Finite Group Databasebeta

← C2×#11723637 #13178087

Subgroups — C2×#11723637 #13178087

Order 1728.

3500 conjugacy classes of subgroups.

The subgroup lattice has 544 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, complemented
3C22×C32×#67C211yesmaximal, characteristic, complemented
4C22×#6290C211yesmaximal, characteristic, complemented
5#11723637C218yesmaximal, direct product factor
6C2×#88806C211yesmaximal, characteristic, complemented
7C2×#88803C211yesmaximal, characteristic, complemented
8C2×(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
9C2×#64148—31nomaximal
10C2×#63980—34nomaximal
11C24×C33C2211yescharacteristic
12C23×C3×C3⋊D3C2211yescharacteristic
13C23×C32×D3C2211yescharacteristic
14C22×C3⋊(C3⋊Dic3)—24no—
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C22×C3⋊(C3⋊Dic3)C2214yescomplemented
18C2×C3×#638—24no—
19C2×C3×#638C2214yescomplemented
20C22×C3×C3⋊Dic3C2211yescharacteristic
21C2×C32×#67—24no—
22C2×C32×#67C2214yescomplemented
23C22×C32×Dic3C2211yescharacteristic
24C2×#6290—24no—
25C2×#6290C2214yescomplemented
26#88806C2214yescomplemented
27#88803C2214yescomplemented
28(C2×D3)⋊(C3⋊Dic3)C2214yescomplemented
29C22×C32×D4—31no—
30C22×#638—31no—
31C22×#638D311yescharacteristic, complemented
32C23×C3⋊Dic3—31no—
33C23×C3⋊Dic3—34no—
34C23×C3⋊Dic3—64no—
35C22×C3×#67—34no—
36C22×C3×#67D314yescomplemented
37C22×#625—34no—
38C2×#3757—31no—
39#64148—38no—
40C2×#3752—31no—
41C2×#3750—31no—
42#63980—332no—
43C2×#3724—34no—
44C2×(C2×D3)⋊Dic3—34no—
45C2×(C2×D3)⋊Dic3—34no—
46C23×C33—22no—
47C23×C33—22no—
48C23×C33D414yescomplemented
49C23×C33C2312yes—
50C23×C33C2311yescharacteristic
51C22×C3×C3⋊D3—22no—
52C22×C3×C3⋊D3—22no—
53C22×C3×C3⋊D3—22no—
54C22×C3×C3⋊D3C2312yes—
55C22×C32×D3—22no—
56C22×C32×D3—22no—
57C22×C32×D3—22no—
58C22×C32×D3C2312yes—
59C2×C3⋊(C3⋊Dic3)—22no—
60C2×C3⋊(C3⋊Dic3)—28no—
61C2×C3⋊(C3⋊Dic3)D414yescomplemented
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C2×C3⋊(C3⋊Dic3)C2312yes—
64C3×#638—28no—
65C2×C3×C3⋊Dic3—22no—
66C2×C3×C3⋊Dic3C2312yes—
67C32×#67—28no—
68C2×C32×Dic3—22no—
69C2×C32×Dic3C2312yes—
70#6290—28no—
71C2×#939—94no—
72C2×#939—91no—
73C24×C32—24no—
74C24×C32—31no—
75C24×C32C2×D314yes—
76C24×C32C2×D311yescharacteristic
77C23×C3⋊D3C2×D311yescharacteristic
78C23×C3×D3—34no—
79C23×C3×D3C2×D314yes—
80C2×C32×D4—34no—
81C2×C32×D4—64no—
82C22×C32×C4—31no—
83C2×#638—24no—
84C2×#638—34no—
85C2×#638—64no—
86C2×#638C2×D314yescomplemented
87C22×C3⋊Dic3—31no—
88C22×C3⋊Dic3—34no—
89C22×C3⋊Dic3—31no—
90C22×C3⋊Dic3—34no—
91C22×C3⋊Dic3—34no—
92C22×C3⋊Dic3—316no—
93C22×C3⋊Dic3—64no—
94C22×C3⋊Dic3—616no—
