Finite Group Databasebeta

← C2×#11722951 #13172318

Subgroups — C2×#11722951 #13172318

Order 1728.

3202 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#1113C211yesmaximal, characteristic, complemented
2C22×#6290C211yesmaximal, characteristic, complemented
3C22×C3×#626C212yesmaximal, complemented
4C2×#88803C212yesmaximal, complemented
5C2×C3×#3722C211yesmaximal, characteristic, complemented
6#11722951C218yesmaximal, direct product factor
7C2×#63983—31nomaximal
8C2×#63922—32nomaximal
9C23×C3×C3⋊D3C2212yes—
10C23×C3×C3⋊D3C2211yescharacteristic
11C23×C32×D3C2211yescharacteristic, complemented
12C22×#1113—24no—
13C22×#1113—28no—
14C22×#1113C2214yescomplemented
15C22×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
16C22×C32×Dic3C2212yescomplemented
17C2×#6290—24no—
18C2×#6290C2214yescomplemented
19C2×C3×#626—28no—
20C2×C3×#626C2218yescomplemented
21#88803C2218yescomplemented
22C3×#3722C2214yescomplemented
23C23×D32—31no—
24C23×D32—32no—
25C22×#638—31no—
26C22×C3×#67—32no—
27C22×C3×D12—32no—
28C22×#626D312yescomplemented
29C22×#625—32no—
30C22×#625—61no—
31C2×#3752—32no—
32C2×C3×(C2×D3)⋊C4—32no—
33#63983—38no—
34C2×#3722D311yescharacteristic, complemented
35C2×(C2×D3)⋊Dic3—32no—
36C2×(C2×D3)⋊Dic3—32no—
37C2×(C2×D3)⋊Dic3—62no—
38#63922—316no—
39C23×C33C2311yescharacteristic
40C22×C3×C3⋊D3—24no—
41C22×C3×C3⋊D3—24no—
42C22×C3×C3⋊D3—24no—
43C22×C3×C3⋊D3D414yescomplemented
44C22×C3×C3⋊D3D418yescomplemented
45C22×C3×C3⋊D3C2314yes—
46C22×C3×C3⋊D3C2312yes—
47C22×C32×D3—22no—
48C22×C32×D3—24no—
49C22×C32×D3C2312yescomplemented
50C2×#1113—232no—
51C2×#1113—28no—
52C2×#1113—216no—
53C2×C3⋊(C3⋊Dic3)—22no—
54C2×C3⋊(C3⋊Dic3)C2312yescomplemented
55C2×C32×Dic3—24no—
56C2×C32×Dic3C2314yescomplemented
57#6290—28no—
58C3×#626—216no—
59C2×#937—92no—
60C2×#882—91no—
61C24×C32—31no—
62C23×C3⋊D3C2×D311yescharacteristic
63C23×C3⋊D3C2×D312yes—
64C23×C3×D3—21no—
65C23×C3×D3—32no—
66C23×C3×D3—32no—
67C23×C3×D3—32no—
68C23×C3×D3—61no—
69C23×C3×D3—62no—
70C23×C3×D3C2×D312yescomplemented
71C22×D32—38no—
72C22×D32—34no—
73C22×D32—68no—
74C22×D32—68no—
75C22×D32—68no—
76C22×D32—64no—
77C22×D32—68no—
78C22×C32×C4—32no—
79C2×#638—34no—
80C2×#638—64no—
81C22×C3⋊Dic3—32no—
82C22×C3⋊Dic3—31no—
83C22×C3⋊Dic3—61no—
84C22×C3⋊Dic3—62no—
85C22×C3⋊Dic3—121no—
86C2×C3×#67—38no—
87C2×C3×#67—68no—
88C22×C3×Dic3—22no—
89C22×C3×Dic3C2×D312yescomplemented
90C22×C3×Dic3C2×D312yescomplemented
91C2×C3×D12—38no—
92C2×C3×D12—68no—
93C2×#626—28no—
94C2×#626C2×D318yescomplemented
95C2×#625—38no—
96C2×#625—68no—
97C2×#625—64no—
98C2×#625—68no—
99#3752—38no—
100C3×(C2×D3)⋊C4—38no—
101#3722C2×D314yescomplemented
102(C2×D3)⋊Dic3—38no—
103(C2×D3)⋊Dic3—38no—
