—62no—
94C3×C4×D3—62no—
95C3×C4×D3—62no—
96C3×C4×D3—62no—
97#626—124no—
98#625—124no—
99C32×C8—22no—
100C32×C8—22no—
101C3×C3⋊C8—34no—
102C3×C3⋊C8—62no—
103#466—271noSylow
104C2×H(3,𝔽3)#12111yescharacteristic
105C2×H(3,𝔽3)C2×D811yescharacteristic
106C2×H(3,𝔽3)#13511yescharacteristic
107#272—22no—
108(C3⋊D3)⋊C3—82no—
109C2×C3×D4—92no—
110C22×C3×C4—91no—
111C2×#67—182no—
112C2×#67—182no—
113C2×D12—32no—
114C2×D12—92no—
115C2×D12—182no—
116C2×C4×D3—32no—
117C2×C4×D3—61no—
118C2×C3×C8—32no—
119C2×C3×C8—61no—
120C2×C3×C8—91no—
121C2×C3×C8—182no—
122C2×C3×C8D3211yescharacteristic
123C3×C4⋊C4—92no—
124#217—182no—
125#213—184no—
126(C2×D3)⋊C4—182no—
127(C2×D3)⋊C4—182no—
128C4⋊Dic3—32no—
129C4⋊Dic3—92no—
130C2×C3⋊C8—32no—
131C2×C3⋊C8—61no—
132D24—182no—
133D3×C8—64no—
134D3×C8—64no—
135C22×C32—21no—
136C22×C32D3×D412yes—
137C2×C3⋊D3—42no—
138C2×C3⋊D3—42no—
139C2×C3⋊D3—42no—
140C2×C3×D3—34no—
141C2×C3×D3—62no—
142C2×C3×D3—62no—
143C2×C3×D3—62no—
144C2×C3×D3—62no—
145C2×C3×D3—62no—
146C2×C3×D3—122no—
147C2×C3×D3—122no—
148C2×C3×D3—122no—
149C32×C4—21no—
150C32×C4—21no—
151C32×C4#23512yes—
152C32×C4D3×D412yes—
153C3⋊Dic3—42no—
154C3×Dic3—62no—
155C3×Dic3—62no—
156C3×Dic3—122no—
157C2×D8—271no—
158C22×C8—91no—
159#121—272no—
160#107—271no—
161#102—92no—
162H(3,𝔽3)#46611yescharacteristic, complemented, Sylow
163C23×C3—91no—
164C23×C3—182no—
165C22×D3—32no—
166C22×D3—61no—
167C22×D3—62no—
168C22×D3—182no—
169C3×D4—182no—
170C3×D4—182no—
171C2×C3×C4—32no—
172C2×C3×C4—61no—
173C2×C3×C4—91no—
174C2×C3×C4—181no—
175C2×C3×C4—181no—
176C2×C3×C4—182no—
177C2×C3×C4C2×D3211yescharacteristic
178#67—364no—
179#67—364no—
180C2×Dic3—32no—
181C2×Dic3—61no—
182C2×Dic3—62no—
183C2×Dic3—182no—
184D12—62no—
185D12—62no—
186D12—182no—
187D12—182no—
188D12—364no—
189C4×D3—62no—
190C4×D3—62no—
191C4×D3—62no—
192C4×D3—62no—
193C3×C8—21no—
194C3×C8—62no—
195C3×C8—62no—
196C3×C8—92no—
197C3⋊C8—34no—
198C3⋊C8—62no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32#88412yes—
203C2×C32D3×D812yes—
204C2×C32#91612yes—
205C3⋊D3—82no—
206C3×D3—64no—
207C3×D3—64no—
208C3×D3—242no—
209C2×D4—92no—
210C2×D4—542no—
211C22×C4—91no—
212D8—542no—
213C2×C8—91no—
214C2×C8—182no—
215C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
216C4⋊C4—92no—
217C22⋊C4—542no—
218C22×C3—32no—
219C22×C3—61no—
220C22×C3—92no—
221C22×C3—181no—
222C22×C3—181no—
223C22×C3—362no—
224C22×C3—362no—
225C22×C3—362no—
226C22×C3#380211yescharacteristic
227C2×D3—34no—
228C2×D3—62no—
229C2×D3—62no—
230C2×D3—62no—
231C2×D3—62no—
232C2×D3—62no—
233C2×D3—122no—
234C2×D3—122no—
235C2×D3—122no—
236C2×D3—362no—
237C2×D3—362no—
238C2×D3—362no—
239C3×C4—32no—
240C3×C4—32no—
241C3×C4—61no—
242C3×C4—61no—
243C3×C4—181no—
244C3×C4—362no—
245C3×C4#379811yescharacteristic
246C3×C4#381111yescharacteristic
247Dic3—62no—
248Dic3—62no—
249Dic3—122no—
250Dic3—362no—
251C32—21no—
252C32#477512yescomplemented
253C23—91no—
254C23—182no—
255D4—182no—
256D4—182no—
257D4—1084no—
258C2×C4—91no—
259C2×C4—181no—
260C2×C4—181no—
261C2×C4—182no—
262C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
263C8—21no—
264C8—92no—
265C2×C3—32no—
266C2×C3—32no—
267C2×C3—32no—
268C2×C3—61no—
269C2×C3—61no—
270C2×C3—61no—
271C2×C3—182no—
272C2×C3—722no—
273C2×C3#6379711yescharacteristic
274C2×C3#6380911yescharacteristic
275C2×C3#6391211yescharacteristic
276D3—64no—
277D3—64no—
278D3—242no—
279D3—722no—
280C22—92no—
281C22—181no—
282C22—181no—
283C22—362no—
284C22—362no—
285C22—362no—
286C22#8865511yescharacteristic
287C4—181no—
288C4—362no—
289C4#8865011yescharacteristic
290C4#8867711yescharacteristic
291C3—32no—
292C3—61no—
293C3#1059083911yescharacteristic
294C2—182no—
295C2—722no—
296C2#1172083811yescharacteristic
297C2#1172084311yescharacteristic
298C2#1172114211yescharacteristic
subgroupG/H
centerC22#88655
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#466
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×C8×#272C2semidirect
2. #11721159C2semidirect
3. C2×#88432C2semidirect
4. #63648⋊C3C2semidirect
5. #11720315C2semidirect
21. #63648D3semidirect
35. #6180D4semidirect
58. C2×#272D8semidirect
162. H(3,𝔽3)#466semidirect
252. C32#4775semidirect

Maximal subgroups

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

subgroupindex
C2×C8×#2722
#117211592
C2×#884322
#63648⋊C32
#117203152
#105908263