95C22×C3⋊Dic3—64no—
96C22×C3⋊Dic3—64no—
97C22×C3⋊Dic3—616no—
98C22×C3⋊Dic3—632no—
99C22×C3⋊Dic3C2×D311yescharacteristic
100C2×C3×#67—216no—
101C2×C3×#67—316no—
102C2×C3×#67—616no—
103C2×C3×#67C2×D3116yescomplemented
104C22×C3×Dic3—34no—
105C22×C3×Dic3C2×D314yes—
106C2×#625—316no—
107C2×#625—616no—
108#3757—34no—
109#3752—34no—
110#3750—34no—
111#3724—316no—
112(C2×D3)⋊Dic3—316no—
113(C2×D3)⋊Dic3—316no—
114C22×C33—28no—
115C22×C33—22no—
116C22×C33—22no—
117C22×C33C2×D412yes—
118C22×C33C2×D412yes—
119C22×C33C2×D414yescomplemented
120C22×C33#4112yes—
121C22×C33C2411yescharacteristic
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—22no—
125C2×C3×C3⋊D3—44no—
126C2×C32×D3—22no—
127C2×C32×D3—22no—
128C2×C32×D3—22no—
129C2×C32×D3—44no—
130C3⋊(C3⋊Dic3)—22no—
131C3⋊(C3⋊Dic3)C2×D414yescomplemented
132C3×C3⋊Dic3—22no—
133C32×Dic3—22no—
134C22×C3×D4—34no—
135C22×C3×D4—91no—
136C22×#67—34no—
137C22×#67—94no—
138C22×#67—91no—
139C22×#67C3⋊D311yescharacteristic, complemented
140C23×Dic3—94no—
141C23×Dic3—91no—
142C23×Dic3—184no—
143C2×#217—94no—
144C2×#217—91no—
145#939—932no—
146#939—98no—
147C2×(C2×D3)⋊C4—94no—
148C2×(C2×D3)⋊C4—91no—
149C2×Dic3⋊C4—94no—
150C2×Dic3⋊C4—91no—
151C23×C32—216no—
152C23×C32—216no—
153C23×C32—28no—
154C23×C32—22no—
155C23×C32—28no—
156C23×C32—22no—
157C23×C32—216no—
158C23×C32—28no—
159C23×C32—24no—
160C23×C32—32no—
161C23×C32—62no—
162C23×C32—62no—
163C23×C32—62no—
164C23×C32#67116yescomplemented
165C23×C32#6714yescomplemented
166C23×C32C22×D318yes—
167C23×C32C22×D314yes—
168C23×C32C22×D312yes—
169C23×C32C22×D311yescharacteristic
170C22×C3⋊D3—22no—
171C22×C3⋊D3—22no—
172C22×C3⋊D3—22no—
173C22×C3⋊D3C22×D312yes—
174C22×C3×D3—28no—
175C22×C3×D3—28no—
176C22×C3×D3—28no—
177C22×C3×D3—38no—
178C22×C3×D3—68no—
179C22×C3×D3—68no—
180C22×C3×D3—68no—
181C22×C3×D3C22×D318yes—
182C32×D4—68no—
183C2×C32×C4—32no—
184C2×C32×C4—62no—
185#638—28no—
186#638—68no—
187C2×C3⋊Dic3—22no—
188C2×C3⋊Dic3—34no—
189C2×C3⋊Dic3—32no—
190C2×C3⋊Dic3—32no—
191C2×C3⋊Dic3—316no—
192C2×C3⋊Dic3—38no—
193C2×C3⋊Dic3—38no—
194C2×C3⋊Dic3—616no—
195C2×C3⋊Dic3—68no—
196C2×C3⋊Dic3—68no—
197C2×C3⋊Dic3—68no—
198C2×C3⋊Dic3—632no—
199C2×C3⋊Dic3—664no—
200C2×C3⋊Dic3—616no—
201C2×C3⋊Dic3—68no—
202C2×C3⋊Dic3—62no—
203C2×C3⋊Dic3C22×D312yes—
204C3×#67—232no—
205C3×#67—632no—
206C2×C3×Dic3—28no—
207C2×C3×Dic3—38no—
208C2×C3×Dic3—68no—
209C2×C3×Dic3C22×D318yes—
210#625—632no—
211C2×#121—271noSylow
212C2×C33—24no—
213C2×C33#12114yescomplemented
214C2×C33C22×D411yescharacteristic
215C2×C33C22×D411yescharacteristic
216C2×C33C2×#4111yescharacteristic
217C3×C3⋊D3—42no—
218C32×D3—42no—