104(C2×D3)⋊Dic3—68no—
105C22×C33C2×D412yes—
106C22×C33C2×D414yes—
107C22×C33C2411yescharacteristic
108C2×C3×C3⋊D3—216no—
109C2×C3×C3⋊D3—28no—
110C2×C3×C3⋊D3—24no—
111C2×C3×C3⋊D3—24no—
112C2×C3×C3⋊D3—24no—
113C2×C3×C3⋊D3C2×D414yescomplemented
114C2×C3×C3⋊D3C2×D418yescomplemented
115C2×C32×D3—22no—
116C2×C32×D3—24no—
117C2×C32×D3—44no—
118#1113—232no—
119C3⋊(C3⋊Dic3)—22no—
120C32×Dic3—24no—
121C24×D3—91no—
122C24×D3—92no—
123C22×C3×D4—92no—
124C22×#67—32no—
125C22×#67—92no—
126C22×#67—91no—
127C22×#67—181no—
128C22×D12—32no—
129C2×C3×C22⋊C4—91no—
130C2×#217—92no—
131#937—916no—
132C2×(C2×D3)⋊C4—32no—
133C2×(C2×D3)⋊C4—92no—
134C2×(C2×D3)⋊C4—92no—
135C2×(C2×D3)⋊C4—182no—
136#882—98no—
137C23×C32—21no—
138C23×C32—22no—
139C23×C32—32no—
140C23×C32—41no—
141C23×C32—62no—
142C23×C32—64no—
143C23×C32C22×D311yescharacteristic
144C23×C32C22×D312yes—
145C22×C3⋊D3—24no—
146C22×C3⋊D3—24no—
147C22×C3⋊D3—24no—
148C22×C3⋊D3D1214yescomplemented
149C22×C3⋊D3#6718yescomplemented
150C22×C3⋊D3C22×D312yes—
151C22×C3⋊D3C22×D314yes—
152C22×C3×D3—24no—
153C22×C3×D3—24no—
154C22×C3×D3—24no—
155C22×C3×D3—24no—
156C22×C3×D3—22no—
157C22×C3×D3—38no—
158C22×C3×D3—38no—
159C22×C3×D3—38no—
160C22×C3×D3—34no—
161C22×C3×D3—34no—
162C22×C3×D3—34no—
163C22×C3×D3—44no—
164C22×C3×D3—64no—
165C22×C3×D3—68no—
166C22×C3×D3—64no—
167C22×C3×D3—62no—
168C22×C3×D3—64no—
169C22×C3×D3—64no—
170C22×C3×D3—64no—
171C22×C3×D3—64no—
172C22×C3×D3—64no—
173C22×C3×D3—64no—
174C22×C3×D3—124no—
175C22×C3×D3—124no—
176C22×C3×D3—124no—
177C22×C3×D3C22×D314yescomplemented
178C2×D32—616no—
179C2×D32—616no—
180C2×D32—616no—
181C2×D32—632no—
182C2×D32—664no—
183C2×D32—68no—
184C2×D32—616no—
185C2×C32×C4—34no—
186C2×C32×C4—64no—
187#638—68no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—34no—
190C2×C3⋊Dic3—62no—
191C2×C3⋊Dic3—64no—
192C2×C3⋊Dic3—62no—
193C2×C3⋊Dic3—64no—
194C2×C3⋊Dic3—64no—
195C2×C3⋊Dic3—122no—
196C2×C3⋊Dic3—124no—
197C2×C3⋊Dic3—124no—
198C3×#67—616no—
199C2×C3×Dic3—24no—
200C2×C3×Dic3—24no—
201C2×C3×Dic3—24no—
202C2×C3×Dic3—44no—
203C2×C3×Dic3C22×D314yescomplemented
204C2×C3×Dic3C22×D314yescomplemented
205C3×D12—616no—
206#626—216no—
207#625—616no—
208#625—616no—
209C2×#120—271noSylow
210C2×C33#12014yescomplemented
211C2×C33C22×D411yescharacteristic
212C2×C33C22×D412yes—
213C3×C3⋊D3—28no—
214C3×C3⋊D3—24no—
215C32×D3—42no—
216C24×C3—32no—
217C24×C3—61no—
218C24×C3—92no—
219C24×C3—91no—
220C23×D3—32no—
221C23×D3—32no—
222C23×D3—32no—
223C23×D3—62no—
224C23×D3—61no—
225C23×D3—94no—
226C23×D3—98no—
227C23×D3—188no—
228C23×D3—188no—
229C23×D3—188no—
230C23×D3—188no—
231C23×D3—184no—
232C23×D3D3211yescharacteristic, complemented
233C2×C3×D4—98no—
234C2×C3×D4—188no—
