Finite Group Databasebeta

← C2×#11723622 #13178048

Subgroups — C2×#11723622 #13178048

Order 1728.

3500 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6312C211yesmaximal, characteristic
2C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C22×#6289C211yesmaximal, characteristic, complemented
4C2×C32×#217C211yesmaximal, characteristic, complemented
5#11723622C218yesmaximal, direct product factor
6C2×#88806C211yesmaximal, characteristic, complemented
7C2×#88804C211yesmaximal, characteristic, complemented
8C2×Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
9C2×#64090—31nomaximal
10C2×#63952—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C3⋊(C3⋊D3)C2211yescharacteristic
13C2×#6312C418yescomplemented
14C2×#6312C2214yes—
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3×C3⋊Dic3—24no—
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C3×C3⋊Dic3C2211yescharacteristic
19C22×C3×C3⋊Dic3C2214yescomplemented
20C22×C32×Dic3C2211yescharacteristic
21C22×C32×Dic3C2211yescharacteristic
22C2×#6289—24no—
23C2×#6289C2214yescomplemented
24C32×#217C2214yescomplemented
25#88806C2214yescomplemented
26#88804C2214yescomplemented
27Dic3×C3⋊Dic3C2214yescomplemented
28C22×#638—31no—
29C22×#638—34no—
30C22×#638—64no—
31C23×C3⋊Dic3D311yescharacteristic, complemented
32C22×C4×C3⋊D3—31no—
33C23×C3×Dic3—34no—
34C22×#624—34no—
35C2×C32×C22⋊C4—31no—
36C2×#3752—31no—
37C2×#3750—31no—
38C2×C4×C3⋊Dic3—31no—
39#64090—38no—
40C2×C3×#217D314yescomplemented
41C2×#3724—34no—
42C2×#3722—34no—
43C2×Dic32—34no—
44#63952—332no—
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⋊D3)C2×C414yescomplemented
53C22×C3⋊(C3⋊D3)C2312yes—
54#6312C2×C4116yescomplemented
55C2×C3⋊(C3⋊Dic3)C2×C414yescomplemented
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C2×C3×C3⋊Dic3—22no—
58C2×C3×C3⋊Dic3—28no—
59C2×C3×C3⋊Dic3D414yescomplemented
60C2×C3×C3⋊Dic3C2312yes—
61C2×C3×C3⋊Dic3C2312yes—
62C2×C32×Dic3—22no—
63C2×C32×Dic3—22no—
64C2×C32×Dic3C2312yes—
65C2×C32×Dic3C2312yes—
66#6289—28no—
67C2×C4×#67—91no—
68C2×#67⋊C4—94no—
69C24×C32—24no—
70C24×C32C2×D314yes—
71C24×C32C2×D311yescharacteristic
72C23×C3⋊D3—31no—
73C23×C3⋊D3—34no—
74C23×C3⋊D3—64no—
75C22×C32×C4—31no—
76C22×C32×C4—31no—
77C2×#638—38no—
78C2×#638—332no—
79C2×#638—34no—
80C2×#638—316no—
81C2×#638—632no—
82C2×#638—616no—
83C22×C3⋊Dic3—24no—
84C22×C3⋊Dic3—31no—
85C22×C3⋊Dic3—34no—
86C22×C3⋊Dic3—64no—
87C22×C3⋊Dic3C2×D311yescharacteristic
88C22×C3⋊Dic3C2×D311yescharacteristic
89C22×C3⋊Dic3C2×D314yescomplemented
90C2×C4×C3⋊D3—34no—
