Finite Group Databasebeta

← #10595993⋊C3 #13152516

Subgroups — #10595993⋊C3 #13152516

Order 1728.

1161 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
3C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#64102⋊C3C212yesmaximal, complemented
5#64096⋊C3C212yesmaximal, complemented
6#64090⋊C3C214yesmaximal, complemented
7#64081⋊C3C212yesmaximal, complemented
8C42×(C3⋊D3)⋊C3C212yesmaximal, complemented
9#10595993C311yesmaximal, characteristic, complemented
10C3×#5522—31nomaximal
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3C2212yes—
13C22×(C3⋊Dic3)⋊C3C2212yes—
14#3828⋊C3C418yescomplemented
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2214yes—
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×#634⋊C3C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2212yes—
20C22⋊C4×H(3,𝔽3)C2212yes—
21C42×H(3,𝔽3)C2212yes—
22#3752⋊C3C2214yes—
23#3750⋊C3C2214yes—
24C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
25C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
26C4×(C3⋊Dic3)⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2214yes—
28C2×#3828C2×C311yescharacteristic
29C2×C3×#235—31no—
30C32×#117—21no—
31C32×#117D311yescharacteristic, complemented
32C32×#117C2×C311yescharacteristic, complemented
33C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
34#64102C2×C312yescomplemented
35#64096C2×C312yescomplemented
36#64090C2×C314yescomplemented
37#64081C2×C312yescomplemented
38C42×C3⋊D3C2×C312yescomplemented
39C2×C3×C4×Dic3—31no—
40C3×#893—32no—
41C3×#887—32no—
42C3×#67⋊C4—34no—
43C3×#872—32no—
44C3×C42×D3—32no—
45C23×H(3,𝔽3)C2311yescharacteristic
46C22×(C3⋊D3)⋊C3C2312yes—
47C2×C4×H(3,𝔽3)C2×C414yescomplemented
48C2×C4×H(3,𝔽3)C2314yes—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51#638⋊C3C2×C418yescomplemented
52C2×(C3⋊Dic3)⋊C3—24no—
53C2×(C3⋊Dic3)⋊C3C2312yes—
54C2×(C3⋊Dic3)⋊C3C2312yes—
55C2×(C3⋊Dic3)⋊C3C2312yes—
56C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
57C4×(C3⋊D3)⋊C3—28no—
58C4×(C3⋊D3)⋊C3C2×C418yescomplemented
59#634⋊C3C2×C414yescomplemented
60C3×C4×#41—91no—
61#5522—31no—
62#5522—91no—
63C22×C32×C4—21no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C22×C311yescharacteristic
66C2×#638C22×C312yes—
67C22×C3⋊Dic3C22×C312yes—
68#3828C3×C418yescomplemented
69C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
70C2×C4×C3⋊D3C22×C314yes—
71C2×C4×C3⋊D3C22×C312yes—
72C2×#634C22×C311yescharacteristic
73C2×C3×#67—32no—
74C22×C3×Dic3—32no—
75C3×#235—38no—
76C2×C3×D12—31no—
77C2×C3×C4×D3—32no—
78C2×C3×C4×D3—34no—
79C2×C3×Dic6—31no—
80C32×C4⋊C4—22no—
81C32×C4⋊C4C2×D312yes—
82C32×C4⋊C4C22×C312yes—
83C32×C22⋊C4—22no—
84C32×C22⋊C4C2×D312yes—
85C32×C22⋊C4C22×C312yes—
86C32×C42—22no—
87C32×C42C2×D312yes—
88C32×C42C22×C312yes—
89#3752C22×C314yes—
90#3750C22×C314yes—
91C4×C3⋊Dic3C22×C311yescharacteristic
92C4×C3⋊Dic3C22×C314yes—
