Finite Group Databasebeta

← C2×#11720982 #13150449

Subgroups — C2×#11720982 #13150449

Order 1728.

1370 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D36C211yesmaximal, characteristic, complemented
2C22×#6194C211yesmaximal, characteristic, complemented
3C22×#6194C211yesmaximal, characteristic, complemented
4C22×Dic3×D9C211yesmaximal, characteristic, complemented
5C2×#88534C211yesmaximal, characteristic, complemented
6C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
8#11720982C218yesmaximal, direct product factor
9C2×#63885—31nomaximal
10C2×#63446—31nomaximal
11C23×D3×C9C2211yescharacteristic
12C23×C3×D9C2211yescharacteristic
13C23×C3×D9C2211yescharacteristic
14C22×C3×C4×C9C2211yescharacteristic
15C22×C3⋊Dic9C2211yescharacteristic
16C22×C3⋊Dic9C2211yescharacteristic
17C22×C9×Dic3C2211yescharacteristic
18C2×C3×D36—24no—
19C2×C3×D36C2214yescomplemented
20C2×#6194—24no—
21C2×#6194—24no—
22C2×#6194C2214yescomplemented
23C2×#6194C2214yescomplemented
24C2×Dic3×D9—24no—
25C2×Dic3×D9C2214yescomplemented
26#88534C2214yescomplemented
27C9×(C2×D3)⋊C4C2214yescomplemented
28(C2×D9)⋊Dic3C2214yescomplemented
29C22×C3×D12—31no—
30C22×#625—31no—
31C22×#625—31no—
32C22×D3×Dic3—31no—
33C2×#3750—31no—
34C2×C3×(C2×D3)⋊C4D311yescharacteristic
35C2×(C2×D3)⋊Dic3—31no—
36#63885—38no—
37C22×#611—31no—
38C22×#611—31no—
39C22×D36D311yescharacteristic, complemented
40C22×C4×D9—31no—
41C2×C9×C22⋊C4—31no—
42C2×(C2×D9)⋊C4—31no—
43C2×Dic9⋊C4—31no—
44C2×Dic9⋊C4—61no—
45#63446—38no—
46C23×C3×C9C2311yescharacteristic
47C22×D3×C9—22no—
48C22×D3×C9—22no—
49C22×D3×C9—22no—
50C22×D3×C9C2312yes—
51C22×C3×D9—22no—
52C22×C3×D9—22no—
53C22×C3×D9—22no—
54C22×C3×D9—22no—
55C22×C3×D9—22no—
56C22×C3×D9D414yescomplemented
57C22×C3×D9C2312yes—
58C22×C3×D9C2312yes—
59C2×C3×C4×C9—22no—
60C2×C3×C4×C9C2312yes—
61C2×C3⋊Dic9—22no—
62C2×C3⋊Dic9D414yescomplemented
63C2×C3⋊Dic9C2312yes—
64C2×C3⋊Dic9C2312yes—
65C2×C9×Dic3—22no—
66C2×C9×Dic3C2312yes—
67C3×D36—28no—
68#6194—28no—
69#6194—28no—
70Dic3×D9—28no—
71C2×#939—91no—
72C2×#884—91no—
73C23×C3×D3—31no—
74C23×C3×D3—31no—
75C23×C3×D3C2×D311yescharacteristic
76C22×C32×C4C2×D311yescharacteristic
77C22×C3⋊Dic3—31no—
78C22×C3⋊Dic3—31no—
79C22×C3×Dic3C2×D311yescharacteristic
80C2×C3×D12—34no—
81C2×C3×D12—64no—
82C2×#625—34no—
83C2×#625—34no—
84C2×#625—64no—
85C2×#625—64no—
86C2×D3×Dic3—34no—
87C2×D3×Dic3—64no—
88C24×C9—31no—
89C23×D9C2×D311yescharacteristic
90C23×D9C2×D311yescharacteristic
91#3750—34no—
92C3×(C2×D3)⋊C4C2×D314yes—
93(C2×D3)⋊Dic3—34no—
94C22×C4×C9—21no—
95C22×C4×C9—31no—
96C22×C4×C9C2×D311yescharacteristic
97C2×#611—34no—
98C2×#611—34no—
99C2×#611—64no—
100C2×#611—64no—
101C22×Dic9—31no—
102C22×Dic9—31no—
103C22×Dic9—61no—
104C22×Dic9—61no—
105C2×D36—24no—
106C2×D36C2×D314yescomplemented
107C2×C4×D9—34no—
108C2×C4×D9—64no—
109C9×C22⋊C4—34no—
110(C2×D9)⋊C4—34no—
111Dic9⋊C4—34no—
112Dic9⋊C4—64no—
113C22×C3×C9C2×D412yes—
114C22×C3×C9C2×D412yes—
115C22×C3×C9#4112yes—
116C22×C3×C9C2411yescharacteristic
