Finite Group Databasebeta

← C2×#11723626 #13178059

Subgroups — C2×#11723626 #13178059

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×C3×#638C211yesmaximal, characteristic
3C22×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
4C2×C32×#217C211yesmaximal, characteristic, complemented
5#11723626C218yesmaximal, direct product factor
6C2×#88807C211yesmaximal, characteristic, complemented
7C2×#88803C211yesmaximal, characteristic, complemented
8C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
9C2×#64090—31nomaximal
10C2×#67⋊Dic3—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C3×C3⋊D3C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)—24no—
14C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3⋊(C3⋊Dic3)C2214yescomplemented
17C2×C3×#638C418yescomplemented
18C2×C3×#638C2214yes—
19C22×C3×C3⋊Dic3C2211yescharacteristic
20C22×C32×Dic3C2211yescharacteristic
21C22×C32×Dic3C2211yescharacteristic
22C2×Dic3×C3⋊D3—24no—
23C2×Dic3×C3⋊D3C2214yescomplemented
24C32×#217C2214yescomplemented
25#88807C2214yescomplemented
26#88803C2214yescomplemented
27Dic3×C3⋊Dic3C2214yescomplemented
28C22×#638D311yescharacteristic
29C23×C3⋊Dic3—31no—
30C23×C3⋊Dic3—34no—
31C23×C3⋊Dic3—64no—
32C22×C4×C3⋊D3—31no—
33C22×C3×#67—34no—
34C22×D3×Dic3—34no—
35C2×C32×C22⋊C4—31no—
36C2×#3752—31no—
37C2×#3750—31no—
38C2×C4×C3⋊Dic3—31no—
39#64090—38no—
40C2×C3×#217D314yescomplemented
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×C3×C3⋊D3—22no—
51C22×C3×C3⋊D3—22no—
52C22×C3×C3⋊D3C2×C414yescomplemented
53C22×C3×C3⋊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—
59C3×#638C2×C4116yescomplemented
60C2×C3×C3⋊Dic3C2×C414yescomplemented
61C2×C3×C3⋊Dic3C2312yes—
62C2×C32×Dic3—22no—
63C2×C32×Dic3—22no—
64C2×C32×Dic3C2312yes—
65C2×C32×Dic3C2312yes—
66Dic3×C3⋊D3—28no—
67C2×D4×Dic3—91no—
68C2×#67⋊C4—94no—
69C24×C32—24no—
70C24×C32C2×D314yes—
71C24×C32C2×D311yescharacteristic
72C23×C3⋊D3C2×D311yescharacteristic
73C23×C3×D3—34no—
74C22×C32×C4—31no—
75C22×C32×C4—31no—
76C2×#638Dic318yescomplemented
77C2×#638C2×D314yes—
78C22×C3⋊Dic3—31no—
79C22×C3⋊Dic3—34no—
80C22×C3⋊Dic3—34no—
81C22×C3⋊Dic3—31no—
82C22×C3⋊Dic3—34no—
83C22×C3⋊Dic3—316no—
84C22×C3⋊Dic3—64no—
85C22×C3⋊Dic3—616no—
86C22×C3⋊Dic3—64no—
87C22×C3⋊Dic3—64no—
88C22×C3⋊Dic3—616no—
89C22×C3⋊Dic3—632no—
90C22×C3⋊Dic3C2×D311yescharacteristic
91C2×C4×C3⋊D3—34no—
92C2×C4×C3⋊D3—64no—
93C2×C3×#67—332no—
94C2×C3×#67—316no—
95C22×C3×Dic3—34no—
96C22×C3×Dic3C2×D314yes—
97C22×C3×Dic3C2×D314yes—
98C2×D3×Dic3—316no—
99C2×D3×Dic3—616no—
100C32×C22⋊C4—34no—
101#3752—34no—
102#3750—34no—
103C4×C3⋊Dic3—34no—
104C3×#217C2×D3116yescomplemented
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×C3×C3⋊D3—28no—
117C2×C3×C3⋊D3—22no—
118C2×C3×C3⋊D3—22no—
119C2×C3×C3⋊D3C22×C414yescomplemented
120C3⋊(C3⋊Dic3)—22no—
121C3⋊(C3⋊Dic3)C2×D414yescomplemented