91C2×C4×C3⋊D3—64no—
92C22×C3×Dic3—34no—
93C22×C3×Dic3—316no—
94C22×C3×Dic3—34no—
95C22×C3×Dic3—616no—
96C22×C3×Dic3C2×D314yes—
97C22×C3×Dic3C2×D314yes—
98C2×#624—316no—
99C2×#624—616no—
100C32×C22⋊C4—34no—
101#3752—34no—
102#3750—34no—
103C4×C3⋊Dic3—34no—
104C3×#217C2×D3116yescomplemented
105#3724—316no—
106#3722—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⋊D3)C22×C414yescomplemented
120C3⋊(C3⋊Dic3)C22×C414yescomplemented
121C3×C3⋊Dic3—22no—
122C3×C3⋊Dic3C2×D414yescomplemented
123C32×Dic3—22no—
124C32×Dic3—22no—
125C23×C3×C4—91no—
126C22×#67—94no—
127C22×#67—91no—
128C22×#67—184no—
129C23×Dic3—34no—
130C22×C4×D3—94no—
131C22×C4×D3—91no—
132C2×C3×C22⋊C4—34no—
133C2×#217C3⋊D311yescharacteristic, complemented
134C4×#67—98no—
135C2×(C2×D3)⋊C4—94no—
136C2×(C2×D3)⋊C4—91no—
137C2×Dic3⋊C4—94no—
138C2×Dic3⋊C4—91no—
139C2×C4×Dic3—94no—
140C2×C4×Dic3—91no—
141#67⋊C4—932no—
142C23×C32—22no—
143C23×C32—22no—
144C23×C32—216no—
145C23×C32—24no—
146C23×C32—28no—
147C23×C32—28no—
148C23×C32—28no—
149C23×C32—48no—
150C23×C32—48no—
151C23×C32C4×D3116yescomplemented
152C23×C32C4×D314yescomplemented
153C23×C32C22×D318yes—
154C23×C32C22×D314yes—
155C23×C32C22×D312yes—
156C23×C32C22×D311yescharacteristic
157C22×C3⋊D3—34no—
158C22×C3⋊D3—32no—
159C22×C3⋊D3—316no—
160C22×C3⋊D3—38no—
161C22×C3⋊D3—616no—
162C22×C3⋊D3—68no—
163C22×C3⋊D3—62no—
164C22×C3⋊D3—62no—
165C22×C3⋊D3—68no—
166C22×C3⋊D3—68no—
167C22×C3⋊D3—128no—
168C22×C3⋊D3—128no—
169C2×C32×C4—32no—
170C2×C32×C4—32no—
171C2×C32×C4—62no—
172C2×C32×C4—62no—
173#638—316no—
174#638—364no—
175#638—664no—
176C2×C3⋊Dic3—22no—
177C2×C3⋊Dic3—28no—
178C2×C3⋊Dic3—34no—
179C2×C3⋊Dic3—32no—
180C2×C3⋊Dic3—316no—
181C2×C3⋊Dic3—38no—
182C2×C3⋊Dic3—616no—
183C2×C3⋊Dic3—68no—
184C2×C3⋊Dic3#6714yescomplemented
185C2×C3⋊Dic3C22×D312yes—
186C2×C3⋊Dic3C22×D312yes—
187C4×C3⋊D3—68no—
188C2×C3×Dic3—28no—
189C2×C3×Dic3—28no—
190C2×C3×Dic3—316no—
191C2×C3×Dic3—38no—
192C2×C3×Dic3—38no—
193C2×C3×Dic3—632no—
194C2×C3×Dic3—68no—
195C2×C3×Dic3C22×D318yes—
196C2×C3×Dic3C22×D318yes—
197#624—632no—
198C2×C4×D4—271noSylow
199C2×C33—24no—
200C2×C33C4×D414yescomplemented
201C2×C33C23×C411yescharacteristic
202C2×C33C22×D411yescharacteristic
203C2×C33C2×#4111yescharacteristic
204C3⋊(C3⋊D3)—24no—
205C24×C3—24no—
206C24×C3D3214yes—
207C24×C3C2×C3⋊D311yescharacteristic
208C23×D3—94no—
209C23×D3—91no—
210C23×D3—184no—
211C22×C3×C4—34no—
212C22×C3×C4—34no—
213C22×C3×C4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—94no—
216C22×C3×C4—184no—
217C2×#67—932no—
218C2×#67—98no—
