Finite Group Databasebeta

← C3×#10595170 #13174733

Subgroups — C3×#10595170 #13174733

Order 1728.

1762 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3C32×#936C212yesmaximal, complemented
4C3×#63974C214yesmaximal, complemented
5C3×#63953C214yesmaximal, complemented
6C3×C4×#624C211yesmaximal, characteristic, complemented
7C3×#63864C212yesmaximal, complemented
8C3×D3×#124—32nomaximal
9#10595170C311yesmaximal, characteristic, direct product factor
10C23×C3×C3⋊D3C2212yes—
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638C2212yes—
13C3×D4×C3⋊D3—24no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C32×Dic6C2212yescomplemented
17C2×C3×#627C2212yescomplemented
18C2×C3×#624C2212yes—
19C2×C3×#624C2212yes—
20C3×#3797—24no—
21C32×#217C2218yescomplemented
22C32×C4×Dic3C2212yescomplemented
23C3×#3722C2218yescomplemented
24C3×Dic32C2212yescomplemented
25C2×D4×C3⋊D3C2×C311yescharacteristic
26C2×C3×D3×Q8—32no—
27C2×C3×D3×D4—32no—
28C2×C3×D3×D4—61no—
29C2×#3797C2×C311yescharacteristic, complemented
30C32×#124—32no—
31C3×#947—32no—
32C3×#936—22no—
33C3×#936D312yescomplemented
34C3×#936C2×C312yescomplemented
35C3×#885—38no—
36C3×D3×C22⋊C4—38no—
37C3×#875—32no—
38C3×C42×D3—32no—
39#63974C2×C314yescomplemented
40#63953C2×C314yescomplemented
41C4×#624C2×C311yescharacteristic, complemented
42#63864C2×C312yescomplemented
43C23×C33C2312yes—
44C22×C3×C3⋊D3—24no—
45C22×C3×C3⋊D3—28no—
46C22×C3×C3⋊D3C2312yes—
47C22×C3×C3⋊D3C2311yescharacteristic
48C33×D4—22no—
49C2×C33×C4C2311yescharacteristic
50C3×#638—24no—
51C2×C3×C3⋊Dic3C2311yescharacteristic
52C3×C4⋊(C3⋊D3)—22no—
53C3×C4×C3⋊D3D414yes—
54C2×C32×Dic3C2314yes—
55C2×C32×Dic3C2314yes—
56C32×Dic6—24no—
57C3×#627—24no—
58C3×#624—24no—
59C3×#624—24no—
60C2×C3×#124—91no—
61D3×#124—32no—
62C23×C3⋊D3C22×C312yes—
63C23×C3×D3—34no—
64C23×C3×D3—62no—
65C2×C32×Q8—32no—
66C2×C32×D4—21no—
67C2×C32×D4—22no—
68C2×C32×D4—41no—
69C2×C32×D4C2×D312yes—
70C2×C32×D4C22×C311yescharacteristic
71C2×#638C22×C312yes—
72D4×C3⋊D3—24no—
73C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
74C2×C4×C3⋊D3C22×C311yescharacteristic
75C2×C3×#67—34no—
76C2×C3×#67—62no—
77C3×D3×Q8—68no—
78C3×D3×D4—68no—
79C3×D3×D4—68no—
80C2×C3×D12—32no—
81C2×C3×D12—61no—
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—34no—
84C2×C3×C4×D3—34no—
85C2×C3×C4×D3—61no—
86C2×C3×Dic6—22no—
87C2×C3×Dic6—34no—
88C2×C3×Dic6C2×D312yescomplemented
89C2×C3×Dic6C22×C312yescomplemented
90C2×#627C22×C312yescomplemented
91C2×#624C22×C312yes—
92C2×#624C22×C312yes—
93#3797—24no—
94C32×C22⋊C4—38no—
95C32×C42—32no—
96C3×#217—28no—
97C3×#217C2×D318yescomplemented
98C3×#217C22×C318yescomplemented
99C3×(C2×D3)⋊C4—38no—
100C3×(C2×D3)⋊C4—38no—
101C3×C4×Dic3—22no—
102C3×C4×Dic3—34no—
103C3×C4×Dic3C2×D312yescomplemented
104C3×C4×Dic3C22×C312yescomplemented
105#3722C22×C318yescomplemented
106Dic32C22×C312yescomplemented
107C22×C33—22no—
108C22×C33—24no—
109C22×C33C2411yescharacteristic