122C3×C3⋊Dic3C22×C414yescomplemented
123C32×Dic3—22no—
124C32×Dic3—22no—
125C22×C3×D4—91no—
126C22×#67—34no—
127C23×Dic3—94no—
128C23×Dic3—91no—
129C23×Dic3—91no—
130C23×Dic3—184no—
131C22×C4×D3—94no—
132C2×C3×C22⋊C4—34no—
133C2×#217—91no—
134C2×#217C3⋊D311yescharacteristic, complemented
135D4×Dic3—98no—
136C2×(C2×D3)⋊C4—94no—
137C2×C4⋊Dic3—91no—
138C2×Dic3⋊C4—94no—
139C2×C4×Dic3—94no—
140C2×C4×Dic3—91no—
141#67⋊C4—932no—
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×C32C4×D3116yescomplemented
152C23×C32C2×Dic314yescomplemented
153C23×C32C22×D318yes—
154C23×C32C22×D314yes—
155C23×C32C22×D312yes—
156C23×C32C22×D311yescharacteristic
157C22×C3⋊D3—22no—
158C22×C3⋊D3—22no—
159C22×C3⋊D3C2×Dic314yescomplemented
160C22×C3⋊D3C22×D312yes—
161C22×C3×D3—316no—
162C22×C3×D3—38no—
163C22×C3×D3—68no—
164C22×C3×D3—68no—
165C2×C32×C4—32no—
166C2×C32×C4—32no—
167C2×C32×C4—62no—
168C2×C32×C4—62no—
169#638C2×Dic3116yescomplemented
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—632no—
181C2×C3⋊Dic3—664no—
182C2×C3⋊Dic3—616no—
183C2×C3⋊Dic3—68no—
184C2×C3⋊Dic3—62no—
185C2×C3⋊Dic3C2×Dic314yescomplemented
186C2×C3⋊Dic3C22×D312yes—
187C4×C3⋊D3—68no—
188C3×#67—364no—
189C2×C3×Dic3—28no—
190C2×C3×Dic3—28no—
191C2×C3×Dic3—316no—
192C2×C3×Dic3—38no—
193C2×C3×Dic3C22×D318yes—
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
202C3×C3⋊D3—24no—
203C24×C3—24no—
204C24×C3—91no—
205C24×C3D3214yes—
206C24×C3C2×C3⋊D311yescharacteristic
207C23×D3—34no—
208C2×C3×D4—98no—
209C2×C3×D4—94no—
210C22×C3×C4—34no—
211C22×C3×C4—34no—
212C22×C3×C4—91no—
213C2×#67—332no—
214C2×#67—316no—
215C22×Dic3—34no—
216C22×Dic3—94no—
217C22×Dic3—91no—
218C22×Dic3—916no—
219C22×Dic3—94no—
220C22×Dic3—91no—
221C22×Dic3—94no—
222C22×Dic3—94no—
223C22×Dic3—184no—
224C22×Dic3—1816no—
225C22×Dic3—184no—
226C22×Dic3—184no—
227C22×Dic3—1816no—
228C22×Dic3—184no—
229C22×Dic3—1832no—
230C22×Dic3C2×C3⋊D311yescharacteristic
231C22×Dic3C2×C3⋊D311yescharacteristic
232C2×C4×D3—916no—
233C2×C4×D3—1816no—
234C3×C22⋊C4—316no—
235#217—94no—
236#217C2×C3⋊D314yescomplemented
237(C2×D3)⋊C4—916no—
238C4⋊Dic3—94no—
239Dic3⋊C4—916no—
240C4×Dic3—916no—
241C4×Dic3—94no—
242C22×C32—216no—
243C22×C32—216no—
244C22×C32—264no—
245C22×C32—232no—
246C22×C32—28no—
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×C32C2×C4×D318yes—
257C22×C32C2×C4×D3116yescomplemented
258C22×C32D3×D418yes—
259C22×C32D3×D412yes—
260C22×C32#23718yes—
261C22×C32#23712yes—
262C22×C32C22×Dic312yes—
263C22×C32C22×Dic314yescomplemented
264C22×C32C23×D314yes—
265C22×C32C23×D311yescharacteristic
266C2×C3⋊D3—28no—
267C2×C3⋊D3—22no—
268C2×C3⋊D3—22no—
269C2×C3⋊D3C22×Dic314yescomplemented
270C2×C3×D3—316no—
271C2×C3×D3—632no—
272C2×C3×D3—68no—
273C2×C3×D3—68no—
274C32×C4—62no—
275C32×C4—62no—
276C3⋊Dic3—34no—
277C3⋊Dic3—316no—
278C3⋊Dic3—616no—