219C2×#67—916no—
220C2×#67—94no—
221C2×#67—1832no—
222C2×#67—1816no—
223C22×Dic3—34no—
224C22×Dic3—316no—
225C22×Dic3—34no—
226C22×Dic3—616no—
227C22×Dic3—94no—
228C22×Dic3—91no—
229C22×Dic3—184no—
230C22×Dic3C2×C3⋊D311yescharacteristic
231C22×Dic3C2×C3⋊D311yescharacteristic
232C2×C4×D3—916no—
233C2×C4×D3—94no—
234C2×C4×D3—184no—
235C2×C4×D3—1816no—
236C3×C22⋊C4—316no—
237#217C2×C3⋊D314yescomplemented
238(C2×D3)⋊C4—916no—
239(C2×D3)⋊C4—94no—
240Dic3⋊C4—916no—
241Dic3⋊C4—94no—
242C4×Dic3—916no—
243C4×Dic3—94no—
244C22×C32—28no—
245C22×C32—22no—
246C22×C32—22no—
247C22×C32—24no—
248C22×C32—28no—
249C22×C32—28no—
250C22×C32—216no—
251C22×C32—28no—
252C22×C32—232no—
253C22×C32—28no—
254C22×C32—28no—
255C22×C32—48no—
256C22×C32—48no—
257C22×C32—432no—
258C22×C32C2×C4×D318yes—
259C22×C32C2×C4×D312yes—
260C22×C32C2×C4×D3116yescomplemented
261C22×C32C2×C4×D314yescomplemented
262C22×C32#23512yes—
263C22×C32D3×D418yes—
264C22×C32#23718yes—
265C22×C32C2×#6712yes—
266C22×C32C23×D314yes—
267C22×C32C23×D311yescharacteristic
268C2×C3⋊D3—34no—
269C2×C3⋊D3—316no—
270C2×C3⋊D3—616no—
271C2×C3⋊D3—68no—
272C2×C3⋊D3—62no—
273C2×C3⋊D3—62no—
274C2×C3⋊D3—632no—
275C2×C3⋊D3—68no—
276C2×C3⋊D3—68no—
277C2×C3⋊D3—128no—
278C2×C3⋊D3—128no—
279C2×C3⋊D3—1232no—
280C32×C4—62no—
281C32×C4—62no—
282C3⋊Dic3—22no—
283C3⋊Dic3—34no—
284C3⋊Dic3—316no—
285C3⋊Dic3—616no—
286C3⋊Dic3C2×#6714yescomplemented
287C3×Dic3—28no—
288C3×Dic3—28no—
289C3×Dic3—316no—
290C3×Dic3—68no—
291C22×D4—271no—
292C23×C4—91no—
293C23×C4—271no—
294C4×D4—278no—
295C2×C4⋊C4—271no—
296C2×C22⋊C4—31no—
297C2×C22⋊C4—271no—
298C2×C42—271no—
299C33C2×C4×D411yescharacteristic, complemented, Sylow
300C23×C3—28no—
301C23×C3—28no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—216no—
305C23×C3—28no—
306C23×C3—24no—
307C23×C3—48no—
308C23×C3—48no—
309C23×C3#624116yescomplemented
310C23×C3C4×C3⋊D314yescomplemented
311C23×C3C2×D3218yes—
312C23×C3C2×D3214yes—
313C23×C3C22×C3⋊D312yes—
314C23×C3C22×C3⋊D311yescharacteristic
315C22×D3—98no—
316C22×D3—92no—
317C22×D3—916no—
318C22×D3—94no—
319C22×D3—188no—
320C22×D3—1816no—
321C22×D3—188no—
322C22×D3—188no—
323C22×D3—182no—
324C22×D3—182no—
325C22×D3—368no—
326C22×D3—368no—
327C2×C3×C4—38no—
328C2×C3×C4—38no—
329C2×C3×C4—68no—
330C2×C3×C4—68no—
331C2×C3×C4—92no—
332C2×C3×C4—94no—
333C2×C3×C4—92no—
334C2×C3×C4—188no—
335C2×C3×C4—182no—
336#67—964no—
337#67—916no—
338#67—1864no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—38no—
342C2×Dic3—316no—
343C2×Dic3—38no—
344C2×Dic3—632no—