219C24×C3—24no—
220C24×C3—34no—
221C24×C3—91no—
222C24×C3D3214yes—
223C24×C3C2×C3⋊D311yescharacteristic
224C23×D3—34no—
225C23×D3C2×C3⋊D311yescharacteristic
226C2×C3×D4—316no—
227C2×C3×D4—616no—
228C2×C3×D4—94no—
229C2×C3×D4—184no—
230C22×C3×C4—34no—
231C22×C3×C4—91no—
232C2×#67—24no—
233C2×#67—316no—
234C2×#67—616no—
235C2×#67—916no—
236C2×#67—94no—
237C2×#67—184no—
238C2×#67—1816no—
239C2×#67C2×C3⋊D314yescomplemented
240C22×Dic3—34no—
241C22×Dic3—94no—
242C22×Dic3—91no—
243C22×Dic3—94no—
244C22×Dic3—916no—
245C22×Dic3—91no—
246C22×Dic3—94no—
247C22×Dic3—184no—
248C22×Dic3—1816no—
249C22×Dic3—184no—
250C22×Dic3—1816no—
251C22×Dic3—184no—
252C22×Dic3—1832no—
253C22×Dic3C2×C3⋊D311yescharacteristic
254#217—916no—
255#217—94no—
256(C2×D3)⋊C4—916no—
257(C2×D3)⋊C4—94no—
258Dic3⋊C4—916no—
259Dic3⋊C4—94no—
260C22×C32—216no—
261C22×C32—216no—
262C22×C32—264no—
263C22×C32—232no—
264C22×C32—28no—
265C22×C32—28no—
266C22×C32—22no—
267C22×C32—28no—
268C22×C32—22no—
269C22×C32—24no—
270C22×C32—28no—
271C22×C32—28no—
272C22×C32—216no—
273C22×C32—28no—
274C22×C32—62no—
275C22×C32—62no—
276C22×C32—62no—
277C22×C32—124no—
278C22×C32D3×D418yes—
279C22×C32D3×D412yes—
280C22×C32#23718yes—
281C22×C32#23712yes—
282C22×C32C2×#6718yes—
283C22×C32C2×#6712yes—
284C22×C32C2×#67116yescomplemented
285C22×C32C2×#6714yescomplemented
286C22×C32C23×D314yes—
287C22×C32C23×D311yescharacteristic
288C2×C3⋊D3—22no—
289C2×C3⋊D3—22no—
290C2×C3⋊D3—22no—
291C2×C3⋊D3—44no—
292C2×C3×D3—28no—
293C2×C3×D3—28no—
294C2×C3×D3—28no—
295C2×C3×D3—416no—
296C2×C3×D3—68no—
297C2×C3×D3—68no—
298C2×C3×D3—68no—
299C2×C3×D3—1216no—
300C32×C4—62no—
301C3⋊Dic3—22no—
302C3⋊Dic3—34no—
303C3⋊Dic3—316no—
304C3⋊Dic3—616no—
305C3⋊Dic3—616no—
306C3⋊Dic3—68no—
307C3⋊Dic3—62no—
308C3×Dic3—28no—
309C3×Dic3—68no—
310C22×D4—31no—
311C22×D4—91no—
312C22×D4—271no—
313C23×C4—271no—
314#121—278no—
315C2×C4⋊C4—271no—
316C2×C22⋊C4—271no—
317C2×C22⋊C4—271no—
318C33C2×#12111yescharacteristic, complemented, Sylow
319C23×C3—216no—
320C23×C3—216no—
321C23×C3—28no—
322C23×C3—28no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—216no—
326C23×C3—28no—
327C23×C3—24no—
328C23×C3—38no—
329C23×C3—68no—
330C23×C3—68no—
331C23×C3—68no—
332C23×C3—92no—
333C23×C3—182no—
334C23×C3—182no—
335C23×C3—182no—
336C23×C3#625116yescomplemented
337C23×C3#63814yescomplemented
338C23×C3C2×D3218yes—
339C23×C3C2×D3214yes—
340C23×C3C22×C3⋊D312yes—
341C23×C3C22×C3⋊D311yescharacteristic
342C22×D3—22no—
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—38no—
346C22×D3—68no—
347C22×D3—68no—
348C22×D3—68no—
349C22×D3C22×C3⋊D312yes—
350C3×D4—632no—
