Finite Group Databasebeta

← C2×#11722830 #13170534

Subgroups — C2×#11722830 #13170534

Order 1728.

1794 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1794 conjugacy classes of subgroups — too many to draw. Only its 212 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×D12C211yesmaximal, characteristic, complemented
2C2×C3×#3802C211yesmaximal, characteristic, complemented
3C2×#88808C211yesmaximal, characteristic, complemented
4C2×#88790C212yesmaximal, complemented
5C2×C3×#3713C212yesmaximal, complemented
6#11722830C218yesmaximal, direct product factor
7C2×#63926—31nomaximal
8C2×D3×#213—32nomaximal
9C22×D3×C3⋊D3—21no—
10C22×C3×D32—21no—
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic, complemented
14C2×C32×D12C2212yescomplemented
15C2×#6291—21no—
16C3⋊D3×D12—22no—
17C3⋊D3×D12C2214yescomplemented
18C2×C3×#625—21no—
19C3×#3802—22no—
20C3×#3802C2214yescomplemented
21C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
22#88808C2214yescomplemented
23#88790C2218yescomplemented
24C3×#3713C2218yescomplemented
25C2×D4×C3⋊D3—31no—
26C2×C3×D3×D4—32no—
27C2×#3802D311yescharacteristic, complemented
28C2×D3×D12—32no—
29C2×D3×D12—61no—
30C2×#3753—32no—
31C2×C3×#213—32no—
32C2×C3×#213—32no—
33#63926—38no—
34D3×#213—316no—
35C2×#3714—32no—
36C2×#3714—32no—
37C2×#3714—62no—
38C2×#3713D312yescomplemented
39C2×#3710—31no—
40C2×D3×C3⋊C8—32no—
41C22×C3⋊(C3⋊D3)—21no—
42C22×C3×C3⋊D3D411yescharacteristic
43C22×C32×D3—21no—
44C22×C32×D3—22no—
45C2×D3×C3⋊D3—24no—
46C2×D3×C3⋊D3—44no—
47C2×C3×D32—24no—
48C2×C3×D32—44no—
49C2×C33×C4C2311yescharacteristic
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))C2312yescomplemented
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4×C3⋊D3D412yes—
54C3×C4×C3⋊D3C2312yes—
55C32×D12—21no—
56C32×D12—22no—
57C32×D12C2314yescomplemented
58C32×D12C2312yescomplemented
59#6291—24no—
60C3×#625—24no—
61C3×C3⋊(C3⋊C8)C2314yescomplemented
62C22×#213—91no—
63C2×D3×D8—92no—
64C23×C3⋊D3—61no—
65C23×C3×D3—62no—
66C22×D32—21no—
67C22×D32—62no—
68C22×D32—62no—
69C2×C32×D4—31no—
70C2×C32×D4—32no—
71C2×#638—61no—
72D4×C3⋊D3—34no—
73D4×C3⋊D3—62no—
74C2×C4⋊(C3⋊D3)—32no—
75C2×C4⋊(C3⋊D3)—31no—
76C2×C4⋊(C3⋊D3)—61no—
77C2×C4⋊(C3⋊D3)—62no—
78C2×C4⋊(C3⋊D3)—121no—
79C2×C4×C3⋊D3C2×D311yescharacteristic
80C2×C3×#67—62no—
81C3×D3×D4—38no—
82C3×D3×D4—64no—
83C2×C3×D12—21no—
84C2×C3×D12—22no—
85C2×C3×D12C2×D312yescomplemented
86C2×C3×D12C2×D312yescomplemented
87C2×C3×D12C2×D312yescomplemented
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—61no—
90C2×#626—62no—
91C2×#626—62no—
92C2×#625—21no—
93#3802—22no—
94#3802C2×D314yescomplemented
95D3×D12—38no—
96D3×D12—64no—
97D3×D12—64no—
98D3×D12—64no—
99#3753—38no—
100C2×C3⋊(C3⋊C8)C2×D312yescomplemented
101C3×#213—38no—
102C3×#213—38no—
103C2×C3×C3⋊C8—32no—
104C2×C3×C3⋊C8—32no—
105C2×C3×C3⋊C8—62no—
106#3714—38no—
107#3714—38no—
108#3714—68no—
109#3713C2×D318yescomplemented
110#3710—34no—
111D3×C3⋊C8—38no—
112C22×C33C2×D411yescharacteristic
113C2×C3⋊(C3⋊D3)—22no—
114C2×C3⋊(C3⋊D3)—42no—
115C2×C3×C3⋊D3D814yescomplemented