345C2×Dic3—68no—
346C2×Dic3—98no—
347C2×Dic3—92no—
348C2×Dic3—916no—
349C2×Dic3—94no—
350C2×Dic3—188no—
351C2×Dic3—1816no—
352C2×Dic3C22×C3⋊D312yes—
353C2×Dic3C22×C3⋊D312yes—
354C4×D3—188no—
355C4×D3—1832no—
356C2×C32—24no—
357C2×C32—24no—
358C2×C32—24no—
359C2×C32—24no—
360C2×C32—216no—
361C2×C32—216no—
362C2×C32—416no—
363C2×C32#67⋊C4116yescomplemented
364C2×C32C4×#6714yescomplemented
365C2×C32C22×C4×D314yes—
366C2×C32C22×C4×D311yescharacteristic
367C2×C32C2×#23511yescharacteristic
368C2×C32C2×D3×D414yes—
369C2×C32C2×#23714yes—
370C2×C32C22×#6711yescharacteristic
371C3⋊D3—64no—
372C3⋊D3—616no—
373C3⋊D3—1216no—
374C24—271no—
375C24D3×C3⋊D311yescharacteristic
376C2×D4—278no—
377C2×D4—274no—
378C22×C4—31no—
379C22×C4—31no—
380C22×C4—91no—
381C22×C4—94no—
382C22×C4—91no—
383C22×C4—184no—
384C22×C4—274no—
385C22×C4—271no—
386C22×C4—544no—
387C4⋊C4—274no—
388C22⋊C4—34no—
389C22⋊C4—274no—
390C42—274no—
391C22×C3—28no—
392C22×C3—28no—
393C22×C3—28no—
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—232no—
397C22×C3—24no—
398C22×C3—28no—
399C22×C3—28no—
400C22×C3—216no—
401C22×C3—28no—
402C22×C3—48no—
403C22×C3—48no—
404C22×C3—432no—
405C22×C3#380418yes—
406C22×C3C2×#62418yes—
407C22×C3C2×#624116yescomplemented
408C22×C3D3×#6718yes—
409C22×C3C2×C4×C3⋊D312yes—
410C22×C3C2×C4×C3⋊D314yescomplemented
411C22×C3D4×C3⋊D312yes—
412C22×C3#383012yes—
413C22×C3C22×D3214yes—
414C22×C3C23×C3⋊D311yescharacteristic
415C2×D3—916no—
416C2×D3—94no—
417C2×D3—1816no—
418C2×D3—188no—
419C2×D3—188no—
420C2×D3—188no—
421C2×D3—182no—
422C2×D3—182no—
423C2×D3—1832no—
424C2×D3—368no—
425C2×D3—368no—
426C2×D3—3632no—
427C3×C4—68no—
428C3×C4—68no—
429C3×C4—94no—
430C3×C4—182no—
431Dic3—22no—
432Dic3—22no—
433Dic3—316no—
434Dic3—68no—
435Dic3—916no—
436Dic3—94no—
437Dic3—1816no—
438C32—24no—
439C32C2×#67⋊C414yescomplemented
440C32C2×C4×#6711yescharacteristic, complemented
441C23—22no—
442C23—22no—
443C23—272no—
444C23—274no—
445C23—542no—
446C23—542no—
447C23#628914yescomplemented
448C23C2×D3×C3⋊D312yes—
449C23C2×D3×C3⋊D311yescharacteristic
450D4—2716no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—92no—
456C2×C4—94no—
457C2×C4—92no—
458C2×C4—188no—
459C2×C4—182no—
460C2×C4—272no—
461C2×C4—274no—
462C2×C4—548no—
463C2×C3—24no—
464C2×C3—216no—
465C2×C3—24no—
466C2×C3—24no—
467C2×C3—24no—
468C2×C3—216no—
469C2×C3—416no—
470C2×C3#63952116yescomplemented
471C2×C3#6409014yescomplemented
472C2×C3C2×#380414yes—
473C2×C3C22×#62414yes—
474C2×C3C2×D3×#6714yes—
475C2×C3C22×C4×C3⋊D311yescharacteristic