93C4×C3⋊Dic3C22×C311yescharacteristic
94C4×C3⋊Dic3C22×C312yes—
95C3×(C2×D3)⋊C4—34no—
96C3×Dic3⋊C4—34no—
97C3×C4×Dic3—34no—
98C3×C4×Dic3—32no—
99C3×C4×Dic3—31no—
100C3×C4×Dic3—31no—
101C22×H(3,𝔽3)—22no—
102C22×H(3,𝔽3)C22×C412yes—
103C22×H(3,𝔽3)C2411yescharacteristic
104C2×(C3⋊D3)⋊C3—24no—
105C2×(C3⋊D3)⋊C3C22×C414yes—
106C4×H(3,𝔽3)—24no—
107C4×H(3,𝔽3)C22×C412yes—
108C4×H(3,𝔽3)C22×C412yes—
109(C3⋊Dic3)⋊C3—22no—
110(C3⋊Dic3)⋊C3C22×C414yes—
111(C3⋊Dic3)⋊C3#4114yescomplemented
112C2×C3×#41—91no—
113C2×#235—31no—
114C2×#235—91no—
115C3×C4×Q8—92no—
116C3×C4×D4—94no—
117C3×C4×D4—92no—
118C3×#117—31no—
119C3×#117—31no—
120C3×#117—61no—
121C3×#117—92no—
122C3×#117C3×D311yescharacteristic
123C2×C3×C42—92no—
124C2×C3×C42—91no—
125C2×C4×Dic3—31no—
126C2×C4×Dic3—91no—
127#893—32no—
128#893—92no—
129#887—32no—
130#887—92no—
131#67⋊C4—34no—
132#67⋊C4—94no—
133#872—32no—
134#872—92no—
135C42×D3—32no—
136C42×D3—92no—
137C23×C32—21no—
138C23×C32C22×D311yescharacteristic
139C23×C32C23×C311yescharacteristic
140C22×C3⋊D3C23×C312yes—
141C22×C3×D3—32no—
142C2×C32×C4—24no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—24no—
146C2×C32×C4C4×D314yescomplemented
147C2×C32×C4C2×C3×C414yescomplemented
148C2×C32×C4C22×D314yes—
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C32×C4C23×C314yes—
152C2×C32×C4C23×C311yescharacteristic
153C2×C32×C4C23×C311yescharacteristic
154#638C2×C3×C418yescomplemented
155C2×C3⋊Dic3—24no—
156C2×C3⋊Dic3C23×C312yes—
157C2×C3⋊Dic3C23×C312yes—
158C2×C3⋊Dic3C23×C312yes—
159C4⋊(C3⋊D3)C2×C3×C414yescomplemented
160C4×C3⋊D3—28no—
161C4×C3⋊D3C2×C3×C418yescomplemented
162#634C2×C3×C414yescomplemented
163C3×#67—38no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—64no—
168C3×D12—34no—
169C3×C4×D3—38no—
170C3×C4×D3—68no—
171C3×Dic6—34no—
172C4×#41—91noSylow
173C2×H(3,𝔽3)—22no—
174C2×H(3,𝔽3)C23×C411yescharacteristic
175C2×H(3,𝔽3)C2×#4112yes—
176(C3⋊D3)⋊C3—24no—
177C3×#41—98no—
178C2×C3×Q8—91no—
179C2×C3×D4—91no—
180C2×C3×D4—92no—
181C22×C3×C4—31no—
182C22×C3×C4—31no—
183C22×C3×C4—61no—
184C22×C3×C4—92no—
185C22×C3×C4—94no—
186C22×C3×C4—92no—
187C22×C3×C4C2×C3×D311yescharacteristic
188C2×#67—32no—
189C2×#67—92no—
190C22×Dic3—32no—
191C22×Dic3—92no—
192#235—38no—
193#235—98no—
194C2×D12—31no—
195C2×D12—91no—
196C2×C4×D3—32no—
197C2×C4×D3—34no—
198C2×C4×D3—92no—
199C2×C4×D3—94no—
200C2×Dic6—31no—
201C2×Dic6—91no—
202C3×C4⋊C4—32no—
203C3×C4⋊C4—32no—
204C3×C4⋊C4—62no—
205C3×C4⋊C4—94no—
206C3×C4⋊C4C2×C3×D312yes—
207C3×C22⋊C4—32no—
208C3×C22⋊C4—32no—
209C3×C22⋊C4—62no—
210C3×C22⋊C4—94no—
211C3×C22⋊C4C2×C3×D312yes—
212C3×C42—32no—
213C3×C42—32no—
214C3×C42—62no—
215C3×C42—91no—
216C3×C42—91no—
217C3×C42—92no—
218C3×C42—94no—
219C3×C42C2×C3×D312yes—
220(C2×D3)⋊C4—34no—
221(C2×D3)⋊C4—94no—
222Dic3⋊C4—34no—
223Dic3⋊C4—94no—
224C4×Dic3—31no—
225C4×Dic3—34no—
226C4×Dic3—31no—
227C4×Dic3—32no—