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3—24no—
112C2×C3×C3⋊D3—48no—
113C2×C3×C3⋊D3C2×D412yes—
114C2×C3×C3⋊D3#4114yes—
115C33×C4C2×D412yes—
116C3×C3⋊Dic3C2×D412yes—
117C32×Dic3—24no—
118C32×Dic3—24no—
119C22×C3×Q8—91no—
120C22×C3×D4—91no—
121C2×D3×Q8—32no—
122C2×D3×D4—32no—
123C2×D3×D4—61no—
124C3×#124—32no—
125C3×#124—32no—
126C3×#124—62no—
127C3×#124—92no—
128C3×#124—94no—
129C2×C3×C22⋊C4—94no—
130C2×C3×C42—91no—
131#947—32no—
132#936C3×D312yescomplemented
133#885—38no—
134D3×C22⋊C4—38no—
135#875—32no—
136C42×D3—32no—
137C23×C32—22no—
138C23×C32—24no—
139C23×C32—42no—
140C23×C32C22×D314yes—
141C23×C32C23×C312yes—
142C22×C3⋊D3—24no—
143C22×C3⋊D3—28no—
144C22×C3⋊D3C23×C312yes—
145C22×C3⋊D3C23×C311yescharacteristic
146C22×C3×D3—32no—
147C22×C3×D3—34no—
148C22×C3×D3—61no—
149C22×C3×D3—62no—
150C22×C3×D3—616no—
151C22×C3×D3—68no—
152C22×C3×D3—68no—
153C22×C3×D3—128no—
154C32×Q8—64no—
155C32×D4—22no—
156C32×D4—24no—
157C32×D4—24no—
158C32×D4—44no—
159C32×D4—44no—
160C2×C32×C4—21no—
161C2×C32×C4—22no—
162C2×C32×C4—34no—
163C2×C32×C4—34no—
164C2×C32×C4—41no—
165C2×C32×C4C22×D312yes—
166C2×C32×C4C23×C311yescharacteristic
167#638—24no—
168C2×C3⋊Dic3C23×C311yescharacteristic
169C4⋊(C3⋊D3)—22no—
170C4×C3⋊D3C3×D414yes—
171C3×#67—68no—
172C3×#67—68no—
173C2×C3×Dic3—24no—
174C2×C3×Dic3—24no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3C22×D314yes—
178C2×C3×Dic3C22×D314yes—
179C2×C3×Dic3C23×C314yes—
180C2×C3×Dic3C23×C314yes—
181C3×D12—64no—
182C3×D12—64no—
183C3×C4×D3—38no—
184C3×C4×D3—64no—
185C3×C4×D3—68no—
186C3×C4×D3—68no—
187C3×Dic6—24no—
188C3×Dic6—24no—
189C3×Dic6—44no—
190C3×Dic6—68no—
191#627—24no—
192#624—24no—
193#624—24no—
194C2×#124—91noSylow
195C2×C33—42no—
196C2×C33C22×D411yescharacteristic
197C2×C33C2×#4112yes—
198C3×C3⋊D3—42no—
199C3×C3⋊D3#12414yescomplemented
200C24×C3—92no—
201C23×D3—34no—
202C23×D3—62no—
203C2×C3×Q8—32no—
204C2×C3×Q8—32no—
205C2×C3×Q8—62no—
206C2×C3×Q8—92no—
207C2×C3×Q8—184no—
208C2×C3×D4—21no—
209C2×C3×D4—22no—
210C2×C3×D4—41no—
211C2×C3×D4—92no—
212C2×C3×D4—91no—
213C2×C3×D4—184no—
214C2×C3×D4D3211yescharacteristic
215C2×C3×D4C2×C3×D312yes—
216C22×C3×C4—92no—
217C22×C3×C4—92no—
218C22×C3×C4—91no—
219C2×#67—34no—
220C2×#67—62no—
221D3×Q8—68no—
222D3×D4—68no—
223D3×D4—68no—
224C2×D12—32no—
225C2×D12—61no—
226C2×C4×D3—32no—
227C2×C4×D3—34no—
228C2×C4×D3—34no—
229C2×C4×D3—61no—
230C2×Dic6—34no—
231C2×Dic6C2×C3×D312yescomplemented
232C3×C22⋊C4—38no—
233C3×C22⋊C4—38no—
234C3×C22⋊C4—68no—
235C3×C22⋊C4—98no—
236C3×C42—32no—
237C3×C42—32no—
238C3×C42—62no—
239C3×C42—92no—
240#217C2×C3×D318yescomplemented
241(C2×D3)⋊C4—38no—
242(C2×D3)⋊C4—38no—
243C4×Dic3—34no—
244C4×Dic3C2×C3×D312yescomplemented
245C22×C32—22no—
246C22×C32—24no—
247C22×C32—24no—
248C22×C32—28no—
249C22×C32—24no—
250C22×C32—21no—
251C22×C32—22no—
252C22×C32—41no—
253C22×C32—44no—
254C22×C32—44no—
255C22×C32—44no—
256C22×C32—48no—
257C22×C32—84no—
