Finite Group Databasebeta

← #10596187⋊C3 #13153275

Subgroups — #10596187⋊C3 #13153275

Order 1728.

1193 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3830⋊C3C212yesmaximal, complemented
2H(3,𝔽3)×#127C211yesmaximal, characteristic, complemented
3#64147⋊C3C214yesmaximal, complemented
4#64145⋊C3C212yesmaximal, complemented
5D4×(C3⋊Dic3)⋊C3C214yesmaximal, complemented
6C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
7#64078⋊C3C211yesmaximal, characteristic, complemented
8#10596187C311yesmaximal, characteristic, complemented
9C3×#5716—31nomaximal
10C2×D4×H(3,𝔽3)C2212yes—
11C2×D4×H(3,𝔽3)C2214yes—
12C2×#638⋊C3C2214yes—
13C22×(C3⋊Dic3)⋊C3C2214yes—
14#3830⋊C3—28no—
15C2×C4×(C3⋊D3)⋊C3C2212yes—
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×#634⋊C3C2212yes—
18C42×H(3,𝔽3)C2211yescharacteristic
19#3757⋊C3C2218yes—
20C4⋊((C3⋊Dic3)⋊C3)C2214yes—
21C4×(C3⋊Dic3)⋊C3C2212yes—
22C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
23C2×#3830C2×C312yescomplemented
24C2×C3×#237—32no—
25C32×#127—21no—
26C32×#127D311yescharacteristic, complemented
27C32×#127C2×C311yescharacteristic, complemented
28#64147C2×C314yescomplemented
29#64145C2×C312yescomplemented
30D4×C3⋊Dic3C2×C314yescomplemented
31C42×C3⋊D3C2×C311yescharacteristic, complemented
32#64078C2×C311yescharacteristic, complemented
33C3×#938—34no—
34C3×#936—32no—
35C3×D4×Dic3—34no—
36C3×C42×D3—31no—
37C3×#869—31no—
38C23×H(3,𝔽3)C2314yes—
39C22×(C3⋊D3)⋊C3C2311yescharacteristic
40D4×H(3,𝔽3)—24no—
41D4×H(3,𝔽3)—28no—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×C4×H(3,𝔽3)C2312yes—
44#638⋊C3—28no—
45C2×(C3⋊Dic3)⋊C3—28no—
46C2×(C3⋊Dic3)⋊C3C2314yes—
47C2×(C3⋊Dic3)⋊C3C2312yes—
48C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
49C4×(C3⋊D3)⋊C3—24no—
50C4×(C3⋊D3)⋊C3D414yescomplemented
51#634⋊C3—24no—
52C3×#532—91no—
53#5716—31no—
54#5716—91no—
55C2×C32×D4—24no—
56C2×C32×D4—22no—
57C2×C32×D4C2×D314yes—
58C2×C32×D4C2×D312yes—
59C2×C32×D4C22×C312yes—
60C2×C32×D4C22×C314yes—
61C2×#638C22×C314yes—
62C22×C3⋊Dic3C22×C314yes—
63#3830—28no—
64C2×C4×C3⋊D3C22×C312yes—
65C2×C4×C3⋊D3C22×C311yescharacteristic
66C2×#634C22×C312yes—
67C2×C3×#67—34no—
68C22×C3×Dic3—34no—
69C3×#237—68no—
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—32no—
72C2×C3×Dic6—32no—
73C32×C42—21no—
74C32×C42C2×D311yescharacteristic
75C32×C42C22×C311yescharacteristic