228C4×Dic3—91no—
229C4×Dic3—94no—
230C4×Dic3—91no—
231C4×Dic3—92no—
232C22×C32—22no—
233C22×C32—22no—
234C22×C32—21no—
235C22×C32—22no—
236C22×C32—42no—
237C22×C32C2×C4×D312yes—
238C22×C32C22×C3×C412yes—
239C22×C32C23×D311yescharacteristic
240C22×C32C24×C311yescharacteristic
241C2×C3⋊D3—24no—
242C2×C3⋊D3C22×C3×C414yes—
243C2×C3×D3—34no—
244C2×C3×D3—64no—
245C32×C4—24no—
246C32×C4—24no—
247C32×C4—22no—
248C32×C4—22no—
249C32×C4—44no—
250C32×C4C2×C4×D312yes—
251C32×C4C2×C4×D312yes—
252C32×C4C22×C3×C412yes—
253C32×C4C22×C3×C412yes—
254C3⋊Dic3—22no—
255C3⋊Dic3C22×C3×C414yes—
256C3⋊Dic3C3×#4114yescomplemented
257C3×Dic3—34no—
258C3×Dic3—34no—
259C3×Dic3—62no—
260C2×#41—91no—
261C4×Q8—92no—
262C4×D4—92no—
263C4×D4—94no—
264#117—92no—
265#117(C3⋊D3)⋊C311yescharacteristic, complemented
266C2×C42—91no—
267C2×C42—92no—
268H(3,𝔽3)C4×#4111yescharacteristic, complemented, Sylow
269C23×C3—31no—
270C23×C3—31no—
271C23×C3—61no—
272C23×C3—92no—
273C23×C3C22×C3×D311yescharacteristic
274C22×D3—32no—
275C22×D3—92no—
276C3×Q8—94no—
277C3×D4—94no—
278C3×D4—98no—
279C2×C3×C4—34no—
280C2×C3×C4—34no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—34no—
284C2×C3×C4—34no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—64no—
288C2×C3×C4—61no—
289C2×C3×C4—64no—
290C2×C3×C4—61no—
291C2×C3×C4—98no—
292C2×C3×C4—92no—
293C2×C3×C4—92no—
294C2×C3×C4—92no—
295C2×C3×C4—188no—
296C2×C3×C4—184no—
297C2×C3×C4C3×C4×D314yes—
298C2×C3×C4C22×C3×D314yes—
299C2×C3×C4C22×C3×D311yescharacteristic
300C2×C3×C4C22×C3×D311yescharacteristic
301#67—38no—
302#67—98no—
303C2×Dic3—32no—
304C2×Dic3—32no—
305C2×Dic3—32no—
306C2×Dic3—64no—
307C2×Dic3—92no—
308C2×Dic3—92no—
309C2×Dic3—92no—
310C2×Dic3—184no—
311D12—34no—
312D12—94no—
313C4×D3—38no—
314C4×D3—68no—
315C4×D3—98no—
316C4×D3—188no—
317Dic6—34no—
318Dic6—94no—
319C2×C32—22no—
320C2×C32—22no—
321C2×C32—21no—
322C2×C32—22no—
323C2×C32—42no—
324C2×C32C22×C4×D311yescharacteristic
325C2×C32D3×#4112yes—
326C2×C32C23×C3×C411yescharacteristic
327C2×C32C2×C3×#4112yes—
328C3⋊D3—24no—
329C3×D3—64no—
330#41—98no—
331C2×Q8—91no—
332C2×D4—92no—
333C2×D4—91no—
334C22×C4—92no—
335C22×C4—92no—
336C22×C4—94no—
337C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
338C4⋊C4—94no—
339C4⋊C4C2×(C3⋊D3)⋊C312yes—
340C22⋊C4—94no—
341C22⋊C4C2×(C3⋊D3)⋊C312yes—
342C42—94no—
343C42—92no—
344C42—91no—
345C42—91no—
346C42C2×(C3⋊D3)⋊C312yes—
347C22×C3—22no—
348C22×C3—31no—
349C22×C3—31no—
350C22×C3—32no—
351C22×C3—32no—
352C22×C3—62no—
353C22×C3—62no—
354C22×C3—61no—
355C22×C3—62no—
356C22×C3—94no—
357C22×C3—122no—
358C22×C3—184no—
359C22×C3C2×C3×C4×D312yes—
360C22×C3C23×C3×D311yescharacteristic
361C2×D3—34no—
362C2×D3—64no—
363C2×D3—94no—
364C2×D3—184no—
365C3×C4—24no—
366C3×C4—32no—
367C3×C4—32no—
368C3×C4—32no—
369C3×C4—32no—
370C3×C4—64no—
371C3×C4—64no—
372C3×C4—62no—
373C3×C4—62no—
