Finite Group Databasebeta

← C2×#67⋊(C3⋊Dic3) #13178063

Subgroups — C2×#67⋊(C3⋊Dic3) #13178063

Order 1728.

3500 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C22×C32×#67C211yesmaximal, characteristic
3C22×D3×C3⋊Dic3C211yesmaximal, characteristic, complemented
4C2×C3×#3757C211yesmaximal, characteristic, complemented
5#67⋊(C3⋊Dic3)C218yesmaximal, 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×#67⋊Dic3—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C32×D3C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)—24no—
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3⋊(C3⋊Dic3)C2214yescomplemented
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C3×C3⋊Dic3C2211yescharacteristic
19C2×C32×#67C418yescomplemented
20C2×C32×#67C2214yes—
21C22×C32×Dic3C2211yescharacteristic
22C2×D3×C3⋊Dic3—24no—
23C2×D3×C3⋊Dic3C2214yescomplemented
24C3×#3757C2214yescomplemented
25Dic3⋊(C3⋊Dic3)C2214yescomplemented
26(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
27Dic3×C3⋊Dic3C2214yescomplemented
28C22×C32×D4—31no—
29C23×C3⋊Dic3—31no—
30C23×C3⋊Dic3—31no—
31C23×C3⋊Dic3—34no—
32C23×C3⋊Dic3—64no—
33C22×C3×#67D314yes—
34C22×D3×Dic3—34no—
35C2×#3757—31no—
36C2×#3757D311yescharacteristic, complemented
37D4×C3⋊Dic3—38no—
38C2×C4⋊(C3⋊Dic3)—31no—
39C2×C4×C3⋊Dic3—31no—
40C2×C3×#217—34no—
41#67⋊Dic3—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)—28no—
56C2×C3⋊(C3⋊Dic3)D414yescomplemented
57C2×C3⋊(C3⋊Dic3)C2312yes—
58C2×C3⋊(C3⋊Dic3)C2312yes—
59C2×C3×C3⋊Dic3—22no—
60C2×C3×C3⋊Dic3—22no—
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×#67⋊C4—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—31no—
78C22×C3⋊Dic3—34no—
79C22×C3⋊Dic3—31no—
80C22×C3⋊Dic3—34no—
81C22×C3⋊Dic3—34no—
82C22×C3⋊Dic3—316no—
83C22×C3⋊Dic3—34no—
84C22×C3⋊Dic3—64no—
85C22×C3⋊Dic3—64no—
86C22×C3⋊Dic3—616no—
87C22×C3⋊Dic3—64no—
88C22×C3⋊Dic3—64no—
89C22×C3⋊Dic3—616no—
90C22×C3⋊Dic3—632no—
91C22×C3⋊Dic3C2×D311yescharacteristic
92C22×C3⋊Dic3C2×D311yescharacteristic
93C2×C3×#67Dic3132yescomplemented
94C2×C3×#67C2×D3116yes—
95C22×C3×Dic3—34no—
96C22×C3×Dic3—34no—
97C22×C3×Dic3C2×D314yes—
98C2×D3×Dic3—316no—
99C2×D3×Dic3—616no—
100#3757—34no—
101#3757C2×D314yescomplemented
102C4⋊(C3⋊Dic3)—34no—
103C4×C3⋊Dic3—34no—
104C3×#217—316no—
105Dic3⋊Dic3—316no—
106(C2×D3)⋊Dic3—316no—
107Dic32—316no—
108C22×C33—28no—
109C22×C33—22no—
110C22×C33—22no—
111C22×C33C22×C412yes—
112C22×C33C22×C414yescomplemented
113C22×C33C2×D412yes—
114C22×C33#4112yes—
115C22×C33C2411yescharacteristic