76#3757C22×C318yes—
77C4⋊(C3⋊Dic3)C22×C314yes—
78C4×C3⋊Dic3C22×C312yes—
79C4×C3⋊Dic3C22×C311yescharacteristic
80C3×#217—38no—
81C3×C4⋊Dic3—34no—
82C3×C4×Dic3—32no—
83C3×C4×Dic3—31no—
84C22×H(3,𝔽3)—24no—
85C22×H(3,𝔽3)—28no—
86C22×H(3,𝔽3)C2411yescharacteristic
87C2×(C3⋊D3)⋊C3—22no—
88C2×(C3⋊D3)⋊C3C2×D412yes—
89C4×H(3,𝔽3)C2×D412yes—
90C4×H(3,𝔽3)#4114yescomplemented
91(C3⋊Dic3)⋊C3—24no—
92(C3⋊Dic3)⋊C3—22no—
93(C3⋊Dic3)⋊C3C2×D412yes—
94C2×C3×#41—92no—
95C2×#237—32no—
96C2×#237—92no—
97C3×C4⋊Q8—91no—
98C3×#127—31no—
99C3×#127—31no—
100C3×#127—61no—
101C3×#127C3×D311yescharacteristic
102C3×#124—92no—
103C3×#121—94no—
104C3×C4×D4—94no—
105C2×C3×C42—91no—
106#938—34no—
107#938—94no—
108#936—32no—
109#936—92no—
110D4×Dic3—34no—
111D4×Dic3—94no—
112C42×D3—31no—
113C42×D3—91no—
114#869—31no—
115#869—91no—
116C23×C32—24no—
117C23×C32C22×D314yes—
118C23×C32C23×C314yes—
119C22×C3⋊D3C23×C311yescharacteristic
120C22×C3×D3—31no—
121C32×D4—24no—
122C32×D4—28no—
123C32×D4—24no—
124C32×D4—28no—
125C32×D4—44no—
126C32×D4—48no—
127C2×C32×C4—21no—
128C2×C32×C4—22no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D312yes—
131C2×C32×C4C23×C311yescharacteristic
132C2×C32×C4C23×C312yes—
133#638—28no—
134C2×C3⋊Dic3—28no—
135C2×C3⋊Dic3C23×C312yes—
136C2×C3⋊Dic3C23×C314yes—
137C2×C3⋊Dic3C23×C311yescharacteristic
138C4×C3⋊D3—24no—
139C4×C3⋊D3C3×D414yescomplemented
140#634—24no—
141C3×#67—68no—
142C2×C3×Dic3—31no—
143C2×C3×Dic3—32no—
144C2×C3×Dic3—34no—
145C2×C3×Dic3—68no—
146C3×C4×D3—34no—
147C3×C4×D3—64no—
148C3×Dic6—64no—
149#532—91noSylow
150C2×H(3,𝔽3)—44no—
151C2×H(3,𝔽3)C22×D411yescharacteristic
152C2×H(3,𝔽3)C2×#4112yes—
153(C3⋊D3)⋊C3—22no—
154C3×#41—188no—
155C2×C3×Q8—92no—
156C2×C3×D4—34no—
157C2×C3×D4—32no—
158C2×C3×D4—34no—
159C2×C3×D4—32no—
160C2×C3×D4—64no—
161C2×C3×D4—62no—
162C2×C3×D4—94no—
163C2×C3×D4C2×C3×D314yes—
164C2×C3×D4C2×C3×D312yes—
165C22×C3×C4—91no—
166C22×C3×C4—92no—
167C22×C3×C4—94no—
168C2×#67—34no—
169C2×#67—94no—
170C22×Dic3—34no—
171C22×Dic3—94no—
172#237—68no—
173#237—188no—
174C2×C4×D3—31no—
175C2×C4×D3—32no—
176C2×C4×D3—91no—
177C2×C4×D3—92no—
178C2×Dic6—32no—
179C2×Dic6—92no—
180C3×C4⋊C4—94no—
181C3×C22⋊C4—98no—
182C3×C42—31no—
183C3×C42—31no—
184C3×C42—61no—
185C3×C42—92no—
186C3×C42—91no—
187C3×C42C2×C3×D311yescharacteristic
188#217—38no—
189#217—98no—
190C4⋊Dic3—34no—
191C4⋊Dic3—94no—
