Finite Group Databasebeta

← #10596196⋊C3 #13153322

Subgroups — #10596196⋊C3 #13153322

Order 1728.

1153 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88722C212yesmaximal, complemented
2H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
3#64156⋊C3C212yesmaximal, complemented
4#64104⋊C3C214yesmaximal, complemented
5#64102⋊C3C214yesmaximal, complemented
6#11721385C211yesmaximal, characteristic, complemented
7C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
8#10596196C311yesmaximal, characteristic, complemented
9C3×#5725—31nomaximal
10C2×Q8×H(3,𝔽3)C2212yes—
11#88722—28no—
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
15C2×C4×(C3⋊D3)⋊C3C2214yes—
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2214yes—
19C42×H(3,𝔽3)C2211yescharacteristic
20#3752⋊C3C2218yes—
21C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
22C4×(C3⋊Dic3)⋊C3C2212yes—
23C2×#3832C2×C312yescomplemented
24C2×C3×#239—32no—
25C32×C4⋊Q8—21no—
26C32×C4⋊Q8D311yescharacteristic, complemented
27C32×C4⋊Q8C2×C311yescharacteristic, complemented
28#64156C2×C312yescomplemented
29#64104C2×C314yescomplemented
30#64102C2×C314yescomplemented
31#64083C2×C311yescharacteristic, complemented
32C42×C3⋊D3C2×C311yescharacteristic, complemented
33C3×#947—32no—
34C3×#895—34no—
35C3×#893—34no—
36C3×#874—31no—
37C3×C42×D3—31no—
38C22×(C3⋊D3)⋊C3C2314yes—
39C22×(C3⋊D3)⋊C3C2311yescharacteristic
40Q8×H(3,𝔽3)—24no—
41C2×C4×H(3,𝔽3)C2314yes—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×C4×H(3,𝔽3)C2312yes—
44C2×(C3⋊Dic3)⋊C3C2312yes—
45C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
46C4⋊((C3⋊D3)⋊C3)—28no—
47C4⋊((C3⋊D3)⋊C3)—24no—
48C4⋊((C3⋊D3)⋊C3)—28no—
49C4×(C3⋊D3)⋊C3—24no—
50C4×(C3⋊D3)⋊C3—28no—
51C4×(C3⋊D3)⋊C3D414yescomplemented
52C3×#532—91no—
53#5725—31no—
54#5725—91no—
55C2×C32×Q8—22no—
56C2×C32×Q8C2×D312yes—
57C2×C32×Q8C22×C312yes—
58#3832—28no—
59C2×C4⋊(C3⋊D3)C22×C314yes—
60C2×C4⋊(C3⋊D3)C22×C312yes—
61C2×C4⋊(C3⋊D3)C22×C314yes—
62C2×C4×C3⋊D3C22×C312yes—
63C2×C4×C3⋊D3C22×C314yes—
64C2×C4×C3⋊D3C22×C311yescharacteristic
65C3×#239—68no—
66C2×C3×D12—34no—
67C2×C3×D12—34no—
68C2×C3×D12—32no—
69C2×C3×C4×D3—31no—
70C2×C3×C4×D3—34no—
71C2×C3×C4×D3—32no—
72C32×C4⋊C4—24no—
73C32×C4⋊C4C2×D314yes—
74C32×C4⋊C4C22×C314yes—
75C32×C42—21no—
76C32×C42C2×D311yescharacteristic
77C32×C42C22×C311yescharacteristic
78#3752C22×C318yes—
79C4×C3⋊Dic3C22×C312yes—
80C4×C3⋊Dic3C22×C311yescharacteristic
81C3×(C2×D3)⋊C4—38no—
82C3×C4×Dic3—32no—
83C3×C4×Dic3—31no—
84C22×H(3,𝔽3)C2411yescharacteristic
85C2×(C3⋊D3)⋊C3—22no—
86C2×(C3⋊D3)⋊C3—24no—
87C2×(C3⋊D3)⋊C3—28no—
88C2×(C3⋊D3)⋊C3C2×D412yes—
89C4×H(3,𝔽3)—24no—
90C4×H(3,𝔽3)C2×D412yes—
91C4×H(3,𝔽3)#4114yescomplemented
92(C3⋊Dic3)⋊C3—22no—
93(C3⋊Dic3)⋊C3C2×D412yes—
94C2×C3×#41—92no—
95C2×#239—32no—
96C2×#239—92no—
97C3×C4⋊Q8—31no—
98C3×C4⋊Q8—31no—
99C3×C4⋊Q8—61no—
100C3×C4⋊Q8C3×D311yescharacteristic
101C3×#127—91no—
102C3×#124—92no—
103C3×#121—94no—
104C3×C4×D4—94no—
105C2×C3×C42—91no—
106#947—32no—
107#947—92no—
108#895—34no—
109#895—94no—
110#893—34no—
111#893—94no—
112#874—31no—
113#874—91no—
