Finite Group Databasebeta

← C2×#11723702 #13178659

Subgroups — C2×#11723702 #13178659

Order 1728.

1794 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C4⋊#1113C211yesmaximal, characteristic, complemented
3C2×C3×#3753C212yesmaximal, complemented
4#11723702C218yesmaximal, direct product factor
5C2×#88810C212yesmaximal, complemented
6C2×#88808C211yesmaximal, characteristic, complemented
7C2×#63926—31nomaximal
8C2×D3×#213—32nomaximal
9C23×C3×C3⋊D3—21no—
10C22×#1113—21no—
11C2×C33×D4C2211yescharacteristic, complemented
12C2×C3×#638—21no—
13C3×D4×C3⋊D3—22no—
14C3×D4×C3⋊D3C2214yescomplemented
15C2×C3×C4⋊(C3⋊D3)C2212yescomplemented
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×#6295—21no—
19C4⋊#1113—22no—
20C4⋊#1113C2214yescomplemented
21C3×#3753C2218yescomplemented
22C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
23#88810C2218yescomplemented
24#88808C2214yescomplemented
25C2×D4×C3⋊D3D311yescharacteristic, complemented
26C2×C3×D3×D4—32no—
27C2×C3×D3×D4—61no—
28C2×#3802—31no—
29C2×D3×D12—32no—
30C2×#3753D312yescomplemented
31C2×C3×#213—32no—
32C2×C3×#213—32no—
33C2×C3×#213—62no—
34#63926—38no—
35D3×#213—316no—
36C2×#3714—32no—
37C2×#3714—32no—
38C2×#3713—32no—
39C2×#3710—31no—
40C2×D3×C3⋊C8—32no—
41C23×C33—21no—
42C22×C3×C3⋊D3—21no—
43C22×C3×C3⋊D3—22no—
44C22×C3×C3⋊D3—24no—
45C22×C3×C3⋊D3—44no—
46C22×C3×C3⋊D3D411yescharacteristic
47C2×#1113—24no—
48C2×#1113—44no—
49C33×D4—21no—
50C33×D4C2312yescomplemented
51C2×C33×C4C2311yescharacteristic
52C3×#638—24no—
53C2×C3×C3⋊Dic3D411yescharacteristic
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)—22no—
56C3×C4⋊(C3⋊D3)C2314yescomplemented
57C3×C4⋊(C3⋊D3)C2312yescomplemented
58C3×C4×C3⋊D3D412yes—
59C3×C4×C3⋊D3C2312yes—
60#6295—24no—
61C3×C3⋊(C3⋊C8)C2314yescomplemented
62C22×#213—91no—
63C2×D3×D8—92no—
64C23×C3⋊D3—21no—
65C23×C3×D3—62no—
66C23×C3×D3—62no—
67C22×D32—62no—
68C22×D32—61no—
69C2×C32×D4—21no—
70C2×C32×D4—22no—
71C2×C32×D4—41no—
72C2×C32×D4C2×D311yescharacteristic, complemented
73C2×C32×D4C2×D312yescomplemented
74C2×#638—21no—
75D4×C3⋊D3—22no—
76D4×C3⋊D3C2×D314yescomplemented
77C2×C4⋊(C3⋊D3)C2×D312yescomplemented
78C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
79C2×C4×C3⋊D3C2×D311yescharacteristic
80C2×C3×#67—62no—
81C2×C3×#67—62no—
82C3×D3×D4—38no—
83C3×D3×D4—64no—
84C3×D3×D4—64no—
85C3×D3×D4—64no—
86C2×C3×D12—32no—
87C2×C3×D12—32no—
88C2×C3×D12—32no—
89C2×C3×D12—61no—
90C2×C3×D12—62no—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—61no—
93C2×#626—62no—
94C2×#625—61no—
95#3802—34no—
96#3802—62no—
97D3×D12—38no—
98D3×D12—64no—
99#3753C2×D318yescomplemented
100C2×C3⋊(C3⋊C8)C2×D312yescomplemented