351C3×D4—188no—
352C2×C3×C4—38no—
353C2×C3×C4—68no—
354C2×C3×C4—92no—
355C2×C3×C4—182no—
356#67—28no—
357#67—632no—
358#67—188no—
359#67—1832no—
360C2×Dic3—22no—
361C2×Dic3—38no—
362C2×Dic3—68no—
363C2×Dic3—98no—
364C2×Dic3—92no—
365C2×Dic3—916no—
366C2×Dic3—94no—
367C2×Dic3—98no—
368C2×Dic3—92no—
369C2×Dic3—188no—
370C2×Dic3—1816no—
371C2×Dic3—188no—
372C2×Dic3—1832no—
373C2×Dic3—188no—
374C2×Dic3—1864no—
375C2×Dic3—1816no—
376C2×Dic3—188no—
377C2×Dic3—182no—
378C2×Dic3C22×C3⋊D312yes—
379C2×C32—232no—
380C2×C32—216no—
381C2×C32—24no—
382C2×C32—24no—
383C2×C32—24no—
384C2×C32—24no—
385C2×C32—216no—
386C2×C32—122no—
387C2×C32#939116yescomplemented
388C2×C32#93914yescomplemented
389C2×C32C2×D3×D414yes—
390C2×C32C2×D3×D411yescharacteristic
391C2×C32C2×#23714yes—
392C2×C32C2×#23711yescharacteristic
393C2×C32C22×#6714yes—
394C2×C32C22×#6711yescharacteristic
395C3⋊D3—42no—
396C3×D3—48no—
397C3×D3—128no—
398C24—31no—
399C24—91no—
400C24D3×C3⋊D311yescharacteristic
401C2×D4—34no—
402C2×D4—64no—
403C2×D4—94no—
404C2×D4—184no—
405C2×D4—274no—
406C2×D4—544no—
407C22×C4—31no—
408C22×C4—91no—
409C22×C4—271no—
410C22×C4—274no—
411C22×C4—271no—
412C22×C4—544no—
413C4⋊C4—274no—
414C22⋊C4—274no—
415C22⋊C4—274no—
416C22×C3—216no—
417C22×C3—216no—
418C22×C3—264no—
419C22×C3—28no—
420C22×C3—28no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—28no—
424C22×C3—232no—
425C22×C3—24no—
426C22×C3—28no—
427C22×C3—28no—
428C22×C3—216no—
429C22×C3—28no—
430C22×C3—68no—
431C22×C3—68no—
432C22×C3—68no—
433C22×C3—1216no—
434C22×C3—182no—
435C22×C3—182no—
436C22×C3—182no—
437C22×C3—364no—
438C22×C3#380518yes—
439C22×C3C2×#62518yes—
440C22×C3C2×#625116yescomplemented
441C22×C3#381118yes—
442C22×C3D4×C3⋊D312yes—
443C22×C3#383012yes—
444C22×C3C2×#63812yes—
445C22×C3C2×#63814yescomplemented
446C22×C3C22×D3214yes—
447C22×C3C23×C3⋊D311yescharacteristic
448C2×D3—22no—
449C2×D3—22no—
450C2×D3—22no—
451C2×D3—44no—
452C2×D3—68no—
453C2×D3—68no—
454C2×D3—68no—
455C2×D3—1216no—
456C3×C4—68no—
457C3×C4—182no—
458Dic3—22no—
459Dic3—68no—
460Dic3—916no—
461Dic3—94no—
462Dic3—1816no—
463Dic3—1816no—
464Dic3—188no—
465Dic3—182no—
466C32—24no—
467C32C2×#93914yescomplemented
468C32C2×#93911yescharacteristic, complemented
469C23—22no—
470C23—22no—
471C23—32no—
472C23—62no—
473C23—62no—
474C23—62no—
475C23—92no—
476C23—182no—
477C23—182no—
478C23—182no—
479C23#629014yescomplemented
480C23C2×D3×C3⋊D312yes—
481C23C2×D3×C3⋊D311yescharacteristic
482D4—68no—
483D4—188no—
484D4—548no—
485C2×C4—32no—
486C2×C4—62no—
487C2×C4—92no—
488C2×C4—182no—
489C2×C4—272no—
490C2×C4—274no—
491C2×C4—272no—
492C2×C4—548no—
