Finite Group Databasebeta

← C2×#67×C3⋊Dic3 #13178044

Subgroups — C2×#67×C3⋊Dic3 #13178044

Order 1728.

3084 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C22×C32×#67C211yesmaximal, characteristic
3C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×#88864C211yesmaximal, characteristic, complemented
5#67×C3⋊Dic3C218yesmaximal, direct product factor
6C2×Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
7C2×(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
8C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
9C2×D4×C3⋊Dic3—31nomaximal
10C2×Dic3×#67—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C32×D3C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3×C3⋊Dic3—24no—
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C3×C3⋊Dic3C2214yescomplemented
19C2×C32×#67C418yescomplemented
20C2×C32×#67C2214yes—
21C22×C32×Dic3C2211yescharacteristic
22C2×D3×C3⋊Dic3—24no—
23C2×D3×C3⋊Dic3C2214yescomplemented
24#88864C2214yescomplemented
25Dic3⋊(C3⋊Dic3)C2214yescomplemented
26(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
27Dic3×C3⋊Dic3C2214yescomplemented
28C22×C32×D4—31no—
29C23×C3⋊Dic3—31no—
30C23×C3⋊Dic3D311yescharacteristic, complemented
31C22×C3×#67D314yes—
32C23×C3×Dic3—34no—
33C22×D3×Dic3—34no—
34C2×#3757—31no—
35C2×#3757—31no—
36C2×#3757—34no—
37C2×#3757—64no—
38D4×C3⋊Dic3—38no—
39C2×C4⋊(C3⋊Dic3)—31no—
40C2×C4×C3⋊Dic3—31no—
41Dic3×#67—332no—
42C2×Dic3⋊Dic3—34no—
43C2×(C2×D3)⋊Dic3—34no—
44C2×Dic32—34no—
45C23×C33—22no—
46C23×C33—22no—
47C23×C33C2×C414yescomplemented
48C23×C33C2312yes—
49C23×C33C2311yescharacteristic
50C22×C32×D3—22no—
51C22×C32×D3—22no—
52C22×C32×D3C2×C414yescomplemented
53C22×C32×D3C2312yes—
54C2×C3⋊(C3⋊Dic3)—22no—
55C2×C3⋊(C3⋊Dic3)—22no—
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C2×C3⋊(C3⋊Dic3)C2312yes—
58C2×C3×C3⋊Dic3—22no—
59C2×C3×C3⋊Dic3—28no—
60C2×C3×C3⋊Dic3D414yescomplemented
61C2×C3×C3⋊Dic3C2312yes—
62C2×C3×C3⋊Dic3C2312yes—
63C32×#67C2×C4116yescomplemented
64C2×C32×Dic3C2×C414yescomplemented
65C2×C32×Dic3C2312yes—
66D3×C3⋊Dic3—28no—
67C2×D4×Dic3—94no—
68C2×C4×#67—91no—
69C24×C32—24no—
70C24×C32—31no—
71C24×C32C2×D311yescharacteristic
72C24×C32C2×D314yes—
73C23×C3×D3C2×D314yes—
74C2×C32×D4—38no—
75C2×C32×D4—34no—
76C22×C32×C4—31no—
77C22×C3⋊Dic3—24no—
78C22×C3⋊Dic3—31no—
79C22×C3⋊Dic3—31no—
80C22×C3⋊Dic3—34no—
81C22×C3⋊Dic3—34no—
82C22×C3⋊Dic3—34no—
83C22×C3⋊Dic3—64no—
84C22×C3⋊Dic3—64no—
85C22×C3⋊Dic3—64no—
86C22×C3⋊Dic3C2×D311yescharacteristic
87C22×C3⋊Dic3C2×D311yescharacteristic
88C22×C3⋊Dic3C2×D314yescomplemented
89C2×C3×#67Dic3132yescomplemented
90C2×C3×#67C2×D3116yes—
91C22×C3×Dic3—34no—
92C22×C3×Dic3—34no—
93C22×C3×Dic3—316no—