258C22×C32C23×D312yes—
259C22×C32C24×C311yescharacteristic
260C2×C3⋊D3—22no—
261C2×C3⋊D3—24no—
262C2×C3⋊D3—48no—
263C2×C3⋊D3C2×C3×D412yes—
264C2×C3⋊D3C3×#4114yes—
265C2×C3×D3—34no—
266C2×C3×D3—38no—
267C2×C3×D3—62no—
268C2×C3×D3—64no—
269C2×C3×D3—64no—
270C2×C3×D3—64no—
271C2×C3×D3—68no—
272C2×C3×D3—124no—
273C2×C3×D3—1216no—
274C2×C3×D3—1216no—
275C32×C4—22no—
276C32×C4—24no—
277C32×C4—42no—
278C32×C4—64no—
279C32×C4—64no—
280C32×C4D3×D414yes—
281C32×C4C2×C3×D412yes—
282C3⋊Dic3C2×C3×D412yes—
283C3×Dic3—24no—
284C3×Dic3—24no—
285C3×Dic3—24no—
286C3×Dic3—24no—
287C3×Dic3—34no—
288C3×Dic3—44no—
289C3×Dic3—44no—
290C3×Dic3—62no—
291C22×Q8—91no—
292C22×D4—91no—
293#124—32no—
294#124—92no—
295#124—94no—
296C2×C22⋊C4—94no—
297C2×C42—91no—
298C33C2×#12411yescharacteristic, complemented, Sylow
299C23×C3—22no—
300C23×C3—24no—
301C23×C3—42no—
302C23×C3—91no—
303C23×C3—92no—
304C23×C3—184no—
305C23×C3—188no—
306C23×C3C2×D3212yes—
307C23×C3C22×C3×D314yes—
308C22×D3—32no—
309C22×D3—34no—
310C22×D3—61no—
311C22×D3—62no—
312C22×D3—68no—
313C22×D3—616no—
314C22×D3—68no—
315C22×D3—128no—
316C3×Q8—64no—
317C3×Q8—64no—
318C3×Q8—124no—
319C3×Q8—184no—
320C3×D4—24no—
321C3×D4—22no—
322C3×D4—24no—
323C3×D4—44no—
324C3×D4—44no—
325C3×D4—184no—
326C3×D4—182no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—34no—
330C2×C3×C4—34no—
331C2×C3×C4—34no—
332C2×C3×C4—34no—
333C2×C3×C4—41no—
334C2×C3×C4—64no—
335C2×C3×C4—64no—
336C2×C3×C4—91no—
337C2×C3×C4—94no—
338C2×C3×C4—184no—
339C2×C3×C4—184no—
340C2×C3×C4C2×D3211yescharacteristic
341C2×C3×C4C22×C3×D312yes—
342#67—68no—
343#67—68no—
344C2×Dic3—32no—
345C2×Dic3—61no—
346C2×Dic3C22×C3×D314yes—
347C2×Dic3C22×C3×D314yes—
348D12—64no—
349D12—64no—
350C4×D3—38no—
351C4×D3—64no—
352C4×D3—68no—
353C4×D3—68no—
354Dic6—24no—
355Dic6—68no—
356C2×C32—22no—
357C2×C32—21no—
358C2×C32—22no—
359C2×C32—24no—
360C2×C32—41no—
361C2×C32—42no—
362C2×C32—42no—
363C2×C32—44no—
364C2×C32—44no—
365C2×C32—84no—
366C2×C32—84no—
367C2×C32C2×D3×D412yes—
368C2×C32D3×#4114yes—
369C2×C32C22×C3×D411yescharacteristic
370C2×C32C2×C3×#4112yes—
371C3⋊D3—42no—
372C3⋊D3C3×#12414yescomplemented
373C3×D3—38no—
374C3×D3—64no—
375C3×D3—124no—
376C3×D3—124no—
377C24—92no—
378C2×Q8—32no—
379C2×Q8—92no—
380C2×Q8—184no—
381C2×D4—92no—
382C2×D4—91no—
383C2×D4—184no—
384C2×D4C3×D3211yescharacteristic
385C22×C4—92no—
386C22×C4—92no—
387C22×C4—91no—
388C22⋊C4—38no—
389C22⋊C4—98no—
390C42—32no—
391C42—92no—
392C22×C3—24no—
393C22×C3—24no—
394C22×C3—22no—
395C22×C3—24no—
396C22×C3—28no—
397C22×C3—21no—
398C22×C3—22no—
399C22×C3—44no—
400C22×C3—44no—
401C22×C3—48no—
402C22×C3—44no—
403C22×C3—41no—
404C22×C3—84no—
405C22×C3—94no—
406C22×C3—92no—
407C22×C3—184no—
408C22×C3—182no—
409C22×C3—368no—
410C22×C3C22×D3211yescharacteristic
411C22×C3C23×C3×D312yes—
412C2×D3—34no—
413C2×D3—38no—
414C2×D3—62no—
415C2×D3—64no—
416C2×D3—64no—
417C2×D3—64no—
418C2×D3—68no—
419C2×D3—124no—