117C2×D3×C9—22no—
118C2×D3×C9—22no—
119C2×D3×C9—22no—
120C2×D3×C9—44no—
121C2×C3×D9—28no—
122C2×C3×D9—22no—
123C2×C3×D9—22no—
124C2×C3×D9—22no—
125C2×C3×D9—22no—
126C2×C3×D9—22no—
127C2×C3×D9—44no—
128C2×C3×D9C2×D414yescomplemented
129C3×C4×C9—22no—
130C3⋊Dic9—22no—
131C3⋊Dic9C2×D414yescomplemented
132C9×Dic3—22no—
133C22×C3×D4—91no—
134C22×#67—91no—
135C22×#67—91no—
136C22×#67—91no—
137C22×#67—91no—
138C23×Dic3—91no—
139C22×D12—31no—
140C22×C4×D3—91no—
141C2×C3×C22⋊C4—31no—
142C2×#217—91no—
143#939—98no—
144C2×(C2×D3)⋊C4—91no—
145C2×(C2×D3)⋊C4D911yescharacteristic, complemented
146C2×Dic3⋊C4—91no—
147C2×Dic3⋊C4—91no—
148C2×Dic3⋊C4—181no—
149#884—98no—
150C23×C32C22×D311yescharacteristic
151C22×C3×D3—22no—
152C22×C3×D3—22no—
153C22×C3×D3—22no—
154C22×C3×D3—34no—
155C22×C3×D3—32no—
156C22×C3×D3—32no—
157C22×C3×D3—62no—
158C22×C3×D3—62no—
159C22×C3×D3—62no—
160C22×C3×D3—62no—
161C22×C3×D3—62no—
162C22×C3×D3C22×D312yes—
163C2×C32×C4—22no—
164C2×C32×C4C22×D312yes—
165C2×C3⋊Dic3—34no—
166C2×C3⋊Dic3—32no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—62no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3C22×D312yes—
171C3×D12—68no—
172#625—68no—
173#625—68no—
174D3×Dic3—68no—
175C23×C9—21no—
176C23×C9—32no—
177C23×C9—62no—
178C23×C9—62no—
179C23×C9—62no—
180C23×C9C22×D311yescharacteristic
181C22×D9—22no—
182C22×D9—22no—
183C22×D9—22no—
184C22×D9—22no—
185C22×D9—22no—
186C22×D9#6714yescomplemented
187C22×D9C22×D312yes—
188C22×D9C22×D312yes—
189C2×C4×C9—22no—
190C2×C4×C9—22no—
191C2×C4×C9—32no—
192C2×C4×C9—42no—
193C2×C4×C9—62no—
194C2×C4×C9C22×D312yes—
195#611—68no—
196#611—68no—
197C2×Dic9—34no—
198C2×Dic9—32no—
199C2×Dic9—32no—
200C2×Dic9—62no—
201C2×Dic9—64no—
202C2×Dic9—62no—
203C2×Dic9—62no—
204C2×Dic9—122no—
205D36—28no—
206C4×D9—68no—
207C2×#121—271noSylow
208C2×C3×C9#12114yescomplemented
209C2×C3×C9C22×D411yescharacteristic
210C2×C3×C9C22×D411yescharacteristic
211C2×C3×C9C2×#4111yescharacteristic
212D3×C9—42no—
213C3×D9—24no—
214C3×D9—42no—
215C24×C3—31no—
216C24×C3—91no—
217C24×C3—91no—
218C23×D3—31no—
219C23×D3—31no—
220C23×D3C2×D911yescharacteristic
221C2×C3×D4—94no—
222C2×C3×D4—184no—
223C22×C3×C4—21no—
224C22×C3×C4—31no—
225C22×C3×C4C2×D911yescharacteristic
226C22×C3×C4D3211yescharacteristic
227C2×#67—94no—
228C2×#67—94no—
229C2×#67—94no—
230C2×#67—94no—
231C2×#67—184no—
232C2×#67—184no—
233C2×#67—184no—
234C2×#67—184no—
235C22×Dic3—91no—
236C22×Dic3—91no—
237C22×Dic3—91no—
238C22×Dic3—91no—
239C22×Dic3—94no—
240C22×Dic3—181no—
241C22×Dic3—181no—
242C22×Dic3—184no—
243C22×Dic3C2×D911yescharacteristic
244C2×D12—34no—
245C2×D12—64no—
246C2×C4×D3—94no—
247C2×C4×D3—184no—
248C3×C22⋊C4—34no—
249#217—94no—
250(C2×D3)⋊C4—94no—
251(C2×D3)⋊C4C2×D914yescomplemented
252Dic3⋊C4—94no—
253Dic3⋊C4—94no—
254Dic3⋊C4—184no—
255C22×C32#23512yes—
256C22×C32D3×D412yes—
257C22×C32D3×D412yes—
258C22×C32C23×D311yescharacteristic
259C2×C3×D3—22no—
260C2×C3×D3—22no—
261C2×C3×D3—22no—
262C2×C3×D3—34no—
263C2×C3×D3—44no—