114C42×D3—31no—
115C42×D3—91no—
116C22×C3⋊D3C23×C314yes—
117C22×C3⋊D3C23×C311yescharacteristic
118C22×C3×D3—31no—
119C22×C3×D3—34no—
120C32×Q8—24no—
121C32×Q8—24no—
122C32×Q8—44no—
123C2×C32×C4—22no—
124C2×C32×C4—21no—
125C2×C32×C4—24no—
126C2×C32×C4C22×D314yes—
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C22×D312yes—
129C2×C32×C4C23×C314yes—
130C2×C32×C4C23×C311yescharacteristic
131C2×C32×C4C23×C312yes—
132C2×C3⋊Dic3C23×C312yes—
133C2×C3⋊Dic3C23×C311yescharacteristic
134C4⋊(C3⋊D3)—28no—
135C4⋊(C3⋊D3)—24no—
136C4⋊(C3⋊D3)—28no—
137C4×C3⋊D3—24no—
138C4×C3⋊D3—28no—
139C4×C3⋊D3C3×D414yescomplemented
140C2×C3×Dic3—31no—
141C2×C3×Dic3—32no—
142C3×D12—68no—
143C3×D12—64no—
144C3×D12—68no—
145C3×C4×D3—34no—
146C3×C4×D3—68no—
147C3×C4×D3—64no—
148#532—91noSylow
149C2×H(3,𝔽3)C22×D411yescharacteristic
150C2×H(3,𝔽3)C2×#4112yes—
151(C3⋊D3)⋊C3—22no—
152(C3⋊D3)⋊C3—44no—
153C3×#41—188no—
154C2×C3×Q8—32no—
155C2×C3×Q8—32no—
156C2×C3×Q8—62no—
157C2×C3×Q8C2×C3×D312yes—
158C2×C3×D4—94no—
159C2×C3×D4—94no—
160C2×C3×D4—92no—
161C22×C3×C4—91no—
162C22×C3×C4—92no—
163C22×C3×C4—94no—
164#239—68no—
165#239—188no—
166C2×D12—34no—
167C2×D12—32no—
168C2×D12—34no—
169C2×D12—94no—
170C2×D12—92no—
171C2×D12—94no—
172C2×C4×D3—31no—
173C2×C4×D3—32no—
174C2×C4×D3—34no—
175C2×C4×D3—91no—
176C2×C4×D3—92no—
177C2×C4×D3—94no—
178C3×C4⋊C4—34no—
179C3×C4⋊C4—34no—
180C3×C4⋊C4—64no—
181C3×C4⋊C4C2×C3×D314yes—
182C3×C22⋊C4—98no—
183C3×C42—31no—
184C3×C42—31no—
185C3×C42—61no—
186C3×C42—92no—
187C3×C42—91no—
188C3×C42C2×C3×D311yescharacteristic
189(C2×D3)⋊C4—38no—
190(C2×D3)⋊C4—98no—
191C4×Dic3—32no—
192C4×Dic3—31no—
193C4×Dic3—92no—
194C4×Dic3—91no—
195C22×C32—21no—
196C22×C32C23×D311yescharacteristic
197C22×C32C24×C311yescharacteristic
198C2×C3⋊D3—22no—
199C2×C3⋊D3—24no—
200C2×C3⋊D3—28no—
201C2×C3⋊D3C2×C3×D412yes—
202C2×C3×D3—32no—
203C2×C3×D3—62no—
204C2×C3×D3—64no—
205C2×C3×D3—68no—
206C32×C4—24no—
207C32×C4—24no—
208C32×C4—22no—
209C32×C4—24no—
210C32×C4—44no—
211C32×C4D3×D412yes—
212C32×C4#23914yescomplemented
213C32×C4C2×C3×D412yes—
214C32×C4C3×#4114yescomplemented
215C3⋊Dic3—22no—
216C3⋊Dic3C2×C3×D412yes—
217C3×Dic3—32no—
218C3×Dic3—62no—
219C2×#41—92no—
220C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
221#127—91no—
222#124—92no—
223#121—94no—
224C4×D4—94no—
225C2×C42—91no—
226H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
227C23×C3—91no—
228C23×C3—94no—
229C22×D3—31no—
230C22×D3—34no—
231C22×D3—91no—
232C22×D3—94no—
233C3×Q8—24no—
234C3×Q8—64no—
235C3×Q8—64no—
236C3×Q8—124no—
237C3×D4—188no—
238C3×D4—184no—
239C3×D4—188no—
240C2×C3×C4—32no—
241C2×C3×C4—31no—
242C2×C3×C4—32no—
243C2×C3×C4—31no—
244C2×C3×C4—34no—
245C2×C3×C4—34no—
246C2×C3×C4—62no—
247C2×C3×C4—61no—
248C2×C3×C4—64no—
249C2×C3×C4—94no—
250C2×C3×C4—91no—
251C2×C3×C4—92no—
252C2×C3×C4—184no—
253C2×C3×C4—188no—
254C2×C3×C4C22×C3×D314yes—
255C2×C3×C4C22×C3×D311yescharacteristic
256C2×C3×C4C22×C3×D312yes—
257C2×Dic3—31no—
258C2×Dic3—32no—
259C2×Dic3—91no—
260C2×Dic3—92no—
261D12—68no—
262D12—64no—
263D12—68no—
264D12—188no—
265D12—184no—