235C22×C3×C4—32no—
236C22×C3×C4—32no—
237C22×C3×C4—62no—
238C2×#67—38no—
239C2×#67—68no—
240C2×#67—98no—
241C2×#67—94no—
242C2×#67—184no—
243C2×#67—184no—
244C2×#67—188no—
245C2×#67—188no—
246C22×Dic3—91no—
247C22×Dic3—92no—
248C22×Dic3—181no—
249C22×Dic3—182no—
250C22×Dic3—361no—
251C22×Dic3D3212yescomplemented
252C2×D12—38no—
253C2×D12—68no—
254C3×C22⋊C4—94no—
255#217—98no—
256(C2×D3)⋊C4—38no—
257(C2×D3)⋊C4—98no—
258(C2×D3)⋊C4—98no—
259(C2×D3)⋊C4—188no—
260C22×C32—22no—
261C22×C32—21no—
262C22×C32—24no—
263C22×C32—24no—
264C22×C32—24no—
265C22×C32—24no—
266C22×C32—22no—
267C22×C32—42no—
268C22×C32—44no—
269C22×C32—41no—
270C22×C32—62no—
271C22×C32—64no—
272C22×C32—124no—
273C22×C32C2×D1212yes—
274C22×C32D3×D414yes—
275C22×C32D3×D414yes—
276C22×C32D3×D414yes—
277C22×C32C2×#6714yes—
278C22×C32C23×D311yescharacteristic
279C22×C32C23×D312yes—
280C2×C3⋊D3—28no—
281C2×C3⋊D3—216no—
282C2×C3⋊D3—24no—
283C2×C3⋊D3—24no—
284C2×C3⋊D3—24no—
285C2×C3⋊D3C2×D1214yescomplemented
286C2×C3⋊D3C2×#6718yescomplemented
287C2×C3×D3—24no—
288C2×C3×D3—24no—
289C2×C3×D3—24no—
290C2×C3×D3—24no—
291C2×C3×D3—38no—
292C2×C3×D3—38no—
293C2×C3×D3—38no—
294C2×C3×D3—44no—
295C2×C3×D3—48no—
296C2×C3×D3—48no—
297C2×C3×D3—64no—
298C2×C3×D3—68no—
299C2×C3×D3—616no—
300C2×C3×D3—616no—
301C2×C3×D3—616no—
302C2×C3×D3—64no—
303C2×C3×D3—64no—
304C2×C3×D3—64no—
305C2×C3×D3—64no—
306C2×C3×D3—64no—
307C2×C3×D3—64no—
308C2×C3×D3—124no—
309C2×C3×D3—124no—
310C2×C3×D3—124no—
311C2×C3×D3—1216no—
312C2×C3×D3—128no—
313D32—632no—
314D32—664no—
315C32×C4—64no—
316C3⋊Dic3—62no—
317C3⋊Dic3—64no—
318C3⋊Dic3—64no—
319C3⋊Dic3—124no—
320C3⋊Dic3—124no—
321C3×Dic3—24no—
322C3×Dic3—24no—
323C3×Dic3—44no—
324C25—271no—
325C22×D4—92no—
326C22×D4—271no—
327#120—278no—
328C2×C22⋊C4—91no—
329C2×C22⋊C4—272no—
330C33C2×#12011yescharacteristic, complemented, Sylow
331C23×C3—22no—
332C23×C3—21no—
333C23×C3—34no—
334C23×C3—41no—
335C23×C3—62no—
336C23×C3—64no—
337C23×C3—64no—
338C23×C3—64no—
339C23×C3—64no—
340C23×C3—94no—
341C23×C3—92no—
342C23×C3—98no—
343C23×C3—94no—
344C23×C3—124no—
345C23×C3—184no—
346C23×C3—184no—
347C23×C3—184no—
348C23×C3C2×D3212yes—
349C23×C3C2×D3211yescharacteristic
350C22×D3—24no—
351C22×D3—22no—
352C22×D3—34no—
353C22×D3—34no—
354C22×D3—34no—
355C22×D3—38no—
356C22×D3—38no—
357C22×D3—38no—
358C22×D3—64no—
359C22×D3—68no—
360C22×D3—64no—
361C22×D3—62no—
362C22×D3—64no—
363C22×D3—64no—
364C22×D3—64no—
365C22×D3—64no—
366C22×D3—64no—
367C22×D3—64no—
368C22×D3—124no—
369C22×D3—124no—
370C22×D3—124no—
371C22×D3—1816no—
372C22×D3—1816no—
373C22×D3—1816no—
374C22×D3—1816no—
375C22×D3—188no—
376C22×D3—1864no—
377C22×D3—1832no—
378C22×D3C2×D3212yescomplemented