116C2×C32×D3—28no—
117C2×C32×D3—22no—
118C2×C32×D3—22no—
119C2×C32×D3C22×C414yescomplemented
120C3⋊(C3⋊Dic3)—22no—
121C3⋊(C3⋊Dic3)C2×D414yescomplemented
122C3×C3⋊Dic3—22no—
123C3×C3⋊Dic3—22no—
124C32×Dic3C22×C414yescomplemented
125C22×C3×D4—34no—
126C22×#67C3⋊D311yescharacteristic
127C23×Dic3—94no—
128C23×Dic3—94no—
129C23×Dic3—91no—
130C23×Dic3—184no—
131C22×C4×D3—91no—
132C2×C3×C22⋊C4—91no—
133C2×#217—34no—
134C2×#217—94no—
135D4×Dic3—932no—
136C2×(C2×D3)⋊C4—91no—
137C2×C4⋊Dic3—94no—
138C2×Dic3⋊C4—91no—
139C2×C4×Dic3—94no—
140C2×C4×Dic3—91no—
141#67⋊C4—98no—
142C23×C32—216no—
143C23×C32—216no—
144C23×C32—28no—
145C23×C32—22no—
146C23×C32—28no—
147C23×C32—22no—
148C23×C32—216no—
149C23×C32—28no—
150C23×C32—24no—
151C23×C32—34no—
152C23×C32—32no—
153C23×C32—62no—
154C23×C32—62no—
155C23×C32C4×D314yescomplemented
156C23×C32C2×Dic3116yescomplemented
157C23×C32C22×D318yes—
158C23×C32C22×D314yes—
159C23×C32C22×D312yes—
160C23×C32C22×D311yescharacteristic
161C22×C3×D3—28no—
162C22×C3×D3—28no—
163C22×C3×D3C2×Dic3116yescomplemented
164C22×C3×D3C22×D318yes—
165C32×D4—316no—
166C2×C32×C4—34no—
167C2×C32×C4—32no—
168C2×C3⋊Dic3—22no—
169C2×C3⋊Dic3—22no—
170C2×C3⋊Dic3—34no—
171C2×C3⋊Dic3—32no—
172C2×C3⋊Dic3—32no—
173C2×C3⋊Dic3—316no—
174C2×C3⋊Dic3—38no—
175C2×C3⋊Dic3—38no—
176C2×C3⋊Dic3—616no—
177C2×C3⋊Dic3—68no—
178C2×C3⋊Dic3—68no—
179C2×C3⋊Dic3—68no—
180C2×C3⋊Dic3—68no—
181C2×C3⋊Dic3—632no—
182C2×C3⋊Dic3—664no—
183C2×C3⋊Dic3—616no—
184C2×C3⋊Dic3—68no—
185C2×C3⋊Dic3—62no—
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D312yes—
188C3×#67C2×Dic3164yescomplemented
189C2×C3×Dic3—38no—
190C2×C3×Dic3—38no—
191C2×C3×Dic3—68no—
192C2×C3×Dic3—68no—
193C2×C3×Dic3C2×Dic3116yescomplemented
194C2×C3×Dic3C22×D318yes—
195D3×Dic3—632no—
196C2×C4×D4—271noSylow
197C2×C33—24no—
198C2×C33C4×D414yescomplemented
199C2×C33C23×C411yescharacteristic
200C2×C33C22×D411yescharacteristic
201C2×C33C2×#4111yescharacteristic
202C32×D3—24no—
203C24×C3—24no—
204C24×C3—34no—
205C24×C3D3214yes—
206C24×C3C2×C3⋊D311yescharacteristic
207C23×D3C2×C3⋊D311yescharacteristic
208C2×C3×D4—332no—
209C2×C3×D4—316no—
210C22×C3×C4—34no—
211C22×C3×C4—91no—
212C22×C3×C4—91no—
213C2×#67C3⋊Dic318yescomplemented
214C2×#67C2×C3⋊D314yes—
215C22×Dic3—34no—
216C22×Dic3—34no—
217C22×Dic3—916no—
218C22×Dic3—94no—
219C22×Dic3—91no—
220C22×Dic3—94no—
221C22×Dic3—916no—
222C22×Dic3—94no—
223C22×Dic3—91no—
224C22×Dic3—184no—
225C22×Dic3—1816no—
226C22×Dic3—184no—
227C22×Dic3—1816no—
228C22×Dic3—1816no—
