Finite Group Databasebeta

← #10592773⋊C3 #13142981

Subgroups — #10592773⋊C3 #13142981

Order 1728.

612 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 100 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
1H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C212yesmaximal, complemented
3C2×#3754⋊C3C212yesmaximal, complemented
4#64084⋊C3C211yesmaximal, characteristic, complemented
5#63630⋊C3C211yesmaximal, characteristic, complemented
6#10592773C311yesmaximal, characteristic, complemented
7C3×#5074—31nomaximal
8C2×D4×H(3,𝔽3)—22no—
9C2×D4×H(3,𝔽3)C2212yes—
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×#634⋊C3C2211yescharacteristic
12C42×H(3,𝔽3)C2211yescharacteristic
13#3753⋊C3—24no—
14#3754⋊C3—24no—
15#3752⋊C3—22no—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C32×#127—21no—
18C32×#127D311yescharacteristic, complemented
19C32×#127C2×C311yescharacteristic, complemented
20C2×#3754C2×C312yescomplemented
21C2×#3753C2×C312yescomplemented
22#64084C2×C311yescharacteristic, complemented
23C2×C3×#214—32no—
24C2×C3×#213—32no—
25C3×#875—31no—
26#63630C2×C311yescharacteristic, complemented
27C3×#803—31no—
28C23×H(3,𝔽3)—22no—
29C22×(C3⋊D3)⋊C3—21no—
30D4×H(3,𝔽3)—22no—
31D4×H(3,𝔽3)—22no—
32D4×H(3,𝔽3)—44no—
33C2×C4×H(3,𝔽3)D412yes—
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35C2×(C3⋊Dic3)⋊C3—21no—
36C4⋊((C3⋊D3)⋊C3)—22no—
37#634⋊C3—22no—
38(C3⋊(C3⋊C8))⋊C3D414yescomplemented
39C3×#496—91no—
40#5074—31no—
41#5074—91no—
42C2×C32×D4—24no—
43C2×C32×D4—22no—
44C2×C32×D4—22no—
45C2×C32×D4—22no—
46C2×C32×D4C2×D312yes—
47C2×C32×D4C22×C312yes—
48C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
49C2×#634C22×C311yescharacteristic
50C2×C3×D12—31no—
51C2×C3×Dic6—31no—
52C32×C42—21no—
53C32×C42C2×D311yescharacteristic
54C32×C42C22×C311yescharacteristic
55#3754—24no—
56#3753—24no—
57#3752—22no—
58C2×C3⋊(C3⋊C8)C22×C312yes—
59C3×#214—64no—
60C3×#213—64no—
61C3×(C2×D3)⋊C4—62no—
62C2×C3×C3⋊C8—32no—
63C22×H(3,𝔽3)—42no—
64C22×H(3,𝔽3)—44no—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C2×(C3⋊D3)⋊C3—41no—
67C2×(C3⋊D3)⋊C3—42no—
68C4×H(3,𝔽3)—22no—
69C4×H(3,𝔽3)C2×D412yes—
70(C3⋊Dic3)⋊C3—41no—
71C2×C3×SD16—92no—
72C2×C3×D8—92no—
73C3×#127—31no—
74C3×#127—31no—
75C3×#127—61no—
76C3×#127C3×D311yescharacteristic
77C3×#124—91no—
78C2×#214—32no—
79C2×#214—92no—
80C2×#213—32no—
81C2×#213—92no—
82#875—31no—
83#875—91no—
84C3×#97—91no—
85#803—31no—
86#803—91no—
87C23×C32—24no—
88C23×C32—22no—
89C23×C32—22no—
90C22×C3⋊D3—21no—
91C22×C3×D3—61no—
92C32×D4—22no—
93C32×D4—22no—
94C32×D4—22no—
95C32×D4—22no—
96C32×D4—42no—
97C32×D4—42no—
98C32×D4—48no—
99C32×D4—44no—
100C32×D4—44no—
101C2×C32×C4—22no—
102C2×C32×C4—21no—
103C2×C32×C4#6712yes—
104C2×C32×C4C3×D412yes—
105C2×C32×C4C22×D311yescharacteristic
106C2×C32×C4C23×C311yescharacteristic
107C2×C3⋊Dic3—21no—
108C4⋊(C3⋊D3)—22no—
109#634—22no—
110C2×C3×Dic3—61no—
111C3×D12—62no—
112C3×Dic6—62no—
113C3⋊(C3⋊C8)C3×D414yescomplemented
114C3×C3⋊C8—34no—
115#496—91noSylow
116C2×H(3,𝔽3)—82no—
117C2×H(3,𝔽3)#12711yescharacteristic
118C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
119(C3⋊D3)⋊C3—81no—
120C2×C3×Q8—91no—
121C2×C3×D4—22no—
122C2×C3×D4—32no—
123C2×C3×D4—32no—
124C2×C3×D4—62no—
125C2×C3×D4—62no—
126C2×C3×D4—64no—
127C2×C3×D4—62no—
128C2×C3×D4—91no—
129C2×C3×D4C2×C3×D312yes—
130C2×D12—31no—
131C2×D12—91no—
132C2×Dic6—31no—
133C2×Dic6—91no—
134C3×SD16—184no—
135C3×D8—184no—
136C2×C3×C8—92no—
137C3×C22⋊C4—182no—
138C3×C42—31no—
139C3×C42—31no—
140C3×C42—61no—
141C3×C42C2×C3×D311yescharacteristic
142#214—64no—
143#214—184no—
144#213—64no—
145#213—184no—
146(C2×D3)⋊C4—62no—
147(C2×D3)⋊C4—182no—
148C2×C3⋊C8—32no—
149C2×C3⋊C8—92no—
150C22×C32—21no—
151C22×C32—48no—
152C22×C32—44no—
153C22×C32—44no—
154C22×C32—42no—