279C3⋊Dic3—616no—
280C3⋊Dic3—68no—
281C3⋊Dic3—62no—
282C3⋊Dic3C22×Dic314yescomplemented
283C3×Dic3—28no—
284C3×Dic3—28no—
285C3×Dic3—316no—
286C22×D4—91no—
287C23×C4—271no—
288C23×C4—271no—
289C4×D4—278no—
290C2×C4⋊C4—271no—
291C2×C22⋊C4—31no—
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—92no—
305C23×C3—94no—
306C23×C3—182no—
307C23×C3—182no—
308C23×C3D3×Dic3116yescomplemented
309C23×C3C4×C3⋊D314yescomplemented
310C23×C3C2×D3218yes—
311C23×C3C2×D3214yes—
312C23×C3C22×C3⋊D312yes—
313C23×C3C22×C3⋊D311yescharacteristic
314C22×D3—38no—
315C22×D3—316no—
316C22×D3—68no—
317C22×D3—68no—
318C3×D4—916no—
319C2×C3×C4—38no—
320C2×C3×C4—38no—
321C2×C3×C4—68no—
322C2×C3×C4—68no—
323C2×C3×C4—92no—
324C2×C3×C4—94no—
325#67—364no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—38no—
329C2×Dic3—316no—
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—188no—
340C2×Dic3—1832no—
341C2×Dic3—188no—
342C2×Dic3—1864no—
343C2×Dic3—1816no—
344C2×Dic3—188no—
345C2×Dic3—182no—
346C2×Dic3C22×C3⋊D312yes—
347C2×Dic3C22×C3⋊D312yes—
348C4×D3—1832no—
349C2×C32—232no—
350C2×C32—216no—
351C2×C32—24no—
352C2×C32—24no—
353C2×C32—24no—
354C2×C32—24no—
355C2×C32—216no—
356C2×C32#67⋊C4116yescomplemented
357C2×C32D4×Dic314yescomplemented
358C2×C32C22×C4×D314yes—
359C2×C32C2×D3×D414yes—
360C2×C32C2×D3×D411yescharacteristic
361C2×C32C2×#23714yes—
362C2×C32C2×#23711yescharacteristic
363C2×C32C23×Dic311yescharacteristic
364C3⋊D3—24no—
365C3×D3—616no—
366C24—91no—
367C24D3×C3⋊D311yescharacteristic
368C2×D4—98no—
369C2×D4—94no—
370C22×C4—31no—
371C22×C4—31no—
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—34no—
381C22⋊C4—274no—
382C42—274no—
383C22×C3—216no—
384C22×C3—216no—
385C22×C3—264no—
386C22×C3—28no—
387C22×C3—28no—
388C22×C3—28no—
389C22×C3—22no—
390C22×C3—22no—
391C22×C3—232no—
392C22×C3—24no—
393C22×C3—28no—
394C22×C3—28no—
395C22×C3—216no—
396C22×C3—28no—
397C22×C3—94no—
398C22×C3—188no—
399C22×C3—182no—
400C22×C3—182no—
401C22×C3C2×D3×Dic318yes—
402C22×C3C2×D3×Dic3116yescomplemented
403C22×C3#380518yes—
404C22×C3#381118yes—
405C22×C3C2×C4×C3⋊D312yes—
406C22×C3C2×C4×C3⋊D314yescomplemented
407C22×C3D4×C3⋊D312yes—
408C22×C3#383012yes—
409C22×C3C22×D3214yes—
410C22×C3C23×C3⋊D311yescharacteristic
411C2×D3—316no—
412C2×D3—68no—
413C2×D3—68no—
414C2×D3—632no—
415C3×C4—68no—
416C3×C4—68no—
417C3×C4—94no—
418Dic3—22no—
419Dic3—22no—
420Dic3—316no—
421Dic3—916no—
422Dic3—94no—
423Dic3—1816no—
424Dic3—1816no—
425Dic3—188no—
426Dic3—182no—
427C32—24no—
428C32C2×#67⋊C414yescomplemented
429C32C2×D4×Dic311yescharacteristic, complemented
430C23—22no—
431C23—22no—
432C23—92no—
433C23—94no—
434C23—182no—
435C23—182no—
436C23Dic3×C3⋊D314yescomplemented
437C23C2×D3×C3⋊D312yes—
438C23C2×D3×C3⋊D311yescharacteristic
439D4—916no—
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—92no—
