Finite Group Databasebeta

← C2×#11721256 #13151989

Subgroups — C2×#11721256 #13151989

Order 1728.

1514 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6233C211yesmaximal, characteristic, complemented
2C22×C3×#611C211yesmaximal, characteristic, complemented
3C22×#6192C211yesmaximal, characteristic, complemented
4C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
5C2×C9×#217C211yesmaximal, characteristic, complemented
6C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
7#11721256C218yesmaximal, direct product factor
8C2×Dic9⋊Dic3C211yesmaximal, characteristic, complemented
9C2×#63958—31nomaximal
10C2×#63446—31nomaximal
11C24×C3×C9C2211yescharacteristic
12C23×C3⋊D9C2211yescharacteristic
13C23×C3×D9C2211yescharacteristic
14C2×#6233—24no—
15C2×#6233C2214yescomplemented
16C22×C3⋊Dic9C2211yescharacteristic
17C2×C3×#611—24no—
18C2×C3×#611C2214yescomplemented
19C22×C9×Dic3C2211yescharacteristic
20C22×C9×Dic3C2211yescharacteristic
21C22×C3×Dic9C2211yescharacteristic
22C2×#6192—24no—
23C2×#6192C2214yescomplemented
24C2×(C3⋊D9)⋊C4—24no—
25C2×(C3⋊D9)⋊C4C2214yescomplemented
26C9×#217C2214yescomplemented
27(C2×D9)⋊Dic3C2214yescomplemented
28Dic9⋊Dic3C2214yescomplemented
29C22×#638—31no—
30C22×C3×#67—31no—
31C22×#626—31no—
32C22×#624—31no—
33C2×C3×#217D311yescharacteristic
34C2×Dic3⋊Dic3—31no—
35C2×(C2×D3)⋊Dic3—31no—
36#63958—38no—
37C22×#611—31no—
38C22×#611—61no—
39C22×#611D311yescharacteristic, complemented
40C22×D36—31no—
41C22×C4×D9—31no—
42C2×C9×C22⋊C4—31no—
43C2×(C2×D9)⋊C4—31no—
44C2×Dic9⋊C4—31no—
45#63446—38no—
46C23×C3×C9—22no—
47C23×C3×C9—22no—
48C23×C3×C9—22no—
49C23×C3×C9C2312yes—
50C23×C3×C9C2311yescharacteristic
51C22×C3⋊D9—22no—
52C22×C3⋊D9—22no—
53C22×C3⋊D9D414yescomplemented
54C22×C3⋊D9C2312yes—
55C22×C3×D9—22no—
56C22×C3×D9—22no—
57C22×C3×D9—22no—
58C22×C3×D9C2312yes—
59#6233—28no—
60C2×C3⋊Dic9—22no—
61C2×C3⋊Dic9C2312yes—
62C3×#611—28no—
63C2×C9×Dic3—22no—
64C2×C9×Dic3—22no—
65C2×C9×Dic3C2312yes—
66C2×C9×Dic3C2312yes—
67C2×C3×Dic9D414yescomplemented
68C2×C3×Dic9C2312yes—
69#6192—28no—
70(C3⋊D9)⋊C4—28no—
71C2×#938—91no—
72C2×#884—91no—
73C24×C32C2×D311yescharacteristic
74C23×C3⋊D3—31no—
75C23×C3×D3—31no—
76C2×#638—34no—
77C2×#638—64no—
78C22×C3⋊Dic3—31no—
79C2×C3×#67—34no—
80C2×C3×#67—64no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3C2×D311yescharacteristic
83C22×C3×Dic3C2×D311yescharacteristic
84C2×#626—34no—
85C2×#626—64no—
86C2×#624—34no—
87C2×#624—64no—
88C24×C9—21no—
89C24×C9C2×D311yescharacteristic
90C23×D9—31no—
91C23×D9—61no—
92C23×D9C2×D311yescharacteristic
93C3×#217C2×D314yes—
94Dic3⋊Dic3—34no—
95(C2×D3)⋊Dic3—34no—
96C22×C4×C9—31no—
97C22×C4×C9—31no—
98C2×#611—24no—
99C2×#611—34no—
100C2×#611—64no—
101C2×#611—64no—
102C2×#611—68no—
103C2×#611C2×D314yescomplemented
104C22×Dic9—31no—
105C22×Dic9—61no—
106C22×Dic9C2×D311yescharacteristic
107C2×D36—34no—
108C2×D36—64no—
109C2×C4×D9—34no—
110C2×C4×D9—64no—
111C9×C22⋊C4—34no—
112(C2×D9)⋊C4—34no—
113Dic9⋊C4—34no—
114C22×C3×C9—22no—
115C22×C3×C9—22no—
116C22×C3×C9—22no—
117C22×C3×C9—44no—
118C22×C3×C9C2×D412yes—
119C22×C3×C9C2×D412yes—
120C22×C3×C9#4112yes—
121C22×C3×C9C2411yescharacteristic
122C2×C3⋊D9—28no—
123C2×C3⋊D9—22no—
124C2×C3⋊D9—22no—