379C3×D4—1816no—
380C2×C3×C4—34no—
381C2×C3×C4—34no—
382C2×C3×C4—64no—
383C2×C3×C4—64no—
384C2×C3×C4—64no—
385C2×C3×C4—124no—
386#67—616no—
387#67—188no—
388#67—1816no—
389#67—1816no—
390C2×Dic3—24no—
391C2×Dic3—92no—
392C2×Dic3—94no—
393C2×Dic3—182no—
394C2×Dic3—184no—
395C2×Dic3—184no—
396C2×Dic3—184no—
397C2×Dic3—182no—
398C2×Dic3—362no—
399C2×Dic3—364no—
400C2×Dic3—364no—
401C2×Dic3C2×D3214yescomplemented
402D12—616no—
403C2×C32—22no—
404C2×C32—22no—
405C2×C32—22no—
406C2×C32—22no—
407C2×C32—21no—
408C2×C32—28no—
409C2×C32—24no—
410C2×C32—44no—
411C2×C32—41no—
412C2×C32—42no—
413C2×C32—122no—
414C2×C32#88214yescomplemented
415C2×C32#93718yescomplemented
416C2×C32C22×D1211yescharacteristic
417C2×C32C2×D3×D412yes—
418C2×C32C2×D3×D412yes—
419C2×C32C2×D3×D412yes—
420C2×C32C22×#6712yes—
421C3⋊D3—24no—
422C3⋊D3—28no—
423C3×D3—44no—
424C3×D3—44no—
425C3×D3—68no—
426C3×D3—68no—
427C3×D3—68no—
428C3×D3—124no—
429C3×D3—128no—
430C24—31no—
431C24—91no—
432C24—92no—
433C24—274no—
434C24—548no—
435C24—544no—
436C2×D4—98no—
437C2×D4—188no—
438C2×D4—274no—
439C2×D4—544no—
440C22×C4—32no—
441C22×C4—271no—
442C22⋊C4—94no—
443C22⋊C4—278no—
444C22×C3—22no—
445C22×C3—21no—
446C22×C3—24no—
447C22×C3—24no—
448C22×C3—24no—
449C22×C3—22no—
450C22×C3—24no—
451C22×C3—41no—
452C22×C3—44no—
453C22×C3—42no—
454C22×C3—64no—
455C22×C3—64no—
456C22×C3—64no—
457C22×C3—64no—
458C22×C3—98no—
459C22×C3—94no—
460C22×C3—124no—
461C22×C3—128no—
462C22×C3—128no—
463C22×C3—184no—
464C22×C3—184no—
465C22×C3—184no—
466C22×C3—1816no—
467C22×C3—188no—
468C22×C3D3×D1214yes—
469C22×C3#380214yes—
470C22×C3D3×#6714yes—
471C22×C3D3×#6714yes—
472C22×C3#381112yes—
473C22×C3C22×D3212yes—
474C22×C3C22×D3211yescharacteristic
475C2×D3—24no—
476C2×D3—22no—
477C2×D3—38no—
478C2×D3—38no—
479C2×D3—38no—
480C2×D3—44no—
481C2×D3—64no—
482C2×D3—68no—
483C2×D3—64no—
484C2×D3—64no—
485C2×D3—64no—
486C2×D3—64no—
487C2×D3—64no—
488C2×D3—64no—
489C2×D3—616no—
490C2×D3—616no—
491C2×D3—616no—
492C2×D3—124no—
493C2×D3—124no—
494C2×D3—124no—
495C2×D3—128no—
496C2×D3—1216no—
497C2×D3—1832no—
498C2×D3—1864no—
499C3×C4—64no—
500C3×C4—64no—
501C3×C4—124no—
502Dic3—24no—
503Dic3—184no—
504Dic3—184no—
505Dic3—182no—
506Dic3—364no—
507Dic3—364no—
508C32—21no—
509C32—22no—
510C32—41no—
511C32C2×#88211yescharacteristic, complemented
512C32C2×#93712yescomplemented
513C23—32no—
514C23—64no—
515C23—62no—
516C23—92no—
517C23—94no—
518C23—98no—
519C23—94no—
520C23—184no—
521C23—184no—
522C23—184no—
523C23—5416no—
524C23—548no—
525C23—5432no—
526C23D3311yescharacteristic
527D4—1816no—
528D4—548no—
529C2×C4—34no—
530C2×C4—64no—
531C2×C4—272no—
532C2×C4—542no—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—22no—