229C22×Dic3—184no—
230C22×Dic3—1832no—
231C22×Dic3C2×C3⋊D311yescharacteristic
232C2×C4×D3—94no—
233C2×C4×D3—184no—
234C3×C22⋊C4—94no—
235#217—316no—
236#217—916no—
237(C2×D3)⋊C4—94no—
238C4⋊Dic3—916no—
239Dic3⋊C4—94no—
240C4×Dic3—916no—
241C4×Dic3—94no—
242C22×C32—216no—
243C22×C32—216no—
244C22×C32—264no—
245C22×C32—28no—
246C22×C32—232no—
247C22×C32—28no—
248C22×C32—22no—
249C22×C32—28no—
250C22×C32—22no—
251C22×C32—24no—
252C22×C32—28no—
253C22×C32—216no—
254C22×C32—28no—
255C22×C32—28no—
256C22×C32—34no—
257C22×C32—68no—
258C22×C32—62no—
259C22×C32—62no—
260C22×C32C2×C4×D312yes—
261C22×C32C2×C4×D314yescomplemented
262C22×C32D3×D418yes—
263C22×C32D3×D412yes—
264C22×C32#23718yes—
265C22×C32#23712yes—
266C22×C32C22×Dic318yes—
267C22×C32C22×Dic3116yescomplemented
268C22×C32C23×D314yes—
269C22×C32C23×D311yescharacteristic
270C2×C3×D3—232no—
271C2×C3×D3—28no—
272C2×C3×D3—28no—
273C2×C3×D3C22×Dic3116yescomplemented
274C32×C4—34no—
275C3⋊Dic3—22no—
276C3⋊Dic3—22no—
277C3⋊Dic3—34no—
278C3⋊Dic3—316no—
279C3⋊Dic3—616no—
280C3⋊Dic3—616no—
281C3⋊Dic3—68no—
282C3⋊Dic3—62no—
283C3×Dic3—68no—
284C3×Dic3—68no—
285C3×Dic3C22×Dic3116yescomplemented
286C22×D4—31no—
287C23×C4—271no—
288C23×C4—271no—
289C4×D4—278no—
290C2×C4⋊C4—271no—
291C2×C22⋊C4—91no—
292C2×C22⋊C4—271no—
293C2×C42—271no—
294C33C2×C4×D411yescharacteristic, complemented, Sylow
295C23×C3—216no—
296C23×C3—216no—
297C23×C3—28no—
298C23×C3—28no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—216no—
302C23×C3—28no—
303C23×C3—24no—
304C23×C3—38no—
305C23×C3—316no—
306C23×C3—68no—
307C23×C3—68no—
308C23×C3D3×Dic3116yescomplemented
309C23×C3C2×C3⋊Dic314yescomplemented
310C23×C3C2×D3218yes—
311C23×C3C2×D3214yes—
312C23×C3C22×C3⋊D312yes—
313C23×C3C22×C3⋊D311yescharacteristic
314C22×D3—22no—
315C22×D3—22no—
316C22×D3C2×C3⋊Dic314yescomplemented
317C22×D3C22×C3⋊D312yes—
318C3×D4—364no—
319C2×C3×C4—38no—
320C2×C3×C4—316no—
321C2×C3×C4—92no—
322C2×C3×C4—92no—
323C2×C3×C4—182no—
324C2×C3×C4—182no—
325#67C2×C3⋊Dic3116yescomplemented
326C2×Dic3—38no—
327C2×Dic3—38no—
328C2×Dic3—68no—
329C2×Dic3—68no—
330C2×Dic3—98no—
331C2×Dic3—92no—
332C2×Dic3—916no—
333C2×Dic3—94no—
334C2×Dic3—98no—
335C2×Dic3—92no—
336C2×Dic3—188no—
337C2×Dic3—1816no—
338C2×Dic3—188no—
339C2×Dic3—1832no—
340C2×Dic3—1832no—
341C2×Dic3—188no—
342C2×Dic3—1864no—
343C2×Dic3—1816no—
344C2×Dic3—188no—
345C2×Dic3—182no—
346C2×Dic3C2×C3⋊Dic314yescomplemented
347C2×Dic3C22×C3⋊D312yes—
348C4×D3—188no—
349C2×C32—232no—
350C2×C32—24no—