94C22×C3×Dic3—616no—
95C22×C3×Dic3C2×D314yes—
96C2×D3×Dic3—316no—
97C2×D3×Dic3—616no—
98#3757—34no—
99#3757—316no—
100#3757—34no—
101#3757—616no—
102C4⋊(C3⋊Dic3)—34no—
103C4×C3⋊Dic3—34no—
104Dic3⋊Dic3—316no—
105(C2×D3)⋊Dic3—316no—
106Dic32—316no—
107C22×C33—28no—
108C22×C33—22no—
109C22×C33—22no—
110C22×C33C22×C412yes—
111C22×C33C22×C414yescomplemented
112C22×C33C2×D412yes—
113C22×C33#4112yes—
114C22×C33C2411yescharacteristic
115C2×C32×D3—28no—
116C2×C32×D3—22no—
117C2×C32×D3—22no—
118C2×C32×D3C22×C414yescomplemented
119C3⋊(C3⋊Dic3)—22no—
120C3⋊(C3⋊Dic3)—22no—
121C3×C3⋊Dic3—22no—
122C3×C3⋊Dic3C2×D414yescomplemented
123C32×Dic3C22×C414yescomplemented
124C22×C3×D4—34no—
125C23×C3×C4—91no—
126C22×#67C3⋊D311yescharacteristic
127C23×Dic3—34no—
128C23×Dic3—94no—
129C22×C4×D3—91no—
130C2×#217—94no—
131C2×#217—94no—
132C2×#217—91no—
133C2×#217—184no—
134D4×Dic3—932no—
135C4×#67—98no—
136C2×(C2×D3)⋊C4—91no—
137C2×C4⋊Dic3—94no—
138C2×Dic3⋊C4—91no—
139C2×C4×Dic3—94no—
140C2×C4×Dic3—91no—
141C23×C32—22no—
142C23×C32—22no—
143C23×C32—216no—
144C23×C32—24no—
145C23×C32—28no—
146C23×C32—28no—
147C23×C32—28no—
148C23×C32—34no—
149C23×C32—32no—
150C23×C32—48no—
151C23×C32—48no—
152C23×C32—62no—
153C23×C32—62no—
154C23×C32C4×D314yescomplemented
155C23×C32C2×Dic3116yescomplemented
156C23×C32C22×D318yes—
157C23×C32C22×D314yes—
158C23×C32C22×D312yes—
159C23×C32C22×D311yescharacteristic
160C22×C3×D3—28no—
161C22×C3×D3—28no—
162C22×C3×D3C2×Dic3116yescomplemented
163C22×C3×D3C22×D318yes—
164C32×D4—316no—
165C2×C32×C4—34no—
166C2×C32×C4—32no—
167C2×C3⋊Dic3—22no—
168C2×C3⋊Dic3—28no—
169C2×C3⋊Dic3—32no—
170C2×C3⋊Dic3—32no—
171C2×C3⋊Dic3—38no—
172C2×C3⋊Dic3—38no—
173C2×C3⋊Dic3—68no—
174C2×C3⋊Dic3—68no—
175C2×C3⋊Dic3—62no—
176C2×C3⋊Dic3—62no—
177C2×C3⋊Dic3—68no—
178C2×C3⋊Dic3—68no—
179C2×C3⋊Dic3—68no—
180C2×C3⋊Dic3—128no—
181C2×C3⋊Dic3—128no—
182C2×C3⋊Dic3#6714yesdirect product factor
183C2×C3⋊Dic3C22×D312yes—
184C2×C3⋊Dic3C22×D312yes—
185C3×#67C2×Dic3164yescomplemented
186C2×C3×Dic3—316no—
187C2×C3×Dic3—38no—
188C2×C3×Dic3—38no—
189C2×C3×Dic3—632no—
190C2×C3×Dic3—68no—
191C2×C3×Dic3C2×Dic3116yescomplemented
192C2×C3×Dic3C22×D318yes—
193D3×Dic3—632no—
194C2×C4×D4—271noSylow
195C2×C33—24no—
196C2×C33C4×D414yescomplemented
197C2×C33C23×C411yescharacteristic
198C2×C33C22×D411yescharacteristic
199C2×C33C2×#4111yescharacteristic
200C32×D3—24no—
201C24×C3—24no—
202C24×C3—34no—
203C24×C3D3214yes—
204C24×C3C2×C3⋊D311yescharacteristic
205C23×D3C2×C3⋊D311yescharacteristic
206C2×C3×D4—332no—
207C2×C3×D4—316no—
208C22×C3×C4—34no—
209C22×C3×C4—91no—
210C22×C3×C4—91no—
211C22×C3×C4—94no—
212C22×C3×C4—184no—
213C2×#67C3⋊Dic318yesdirect product factor