445C2×C4—94no—
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—184no—
460C2×C3#67⋊Dic3116yescomplemented
461C2×C3#6409014yescomplemented
462C2×C3C22×D3×Dic314yes—
463C2×C3C2×#380514yes—
464C2×C3C2×#381114yes—
465C2×C3C22×C4×C3⋊D311yescharacteristic
466C2×C3C2×D4×C3⋊D311yescharacteristic
467C2×C3C2×#383011yescharacteristic
468D3—616no—
469C22—22no—
470C22—22no—
471C22—28no—
472C22—94no—
473C22—182no—
474C22—182no—
475C22—188no—
476C22#8902912yes—
477C22C2×Dic3×C3⋊D312yes—
478C22C2×Dic3×C3⋊D314yescomplemented
479C22#8903912yes—
480C22C22×D3×C3⋊D311yescharacteristic
481C4—62no—
482C4—62no—
483C4—94no—
484C4—274no—
485C4—542no—
486C3—24no—
487C3C2×#67⋊Dic314yescomplemented
488C3C2×#6409011yescharacteristic, complemented
489C2—24no—
490C2—184no—
491C2#1172362614yesdirect product factor
492C2C2×#8902911yescharacteristic
493C2C22×Dic3×C3⋊D311yescharacteristic
494C2C2×#8903911yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutatorC2×C33C23×C4
FrattiniC22C22×D3×C3⋊D3
FittingC24×C33C22
socleC23×C33C23
prime radicalC22×C3×#638C2
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×Dic3×C3⋊D3C2semidirect
4. C2×C32×#217C2semidirect
5. #11723626C2direct (N × H)
6. C2×#88807C2semidirect
7. C2×#88803C2semidirect
8. C2×Dic3×C3⋊Dic3C2semidirect
16. C22×C3⋊(C3⋊Dic3)C22semidirect
17. C2×C3×#638C4semidirect
23. C2×Dic3×C3⋊D3C22semidirect
24. C32×#217C22semidirect
25. #88807C22semidirect
26. #88803C22semidirect
27. Dic3×C3⋊Dic3C22semidirect
40. C2×C3×#217D3semidirect
47. C23×C33C2×C4semidirect
52. C22×C3×C3⋊D3C2×C4semidirect
56. C2×C3⋊(C3⋊Dic3)D4semidirect
59. C3×#638C2×C4semidirect
60. C2×C3×C3⋊Dic3C2×C4semidirect
76. C2×#638Dic3semidirect
104. C3×#217C2×D3semidirect
112. C22×C33C22×C4semidirect
119. C2×C3×C3⋊D3C22×C4semidirect
121. C3⋊(C3⋊Dic3)C2×D4semidirect
122. C3×C3⋊Dic3C22×C4semidirect
134. C2×#217C3⋊D3semidirect
151. C23×C32C4×D3semidirect
152. C23×C32C2×Dic3semidirect
159. C22×C3⋊D3C2×Dic3semidirect
169. #638C2×Dic3semidirect
185. C2×C3⋊Dic3C2×Dic3semidirect
198. C2×C33C4×D4semidirect
236. #217C2×C3⋊D3semidirect
257. C22×C32C2×C4×D3semidirect
263. C22×C32C22×Dic3semidirect
269. C2×C3⋊D3C22×Dic3semidirect
282. C3⋊Dic3C22×Dic3semidirect
294. C33C2×C4×D4semidirect
308. C23×C3D3×Dic3semidirect
309. C23×C3C4×C3⋊D3semidirect
356. C2×C32#67⋊C4semidirect
357. C2×C32D4×Dic3semidirect
402. C22×C3C2×D3×Dic3semidirect
406. C22×C3C2×C4×C3⋊D3semidirect
428. C32C2×#67⋊C4semidirect
429. C32C2×D4×Dic3semidirect
436. C23Dic3×C3⋊D3semidirect
460. C2×C3#67⋊Dic3semidirect
461. C2×C3#64090semidirect
478. C22C2×Dic3×C3⋊D3semidirect
487. C3C2×#67⋊Dic3semidirect
488. C3C2×#64090semidirect
491. C2#11723626direct (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×C3×#6382
C22×Dic3×C3⋊D32
C2×C32×#2172
#117236262
C2×#888072
C2×#888032
C2×Dic3×C3⋊Dic32
C2×#640903
C2×#67⋊Dic33