374C3×C4—94no—
375C3×C4—94no—
376C3×C4—124no—
377C3×C4—182no—
378C3×C4C2×C3×C4×D312yes—
379C3×C4C2×C3×C4×D312yes—
380Dic3—34no—
381Dic3—34no—
382Dic3—62no—
383Dic3—94no—
384Dic3—94no—
385Dic3—182no—
386C32—21no—
387C32#552211yescharacteristic, complemented
388C32C3×C4×#4111yescharacteristic, complemented
389C23—92no—
390C23C22×(C3⋊D3)⋊C311yescharacteristic
391Q8—94no—
392D4—98no—
393D4—94no—
394C2×C4—92no—
395C2×C4—92no—
396C2×C4—98no—
397C2×C4—92no—
398C2×C4—188no—
399C2×C4—184no—
400C2×C4C4×(C3⋊D3)⋊C314yescomplemented
401C2×C4C22×(C3⋊D3)⋊C314yes—
402C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
403C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
404C2×C3—22no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—32no—
408C2×C3—32no—
409C2×C3—62no—
410C2×C3—62no—
411C2×C3—61no—
412C2×C3—62no—
413C2×C3—122no—
414C2×C3—184no—
415C2×C3C22×C3×C4×D311yescharacteristic
416C2×C3C3×D3×#4112yes—
417D3—64no—
418D3—184no—
419C22—22no—
420C22—94no—
421C22—184no—
422C22C2×C4×(C3⋊D3)⋊C312yes—
423C22C23×(C3⋊D3)⋊C311yescharacteristic
424C4—24no—
425C4—94no—
426C4—94no—
427C4—182no—
428C4C2×C4×(C3⋊D3)⋊C312yes—
429C4C2×C4×(C3⋊D3)⋊C312yes—
430C3—31no—
431C3—31no—
432C3—61no—
433C3C3×#552211yescharacteristic
434C2—22no—
435C2—184no—
436C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
437C2#41×(C3⋊D3)⋊C312yes—
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC2×C32C23×C3×C4
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#117C2
prime radicalC2×#3828⋊C3C2
cosocleC22×C32C24×C3
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×C4×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#117C2semidirect
3. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
4. #64102⋊C3C2semidirect
5. #64096⋊C3C2semidirect
6. #64090⋊C3C2semidirect
7. #64081⋊C3C2semidirect
8. C42×(C3⋊D3)⋊C3C2semidirect
9. #10595993C3semidirect
14. #3828⋊C3C4semidirect
31. C32×#117D3semidirect
32. C32×#117C2×C3semidirect
33. C2×C4×C3⋊Dic3C2×C3semidirect
34. #64102C2×C3semidirect
35. #64096C2×C3semidirect
36. #64090C2×C3semidirect
37. #64081C2×C3semidirect
38. C42×C3⋊D3C2×C3semidirect
47. C2×C4×H(3,𝔽3)C2×C4semidirect
51. #638⋊C3C2×C4semidirect
56. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
58. C4×(C3⋊D3)⋊C3C2×C4semidirect
59. #634⋊C3C2×C4semidirect
68. #3828C3×C4semidirect
111. (C3⋊Dic3)⋊C3#41semidirect
146. C2×C32×C4C4×D3semidirect
147. C2×C32×C4C2×C3×C4semidirect
154. #638C2×C3×C4semidirect
159. C4⋊(C3⋊D3)C2×C3×C4semidirect
161. C4×C3⋊D3C2×C3×C4semidirect
162. #634C2×C3×C4semidirect
256. C3⋊Dic3C3×#41semidirect
265. #117(C3⋊D3)⋊C3semidirect
268. H(3,𝔽3)C4×#41semidirect
387. C32#5522semidirect
388. C32C3×C4×#41semidirect
400. C2×C4C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
H(3,𝔽3)×#1172
C2×C4×(C3⋊Dic3)⋊C32
#64102⋊C32
#64096⋊C32
#64090⋊C32
#64081⋊C32
C42×(C3⋊D3)⋊C32
#105959933
C3×#55223