476C2×C3C2×D4×C3⋊D311yescharacteristic
477C2×C3C2×#383011yescharacteristic
478D3—184no—
479D3—1816no—
480D3—3616no—
481C22—22no—
482C22—22no—
483C22—28no—
484C22—274no—
485C22—542no—
486C22—542no—
487C22—548no—
488C22#8902812yes—
489C22C2×#628912yes—
490C22C2×#628914yescomplemented
491C22C3⋊D3×#6712yes—
492C22C22×D3×C3⋊D311yescharacteristic
493C4—62no—
494C4—62no—
495C4—94no—
496C4—182no—
497C4—274no—
498C3—24no—
499C3C2×#6395214yescomplemented
500C3C2×#6409011yescharacteristic, complemented
501C2—24no—
502C2—544no—
503C2#1172362214yesdirect product factor
504C2C2×#8902811yescharacteristic
505C2C22×#628911yescharacteristic
506C2C2×C3⋊D3×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutatorC2×C33C23×C4
FrattiniC22C22×D3×C3⋊D3
FittingC24×C33C22
socleC23×C33C23
prime radicalC22×#6312C2
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
2. C23×C3×C3⋊Dic3C2semidirect
3. C22×#6289C2semidirect
4. C2×C32×#217C2semidirect
5. #11723622C2direct (N × H)
6. C2×#88806C2semidirect
7. C2×#88804C2semidirect
8. C2×Dic3×C3⋊Dic3C2semidirect
13. C2×#6312C4semidirect
19. C22×C3×C3⋊Dic3C22semidirect
23. C2×#6289C22semidirect
24. C32×#217C22semidirect
25. #88806C22semidirect
26. #88804C22semidirect
27. Dic3×C3⋊Dic3C22semidirect
31. C23×C3⋊Dic3D3semidirect
40. C2×C3×#217D3semidirect
47. C23×C33C2×C4semidirect
52. C22×C3⋊(C3⋊D3)C2×C4semidirect
54. #6312C2×C4semidirect
55. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
59. C2×C3×C3⋊Dic3D4semidirect
89. C22×C3⋊Dic3C2×D3semidirect
104. C3×#217C2×D3semidirect
112. C22×C33C22×C4semidirect
119. C2×C3⋊(C3⋊D3)C22×C4semidirect
120. C3⋊(C3⋊Dic3)C22×C4semidirect
122. C3×C3⋊Dic3C2×D4semidirect
133. C2×#217C3⋊D3semidirect
151. C23×C32C4×D3semidirect
152. C23×C32C4×D3semidirect
184. C2×C3⋊Dic3#67semidirect
200. C2×C33C4×D4semidirect
237. #217C2×C3⋊D3semidirect
260. C22×C32C2×C4×D3semidirect
261. C22×C32C2×C4×D3semidirect
286. C3⋊Dic3C2×#67semidirect
299. C33C2×C4×D4semidirect
309. C23×C3#624semidirect
310. C23×C3C4×C3⋊D3semidirect
363. C2×C32#67⋊C4semidirect
364. C2×C32C4×#67semidirect
407. C22×C3C2×#624semidirect
410. C22×C3C2×C4×C3⋊D3semidirect
439. C32C2×#67⋊C4semidirect
440. C32C2×C4×#67semidirect
447. C23#6289semidirect
470. C2×C3#63952semidirect
471. C2×C3#64090semidirect
490. C22C2×#6289semidirect
499. C3C2×#63952semidirect
500. C3C2×#64090semidirect
503. C2#11723622direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#63122
C23×C3×C3⋊Dic32
C22×#62892
C2×C32×#2172
#117236222
C2×#888062
C2×#888042
C2×Dic3×C3⋊Dic32
C2×#640903
C2×#639523