264C2×C3×D3—68no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—124no—
271C32×C4—22no—
272C3⋊Dic3—34no—
273C3⋊Dic3—62no—
274C3×Dic3—22no—
275C22×C9—21no—
276C22×C9—22no—
277C22×C9—22no—
278C22×C9—22no—
279C22×C9—62no—
280C22×C9—62no—
281C22×C9—62no—
282C22×C9—124no—
283C22×C9D3×D412yes—
284C22×C9#23712yes—
285C22×C9C2×#6712yes—
286C22×C9C23×D311yescharacteristic
287C2×D9—28no—
288C2×D9—22no—
289C2×D9—22no—
290C2×D9—22no—
291C2×D9—22no—
292C2×D9—22no—
293C2×D9—44no—
294C2×D9C2×#6714yescomplemented
295C4×C9—22no—
296C4×C9—42no—
297C4×C9—62no—
298Dic9—34no—
299Dic9—64no—
300Dic9—62no—
301Dic9—122no—
302C22×D4—91no—
303C22×D4—271no—
304C22×D4—271no—
305C23×C4—271no—
306#121—278no—
307C2×C4⋊C4—271no—
308C2×C22⋊C4—31no—
309C2×C22⋊C4—271no—
310C3×C9C2×#12111yescharacteristic, complemented, Sylow
311C23×C3—21no—
312C23×C3—32no—
313C23×C3—62no—
314C23×C3—62no—
315C23×C3—62no—
316C23×C3—94no—
317C23×C3—92no—
318C23×C3—92no—
319C23×C3—182no—
320C23×C3—182no—
321C23×C3—182no—
322C23×C3—182no—
323C23×C3—182no—
324C23×C3C22×D911yescharacteristic
325C23×C3C2×D3211yescharacteristic
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—32no—
330C22×D3—34no—
331C22×D3—32no—
332C22×D3—62no—
333C22×D3—62no—
334C22×D3—62no—
335C22×D3—62no—
336C22×D3—62no—
337C22×D3C22×D912yes—
338C3×D4—188no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—32no—
343C2×C3×C4—42no—
344C2×C3×C4—62no—
345C2×C3×C4C22×D912yes—
346C2×C3×C4C2×D3212yes—
347#67—188no—
348#67—188no—
349#67—188no—
350#67—188no—
351C2×Dic3—22no—
352C2×Dic3—92no—
353C2×Dic3—94no—
354C2×Dic3—92no—
355C2×Dic3—94no—
356C2×Dic3—92no—
357C2×Dic3—92no—
358C2×Dic3—182no—
359C2×Dic3—184no—
360C2×Dic3—182no—
361C2×Dic3—188no—
362C2×Dic3—182no—
363C2×Dic3—182no—
364C2×Dic3—362no—
365C2×Dic3C22×D912yes—
366D12—68no—
367C4×D3—188no—
368C2×C32#88414yes—
369C2×C32C2×#23511yescharacteristic
370C2×C32C2×D3×D411yescharacteristic
371C2×C32C2×D3×D411yescharacteristic
372C3×D3—42no—
373C3×D3—64no—
374C3×D3—122no—
375C2×C9—21no—
376C2×C9—21no—
377C2×C9—21no—
378C2×C9—24no—
379C2×C9—122no—
380C2×C9#93914yescomplemented
381C2×C9C2×D3×D411yescharacteristic
382C2×C9C2×#23711yescharacteristic
383C2×C9C22×#6711yescharacteristic
384D9—24no—
385D9—42no—
386C24—31no—
387C24—91no—
388C24—91no—
389C2×D4—94no—
390C2×D4—184no—
391C2×D4—274no—
392C2×D4—274no—
393C2×D4—544no—
394C2×D4—544no—
395C22×C4—31no—
396C22×C4—274no—
397C22×C4—271no—
398C22×C4—271no—
399C22×C4—544no—
400C22×C4D3×D911yescharacteristic
401C4⋊C4—274no—
402C22⋊C4—34no—
403C22⋊C4—274no—
404C22×C3—22no—
405C22×C3—21no—
406C22×C3—22no—
407C22×C3—22no—
408C22×C3—62no—
409C22×C3—62no—
410C22×C3—62no—
411C22×C3—94no—
412C22×C3—124no—
413C22×C3—188no—
414C22×C3—182no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3—182no—
419C22×C3—364no—
420C22×C3#369712yes—
421C22×C3D4×D912yes—
422C22×C3D4×D912yes—
423C22×C3C23×D911yescharacteristic
424C22×C3#379512yes—
425C22×C3#380212yes—
426C22×C3D3×#6712yes—
427C22×C3C22×D3211yescharacteristic
428C2×D3—22no—
429C2×D3—22no—