101C3×#213—38no—
102C3×#213—38no—
103C3×#213—68no—
104C2×C3×C3⋊C8—32no—
105C2×C3×C3⋊C8—32no—
106C2×C3×C3⋊C8—62no—
107#3714—38no—
108#3714—38no—
109#3713—38no—
110#3710—34no—
111D3×C3⋊C8—38no—
112C22×C33—22no—
113C22×C33—42no—
114C22×C33C2×D411yescharacteristic
115C2×C3×C3⋊D3—22no—
116C2×C3×C3⋊D3—24no—
117C2×C3×C3⋊D3—42no—
118C2×C3×C3⋊D3—44no—
119C2×C3×C3⋊D3—48no—
120C2×C3×C3⋊D3D814yescomplemented
121C2×C3×C3⋊D3C2×D412yes—
122#1113—48no—
123C33×C4C2×D411yescharacteristic
124C33×C4C2411yescharacteristic
125C3×C3⋊Dic3C2×D412yes—
126C22×C3×D4—91no—
127C2×D3×D4—32no—
128C2×D3×D4—61no—
129C2×D3×D4—92no—
130C22×D12—91no—
131C2×C3×D8—92no—
132C2×#213—32no—
133C2×#213—32no—
134C2×#213—62no—
135C2×#213—92no—
136C2×#213—92no—
137C2×#213—98no—
138C22×C3⋊C8—91no—
139D3×D8—916no—
140C2×D24—92no—
141C2×D3×C8—92no—
142C23×C32—21no—
143C23×C32—22no—
144C23×C32—22no—
145C23×C32—42no—
146C23×C32—42no—
147C22×C3⋊D3—21no—
148C22×C3⋊D3—24no—
149C22×C3⋊D3—22no—
150C22×C3⋊D3—44no—
151C22×C3⋊D3#6711yescharacteristic
152C22×C3×D3—32no—
153C22×C3×D3—61no—
154C22×C3×D3—68no—
155C22×C3×D3—62no—
156C22×C3×D3—68no—
157C22×C3×D3—62no—
158C22×C3×D3—62no—
159C22×C3×D3—62no—
160C22×C3×D3—122no—
161C22×C3×D3—128no—
162C22×C3×D3—128no—
163C2×D32—68no—
164C2×D32—64no—
165C2×D32—128no—
166C2×D32—124no—
167C32×D4—21no—
168C32×D4—22no—
169C32×D4—22no—
170C32×D4—22no—
171C32×D4—24no—
172C32×D4—42no—
173C32×D4—42no—
174C32×D4—42no—
175C32×D4C22×D312yescomplemented
176C32×D4C22×D314yescomplemented
177C2×C32×C4—22no—
178C2×C32×C4—21no—
179C2×C32×C4—41no—
180C2×C32×C4C22×D312yes—
181C2×C32×C4C22×D311yescharacteristic
182#638—24no—
183C2×C3⋊Dic3#6711yescharacteristic
184C4⋊(C3⋊D3)—21no—
185C4⋊(C3⋊D3)—22no—
186C4⋊(C3⋊D3)C22×D312yescomplemented
187C4⋊(C3⋊D3)C22×D314yescomplemented
188C4×C3⋊D3#6712yes—
189C4×C3⋊D3C22×D312yes—
190C3×#67—68no—
191C3×#67—68no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—61no—
194C3×D12—34no—
195C3×D12—34no—
196C3×D12—34no—
197C3×D12—62no—
198C3×D12—64no—
199C3×D12—62no—
200C3×D12—62no—
201C3×D12—62no—
202C3×D12—62no—
203C3×D12—122no—
204C3×C4×D3—34no—
205C3×C4×D3—34no—
206C3×C4×D3—62no—
207C3×C4×D3—62no—
208#626—68no—
209#625—64no—
210C3⋊(C3⋊C8)C22×D314yescomplemented
211C3×C3⋊C8—34no—
212C3×C3⋊C8—34no—
213C3×C3⋊C8—64no—
214C22×D8—271noSylow
215C2×C33—42no—
216C2×C33C2×D812yescomplemented
217C2×C33C22×D411yescharacteristic
218C3×C3⋊D3—42no—
219C3×C3⋊D3—44no—
220C3×C3⋊D3C2×D814yescomplemented
221C24×C3—181no—
222C23×D3—62no—
223C23×D3—62no—
224C23×D3—182no—
225C23×D3—181no—
226C2×C3×D4—21no—
227C2×C3×D4—22no—
228C2×C3×D4—41no—
229C2×C3×D4—94no—
230C2×C3×D4—91no—
231C2×C3×D4—92no—
232C2×C3×D4—182no—