493C2×C4—542no—
494C2×C3—232no—
495C2×C3—24no—
496C2×C3—216no—
497C2×C3—24no—
498C2×C3—24no—
499C2×C3—24no—
500C2×C3—216no—
501C2×C3—128no—
502C2×C3—362no—
503C2×C3#63980116yescomplemented
504C2×C3#6414814yescomplemented
505C2×C3C2×#380514yes—
506C2×C3C22×#62514yes—
507C2×C3C2×#381114yes—
508C2×C3C2×D4×C3⋊D311yescharacteristic
509C2×C3C2×#383011yescharacteristic
510C2×C3C22×#63811yescharacteristic
511D3—42no—
512D3—128no—
513C22—22no—
514C22—22no—
515C22—28no—
516C22—62no—
517C22—62no—
518C22—62no—
519C22—124no—
520C22—182no—
521C22—182no—
522C22—182no—
523C22—364no—
524C22#8902912yes—
525C22C2×#629012yes—
526C22C2×#629014yescomplemented
527C22#8903912yes—
528C22C22×D3×C3⋊D311yescharacteristic
529C4—62no—
530C4—182no—
531C4—274no—
532C4—542no—
533C3—24no—
534C3C2×#6398014yescomplemented
535C3C2×#6414811yescharacteristic, complemented
536C2—24no—
537C2—122no—
538C2—362no—
539C2#1172363714yesdirect product factor
540C2C2×#8902911yescharacteristic
541C2C22×#629011yescharacteristic
542C2C2×#8903911yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC24×C33C22
socleC23×C33C23
hypercenterC24D3×C3⋊D3
hypocenterC33C2×#121
LCSG ▷ C22×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
2. C22×C3×#638C2semidirect
3. C22×C32×#67C2semidirect
4. C22×#6290C2semidirect
5. #11723637C2direct (N × H)
6. C2×#88806C2semidirect
7. C2×#88803C2semidirect
8. C2×(C2×D3)⋊(C3⋊Dic3)C2semidirect
17. C22×C3⋊(C3⋊Dic3)C22semidirect
19. C2×C3×#638C22semidirect
22. C2×C32×#67C22semidirect
25. C2×#6290C22semidirect
26. #88806C22semidirect
27. #88803C22semidirect
28. (C2×D3)⋊(C3⋊Dic3)C22semidirect
31. C22×#638D3semidirect
36. C22×C3×#67D3semidirect
48. C23×C33D4semidirect
61. C2×C3⋊(C3⋊Dic3)D4semidirect
86. C2×#638C2×D3semidirect
103. C2×C3×#67C2×D3semidirect
119. C22×C33C2×D4semidirect
131. C3⋊(C3⋊Dic3)C2×D4semidirect
139. C22×#67C3⋊D3semidirect
164. C23×C32#67semidirect
165. C23×C32#67semidirect
213. C2×C33#121semidirect
239. C2×#67C2×C3⋊D3semidirect
284. C22×C32C2×#67semidirect
285. C22×C32C2×#67semidirect
318. C33C2×#121semidirect
336. C23×C3#625semidirect
337. C23×C3#638semidirect
387. C2×C32#939semidirect
388. C2×C32#939semidirect
440. C22×C3C2×#625semidirect
445. C22×C3C2×#638semidirect
467. C32C2×#939semidirect
468. C32C2×#939semidirect
479. C23#6290semidirect
503. C2×C3#63980semidirect
504. C2×C3#64148semidirect
526. C22C2×#6290semidirect
534. C3C2×#63980semidirect
535. C3C2×#64148semidirect
539. C2#11723637direct (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×C32×#672
C22×#62902
#117236372
C2×#888062
C2×#888032
C2×(C2×D3)⋊(C3⋊Dic3)2
C2×#641483
C2×#639803