351C2×C32—216no—
352C2×C32—24no—
353C2×C32—24no—
354C2×C32—24no—
355C2×C32—216no—
356C2×C32—64no—
357C2×C32#67⋊C414yescomplemented
358C2×C32D4×Dic3116yescomplemented
359C2×C32C22×C4×D311yescharacteristic
360C2×C32C2×D3×D414yes—
361C2×C32C2×D3×D411yescharacteristic
362C2×C32C2×#23714yes—
363C2×C32C2×#23711yescharacteristic
364C2×C32C23×Dic314yes—
365C3×D3—216no—
366C24—31no—
367C24D3×C3⋊D311yescharacteristic
368C2×D4—38no—
369C2×D4—34no—
370C22×C4—31no—
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—274no—
374C22×C4—271no—
375C22×C4—274no—
376C22×C4—271no—
377C22×C4—544no—
378C22×C4—544no—
379C4⋊C4—274no—
380C22⋊C4—94no—
381C22⋊C4—274no—
382C42—274no—
383C22×C3—216no—
384C22×C3—216no—
385C22×C3—264no—
386C22×C3—28no—
387C22×C3—28no—
388C22×C3—22no—
389C22×C3—22no—
390C22×C3—28no—
391C22×C3—232no—
392C22×C3—24no—
393C22×C3—28no—
394C22×C3—216no—
395C22×C3—28no—
396C22×C3—28no—
397C22×C3—316no—
398C22×C3—68no—
399C22×C3—68no—
400C22×C3—632no—
401C22×C3C2×D3×Dic318yes—
402C22×C3C2×D3×Dic3116yescomplemented
403C22×C3#380518yes—
404C22×C3#381118yes—
405C22×C3D4×C3⋊D312yes—
406C22×C3#383012yes—
407C22×C3C22×C3⋊Dic312yes—
408C22×C3C22×C3⋊Dic314yescomplemented
409C22×C3C22×D3214yes—
410C22×C3C23×C3⋊D311yescharacteristic
411C2×D3—22no—
412C2×D3—22no—
413C2×D3—28no—
414C2×D3C22×C3⋊Dic314yescomplemented
415C3×C4—316no—
416C3×C4—182no—
417C3×C4—182no—
418Dic3—68no—
419Dic3—68no—
420Dic3—916no—
421Dic3—94no—
422Dic3—1816no—
423Dic3—1816no—
424Dic3—188no—
425Dic3—182no—
426Dic3C22×C3⋊Dic314yescomplemented
427C32—24no—
428C32C2×#67⋊C411yescharacteristic, complemented
429C32C2×D4×Dic314yescomplemented
430C23—22no—
431C23—22no—
432C23—32no—
433C23—34no—
434C23—62no—
435C23—62no—
436C23D3×C3⋊Dic314yescomplemented
437C23C2×D3×C3⋊D312yes—
438C23C2×D3×C3⋊D311yescharacteristic
439D4—316no—
440C2×C4—32no—
441C2×C4—34no—
442C2×C4—92no—
443C2×C4—92no—
444C2×C4—182no—
445C2×C4—182no—
446C2×C4—272no—
447C2×C4—274no—
448C2×C4—272no—
449C2×C4—548no—
450C2×C4—548no—
451C2×C4—542no—
452C2×C3—232no—
453C2×C3—24no—
454C2×C3—216no—
455C2×C3—24no—
456C2×C3—24no—
457C2×C3—24no—
458C2×C3—216no—
459C2×C3—616no—
460C2×C3#67⋊Dic3116yescomplemented
461C2×C3D4×C3⋊Dic314yescomplemented
462C2×C3C22×D3×Dic314yes—
463C2×C3C2×#380514yes—
464C2×C3C2×#381114yes—
465C2×C3C2×D4×C3⋊D311yescharacteristic
466C2×C3C2×#383011yescharacteristic
467C2×C3C23×C3⋊Dic311yescharacteristic
468D3—24no—
469C22—22no—
470C22—22no—
471C22—28no—
472C22—34no—
473C22—62no—
474C22—62no—
475C22—68no—
