Finite Group Databasebeta

← #10595994⋊C3 #13152517

Subgroups — #10595994⋊C3 #13152517

Order 1728.

1105 conjugacy classes of subgroups.

The subgroup lattice has 470 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)⋊C3)C211yesmaximal, characteristic, complemented
4#64101⋊C3C212yesmaximal, complemented
5C4⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
6#64090⋊C3C214yesmaximal, complemented
7C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
8C4×#634⋊C3C212yesmaximal, complemented
9#10595994C311yesmaximal, characteristic, complemented
10C3×#5523—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)C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C22⋊C4×H(3,𝔽3)C2212yes—
22C42×H(3,𝔽3)C2212yes—
23#3752⋊C3C2214yes—
24C4⋊((C3⋊Dic3)⋊C3)C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2212yes—
26#3750⋊C3C2214yes—
27C4×(C3⋊Dic3)⋊C3C2214yes—
28C2×#3828C2×C311yescharacteristic
29C2×C3×#235—31no—
30C32×#117—21no—
31C32×#117D311yescharacteristic, complemented
32C32×#117C2×C311yescharacteristic, complemented
33C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
34#64101C2×C312yescomplemented
35C4⋊C4×C3⋊D3C2×C312yescomplemented
36#64090C2×C314yescomplemented
37C4×C4⋊(C3⋊D3)C2×C312yescomplemented
38C4×#634C2×C312yescomplemented
39C2×C3×C4⋊Dic3—31no—
40C3×#892—32no—
41C3×D3×C4⋊C4—32no—
42C3×#67⋊C4—34no—
43C3×C4×D12—32no—
44C3×C4×Dic6—32no—
45C23×H(3,𝔽3)C2311yescharacteristic
46C22×(C3⋊D3)⋊C3C2312yes—
47C2×C4×H(3,𝔽3)C2×C414yescomplemented
48C2×C4×H(3,𝔽3)C2312yes—
49C2×C4×H(3,𝔽3)C2312yes—
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52#638⋊C3C2×C418yescomplemented
53C2×(C3⋊Dic3)⋊C3—42no—
54C2×(C3⋊Dic3)⋊C3C2314yes—
55C2×(C3⋊Dic3)⋊C3C2312yes—
56C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
57C4×(C3⋊D3)⋊C3—44no—
58C4×(C3⋊D3)⋊C3C2×C418yescomplemented
59#634⋊C3C2×C414yescomplemented
60C3×#520—91no—
61#5523—31no—
62#5523—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—21no—
81C32×C4⋊C4—21no—
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C22×C311yescharacteristic
85C32×C4⋊C4C22×C311yescharacteristic
86C32×C22⋊C4—22no—
87C32×C22⋊C4C2×D312yes—
88C32×C22⋊C4C22×C312yes—
89C32×C42—22no—
90C32×C42C2×D312yes—
91C32×C42C22×C312yes—
92#3752C22×C314yes—
93C4⋊(C3⋊Dic3)C22×C312yes—
94C4⋊(C3⋊Dic3)C22×C312yes—
95#3750C22×C314yes—
96C4×C3⋊Dic3C22×C314yes—
97C3×(C2×D3)⋊C4—34no—
98C3×C4⋊Dic3—32no—
99C3×C4⋊Dic3—32no—
100C3×Dic3⋊C4—34no—
101C3×C4×Dic3—34no—
102C22×H(3,𝔽3)—21no—
103C22×H(3,𝔽3)—21no—
104C22×H(3,𝔽3)C22×C412yes—
105C22×H(3,𝔽3)C2411yescharacteristic
106C2×(C3⋊D3)⋊C3—22no—
107C2×(C3⋊D3)⋊C3—22no—
108C2×(C3⋊D3)⋊C3C22×C414yes—
109C4×H(3,𝔽3)—22no—
110C4×H(3,𝔽3)—22no—
111C4×H(3,𝔽3)C22×C412yes—
112C4×H(3,𝔽3)C22×C412yes—
113(C3⋊Dic3)⋊C3—24no—
114(C3⋊Dic3)⋊C3C22×C414yes—
115C2×C3×#41—91no—
116C2×#235—31no—
117C2×#235—91no—
118C3×C4×Q8—92no—
119C3×C4×D4—92no—
120C3×C4×D4—94no—
121C3×#117—31no—
122C3×#117—31no—