420C2×D3—1216no—
421C2×D3—1216no—
422C3×C4—22no—
423C3×C4—24no—
424C3×C4—42no—
425C3×C4—64no—
426C3×C4—64no—
427C3×C4—64no—
428C3×C4—64no—
429C3×C4—92no—
430C3×C4—124no—
431C3×C4—124no—
432C3×C4#381112yes—
433C3×C4C3×D3×D414yes—
434Dic3—24no—
435Dic3—24no—
436Dic3—34no—
437Dic3—62no—
438C32—21no—
439C32—22no—
440C32—41no—
441C32D3×#12412yescomplemented
442C32C2×C3×#12411yescharacteristic, complemented
443C23—91no—
444C23—92no—
445C23—184no—
446C23—188no—
447C23C2×C3×D3212yes—
448Q8—64no—
449Q8—184no—
450D4—22no—
451D4—184no—
452D4—182no—
453C2×C4—34no—
454C2×C4—34no—
455C2×C4—91no—
456C2×C4—94no—
457C2×C4—184no—
458C2×C4—184no—
459C2×C4C2×C3×D3211yescharacteristic
460C2×C3—21no—
461C2×C3—22no—
462C2×C3—24no—
463C2×C3—22no—
464C2×C3—44no—
465C2×C3—42no—
466C2×C3—44no—
467C2×C3—41no—
468C2×C3—42no—
469C2×C3—84no—
470C2×C3—84no—
471C2×C3—94no—
472C2×C3—362no—
473C2×C3#6431212yes—
474C2×C3C2×#381111yescharacteristic
475C2×C3C2×C3×D3×D412yes—
476C2×C3C3×D3×#4114yes—
477D3—38no—
478D3—64no—
479D3—124no—
480D3—124no—
481C22—22no—
482C22—24no—
483C22—92no—
484C22—94no—
485C22—182no—
486C22—184no—
487C22—368no—
488C22C22×C3×D3211yescharacteristic
489C4—64no—
490C4—64no—
491C4—92no—
492C4C3×#381112yes—
493C3—21no—
494C3—22no—
495C3—41no—
496C3#1059517011yescharacteristic, direct product factor
497C3C3×D3×#12412yescomplemented
498C2—42no—
499C2—94no—
500C2—362no—
501C2C3×#6431212yes—
502C2C2×C3×#381111yescharacteristic
subgroupG/H
centerC22×C3C22×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC2×C33×D4C22
socleC22×C33C24
prime radicalC2×C3×D4×C3⋊D3C2
hypercenterC2×C3×D4D32
hypocenterC32C2×C3×#124
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3797C2semidirect
3. C32×#936C2semidirect
4. C3×#63974C2semidirect
5. C3×#63953C2semidirect
6. C3×C4×#624C2semidirect
7. C3×#63864C2semidirect
9. #10595170C3direct (N × H)
16. C2×C32×Dic6C22semidirect
17. C2×C3×#627C22semidirect
21. C32×#217C22semidirect
22. C32×C4×Dic3C22semidirect
23. C3×#3722C22semidirect
24. C3×Dic32C22semidirect
29. C2×#3797C2×C3semidirect
33. C3×#936D3semidirect
34. C3×#936C2×C3semidirect
39. #63974C2×C3semidirect
40. #63953C2×C3semidirect
41. C4×#624C2×C3semidirect
42. #63864C2×C3semidirect
88. C2×C3×Dic6C2×D3semidirect
89. C2×C3×Dic6C22×C3semidirect
90. C2×#627C22×C3semidirect
97. C3×#217C2×D3semidirect
98. C3×#217C22×C3semidirect
103. C3×C4×Dic3C2×D3semidirect
104. C3×C4×Dic3C22×C3semidirect
105. #3722C22×C3semidirect
106. Dic32C22×C3semidirect
132. #936C3×D3semidirect
199. C3×C3⋊D3#124semidirect
231. C2×Dic6C2×C3×D3semidirect
240. #217C2×C3×D3semidirect
244. C4×Dic3C2×C3×D3semidirect
298. C33C2×#124semidirect
372. C3⋊D3C3×#124semidirect
441. C32D3×#124semidirect
442. C32C2×C3×#124semidirect
496. C3#10595170direct (N × H)
497. C3C3×D3×#124semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×C3×#37972
C32×#9362
C3×#639742
C3×#639532
C3×C4×#6242
C3×#638642
C3×D3×#1243
#105951703