233C2×C3×D4—181no—
234C2×C3×D4D3211yescharacteristic, complemented
235C2×C3×D4D3212yescomplemented
236C22×C3×C4—91no—
237C2×#67—62no—
238C2×#67—62no—
239C2×#67—182no—
240D3×D4—38no—
241D3×D4—64no—
242D3×D4—64no—
243D3×D4—64no—
244D3×D4—98no—
245D3×D4—184no—
246C2×D12—32no—
247C2×D12—32no—
248C2×D12—32no—
249C2×D12—61no—
250C2×D12—62no—
251C2×D12—94no—
252C2×D12—182no—
253C2×D12—181no—
254C2×C4×D3—32no—
255C2×C4×D3—61no—
256C3×D8—98no—
257C2×C3×C8—92no—
258#213—38no—
259#213—38no—
260#213—68no—
261#213—98no—
262#213—98no—
263C2×C3⋊C8—32no—
264C2×C3⋊C8—32no—
265C2×C3⋊C8—62no—
266C2×C3⋊C8—94no—
267D24—98no—
268D3×C8—98no—
269C22×C32—22no—
270C22×C32—24no—
271C22×C32—24no—
272C22×C32—22no—
273C22×C32—21no—
274C22×C32—41no—
275C22×C32—44no—
276C22×C32—44no—
277C22×C32—42no—
278C22×C32—44no—
279C22×C32—44no—
280C22×C32—84no—
281C22×C32—84no—
282C22×C32D3×D412yes—
283C22×C32C2×#6711yescharacteristic
284C2×C3⋊D3—22no—
285C2×C3⋊D3—24no—
286C2×C3⋊D3—42no—
287C2×C3⋊D3—48no—
288C2×C3⋊D3—44no—
289C2×C3⋊D3#21314yescomplemented
290C2×C3⋊D3C2×#6712yes—
291C2×C3×D3—38no—
292C2×C3×D3—34no—
293C2×C3×D3—64no—
294C2×C3×D3—62no—
295C2×C3×D3—64no—
296C2×C3×D3—64no—
297C2×C3×D3—64no—
298C2×C3×D3—64no—
299C2×C3×D3—124no—
300C2×C3×D3—1216no—
301C2×C3×D3—1216no—
302C2×C3×D3—124no—
303C2×C3×D3—124no—
304C2×C3×D3—124no—
305C2×C3×D3—124no—
306C2×C3×D3—244no—
307D32—1216no—
308D32—128no—
309C32×C4—22no—
310C32×C4—21no—
311C32×C4—22no—
312C32×C4—21no—
313C32×C4—41no—
314C32×C4—41no—
315C32×C4D3×D412yes—
316C32×C4C2×#6711yescharacteristic
317C32×C4C23×D312yes—
318C32×C4C23×D311yescharacteristic
319C3⋊Dic3C2×#6712yes—
320C3×Dic3—34no—
321C3×Dic3—62no—
322C22×D4—91no—
323C22×D4—271no—
324C2×D8—92no—
325C2×D8—272no—
326C2×D8—278no—
327C22×C8—271no—
328C33C22×D811yescharacteristic, complemented, Sylow
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—21no—
332C23×C3—42no—
333C23×C3—42no—
334C23×C3—91no—
335C23×C3—184no—
336C23×C3—182no—
337C23×C3—181no—
338C23×C3—364no—
339C22×D3—32no—
340C22×D3—61no—
341C22×D3—68no—
342C22×D3—62no—
343C22×D3—68no—
344C22×D3—62no—
345C22×D3—62no—
346C22×D3—62no—
347C22×D3—122no—
348C22×D3—128no—
349C22×D3—128no—
350C22×D3—188no—
351C22×D3—184no—
352C22×D3—368no—
353C22×D3—364no—
354C3×D4—22no—
355C3×D4—22no—
356C3×D4—21no—
357C3×D4—22no—
358C3×D4—24no—
359C3×D4—42no—
360C3×D4—42no—
361C3×D4—42no—
362C3×D4—92no—
363C3×D4—94no—
364C3×D4—184no—
365C3×D4—182no—
366C3×D4—181no—
367C3×D4C2×D3212yescomplemented
368C3×D4C2×D3214yescomplemented
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—41no—
372C2×C3×C4—91no—
373C2×C3×C4—92no—
374C2×C3×C4—92no—
375C2×C3×C4C2×D3212yes—