123C3×#117—61no—
124C3×#117—92no—
125C3×#117C3×D311yescharacteristic
126C2×C3×C4⋊C4—92no—
127C2×C3×C4⋊C4—91no—
128C2×C4⋊Dic3—31no—
129C2×C4⋊Dic3—91no—
130#892—32no—
131#892—92no—
132D3×C4⋊C4—32no—
133D3×C4⋊C4—92no—
134#67⋊C4—34no—
135#67⋊C4—94no—
136C4×D12—32no—
137C4×D12—92no—
138C4×Dic6—32no—
139C4×Dic6—92no—
140C23×C32—21no—
141C23×C32C22×D311yescharacteristic
142C23×C32C23×C311yescharacteristic
143C22×C3⋊D3C23×C312yes—
144C22×C3×D3—32no—
145C2×C32×C4—24no—
146C2×C32×C4—21no—
147C2×C32×C4—21no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4C4×D314yescomplemented
151C2×C32×C4C2×C3×C414yescomplemented
152C2×C32×C4C22×D312yes—
153C2×C32×C4C22×D312yes—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C23×C312yes—
157C2×C32×C4C23×C312yes—
158C2×C32×C4C23×C311yescharacteristic
159C2×C32×C4C23×C311yescharacteristic
160#638C2×C3×C418yescomplemented
161C2×C3⋊Dic3—42no—
162C2×C3⋊Dic3C23×C314yes—
163C2×C3⋊Dic3C23×C312yes—
164C4⋊(C3⋊D3)C2×C3×C414yescomplemented
165C4×C3⋊D3—44no—
166C4×C3⋊D3C2×C3×C418yescomplemented
167#634C2×C3×C414yescomplemented
168C3×#67—38no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—34no—
171C2×C3×Dic3—122no—
172C3×D12—34no—
173C3×C4×D3—38no—
174C3×C4×D3—124no—
175C3×Dic6—34no—
176#520—91noSylow
177C2×H(3,𝔽3)—22no—
178C2×H(3,𝔽3)C23×C411yescharacteristic
179C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
180C2×H(3,𝔽3)2−1+411yescharacteristic
181(C3⋊D3)⋊C3—24no—
182C3×#41—98no—
183C2×C3×Q8—91no—
184C2×C3×D4—91no—
185C2×C3×D4—92no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—61no—
189C22×C3×C4—92no—
190C22×C3×C4—94no—
191C22×C3×C4—92no—
192C22×C3×C4C2×C3×D311yescharacteristic
193C2×#67—32no—
194C2×#67—92no—
195C22×Dic3—32no—
196C22×Dic3—92no—
197#235—38no—
198#235—98no—
199C2×D12—31no—
200C2×D12—91no—
201C2×C4×D3—32no—
202C2×C4×D3—34no—
203C2×C4×D3—92no—
204C2×C4×D3—94no—
205C2×Dic6—31no—
206C2×Dic6—91no—
207C3×C4⋊C4—31no—
208C3×C4⋊C4—31no—
209C3×C4⋊C4—31no—
210C3×C4⋊C4—31no—
211C3×C4⋊C4—61no—
212C3×C4⋊C4—61no—
213C3×C4⋊C4—94no—
214C3×C4⋊C4—92no—
215C3×C4⋊C4—92no—
216C3×C4⋊C4C2×C3×D311yescharacteristic
217C3×C4⋊C4C2×C3×D311yescharacteristic
218C3×C22⋊C4—32no—
219C3×C22⋊C4—32no—
220C3×C22⋊C4—62no—
221C3×C22⋊C4—94no—
222C3×C22⋊C4C2×C3×D312yes—
223C3×C42—32no—
224C3×C42—32no—
225C3×C42—62no—
226C3×C42—94no—
227C3×C42C2×C3×D312yes—
228(C2×D3)⋊C4—34no—
229(C2×D3)⋊C4—94no—
230C4⋊Dic3—32no—
231C4⋊Dic3—32no—
232C4⋊Dic3—92no—
233C4⋊Dic3—92no—
234Dic3⋊C4—34no—
235Dic3⋊C4—94no—
236C4×Dic3—34no—
237C4×Dic3—94no—
238C22×C32—21no—
239C22×C32—21no—
240C22×C32—21no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—22no—
244C22×C32—41no—
245C22×C32—41no—
246C22×C32C2×C4×D312yes—
247C22×C32C22×C3×C412yes—
248C22×C32C23×D311yescharacteristic
249C22×C32C24×C311yescharacteristic
250C2×C3⋊D3—22no—
251C2×C3⋊D3—22no—
252C2×C3⋊D3C22×C3×C414yes—
253C2×C3×D3—34no—
254C2×C3×D3—62no—
255C2×C3×D3—62no—
