Finite Group Databasebeta

← #13137978

Subgroups — #13137978

Order 1728.

493 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 69 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
1#11721763C211yesmaximal, characteristic
2#11721152C211yesmaximal, characteristic, complemented
3#63635⋊C3C212yesmaximal, complemented
4#11720324C212yesmaximal, complemented
5#11720323C211yesmaximal, characteristic, complemented
6#10590441—32nomaximal
7C2×C4⋊#272—21no—
8C2×C4⋊#272C2211yescharacteristic
9C2×C4×#272C2211yescharacteristic
10C2×#6199—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12#88542—21no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#88449—21no—
16C3×#895—32no—
17C3×#895—61no—
18D3×C4⋊Dic3—32no—
19#63635D312yescomplemented
20C3×#808—32no—
21#63563—32no—
22#63562—32no—
23(C2×D3)⋊(C3⋊C8)—32no—
24C22×#272—21no—
25C22×#272D411yescharacteristic
26C2×C4×H(3,𝔽3)—21no—
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084D411yescharacteristic
29C4⋊#272—22no—
30C4⋊#272—42no—
31C4×#272—22no—
32C2×(C3⋊Dic3)⋊C3—22no—
33#6199—42no—
34(C3⋊(C3⋊C8))⋊C3—22no—
35#5034—91no—
36#4863—92no—
37C2×C3×D12—32no—
38C2×C3×D12—62no—
39C2×C3×D12—61no—
40C2×C3×D12—62no—
41C2×C3×C4×D3—32no—
42C2×C3×C4×D3—61no—
43C2×D3×Dic3—62no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4C2×D312yes—
46C4⋊(C3⋊Dic3)C2×D312yes—
47C2×C3⋊(C3⋊C8)C2×D312yes—
48C3×(C2×D3)⋊C4—62no—
49C3×(C2×D3)⋊C4—62no—
50C3×C4⋊Dic3—32no—
51C2×C3×C3⋊C8—32no—
52Dic3⋊Dic3—62no—
53C22×H(3,𝔽3)C2×D411yescharacteristic
54C2×#272—21no—
55C2×#272—21no—
56C2×#272—41no—
57C2×#272—41no—
58C2×#272—41no—
59C2×#272SD1612yes—
60C4×H(3,𝔽3)—41no—
61C4×H(3,𝔽3)#4112yes—
62#1084—21no—
63(C3⋊Dic3)⋊C3—42no—
64C3×#121—91no—
65C2×C4⋊Dic3—91no—
66#895—32no—
67#895—61no—
68D3×C4⋊C4—92no—
69C3×#106—92no—
70#832—92no—
71#830—91no—
72#821—92no—
73(C2×D3)⋊C8—92no—
74#809—92no—
75#808—32no—
76#808—92no—
77C22×C3×D3—32no—
78C22×C3×D3—61no—
79C22×C3×D3—62no—
80C22×C3×D3—62no—
81C2×C32×C4—22no—
82C2×C32×C4—22no—
83C2×C32×C4—21no—
84C2×C32×C4C22×D312yes—
85C2×C3⋊Dic3—22no—
86C2×C3×Dic3—32no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—62no—
89C3×D12—62no—
90C3×D12—62no—
91C3×D12—122no—
92C3×D12—124no—
93C3×D12—124no—
94C3×C4×D3—62no—
95C3×C4×D3—62no—
96C3×C4×D3—122no—
97D3×Dic3—124no—
98C3⋊(C3⋊C8)—22no—
99C3×C3⋊C8—62no—
100#481—271noSylow
101C2×H(3,𝔽3)#12311yescharacteristic
102C2×H(3,𝔽3)C2×SD1611yescharacteristic
103C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
104#272—22no—
105#272—81no—
106C2×C3×D4—91no—
107C2×C3×D4—181no—
108C22×C3×C4—91no—
109C22×Dic3—181no—
110C2×D12—32no—
111C2×D12—61no—
112C2×D12—62no—
113C2×D12—62no—
114C2×C4×D3—32no—
115C2×C4×D3—61no—
116C2×C4×D3—182no—
117C2×C3×C8—92no—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—61no—
120C3×C4⋊C4—92no—
121C3×C4⋊C4D3211yescharacteristic
122C3×C22⋊C4—181no—
123(C2×D3)⋊C4—62no—
124(C2×D3)⋊C4—62no—
125C4⋊Dic3—32no—
126C4⋊Dic3—92no—
127C4⋊Dic3—181no—
128Dic3⋊C4—182no—
129C2×C3⋊C8—32no—
130C2×C3⋊C8—92no—
131C22×C32—21no—
132C22×C32D3×D412yes—
133C2×C3×D3—34no—
134C2×C3×D3—62no—
135C2×C3×D3—62no—
136C2×C3×D3—62no—
137C2×C3×D3—121no—
138C2×C3×D3—121no—
139C2×C3×D3—122no—
140C2×C3×D3—122no—
141C2×C3×D3—122no—
142C2×C3×D3—122no—
143C2×C3×D3—122no—