155C22×C32—42no—
156C22×C32—44no—
157C22×C32C2×#6711yescharacteristic
158C22×C32C2×C3×D411yescharacteristic
159C2×C3⋊D3—41no—
160C2×C3⋊D3—42no—
161C2×C3×D3—122no—
162C2×C3×D3—121no—
163C32×C4—24no—
164C32×C4—22no—
165C32×C4—22no—
166C32×C4—22no—
167C32×C4D3×D412yes—
168C32×C4C2×C3×D412yes—
169C3⋊Dic3—41no—
170C3×Dic3—121no—
171C2×SD16—92no—
172C2×D8—92no—
173#127(C3⋊D3)⋊C311yescharacteristic, complemented
174#124—91no—
175#97—91no—
176H(3,𝔽3)#49611yescharacteristic, complemented, Sylow
177C23×C3—22no—
178C23×C3—62no—
179C23×C3—62no—
180C23×C3—64no—
181C23×C3—181no—
182C22×D3—61no—
183C22×D3—181no—
184C3×Q8—182no—
185C3×D4—22no—
186C3×D4—22no—
187C3×D4—44no—
188C3×D4—62no—
189C3×D4—62no—
190C3×D4—62no—
191C3×D4—62no—
192C3×D4—122no—
193C3×D4—122no—
194C3×D4—124no—
195C3×D4—124no—
196C3×D4—128no—
197C3×D4—182no—
198C2×C3×C4—32no—
199C2×C3×C4—32no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—62no—
203C2×C3×C4—61no—
204C2×C3×C4—181no—
205C2×C3×C4C3×#6712yes—
206C2×C3×C4C22×C3×D311yescharacteristic
207C2×Dic3—61no—
208C2×Dic3—181no—
209D12—62no—
210D12—182no—
211Dic6—62no—
212Dic6—182no—
213C3×C8—94no—
214C3⋊C8—34no—
215C3⋊C8—94no—
216C2×C32—22no—
217C2×C32—21no—
218C2×C32—84no—
219C2×C32—82no—
220C2×C32—82no—
221C2×C32#93212yes—
222C2×C32#94011yescharacteristic
223C2×C32C3×#12711yescharacteristic
224C2×C32C3×AGL(1,ℤ/8)12yes—
225C3⋊D3—81no—
226C3×D3—241no—
227C2×Q8—91no—
228C2×D4—22no—
229C2×D4—91no—
230C2×D4C2×(C3⋊D3)⋊C312yes—
231SD16—184no—
232D8—184no—
233C2×C8—92no—
234C22⋊C4—182no—
235C42C2×(C3⋊D3)⋊C311yescharacteristic
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—44no—
239C22×C3—42no—
240C22×C3—61no—
241C22×C3—122no—
242C22×C3—122no—
243C22×C3—124no—
244C22×C3—124no—
245C22×C3—128no—
246C22×C3—124no—
247C22×C3—361no—
248C22×C3—362no—
249C22×C3C2×C3×#6711yescharacteristic
250C2×D3—122no—
251C2×D3—121no—
252C2×D3—361no—
253C2×D3—362no—
254C3×C4—22no—
255C3×C4—32no—
256C3×C4—32no—
257C3×C4—62no—
258C3×C4—62no—
259C3×C4—64no—
260C3×C4—62no—
261C3×C4—361no—
262C3×C4C3×D3×D412yes—
263Dic3—121no—
264Dic3—361no—
265C32—21no—
266C32#507411yescharacteristic, complemented
267C32C3×#49611yescharacteristic, complemented
268C23—22no—
269C23—181no—
270Q8—182no—
271D4—22no—
272D4—22no—
273D4—44no—
274D4—182no—
275C2×C4—181no—
276C2×C4#638⋊C312yes—
277C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
278C8—94no—
279C2×C3—32no—
280C2×C3—32no—
281C2×C3—31no—
282C2×C3—31no—
283C2×C3—62no—
284C2×C3—61no—
285C2×C3—82no—
286C2×C3—242no—
287C2×C3—242no—
288C2×C3—244no—
289C2×C3—721no—
290C2×C3C3×#93212yes—
291C2×C3C3×#94011yescharacteristic
292D3—241no—
293D3—721no—
294C22—44no—
295C22—42no—
296C22—362no—
297C22—361no—
298C22C2×#638⋊C311yescharacteristic
299C4—22no—
300C4—361no—
301C4D4×(C3⋊D3)⋊C312yes—
302C3—31no—
303C3—31no—
304C3—61no—
305C3C3×#507411yescharacteristic
306C2—82no—
307C2—721no—
308C2#64141⋊C312yes—
309C2#64149⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#127C2
socleC22×C3C2×C3×#67
hypercenter#127(C3⋊D3)⋊C3
hypocenterC32C3×#496
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #127

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#127C2semidirect
2. C2×#3753⋊C3C2semidirect
3. C2×#3754⋊C3C2semidirect
4. #64084⋊C3C2semidirect
5. #63630⋊C3C2semidirect
6. #10592773C3semidirect
18. C32×#127D3semidirect
19. C32×#127C2×C3semidirect
20. C2×#3754C2×C3semidirect
21. C2×#3753C2×C3semidirect
22. #64084C2×C3semidirect
26. #63630C2×C3semidirect
38. (C3⋊(C3⋊C8))⋊C3D4semidirect
113. C3⋊(C3⋊C8)C3×D4semidirect
173. #127(C3⋊D3)⋊C3semidirect
176. H(3,𝔽3)#496semidirect
266. C32#5074semidirect
267. C32C3×#496semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1272
C2×#3753⋊C32
C2×#3754⋊C32
#64084⋊C32
#63630⋊C32
#105927733
C3×#50743