256C32×C4—22no—
257C32×C4—22no—
258C32×C4—22no—
259C32×C4—22no—
260C32×C4—22no—
261C32×C4—22no—
262C32×C4—42no—
263C32×C4—42no—
264C32×C4C2×C4×D312yes—
265C32×C4C2×C4×D312yes—
266C32×C4C22×C3×C412yes—
267C32×C4C22×C3×C412yes—
268C3⋊Dic3—24no—
269C3⋊Dic3C22×C3×C414yes—
270C3×Dic3—34no—
271C3×Dic3—64no—
272C2×#41—91no—
273C4×Q8—92no—
274C4×D4—94no—
275C4×D4—92no—
276#117—92no—
277#117(C3⋊D3)⋊C311yescharacteristic, complemented
278C2×C4⋊C4—91no—
279C2×C4⋊C4—92no—
280H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
281C23×C3—31no—
282C23×C3—31no—
283C23×C3—61no—
284C23×C3—92no—
285C23×C3C22×C3×D311yescharacteristic
286C22×D3—32no—
287C22×D3—92no—
288C3×Q8—94no—
289C3×D4—94no—
290C3×D4—98no—
291C2×C3×C4—34no—
292C2×C3×C4—34no—
293C2×C3×C4—31no—
294C2×C3×C4—31no—
295C2×C3×C4—31no—
296C2×C3×C4—31no—
297C2×C3×C4—32no—
298C2×C3×C4—32no—
299C2×C3×C4—32no—
300C2×C3×C4—32no—
301C2×C3×C4—64no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—62no—
305C2×C3×C4—62no—
306C2×C3×C4—98no—
307C2×C3×C4—92no—
308C2×C3×C4—94no—
309C2×C3×C4—364no—
310C2×C3×C4—362no—
311C2×C3×C4C3×C4×D314yes—
312C2×C3×C4C22×C3×D312yes—
313C2×C3×C4C22×C3×D312yes—
314C2×C3×C4C22×C3×D311yescharacteristic
315C2×C3×C4C22×C3×D311yescharacteristic
316#67—38no—
317#67—98no—
318C2×Dic3—32no—
319C2×Dic3—34no—
320C2×Dic3—92no—
321C2×Dic3—94no—
322C2×Dic3—122no—
323C2×Dic3—362no—
324D12—34no—
325D12—94no—
326C4×D3—38no—
327C4×D3—98no—
328C4×D3—124no—
329C4×D3—364no—
330Dic6—34no—
331Dic6—94no—
332C2×C32—22no—
333C2×C32—22no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32—21no—
337C2×C32—42no—
338C2×C32C22×C4×D311yescharacteristic
339C2×C32#100911yescharacteristic
340C2×C32#101011yescharacteristic
341C2×C32C23×C3×C411yescharacteristic
342C2×C32C3×H(5,𝔽2)11yescharacteristic
343C2×C32C3×2−1+411yescharacteristic
344C3⋊D3—24no—
345C3×D3—64no—
346#41—98no—
347C2×Q8—91no—
348C2×D4—92no—
349C2×D4—91no—
350C22×C4—92no—
351C22×C4—92no—
352C22×C4—94no—
353C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
354C4⋊C4—92no—
355C4⋊C4—94no—
356C4⋊C4—92no—
357C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
358C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
359C22⋊C4—94no—
360C22⋊C4C2×(C3⋊D3)⋊C312yes—
361C42—94no—
362C42C2×(C3⋊D3)⋊C312yes—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—31no—
366C22×C3—31no—
367C22×C3—32no—
368C22×C3—32no—
369C22×C3—61no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3—61no—
373C22×C3—61no—
374C22×C3—62no—
375C22×C3—94no—
376C22×C3—121no—
377C22×C3—121no—
378C22×C3—182no—
379C22×C3—182no—
380C22×C3C2×C3×C4×D312yes—
381C22×C3C23×C3×D311yescharacteristic
382C2×D3—34no—
383C2×D3—62no—
384C2×D3—62no—
385C2×D3—94no—
386C2×D3—182no—
387C2×D3—182no—
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—32no—
391C3×C4—32no—
392C3×C4—32no—
393C3×C4—32no—
394C3×C4—62no—
395C3×C4—62no—
396C3×C4—62no—
397C3×C4—62no—
398C3×C4—62no—