214C2×#67C2×C3⋊D314yes—
215C22×Dic3—34no—
216C22×Dic3—34no—
217C22×Dic3—316no—
218C22×Dic3—616no—
219C22×Dic3—916no—
220C22×Dic3—94no—
221C22×Dic3—94no—
222C22×Dic3—91no—
223C22×Dic3—91no—
224C22×Dic3—184no—
225C22×Dic3—184no—
226C22×Dic3—1816no—
227C22×Dic3C2×C3⋊D311yescharacteristic
228C2×C4×D3—94no—
229C2×C4×D3—184no—
230#217—916no—
231#217—916no—
232#217—94no—
233#217—1816no—
234(C2×D3)⋊C4—94no—
235C4⋊Dic3—916no—
236Dic3⋊C4—94no—
237C4×Dic3—916no—
238C4×Dic3—94no—
239C22×C32—28no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—24no—
243C22×C32—28no—
244C22×C32—216no—
245C22×C32—28no—
246C22×C32—28no—
247C22×C32—232no—
248C22×C32—28no—
249C22×C32—28no—
250C22×C32—34no—
251C22×C32—48no—
252C22×C32—48no—
253C22×C32—432no—
254C22×C32—68no—
255C22×C32—62no—
256C22×C32—62no—
257C22×C32C2×C4×D312yes—
258C22×C32C2×C4×D314yescomplemented
259C22×C32#23512yes—
260C22×C32D3×D418yes—
261C22×C32#23718yes—
262C22×C32C22×Dic318yes—
263C22×C32C22×Dic3116yescomplemented
264C22×C32C2×#6712yes—
265C22×C32C23×D314yes—
266C22×C32C23×D311yescharacteristic
267C2×C3×D3—232no—
268C2×C3×D3—28no—
269C2×C3×D3—28no—
270C2×C3×D3C22×Dic3116yescomplemented
271C32×C4—34no—
272C3⋊Dic3—22no—
273C3⋊Dic3—62no—
274C3⋊Dic3—62no—
275C3⋊Dic3—68no—
276C3⋊Dic3—68no—
277C3⋊Dic3—128no—
278C3⋊Dic3—128no—
279C3⋊Dic3C2×#6714yesdirect product factor
280C3×Dic3—316no—
281C3×Dic3—68no—
282C3×Dic3C22×Dic3116yescomplemented
283C22×D4—31no—
284C23×C4—91no—
285C23×C4—271no—
286C4×D4—278no—
287C2×C4⋊C4—271no—
288C2×C22⋊C4—271no—
289C2×C22⋊C4—271no—
290C2×C42—271no—
291C33C2×C4×D411yescharacteristic, complemented, Sylow
292C23×C3—28no—
293C23×C3—28no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—216no—
297C23×C3—28no—
298C23×C3—24no—
299C23×C3—38no—
300C23×C3—316no—
301C23×C3—48no—
302C23×C3—48no—
303C23×C3—68no—
304C23×C3—68no—
305C23×C3D3×Dic3116yescomplemented
306C23×C3C2×C3⋊Dic314yescomplemented
307C23×C3C2×D3218yes—
308C23×C3C2×D3214yes—
309C23×C3C22×C3⋊D312yes—
310C23×C3C22×C3⋊D311yescharacteristic
311C22×D3—22no—
312C22×D3—22no—
313C22×D3C2×C3⋊Dic314yescomplemented
314C22×D3C22×C3⋊D312yes—
315C3×D4—364no—
316C2×C3×C4—38no—
317C2×C3×C4—316no—
318C2×C3×C4—92no—
319C2×C3×C4—94no—
320C2×C3×C4—92no—
321C2×C3×C4—188no—
322C2×C3×C4—182no—
323#67C2×C3⋊Dic3116yesdirect product factor
324C2×Dic3—38no—
325C2×Dic3—316no—
326C2×Dic3—38no—
327C2×Dic3—632no—
328C2×Dic3—68no—
329C2×Dic3—98no—
330C2×Dic3—98no—
331C2×Dic3—92no—
332C2×Dic3—92no—
333C2×Dic3—188no—
334C2×Dic3—188no—
335C2×Dic3—1832no—
336C2×Dic3—188no—
337C2×Dic3—188no—
338C2×Dic3—182no—
339C2×Dic3—182no—
340C2×Dic3—368no—
341C2×Dic3—368no—
342C2×Dic3C2×C3⋊Dic314yescomplemented