116C2×C3×C3⋊D3C2×D412yes—
117C2×C32×D3—22no—
118C2×C32×D3—24no—
119C2×C32×D3—42no—
120C2×C32×D3—44no—
121D3×C3⋊D3—48no—
122C3×D32—48no—
123C33×C4C2×D411yescharacteristic
124C33×C4C2411yescharacteristic
125C3×C3⋊Dic3C2×D412yes—
126C22×C3×D4—91no—
127C2×D3×D4—32no—
128C2×D3×D4—92no—
129C2×D3×D4—181no—
130C22×D12—91no—
131C2×C3×D8—92no—
132C2×#213—32no—
133C2×#213—32no—
134C2×#213—92no—
135C2×#213—92no—
136C2×#213—98no—
137C2×#213—182no—
138C22×C3⋊C8—91no—
139D3×D8—916no—
140C2×D24—92no—
141C2×D3×C8—92no—
142C23×C32—61no—
143C23×C32—62no—
144C22×C3⋊D3—61no—
145C22×C3⋊D3—64no—
146C22×C3⋊D3—62no—
147C22×C3⋊D3—62no—
148C22×C3⋊D3—122no—
149C22×C3⋊D3—122no—
150C22×C3⋊D3—124no—
151C22×C3⋊D3#6711yescharacteristic
152C22×C3×D3—22no—
153C22×C3×D3—22no—
154C22×C3×D3—22no—
155C22×C3×D3—22no—
156C22×C3×D3—32no—
157C22×C3×D3—42no—
158C22×C3×D3—61no—
159C22×C3×D3—68no—
160C22×C3×D3—128no—
161C2×D32—24no—
162C2×D32—44no—
163C2×D32—68no—
164C2×D32—68no—
165C2×D32—128no—
166C2×D32—128no—
167C32×D4—32no—
168C32×D4—34no—
169C32×D4—61no—
170C32×D4—62no—
171C2×C32×C4—22no—
172C2×C32×C4—21no—
173C2×C32×C4—41no—
174C2×C32×C4C22×D312yes—
175C2×C32×C4C22×D311yescharacteristic
176#638—64no—
177C2×C3⋊Dic3#6711yescharacteristic
178C4⋊(C3⋊D3)—32no—
179C4⋊(C3⋊D3)—34no—
180C4⋊(C3⋊D3)—62no—
181C4⋊(C3⋊D3)—64no—
182C4⋊(C3⋊D3)—61no—
183C4⋊(C3⋊D3)—62no—
184C4⋊(C3⋊D3)—62no—
185C4⋊(C3⋊D3)—122no—
186C4⋊(C3⋊D3)—122no—
187C4⋊(C3⋊D3)—122no—
188C4×C3⋊D3#6712yes—
189C4×C3⋊D3C22×D312yes—
190C3×#67—68no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—61no—
193C3×D12—22no—
194C3×D12—22no—
195C3×D12—22no—
196C3×D12—22no—
197C3×D12—22no—
198C3×D12—24no—
199C3×D12—42no—
200C3×D12C22×D314yescomplemented
201C3×D12C22×D314yescomplemented
202C3×D12C22×D314yescomplemented
203C3×C4×D3—34no—
204C3×C4×D3—34no—
205C3×C4×D3—62no—
206C3×C4×D3—62no—
207#626—68no—
208#626—68no—
209#625—24no—
210C3⋊(C3⋊C8)C22×D314yescomplemented
211C3×C3⋊C8—34no—
212C3×C3⋊C8—34no—
213C3×C3⋊C8—64no—
214C22×D8—271noSylow
215C2×C33C2×D812yescomplemented
216C2×C33C22×D411yescharacteristic
217C3⋊(C3⋊D3)—42no—
218C3×C3⋊D3C2×D814yescomplemented
219C32×D3—42no—
220C32×D3—44no—
221C24×C3—181no—
222C23×D3—62no—
223C23×D3—181no—
224C23×D3—182no—
225C23×D3—182no—
226C2×C3×D4—32no—
227C2×C3×D4—32no—
228C2×C3×D4—32no—
229C2×C3×D4—61no—
230C2×C3×D4—62no—
231C2×C3×D4—94no—
232C2×C3×D4—182no—
233C2×C3×D4—181no—
234C22×C3×C4—91no—
235C2×#67—62no—
236C2×#67—182no—
237C2×#67—182no—
238D3×D4—38no—
239D3×D4—64no—
240D3×D4—98no—
241D3×D4—184no—
242D3×D4—184no—
243D3×D4—184no—
244C2×D12—92no—
245C2×D12—91no—
246C2×D12—94no—
247C2×D12—181no—
248C2×D12—182no—
249C2×D12—182no—
250C2×D12—181no—
251C2×D12—361no—
252C2×D12D3212yescomplemented
253C2×D12D3211yescharacteristic, complemented
254C2×C4×D3—32no—
255C2×C4×D3—61no—
256C3×D8—98no—
257C2×C3×C8—92no—
258#213—38no—
259#213—38no—
260#213—98no—
261#213—98no—
262#213—188no—
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—21no—
271C22×C32—41no—
272C22×C32—62no—
273C22×C32—64no—
274C22×C32—122no—
275C22×C32—124no—