399C3×C4—62no—
400C3×C4—94no—
401C3×C4—122no—
402C3×C4—122no—
403C3×C4—184no—
404C3×C4C2×C3×C4×D312yes—
405C3×C4C2×C3×C4×D312yes—
406Dic3—34no—
407Dic3—64no—
408Dic3—94no—
409Dic3—184no—
410C32—21no—
411C32#552311yescharacteristic, complemented
412C32C3×#52011yescharacteristic, complemented
413C23—92no—
414C23C22×(C3⋊D3)⋊C311yescharacteristic
415Q8—94no—
416D4—98no—
417D4—94no—
418C2×C4—92no—
419C2×C4—98no—
420C2×C4—94no—
421C2×C4—364no—
422C2×C4—362no—
423C2×C4C4×(C3⋊D3)⋊C314yescomplemented
424C2×C4C22×(C3⋊D3)⋊C312yes—
425C2×C4C22×(C3⋊D3)⋊C312yes—
426C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
427C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
428C2×C3—22no—
429C2×C3—31no—
430C2×C3—31no—
431C2×C3—31no—
432C2×C3—31no—
433C2×C3—31no—
434C2×C3—31no—
435C2×C3—62no—
436C2×C3—62no—
437C2×C3—61no—
438C2×C3—61no—
439C2×C3—61no—
440C2×C3—122no—
441C2×C3—184no—
442C2×C3C22×C3×C4×D311yescharacteristic
443C2×C3C3×#100911yescharacteristic
444C2×C3C3×#101011yescharacteristic
445D3—64no—
446D3—184no—
447C22—21no—
448C22—21no—
449C22—94no—
450C22—182no—
451C22—182no—
452C22C2×C4×(C3⋊D3)⋊C312yes—
453C22C23×(C3⋊D3)⋊C311yescharacteristic
454C4—22no—
455C4—22no—
456C4—94no—
457C4—184no—
458C4C2×C4×(C3⋊D3)⋊C312yes—
459C4C2×C4×(C3⋊D3)⋊C312yes—
460C3—31no—
461C3—31no—
462C3—61no—
463C3C3×#552311yescharacteristic
464C2—22no—
465C2—184no—
466C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
467C2#1172263811yescharacteristic
468C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(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×#520
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ #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)⋊C3)C2semidirect
4. #64101⋊C3C2semidirect
5. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
6. #64090⋊C3C2semidirect
7. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
8. C4×#634⋊C3C2semidirect
9. #10595994C3semidirect
14. #3828⋊C3C4semidirect
31. C32×#117D3semidirect
32. C32×#117C2×C3semidirect
33. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
34. #64101C2×C3semidirect
35. C4⋊C4×C3⋊D3C2×C3semidirect
36. #64090C2×C3semidirect
37. C4×C4⋊(C3⋊D3)C2×C3semidirect
38. C4×#634C2×C3semidirect
47. C2×C4×H(3,𝔽3)C2×C4semidirect
52. #638⋊C3C2×C4semidirect
56. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
58. C4×(C3⋊D3)⋊C3C2×C4semidirect
59. #634⋊C3C2×C4semidirect
68. #3828C3×C4semidirect
150. C2×C32×C4C4×D3semidirect
151. C2×C32×C4C2×C3×C4semidirect
160. #638C2×C3×C4semidirect
164. C4⋊(C3⋊D3)C2×C3×C4semidirect
166. C4×C3⋊D3C2×C3×C4semidirect
167. #634C2×C3×C4semidirect
277. #117(C3⋊D3)⋊C3semidirect
280. H(3,𝔽3)#520semidirect
411. C32#5523semidirect
412. C32C3×#520semidirect
423. 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)⋊C3)2
#64101⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#64090⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#105959943
C3×#55233