343C2×Dic3C22×C3⋊D312yes—
344C4×D3—188no—
345C2×C32—24no—
346C2×C32—24no—
347C2×C32—24no—
348C2×C32—24no—
349C2×C32—216no—
350C2×C32—216no—
351C2×C32—416no—
352C2×C32—64no—
353C2×C32C4×#6714yescomplemented
354C2×C32D4×Dic3116yescomplemented
355C2×C32C22×C4×D311yescharacteristic
356C2×C32C2×#23511yescharacteristic
357C2×C32C2×D3×D414yes—
358C2×C32C2×#23714yes—
359C2×C32C23×Dic314yes—
360C2×C32C22×#6711yescharacteristic
361C3×D3—216no—
362C24—31no—
363C24D3×C3⋊D311yescharacteristic
364C2×D4—38no—
365C2×D4—34no—
366C22×C4—31no—
367C22×C4—91no—
368C22×C4—94no—
369C22×C4—91no—
370C22×C4—184no—
371C22×C4—274no—
372C22×C4—271no—
373C22×C4—271no—
374C22×C4—544no—
375C4⋊C4—274no—
376C22⋊C4—274no—
377C22⋊C4—274no—
378C42—274no—
379C22×C3—28no—
380C22×C3—28no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3—28no—
384C22×C3—232no—
385C22×C3—24no—
386C22×C3—28no—
387C22×C3—216no—
388C22×C3—28no—
389C22×C3—28no—
390C22×C3—316no—
391C22×C3—48no—
392C22×C3—48no—
393C22×C3—432no—
394C22×C3—68no—
395C22×C3—68no—
396C22×C3—632no—
397C22×C3C2×D3×Dic318yes—
398C22×C3C2×D3×Dic3116yescomplemented
399C22×C3#380418yes—
400C22×C3D3×#6718yes—
401C22×C3D4×C3⋊D312yes—
402C22×C3#383012yes—
403C22×C3C22×C3⋊Dic312yes—
404C22×C3C22×C3⋊Dic314yescomplemented
405C22×C3C22×D3214yes—
406C22×C3C23×C3⋊D311yescharacteristic
407C2×D3—22no—
408C2×D3—22no—
409C2×D3—28no—
410C2×D3C22×C3⋊Dic314yescomplemented
411C3×C4—316no—
412C3×C4—94no—
413C3×C4—182no—
414Dic3—316no—
415Dic3—68no—
416Dic3—188no—
417Dic3—188no—
418Dic3—182no—
419Dic3—182no—
420Dic3—368no—
421Dic3—368no—
422Dic3C22×C3⋊Dic314yescomplemented
423C32—24no—
424C32C2×C4×#6711yescharacteristic, complemented
425C32C2×D4×Dic314yescomplemented
426C23—22no—
427C23—22no—
428C23—32no—
429C23—34no—
430C23—62no—
431C23—62no—
432C23D3×C3⋊Dic314yescomplemented
433C23C2×D3×C3⋊D312yes—
434C23C2×D3×C3⋊D311yescharacteristic
435D4—316no—
436C2×C4—32no—
437C2×C4—34no—
438C2×C4—92no—
439C2×C4—94no—
440C2×C4—92no—
441C2×C4—188no—
442C2×C4—182no—
443C2×C4—272no—
444C2×C4—272no—
445C2×C4—548no—
446C2×C4—542no—
447C2×C4—542no—
448C2×C3—216no—
449C2×C3—24no—
450C2×C3—24no—
451C2×C3—24no—
452C2×C3—24no—
453C2×C3—216no—
454C2×C3—416no—
455C2×C3—616no—
456C2×C3Dic3×#67116yescomplemented
457C2×C3D4×C3⋊Dic314yescomplemented
458C2×C3C22×D3×Dic314yes—
459C2×C3C2×#380414yes—
460C2×C3C2×D3×#6714yes—
461C2×C3C2×D4×C3⋊D311yescharacteristic
462C2×C3C2×#383011yescharacteristic
463C2×C3C23×C3⋊Dic311yescharacteristic
464D3—24no—
465C22—22no—
466C22—22no—
467C22—28no—
468C22—34no—
469C22—62no—
470C22—62no—
471C22—68no—
472C22C2×D3×C3⋊Dic312yes—
473C22C2×D3×C3⋊Dic314yescomplemented
474C22#8902812yes—
475C22C3⋊D3×#6712yes—