376C2×C3×C4C2×D3211yescharacteristic
377#67—68no—
378#67—68no—
379#67—188no—
380C2×Dic3—32no—
381C2×Dic3—61no—
382D12—34no—
383D12—34no—
384D12—34no—
385D12—62no—
386D12—64no—
387D12—62no—
388D12—62no—
389D12—62no—
390D12—62no—
391D12—122no—
392D12—184no—
393C4×D3—34no—
394C4×D3—34no—
395C4×D3—62no—
396C4×D3—62no—
397C3×C8—94no—
398C3⋊C8—34no—
399C3⋊C8—34no—
400C3⋊C8—64no—
401C2×C32—22no—
402C2×C32—21no—
403C2×C32—24no—
404C2×C32—22no—
405C2×C32—42no—
406C2×C32—41no—
407C2×C32—42no—
408C2×C32—44no—
409C2×C32—44no—
410C2×C32—84no—
411C2×C32—84no—
412C2×C32D3×D814yescomplemented
413C2×C32C2×#21312yescomplemented
414C2×C32C2×D3×D412yes—
415C2×C32C22×#6711yescharacteristic
416C3⋊D3—42no—
417C3⋊D3—44no—
418C3⋊D3C2×#21314yescomplemented
419C3×D3—38no—
420C3×D3—64no—
421C3×D3—124no—
422C3×D3—124no—
423C3×D3—124no—
424C3×D3—124no—
425C3×D3—244no—
426C24—181no—
427C24—541no—
428C2×D4—94no—
429C2×D4—92no—
430C2×D4—91no—
431C2×D4—182no—
432C2×D4—181no—
433C2×D4—274no—
434C2×D4—542no—
435C2×D4—541no—
436C2×D4#111311yescharacteristic, complemented
437C22×C4—91no—
438D8—98no—
439D8—278no—
440C2×C8—92no—
441C2×C8—274no—
442C22×C3—24no—
443C22×C3—24no—
444C22×C3—22no—
445C22×C3—21no—
446C22×C3—22no—
447C22×C3—44no—
448C22×C3—44no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—42no—
452C22×C3—41no—
453C22×C3—84no—
454C22×C3—84no—
455C22×C3—92no—
456C22×C3—94no—
457C22×C3—184no—
458C22×C3—182no—
459C22×C3—368no—
460C22×C3—364no—
461C22×C3—362no—
462C22×C3D3×#6712yes—
463C22×C3#381111yescharacteristic
464C2×D3—34no—
465C2×D3—38no—
466C2×D3—62no—
467C2×D3—64no—
468C2×D3—64no—
469C2×D3—64no—
470C2×D3—64no—
471C2×D3—64no—
472C2×D3—124no—
473C2×D3—1216no—
474C2×D3—1216no—
475C2×D3—124no—
476C2×D3—124no—
477C2×D3—124no—
478C2×D3—124no—
479C2×D3—244no—
480C2×D3—3616no—
481C2×D3—368no—
482C3×C4—22no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—22no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—92no—
489C3×C4D3×#6712yes—
490C3×C4#381111yescharacteristic
491C3×C4C22×D3212yes—
492C3×C4C22×D3211yescharacteristic
493Dic3—34no—
494Dic3—62no—
495C32—22no—
496C32—21no—
497C32—41no—
498C32C2×D3×D812yescomplemented
499C32C22×#21311yescharacteristic, complemented
500C23—21no—
501C23—91no—
502C23—184no—
503C23—181no—
504C23—182no—
505C23—364no—
506C23—544no—
507C23—1084no—
508D4—21no—
509D4—94no—
510D4—92no—
511D4—184no—
512D4—181no—
513D4—182no—
514D4—544no—
515D4C2×#111312yescomplemented
516C2×C4—91no—
517C2×C4—92no—
518C2×C4—92no—
519C2×C4C2×#111311yescharacteristic
520C8—94no—
521C2×C3—21no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—24no—
525C2×C3—44no—
526C2×C3—44no—
527C2×C3—42no—