192C4×Dic3—32no—
193C4×Dic3—31no—
194C4×Dic3—92no—
195C4×Dic3—91no—
196C22×C32—24no—
197C22×C32—28no—
198C22×C32—24no—
199C22×C32—28no—
200C22×C32—21no—
201C22×C32—44no—
202C22×C32—48no—
203C22×C32C23×D311yescharacteristic
204C22×C32C24×C311yescharacteristic
205C2×C3⋊D3—22no—
206C2×C3⋊D3C2×C3×D412yes—
207C2×C3×D3—32no—
208C2×C3×D3—62no—
209C32×C4—22no—
210C32×C4—24no—
211C32×C4D3×D412yes—
212C32×C4#23714yescomplemented
213C32×C4C2×C3×D412yes—
214C32×C4C3×#4114yescomplemented
215C3⋊Dic3—24no—
216C3⋊Dic3—22no—
217C3⋊Dic3C2×C3×D412yes—
218C3×Dic3—32no—
219C3×Dic3—64no—
220C3×Dic3—62no—
221C2×#41—92no—
222C4⋊Q8—91no—
223#127(C3⋊D3)⋊C311yescharacteristic, complemented
224#124—92no—
225#121—94no—
226C4×D4—94no—
227C2×C42—91no—
228H(3,𝔽3)#53211yescharacteristic, complemented, Sylow
229C23×C3—34no—
230C23×C3—34no—
231C23×C3—64no—
232C23×C3—91no—
233C23×C3C22×C3×D314yes—
234C22×D3—31no—
235C22×D3—91no—
236C3×Q8—184no—
237C3×D4—24no—
238C3×D4—28no—
239C3×D4—64no—
240C3×D4—68no—
241C3×D4—68no—
242C3×D4—64no—
243C3×D4—124no—
244C3×D4—128no—
245C3×D4—188no—
246C2×C3×C4—32no—
247C2×C3×C4—31no—
248C2×C3×C4—32no—
249C2×C3×C4—31no—
250C2×C3×C4—61no—
251C2×C3×C4—62no—
252C2×C3×C4—94no—
253C2×C3×C4—91no—
254C2×C3×C4—92no—
255C2×C3×C4—94no—
256C2×C3×C4—184no—
257C2×C3×C4—188no—
258C2×C3×C4C22×C3×D311yescharacteristic
259C2×C3×C4C22×C3×D312yes—
260#67—68no—
261#67—188no—
262C2×Dic3—31no—
263C2×Dic3—32no—
264C2×Dic3—34no—
265C2×Dic3—68no—
266C2×Dic3—91no—
267C2×Dic3—92no—
268C2×Dic3—94no—
269C2×Dic3—188no—
270C4×D3—34no—
271C4×D3—64no—
272C4×D3—94no—
273C4×D3—184no—
274Dic6—64no—
275Dic6—184no—
276C2×C32—21no—
277C2×C32—22no—
278C2×C32—44no—
279C2×C32—44no—
280C2×C32—84no—
281C2×C32C2×D3×D411yescharacteristic
282C2×C32C2×#23712yes—
283C2×C32C22×C3×D411yescharacteristic
284C2×C32C2×C3×#4112yes—
285C3⋊D3—22no—
286C3×D3—62no—
287#41—188no—
288C2×Q8—92no—
289C2×D4—94no—
290C2×D4C2×(C3⋊D3)⋊C314yes—
291C2×D4C2×(C3⋊D3)⋊C312yes—
292C22×C4—91no—
293C22×C4—94no—
294C22×C4—92no—
295C4⋊C4—94no—
296C22⋊C4—98no—
297C42—92no—
298C42—91no—
299C42C2×(C3⋊D3)⋊C311yescharacteristic
300C22×C3—24no—
301C22×C3—28no—
302C22×C3—31no—
303C22×C3—31no—
304C22×C3—64no—
305C22×C3—64no—
306C22×C3—68no—
307C22×C3—68no—
308C22×C3—61no—
309C22×C3—92no—
310C22×C3—124no—
311C22×C3—128no—
312C22×C3—182no—
313C22×C3C23×C3×D311yescharacteristic
314C2×D3—32no—