144C2×C3×D3—122no—
145C32×C4—22no—
146C32×C4—42no—
147C32×C4—42no—
148C32×C4#23512yes—
149C32×C4#23912yes—
150C3⋊Dic3—42no—
151C3×Dic3—62no—
152C3×Dic3—121no—
153C3×Dic3—122no—
154#121—91no—
155C2×C4⋊C4—271no—
156#106—92no—
157#102—272no—
158C22⋊C8—271no—
159H(3,𝔽3)#48111yescharacteristic, complemented, Sylow
160C23×C3—91no—
161C23×C3—181no—
162C22×D3—32no—
163C22×D3—61no—
164C22×D3—62no—
165C22×D3—62no—
166C3×D4—182no—
167C3×D4—362no—
168C2×C3×C4—21no—
169C2×C3×C4—32no—
170C2×C3×C4—62no—
171C2×C3×C4—62no—
172C2×C3×C4—61no—
173C2×C3×C4—91no—
174C2×C3×C4—182no—
175C2×C3×C4—182no—
176C2×C3×C4C2×D3211yescharacteristic
177C2×Dic3—32no—
178C2×Dic3—61no—
179C2×Dic3—62no—
180C2×Dic3—182no—
181C2×Dic3—362no—
182D12—62no—
183D12—62no—
184D12—122no—
185D12—124no—
186D12—124no—
187C4×D3—62no—
188C4×D3—62no—
189C4×D3—122no—
190C4×D3—364no—
191C3×C8—182no—
192C3⋊C8—62no—
193C3⋊C8—182no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—21no—
197C2×C32#89412yes—
198C2×C32D3×SD1612yes—
199C2×C32#91412yes—
200C3×D3—64no—
201C3×D3—122no—
202C3×D3—242no—
203C3×D3—242no—
204C2×D4—91no—
205C2×D4—181no—
206C22×C4—91no—
207C22×C4—541no—
208C2×C8—92no—
209C4⋊C4—92no—
210C4⋊C4—541no—
211C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
212C22⋊C4—181no—
213C22×C3—32no—
214C22×C3—61no—
215C22×C3—92no—
216C22×C3—181no—
217C22×C3—181no—
218C22×C3—361no—
219C22×C3—361no—
220C22×C3—361no—
221C22×C3#381111yescharacteristic
222C2×D3—34no—
223C2×D3—62no—
224C2×D3—62no—
225C2×D3—62no—
226C2×D3—121no—
227C2×D3—121no—
228C2×D3—122no—
229C2×D3—122no—
230C2×D3—122no—
231C2×D3—122no—
232C2×D3—122no—
233C2×D3—122no—
234C3×C4—32no—
235C3×C4—32no—
236C3×C4—41no—
237C3×C4—62no—
238C3×C4—122no—
239C3×C4—122no—
240C3×C4—181no—
241C3×C4—362no—
242C3×C4#379912yes—
243Dic3—62no—
244Dic3—121no—
245Dic3—122no—
246Dic3—362no—
247C32—21no—
248C32#486312yescomplemented
249C23—91no—
250C23—181no—
251D4—182no—
252D4—362no—
253C2×C4—21no—
254C2×C4—91no—
255C2×C4—182no—
256C2×C4—182no—
257C2×C4—1082no—
258C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
259C8—182no—
260C2×C3—32no—
261C2×C3—32no—
262C2×C3—32no—
263C2×C3—61no—
264C2×C3—61no—
265C2×C3—61no—
266C2×C3—182no—
267C2×C3—721no—
268C2×C3#6389711yescharacteristic
269C2×C3#6393011yescharacteristic
270C2×C3#6394111yescharacteristic
271D3—64no—
272D3—122no—
273D3—242no—
274D3—242no—
275C22—92no—
276C22—181no—
277C22—181no—
278C22—361no—
279C22—361no—
280C22—361no—
281C22#8867711yescharacteristic
282C4—41no—
283C4—181no—
284C4—362no—
285C4#8865212yes—
286C3—32no—
287C3—61no—
288C3#1059044511yescharacteristic
289C2—182no—
290C2—721no—
291C2#1172115111yescharacteristic
292C2#1172118811yescharacteristic
293C2#1172122911yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC4⋊C4×H(3,𝔽3)C22
socleC22×C3#3811
prime radical#11721763C2
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#481
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11721152C2semidirect
3. #63635⋊C3C2semidirect
4. #11720324C2semidirect
5. #11720323C2semidirect
19. #63635D3semidirect
159. H(3,𝔽3)#481semidirect
248. C32#4863semidirect

Maximal subgroups

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

subgroupindex
#117217632
#117211522
#63635⋊C32
#117203242
#117203232
#105904413