528C2×C3—41no—
529C2×C3—42no—
530C2×C3—84no—
531C2×C3—84no—
532C2×C3—94no—
533C2×C3—364no—
534C2×C3—362no—
535C2×C3D3×#21314yescomplemented
536C2×C3#6392612yescomplemented
537C2×C3C2×D3×#6712yes—
538C2×C3C2×#381111yescharacteristic
539D3—38no—
540D3—64no—
541D3—124no—
542D3—124no—
543D3—124no—
544D3—124no—
545D3—244no—
546C22—22no—
547C22—42no—
548C22—92no—
549C22—94no—
550C22—182no—
551C22—184no—
552C22—368no—
553C22—362no—
554C22—364no—
555C22—1088no—
556C22#8904911yescharacteristic
557C4—92no—
558C4#8904911yescharacteristic
559C4C22×#111311yescharacteristic
560C3—21no—
561C3—22no—
562C3—41no—
563C3C2×D3×#21312yescomplemented
564C3C2×#6392611yescharacteristic, complemented
565C2—42no—
566C2—94no—
567C2—362no—
568C2—364no—
569C2#1172370212yesdirect product factor
570C2C2×#8904911yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC33×C4C24
FrattiniC4C22×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33C22×D8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×C3⋊D3C2semidirect
2. C2×C4⋊#1113C2semidirect
3. C2×C3×#3753C2semidirect
4. #11723702C2direct (N × H)
5. C2×#88810C2semidirect
6. C2×#88808C2semidirect
11. C2×C33×D4C22semidirect
14. C3×D4×C3⋊D3C22semidirect
15. C2×C3×C4⋊(C3⋊D3)C22semidirect
16. C2×C3×C4⋊(C3⋊D3)C22semidirect
20. C4⋊#1113C22semidirect
21. C3×#3753C22semidirect
22. C2×C3×C3⋊(C3⋊C8)C22semidirect
23. #88810C22semidirect
24. #88808C22semidirect
25. C2×D4×C3⋊D3D3semidirect
30. C2×#3753D3semidirect
50. C33×D4C23semidirect
56. C3×C4⋊(C3⋊D3)C23semidirect
57. C3×C4⋊(C3⋊D3)C23semidirect
61. C3×C3⋊(C3⋊C8)C23semidirect
72. C2×C32×D4C2×D3semidirect
73. C2×C32×D4C2×D3semidirect
76. D4×C3⋊D3C2×D3semidirect
77. C2×C4⋊(C3⋊D3)C2×D3semidirect
78. C2×C4⋊(C3⋊D3)C2×D3semidirect
99. #3753C2×D3semidirect
100. C2×C3⋊(C3⋊C8)C2×D3semidirect
120. C2×C3×C3⋊D3D8semidirect
175. C32×D4C22×D3semidirect
176. C32×D4C22×D3semidirect
186. C4⋊(C3⋊D3)C22×D3semidirect
187. C4⋊(C3⋊D3)C22×D3semidirect
210. C3⋊(C3⋊C8)C22×D3semidirect
216. C2×C33C2×D8semidirect
220. C3×C3⋊D3C2×D8semidirect
234. C2×C3×D4D32semidirect
235. C2×C3×D4D32semidirect
289. C2×C3⋊D3#213semidirect
328. C33C22×D8semidirect
367. C3×D4C2×D32semidirect
368. C3×D4C2×D32semidirect
412. C2×C32D3×D8semidirect
413. C2×C32C2×#213semidirect
418. C3⋊D3C2×#213semidirect
436. C2×D4#1113semidirect
498. C32C2×D3×D8semidirect
499. C32C22×#213semidirect
515. D4C2×#1113semidirect
535. C2×C3D3×#213semidirect
536. C2×C3#63926semidirect
563. C3C2×D3×#213semidirect
564. C3C2×#63926semidirect
569. C2#11723702direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
C2×C4⋊#11132
C2×C3×#37532
#117237022
C2×#888102
C2×#888082
C2×#639263
C2×D3×#2133