276C22×C32D3×D412yes—
277C22×C32C2×#6711yescharacteristic
278C2×C3⋊D3—62no—
279C2×C3⋊D3—64no—
280C2×C3⋊D3—64no—
281C2×C3⋊D3—124no—
282C2×C3⋊D3—124no—
283C2×C3⋊D3—122no—
284C2×C3⋊D3—128no—
285C2×C3⋊D3—124no—
286C2×C3⋊D3—124no—
287C2×C3⋊D3—244no—
288C2×C3⋊D3—244no—
289C2×C3⋊D3#21314yescomplemented
290C2×C3⋊D3C2×#6712yes—
291C2×C3×D3—24no—
292C2×C3×D3—24no—
293C2×C3×D3—24no—
294C2×C3×D3—24no—
295C2×C3×D3—38no—
296C2×C3×D3—34no—
297C2×C3×D3—44no—
298C2×C3×D3—44no—
299C2×C3×D3—44no—
300C2×C3×D3—44no—
301C2×C3×D3—44no—
302C2×C3×D3—64no—
303C2×C3×D3—62no—
304C2×C3×D3—84no—
305C2×C3×D3—1216no—
306D32—48no—
307D32—1216no—
308D32—1216no—
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—62no—
330C23×C3—62no—
331C23×C3—62no—
332C23×C3—62no—
333C23×C3—91no—
334C23×C3—122no—
335C23×C3—184no—
336C23×C3—364no—
337C22×D3—22no—
338C22×D3—21no—
339C22×D3—32no—
340C22×D3—61no—
341C22×D3—68no—
342C22×D3—128no—
343C22×D3—184no—
344C22×D3—188no—
345C22×D3—188no—
346C22×D3—182no—
347C22×D3—182no—
348C22×D3—181no—
349C22×D3—362no—
350C22×D3—362no—
351C22×D3—368no—
352C22×D3—364no—
353C22×D3—368no—
354C3×D4—34no—
355C3×D4—34no—
356C3×D4—34no—
357C3×D4—62no—
358C3×D4—64no—
359C3×D4—62no—
360C3×D4—62no—
361C3×D4—62no—
362C3×D4—62no—
363C3×D4—122no—
364C3×D4—184no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—41no—
368C2×C3×C4—91no—
369C2×C3×C4—92no—
370C2×C3×C4—92no—
371C2×C3×C4C2×D3212yes—
372C2×C3×C4C2×D3211yescharacteristic
373#67—68no—
374#67—188no—
375#67—188no—
376C2×Dic3—32no—
377C2×Dic3—61no—
378D12—22no—
379D12—21no—
380D12—94no—
381D12—92no—
382D12—182no—
383D12—184no—
384D12—182no—
385D12—184no—
386D12—182no—
387D12—181no—
388D12—362no—
389D12—362no—
390D12—362no—
391D12C2×D3214yescomplemented
392D12C2×D3212yescomplemented
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—122no—
408C2×C32—124no—
409C2×C32D3×D814yescomplemented
410C2×C32C2×#21312yescomplemented
411C2×C32C2×D3×D412yes—
412C2×C32C22×#6711yescharacteristic
413C3⋊D3—122no—
414C3⋊D3—124no—
415C3⋊D3—124no—
416C3⋊D3—244no—
417C3⋊D3—244no—
418C3⋊D3C2×#21314yescomplemented
419C3×D3—38no—
420C3×D3—44no—
421C3×D3—44no—
422C3×D3—44no—
423C3×D3—44no—
424C3×D3—64no—
425C3×D3—84no—
426C24—181no—
427C24—541no—
428C2×D4—31no—
429C2×D4—32no—
430C2×D4—94no—
431C2×D4—182no—
432C2×D4—181no—
433C2×D4—274no—
434C2×D4—271no—
435C2×D4—542no—
436C2×D4—541no—
437C22×C4—91no—
438D8—98no—
439D8—278no—
440C2×C8—92no—
441C2×C8—274no—
442C22×C3—21no—
443C22×C3—22no—
444C22×C3—41no—
445C22×C3—64no—
446C22×C3—64no—
447C22×C3—64no—
448C22×C3—64no—
449C22×C3—92no—
450C22×C3—94no—
451C22×C3—124no—
452C22×C3—124no—
453C22×C3—124no—
454C22×C3—124no—
455C22×C3—124no—
456C22×C3—244no—
457C22×C3—368no—
458C22×C3D3×#6712yes—
459C22×C3#381111yescharacteristic
460C2×D3—24no—
461C2×D3—22no—
462C2×D3—34no—
463C2×D3—38no—
464C2×D3—44no—
465C2×D3—42no—
466C2×D3—62no—
467C2×D3—64no—
468C2×D3—1216no—
469C2×D3—184no—
470C2×D3—184no—
471C2×D3—182no—
472C2×D3—364no—
473C2×D3—364no—
474C2×D3—3616no—
475C2×D3—368no—
476C2×D3—3616no—
477C2×D3—364no—