430C2×D3—22no—
431C2×D3—34no—
432C2×D3—44no—
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—68no—
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—62no—
439C2×D3—124no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—42no—
443C3×C4—62no—
444Dic3—22no—
445Dic3—94no—
446Dic3—94no—
447Dic3—184no—
448Dic3—182no—
449Dic3—182no—
450Dic3—362no—
451C32C2×#88411yescharacteristic
452C9—21no—
453C9C2×#93911yescharacteristic, complemented
454C23—32no—
455C23—62no—
456C23—62no—
457C23—62no—
458C23—92no—
459C23—94no—
460C23—92no—
461C23—182no—
462C23—182no—
463C23—182no—
464C23—182no—
465C23—182no—
466C23C2×D3×D911yescharacteristic
467D4—188no—
468D4—548no—
469D4—548no—
470C2×C4—22no—
471C2×C4—32no—
472C2×C4—62no—
473C2×C4—272no—
474C2×C4—274no—
475C2×C4—272no—
476C2×C4—548no—
477C2×C4—542no—
478C2×C4C2×D3×D912yes—
479C2×C3—24no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—122no—
484C2×C3—184no—
485C2×C3—362no—
486C2×C3#6344614yescomplemented
487C2×C3C2×#369711yescharacteristic
488C2×C3C2×D4×D911yescharacteristic
489C2×C3C2×D4×D911yescharacteristic
490C2×C3#6388514yes—
491C2×C3C2×#379511yescharacteristic
492C2×C3C2×#380211yescharacteristic
493C2×C3C2×D3×#6711yescharacteristic
494D3—42no—
495D3—64no—
496D3—122no—
497C22—62no—
498C22—62no—
499C22—62no—
500C22—94no—
501C22—124no—
502C22—188no—
503C22—182no—
504C22—182no—
505C22—182no—
506C22—182no—
507C22—182no—
508C22—364no—
509C22#8864112yes—
510C22#8864612yes—
511C22D9×#6712yes—
512C22C22×D3×D911yescharacteristic
513C4—22no—
514C4—62no—
515C4—274no—
516C4—542no—
517C3—21no—
518C3C2×#6344611yescharacteristic, complemented
519C3C2×#6388511yescharacteristic
520C2—122no—
521C2—184no—
522C2—362no—
523C2#1172098214yesdirect product factor
524C2C2×#8864111yescharacteristic
525C2C2×#8864611yescharacteristic
526C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
hypercenterC22×C4D3×D9
hypocenterC3×C9C2×#121
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×D36C2semidirect
2. C22×#6194C2semidirect
3. C22×#6194C2semidirect
4. C22×Dic3×D9C2semidirect
5. C2×#88534C2semidirect
6. C2×C9×(C2×D3)⋊C4C2semidirect
7. C2×(C2×D9)⋊Dic3C2semidirect
8. #11720982C2direct (N × H)
19. C2×C3×D36C22semidirect
22. C2×#6194C22semidirect
23. C2×#6194C22semidirect
25. C2×Dic3×D9C22semidirect
26. #88534C22semidirect
27. C9×(C2×D3)⋊C4C22semidirect
28. (C2×D9)⋊Dic3C22semidirect
39. C22×D36D3semidirect
56. C22×C3×D9D4semidirect
62. C2×C3⋊Dic9D4semidirect
106. C2×D36C2×D3semidirect
128. C2×C3×D9C2×D4semidirect
131. C3⋊Dic9C2×D4semidirect
145. C2×(C2×D3)⋊C4D9semidirect
186. C22×D9#67semidirect
208. C2×C3×C9#121semidirect
251. (C2×D3)⋊C4C2×D9semidirect
294. C2×D9C2×#67semidirect
310. C3×C9C2×#121semidirect
380. C2×C9#939semidirect
453. C9C2×#939semidirect
486. C2×C3#63446semidirect
518. C3C2×#63446semidirect
523. C2#11720982direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×D362
C22×#61942
C22×Dic3×D92
C2×#885342
C2×C9×(C2×D3)⋊C42
C2×(C2×D9)⋊Dic32
#117209822
C2×#638853
C2×#634463