476C22C2×D3×C3⋊Dic312yes—
477C22C2×D3×C3⋊Dic314yescomplemented
478C22#8902912yes—
479C22#8903912yes—
480C22C22×D3×C3⋊D311yescharacteristic
481C4—34no—
482C4—182no—
483C4—182no—
484C4—274no—
485C4—542no—
486C3—24no—
487C3C2×#67⋊Dic314yescomplemented
488C3C2×D4×C3⋊Dic311yescharacteristic, complemented
489C2—24no—
490C2—64no—
491C2#67⋊(C3⋊Dic3)14yesdirect product factor
492C2C22×D3×C3⋊Dic311yescharacteristic
493C2C2×#8902911yescharacteristic
494C2C2×#8903911yescharacteristic
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⋊Dic3)C2semidirect
3. C22×D3×C3⋊Dic3C2semidirect
4. C2×C3×#3757C2semidirect
5. #67⋊(C3⋊Dic3)C2direct (N × H)
6. C2×Dic3⋊(C3⋊Dic3)C2semidirect
7. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
8. C2×Dic3×C3⋊Dic3C2semidirect
16. C22×C3⋊(C3⋊Dic3)C22semidirect
19. C2×C32×#67C4semidirect
23. C2×D3×C3⋊Dic3C22semidirect
24. C3×#3757C22semidirect
25. Dic3⋊(C3⋊Dic3)C22semidirect
26. (C2×D3)⋊(C3⋊Dic3)C22semidirect
27. Dic3×C3⋊Dic3C22semidirect
36. C2×#3757D3semidirect
47. C23×C33C2×C4semidirect
52. C22×C32×D3C2×C4semidirect
56. C2×C3⋊(C3⋊Dic3)D4semidirect
63. C32×#67C2×C4semidirect
64. C2×C32×Dic3C2×C4semidirect
93. C2×C3×#67Dic3semidirect
101. #3757C2×D3semidirect
112. C22×C33C22×C4semidirect
119. C2×C32×D3C22×C4semidirect
121. C3⋊(C3⋊Dic3)C2×D4semidirect
124. C32×Dic3C22×C4semidirect
155. C23×C32C4×D3semidirect
156. C23×C32C2×Dic3semidirect
163. C22×C3×D3C2×Dic3semidirect
188. C3×#67C2×Dic3semidirect
193. C2×C3×Dic3C2×Dic3semidirect
198. C2×C33C4×D4semidirect
213. C2×#67C3⋊Dic3semidirect
261. C22×C32C2×C4×D3semidirect
267. C22×C32C22×Dic3semidirect
273. C2×C3×D3C22×Dic3semidirect
285. C3×Dic3C22×Dic3semidirect
294. C33C2×C4×D4semidirect
308. C23×C3D3×Dic3semidirect
309. C23×C3C2×C3⋊Dic3semidirect
316. C22×D3C2×C3⋊Dic3semidirect
325. #67C2×C3⋊Dic3semidirect
346. C2×Dic3C2×C3⋊Dic3semidirect
357. C2×C32#67⋊C4semidirect
358. C2×C32D4×Dic3semidirect
402. C22×C3C2×D3×Dic3semidirect
408. C22×C3C22×C3⋊Dic3semidirect
414. C2×D3C22×C3⋊Dic3semidirect
426. Dic3C22×C3⋊Dic3semidirect
428. C32C2×#67⋊C4semidirect
429. C32C2×D4×Dic3semidirect
436. C23D3×C3⋊Dic3semidirect
460. C2×C3#67⋊Dic3semidirect
461. C2×C3D4×C3⋊Dic3semidirect
477. C22C2×D3×C3⋊Dic3semidirect
487. C3C2×#67⋊Dic3semidirect
488. C3C2×D4×C3⋊Dic3semidirect
491. C2#67⋊(C3⋊Dic3)direct (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⋊Dic3)2
C22×C32×#672
C22×D3×C3⋊Dic32
C2×C3×#37572
#67⋊(C3⋊Dic3)2
C2×Dic3⋊(C3⋊Dic3)2
C2×(C2×D3)⋊(C3⋊Dic3)2
C2×Dic3×C3⋊Dic32
C2×D4×C3⋊Dic33
C2×#67⋊Dic33