478C2×D3—364no—
479C2×D3—362no—
480C2×D3—724no—
481C2×D3—724no—
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—62no—
501C23—61no—
502C23—91no—
503C23—184no—
504C23—364no—
505C23—544no—
506C23—541no—
507C23—1084no—
508D4—32no—
509D4—34no—
510D4—62no—
511D4—61no—
512D4—184no—
513D4—272no—
514D4—544no—
515D4—541no—
516C2×C4—91no—
517C2×C4—92no—
518C2×C4—92no—
519C2×C4D3311yescharacteristic
520C8—94no—
521C2×C3—22no—
522C2×C3—24no—
523C2×C3—21no—
524C2×C3—22no—
525C2×C3—42no—
526C2×C3—41no—
527C2×C3—94no—
528C2×C3—124no—
529C2×C3—124no—
530C2×C3—124no—
531C2×C3—124no—
532C2×C3—244no—
533C2×C3D3×#21314yescomplemented
534C2×C3#6392612yescomplemented
535C2×C3C2×D3×#6712yes—
536C2×C3C2×#381111yescharacteristic
537D3—38no—
538D3—44no—
539D3—42no—
540D3—64no—
541D3—364no—
542D3—364no—
543D3—362no—
544D3—724no—
545D3—724no—
546C22—64no—
547C22—62no—
548C22—94no—
549C22—92no—
550C22—124no—
551C22—122no—
552C22—368no—
553C22—542no—
554C22—1088no—
555C22—1082no—
556C22#8895211yescharacteristic
557C4—92no—
558C4#8895211yescharacteristic
559C4C2×D3311yescharacteristic
560C3—21no—
561C3—22no—
562C3—41no—
563C3C2×D3×#21312yescomplemented
564C3C2×#6392611yescharacteristic, complemented
565C2—94no—
566C2—124no—
567C2—122no—
568C2—1082no—
569C2#1172283012yesdirect product factor
570C2C2×#8895211yescharacteristic
subgroupG/H
centerC22#88952
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
hypercenterC2×C4D33
hypocenterC33C22×D8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×D12C2semidirect
2. C2×C3×#3802C2semidirect
3. C2×#88808C2semidirect
4. C2×#88790C2semidirect
5. C2×C3×#3713C2semidirect
6. #11722830C2direct (N × H)
11. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
13. C2×C32×D12C22semidirect
14. C2×C32×D12C22semidirect
17. C3⋊D3×D12C22semidirect
20. C3×#3802C22semidirect
21. C2×C3×C3⋊(C3⋊C8)C22semidirect
22. #88808C22semidirect
23. #88790C22semidirect
24. C3×#3713C22semidirect
27. C2×#3802D3semidirect
38. C2×#3713D3semidirect
51. C4⋊(C3⋊(C3⋊D3))C23semidirect
57. C32×D12C23semidirect
58. C32×D12C23semidirect
61. C3×C3⋊(C3⋊C8)C23semidirect
85. C2×C3×D12C2×D3semidirect
86. C2×C3×D12C2×D3semidirect
87. C2×C3×D12C2×D3semidirect
94. #3802C2×D3semidirect
100. C2×C3⋊(C3⋊C8)C2×D3semidirect
109. #3713C2×D3semidirect
115. C2×C3×C3⋊D3D8semidirect
200. C3×D12C22×D3semidirect
201. C3×D12C22×D3semidirect
202. C3×D12C22×D3semidirect
210. C3⋊(C3⋊C8)C22×D3semidirect
215. C2×C33C2×D8semidirect
218. C3×C3⋊D3C2×D8semidirect
252. C2×D12D32semidirect
253. C2×D12D32semidirect
289. C2×C3⋊D3#213semidirect
328. C33C22×D8semidirect
391. D12C2×D32semidirect
392. D12C2×D32semidirect
409. C2×C32D3×D8semidirect
410. C2×C32C2×#213semidirect
418. C3⋊D3C2×#213semidirect
498. C32C2×D3×D8semidirect
499. C32C22×#213semidirect
533. C2×C3D3×#213semidirect
534. C2×C3#63926semidirect
563. C3C2×D3×#213semidirect
564. C3C2×#63926semidirect
569. C2#11722830direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×D122
C2×C3×#38022
C2×#888082
C2×#887902
C2×C3×#37132
#117228302
C2×#639263
C2×D3×#2133