266D12—188no—
267C4×D3—34no—
268C4×D3—68no—
269C4×D3—64no—
270C4×D3—94no—
271C4×D3—188no—
272C4×D3—184no—
273C2×C32—21no—
274C2×C32—22no—
275C2×C32C2×D3×D411yescharacteristic
276C2×C32C2×#23912yes—
277C2×C32C22×C3×D411yescharacteristic
278C2×C32C2×C3×#4112yes—
279C3⋊D3—22no—
280C3⋊D3—44no—
281C3×D3—62no—
282C3×D3—124no—
283#41—188no—
284C2×Q8C2×(C3⋊D3)⋊C312yes—
285C2×D4—94no—
286C2×D4—94no—
287C2×D4—92no—
288C22×C4—91no—
289C22×C4—92no—
290C22×C4—94no—
291C4⋊C4C2×(C3⋊D3)⋊C314yes—
292C22⋊C4—98no—
293C42—92no—
294C42—91no—
295C42C2×(C3⋊D3)⋊C311yescharacteristic
296C22×C3—31no—
297C22×C3—31no—
298C22×C3—61no—
299C22×C3—92no—
300C22×C3—182no—
301C22×C3—184no—
302C22×C3—188no—
303C22×C3C23×C3×D311yescharacteristic
304C2×D3—32no—
305C2×D3—62no—
306C2×D3—64no—
307C2×D3—68no—
308C2×D3—92no—
309C2×D3—182no—
310C2×D3—184no—
311C2×D3—188no—
312C3×C4—24no—
313C3×C4—32no—
314C3×C4—32no—
315C3×C4—34no—
316C3×C4—34no—
317C3×C4—64no—
318C3×C4—64no—
319C3×C4—62no—
320C3×C4—64no—
321C3×C4—92no—
322C3×C4—124no—
323C3×C4—182no—
324C3×C4C3×D3×D412yes—
325C3×C4C3×#23914yes—
326Dic3—32no—
327Dic3—62no—
328Dic3—92no—
329Dic3—182no—
330C32—21no—
331C32#572511yescharacteristic, complemented
332C32C3×#53211yescharacteristic, complemented
333C23—91no—
334C23—94no—
335Q8—24no—
336D4—188no—
337D4—184no—
338D4—188no—
339C2×C4—91no—
340C2×C4—94no—
341C2×C4—92no—
342C2×C4—188no—
343C2×C4—184no—
344C2×C4C22×(C3⋊D3)⋊C314yes—
345C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
346C2×C4C22×(C3⋊D3)⋊C312yes—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—32no—
350C2×C3—32no—
351C2×C3—61no—
352C2×C3—62no—
353C2×C3—182no—
354C2×C3—364no—
355C2×C3C2×C3×D3×D411yescharacteristic
356C2×C3C2×C3×#23912yes—
357D3—62no—
358D3—124no—
359D3—182no—
360D3—364no—
361C22—92no—
362C22—182no—
363C22—184no—
364C22—188no—
365C22C23×(C3⋊D3)⋊C311yescharacteristic
366C4—24no—
367C4—92no—
368C4—182no—
369C4D4×(C3⋊D3)⋊C312yes—
370C4#8872214yescomplemented
371C3—31no—
372C3—31no—
373C3—61no—
374C3C3×#572511yescharacteristic
375C2—182no—
376C2—364no—
377C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
378C2C2×#8872212yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#532
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88722C2semidirect
2. H(3,𝔽3)×C4⋊Q8C2semidirect
3. #64156⋊C3C2semidirect
4. #64104⋊C3C2semidirect
5. #64102⋊C3C2semidirect
6. #11721385C2semidirect
7. C42×(C3⋊D3)⋊C3C2semidirect
8. #10596196C3semidirect
23. C2×#3832C2×C3semidirect
26. C32×C4⋊Q8D3semidirect
27. C32×C4⋊Q8C2×C3semidirect
28. #64156C2×C3semidirect
29. #64104C2×C3semidirect
30. #64102C2×C3semidirect
31. #64083C2×C3semidirect
32. C42×C3⋊D3C2×C3semidirect
51. C4×(C3⋊D3)⋊C3D4semidirect
91. C4×H(3,𝔽3)#41semidirect
139. C4×C3⋊D3C3×D4semidirect
212. C32×C4#239semidirect
214. C32×C4C3×#41semidirect
220. C4⋊Q8(C3⋊D3)⋊C3semidirect
226. H(3,𝔽3)#532semidirect
331. C32#5725semidirect
332. C32C3×#532semidirect
370. C4#88722semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
H(3,𝔽3)×C4⋊Q82
#64156⋊C32
#64104⋊C32
#64102⋊C32
#117213852
C42×(C3⋊D3)⋊C32
#105961963
C3×#57253