536C2×C3—21no—
537C2×C3—22no—
538C2×C3—24no—
539C2×C3—28no—
540C2×C3—42no—
541C2×C3—41no—
542C2×C3—44no—
543C2×C3—124no—
544C2×C3—124no—
545C2×C3—184no—
546C2×C3—188no—
547C2×C3#6392218yescomplemented
548C2×C3#6398314yescomplemented
549C2×C3C2×D3×D1212yes—
550C2×C3C2×#380212yes—
551C2×C3C2×D3×#6712yes—
552C2×C3C2×D3×#6712yes—
553C2×C3C2×#381111yescharacteristic
554D3—42no—
555D3—68no—
556D3—68no—
557D3—68no—
558D3—124no—
559D3—128no—
560C22—64no—
561C22—62no—
562C22—98no—
563C22—94no—
564C22—124no—
565C22—184no—
566C22—184no—
567C22—184no—
568C22—1816no—
569C22—188no—
570C22—5432no—
571C22#8895314yes—
572C22#8895512yes—
573C22C2×D3311yescharacteristic
574C4—64no—
575C4—542no—
576C3—21no—
577C3—22no—
578C3—41no—
579C3C2×#6392212yescomplemented
580C3C2×#6398311yescharacteristic, complemented
581C2—122no—
582C2—188no—
583C2—184no—
584C2#1172295114yesdirect product factor
585C2C2×#8895312yes—
586C2C2×#8895511yescharacteristic
subgroupG/H
center, hypercenterC23D33
commutator, cosocleC22×C33C24
FrattiniC22C2×D33
Fitting, socleC23×C33C23
hypocenterC33C2×#120
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×#1113C2semidirect
2. C22×#6290C2semidirect
3. C22×C3×#626C2semidirect
4. C2×#88803C2semidirect
5. C2×C3×#3722C2semidirect
6. #11722951C2direct (N × H)
11. C23×C32×D3C22semidirect
14. C22×#1113C22semidirect
15. C22×C3⋊(C3⋊Dic3)C22semidirect
16. C22×C32×Dic3C22semidirect
18. C2×#6290C22semidirect
20. C2×C3×#626C22semidirect
21. #88803C22semidirect
22. C3×#3722C22semidirect
28. C22×#626D3semidirect
34. C2×#3722D3semidirect
43. C22×C3×C3⋊D3D4semidirect
44. C22×C3×C3⋊D3D4semidirect
49. C22×C32×D3C23semidirect
54. C2×C3⋊(C3⋊Dic3)C23semidirect
56. C2×C32×Dic3C23semidirect
70. C23×C3×D3C2×D3semidirect
89. C22×C3×Dic3C2×D3semidirect
90. C22×C3×Dic3C2×D3semidirect
94. C2×#626C2×D3semidirect
101. #3722C2×D3semidirect
113. C2×C3×C3⋊D3C2×D4semidirect
114. C2×C3×C3⋊D3C2×D4semidirect
148. C22×C3⋊D3D12semidirect
149. C22×C3⋊D3#67semidirect
177. C22×C3×D3C22×D3semidirect
203. C2×C3×Dic3C22×D3semidirect
204. C2×C3×Dic3C22×D3semidirect
210. C2×C33#120semidirect
232. C23×D3D32semidirect
251. C22×Dic3D32semidirect
285. C2×C3⋊D3C2×D12semidirect
286. C2×C3⋊D3C2×#67semidirect
330. C33C2×#120semidirect
378. C22×D3C2×D32semidirect
401. C2×Dic3C2×D32semidirect
414. C2×C32#882semidirect
415. C2×C32#937semidirect
511. C32C2×#882semidirect
512. C32C2×#937semidirect
547. C2×C3#63922semidirect
548. C2×C3#63983semidirect
579. C3C2×#63922semidirect
580. C3C2×#63983semidirect
584. C2#11722951direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#11132
C22×#62902
C22×C3×#6262
C2×#888032
C2×C3×#37222
#117229512
C2×#639833
C2×#639223