476C22C22×D3×C3⋊D311yescharacteristic
477C4—34no—
478C4—94no—
479C4—182no—
480C4—542no—
481C4—542no—
482C3—24no—
483C3C2×Dic3×#6714yescomplemented
484C3C2×D4×C3⋊Dic311yescharacteristic, complemented
485C2—24no—
486C2—64no—
487C2#67×C3⋊Dic314yesdirect product factor
488C2C22×D3×C3⋊Dic311yescharacteristic
489C2C2×#8902811yescharacteristic
490C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutatorC2×C33C23×C4
FrattiniC22C22×D3×C3⋊D3
FittingC24×C33C22
socleC23×C33C23
prime radicalC22×C32×#67C2
cosocleC22×C33C24
hypercenterC24D3×C3⋊D3
hypocenterC33C2×C4×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C23×C3×C3⋊Dic3C2semidirect
3. C22×D3×C3⋊Dic3C2semidirect
4. C2×#88864C2semidirect
5. #67×C3⋊Dic3C2direct (N × H)
6. C2×Dic3⋊(C3⋊Dic3)C2semidirect
7. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
8. C2×Dic3×C3⋊Dic3C2semidirect
18. C22×C3×C3⋊Dic3C22semidirect
19. C2×C32×#67C4semidirect
23. C2×D3×C3⋊Dic3C22semidirect
24. #88864C22semidirect
25. Dic3⋊(C3⋊Dic3)C22semidirect
26. (C2×D3)⋊(C3⋊Dic3)C22semidirect
27. Dic3×C3⋊Dic3C22semidirect
30. C23×C3⋊Dic3D3semidirect
47. C23×C33C2×C4semidirect
52. C22×C32×D3C2×C4semidirect
60. C2×C3×C3⋊Dic3D4semidirect
63. C32×#67C2×C4semidirect
64. C2×C32×Dic3C2×C4semidirect
88. C22×C3⋊Dic3C2×D3semidirect
89. C2×C3×#67Dic3semidirect
111. C22×C33C22×C4semidirect
118. C2×C32×D3C22×C4semidirect
122. C3×C3⋊Dic3C2×D4semidirect
123. C32×Dic3C22×C4semidirect
154. C23×C32C4×D3semidirect
155. C23×C32C2×Dic3semidirect
162. C22×C3×D3C2×Dic3semidirect
182. C2×C3⋊Dic3#67direct (N × H)
185. C3×#67C2×Dic3semidirect
191. C2×C3×Dic3C2×Dic3semidirect
196. C2×C33C4×D4semidirect
213. C2×#67C3⋊Dic3direct (N × H)
258. C22×C32C2×C4×D3semidirect
263. C22×C32C22×Dic3semidirect
270. C2×C3×D3C22×Dic3semidirect
279. C3⋊Dic3C2×#67direct (N × H)
282. C3×Dic3C22×Dic3semidirect
291. C33C2×C4×D4semidirect
305. C23×C3D3×Dic3semidirect
306. C23×C3C2×C3⋊Dic3semidirect
313. C22×D3C2×C3⋊Dic3semidirect
323. #67C2×C3⋊Dic3direct (N × H)
342. C2×Dic3C2×C3⋊Dic3semidirect
353. C2×C32C4×#67semidirect
354. C2×C32D4×Dic3semidirect
398. C22×C3C2×D3×Dic3semidirect
404. C22×C3C22×C3⋊Dic3semidirect
410. C2×D3C22×C3⋊Dic3semidirect
422. Dic3C22×C3⋊Dic3semidirect
424. C32C2×C4×#67semidirect
425. C32C2×D4×Dic3semidirect
432. C23D3×C3⋊Dic3semidirect
456. C2×C3Dic3×#67semidirect
457. C2×C3D4×C3⋊Dic3semidirect
473. C22C2×D3×C3⋊Dic3semidirect
483. C3C2×Dic3×#67semidirect
484. C3C2×D4×C3⋊Dic3semidirect
487. C2#67×C3⋊Dic3direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C3⋊Dic32
C22×C32×#672
C22×D3×C3⋊Dic32
C2×#888642
#67×C3⋊Dic32
C2×Dic3⋊(C3⋊Dic3)2
C2×(C2×D3)⋊(C3⋊Dic3)2
C2×Dic3×C3⋊Dic32
C2×D4×C3⋊Dic33
C2×Dic3×#673