315C2×D3—62no—
316C2×D3—92no—
317C2×D3—182no—
318C3×C4—32no—
319C3×C4—32no—
320C3×C4—34no—
321C3×C4—34no—
322C3×C4—62no—
323C3×C4—64no—
324C3×C4—92no—
325C3×C4—184no—
326C3×C4—182no—
327C3×C4C3×D3×D412yes—
328C3×C4C3×#23714yes—
329Dic3—32no—
330Dic3—64no—
331Dic3—62no—
332Dic3—92no—
333Dic3—184no—
334Dic3—182no—
335C32—21no—
336C32#571611yescharacteristic, complemented
337C32C3×#53211yescharacteristic, complemented
338C23—91no—
339C23C22×(C3⋊D3)⋊C314yes—
340Q8—184no—
341D4—24no—
342D4—28no—
343D4—188no—
344C2×C4—91no—
345C2×C4—94no—
346C2×C4—92no—
347C2×C4—94no—
348C2×C4—188no—
349C2×C4—184no—
350C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
351C2×C4C22×(C3⋊D3)⋊C312yes—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—32no—
355C2×C3—32no—
356C2×C3—44no—
357C2×C3—61no—
358C2×C3—62no—
359C2×C3—124no—
360C2×C3—124no—
361C2×C3—182no—
362C2×C3—244no—
363C2×C3C2×C3×D3×D411yescharacteristic
364C2×C3C2×C3×#23712yes—
365D3—62no—
366D3—182no—
367C22—24no—
368C22—28no—
369C22—92no—
370C22—182no—
371C22C23×(C3⋊D3)⋊C311yescharacteristic
372C4—92no—
373C4—184no—
374C4—182no—
375C4D4×(C3⋊D3)⋊C312yes—
376C4#3830⋊C314yescomplemented
377C3—31no—
378C3—31no—
379C3—61no—
380C3C3×#571611yescharacteristic
381C2—44no—
382C2—182no—
383C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
384C2C2×#3830⋊C312yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#127C2
hypercenter#127(C3⋊D3)⋊C3
hypocenterC32C3×#532
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. C2×#3830⋊C3C2semidirect
2. H(3,𝔽3)×#127C2semidirect
3. #64147⋊C3C2semidirect
4. #64145⋊C3C2semidirect
5. D4×(C3⋊Dic3)⋊C3C2semidirect
6. C42×(C3⋊D3)⋊C3C2semidirect
7. #64078⋊C3C2semidirect
8. #10596187C3semidirect
23. C2×#3830C2×C3semidirect
26. C32×#127D3semidirect
27. C32×#127C2×C3semidirect
28. #64147C2×C3semidirect
29. #64145C2×C3semidirect
30. D4×C3⋊Dic3C2×C3semidirect
31. C42×C3⋊D3C2×C3semidirect
32. #64078C2×C3semidirect
50. C4×(C3⋊D3)⋊C3D4semidirect
90. C4×H(3,𝔽3)#41semidirect
139. C4×C3⋊D3C3×D4semidirect
212. C32×C4#237semidirect
214. C32×C4C3×#41semidirect
223. #127(C3⋊D3)⋊C3semidirect
228. H(3,𝔽3)#532semidirect
336. C32#5716semidirect
337. C32C3×#532semidirect
376. C4#3830⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3830⋊C32
H(3,𝔽3)×#1272
#64147⋊C32
#64145⋊C32
D4×(C3⋊Dic3)⋊C32
C42×(C3⋊D3)⋊C32
#64078⋊C32
#105961873
C3×#57163