125C2×C3⋊D9C2×D414yescomplemented
126C2×C3×D9—22no—
127C2×C3×D9—22no—
128C2×C3×D9—22no—
129C2×C3×D9—44no—
130C3⋊Dic9—22no—
131C9×Dic3—22no—
132C9×Dic3—22no—
133C3×Dic9C2×D414yescomplemented
134C22×C3×D4—91no—
135C22×#67—31no—
136C22×#67—91no—
137C22×#67—91no—
138C22×#67—91no—
139C22×#67—181no—
140C22×D12—91no—
141C22×C4×D3—91no—
142C22×C4×D3—91no—
143C2×C3×C22⋊C4—31no—
144C2×#217—91no—
145C2×#217D911yescharacteristic, complemented
146#938—98no—
147C2×(C2×D3)⋊C4—91no—
148C2×C4⋊Dic3—91no—
149C2×Dic3⋊C4—91no—
150#884—98no—
151C23×C32—22no—
152C23×C32—22no—
153C23×C32—22no—
154C23×C32C22×D312yes—
155C23×C32C22×D311yescharacteristic
156C22×C3⋊D3—34no—
157C22×C3⋊D3—32no—
158C22×C3⋊D3—62no—
159C22×C3⋊D3—62no—
160C22×C3×D3—32no—
161C22×C3×D3—62no—
162C22×C3×D3—62no—
163C22×C3×D3—62no—
164#638—68no—
165C2×C3⋊Dic3—32no—
166C2×C3⋊Dic3—62no—
167C3×#67—68no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—34no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3C22×D312yes—
173C2×C3×Dic3C22×D312yes—
174#626—68no—
175#624—68no—
176C23×C9—24no—
177C23×C9—22no—
178C23×C9—22no—
179C23×C9—22no—
180C23×C9—21no—
181C23×C9—22no—
182C23×C9—42no—
183C23×C9—42no—
184C23×C9C22×D312yes—
185C23×C9C22×D311yescharacteristic
186C22×D9—22no—
187C22×D9—22no—
188C22×D9—22no—
189C22×D9—34no—
190C22×D9—32no—
191C22×D9—64no—
192C22×D9—62no—
193C22×D9—62no—
194C22×D9—62no—
195C22×D9—122no—
196C22×D9—122no—
197C22×D9C22×D312yes—
198C2×C4×C9—32no—
199C2×C4×C9—32no—
200C2×C4×C9—62no—
201C2×C4×C9—62no—
202#611—28no—
203#611—68no—
204#611—616no—
205C2×Dic9—32no—
206C2×Dic9—62no—
207C2×Dic9—64no—
208C2×Dic9—62no—
209C2×Dic9#6714yescomplemented
210C2×Dic9C22×D312yes—
211D36—68no—
212C4×D9—68no—
213C2×#121—271noSylow
214C2×C3×C9—42no—
215C2×C3×C9#12114yescomplemented
216C2×C3×C9C22×D411yescharacteristic
217C2×C3×C9C22×D411yescharacteristic
218C2×C3×C9C2×#4111yescharacteristic
219C3⋊D9—24no—
220C3×D9—42no—
221C24×C3—21no—
222C24×C3—91no—
223C24×C3C2×D911yescharacteristic
224C24×C3D3211yescharacteristic
225C23×D3—31no—
226C23×D3—91no—
227C23×D3—91no—
228C23×D3—181no—
229C2×C3×D4—94no—
230C2×C3×D4—184no—
231C22×C3×C4—31no—
232C22×C3×C4—31no—
233C22×C3×C4—91no—
234C2×#67—34no—
235C2×#67—64no—
236C2×#67—94no—
237C2×#67—94no—
238C2×#67—94no—
239C2×#67—184no—
240C2×#67—184no—
241C2×#67—184no—
242C2×#67—184no—
243C2×#67—188no—
244C22×Dic3—31no—
245C22×Dic3—91no—
246C22×Dic3—91no—
247C22×Dic3—181no—
248C22×Dic3C2×D911yescharacteristic
249C22×Dic3C2×D911yescharacteristic
250C2×D12—94no—
251C2×D12—184no—
252C2×C4×D3—94no—
253C2×C4×D3—94no—
254C2×C4×D3—184no—
255C2×C4×D3—184no—
256C3×C22⋊C4—34no—
257#217—94no—
258#217C2×D914yescomplemented
259(C2×D3)⋊C4—94no—
260C4⋊Dic3—94no—
261Dic3⋊C4—94no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—22no—
265C22×C32—44no—
266C22×C32#23512yes—
267C22×C32D3×D412yes—
268C22×C32D3×D412yes—
269C22×C32C23×D311yescharacteristic
270C2×C3⋊D3—34no—
271C2×C3⋊D3—68no—
272C2×C3⋊D3—62no—
273C2×C3⋊D3—62no—
274C2×C3×D3—62no—
275C2×C3×D3—62no—
276C2×C3×D3—62no—
277C2×C3×D3—124no—
278C3⋊Dic3—62no—
279C3×Dic3—22no—
280C3×Dic3—22no—
281C3×Dic3—34no—
282C22×C9—24no—
283C22×C9—22no—
284C22×C9—22no—
285C22×C9—22no—
286C22×C9—21no—
287C22×C9—22no—
288C22×C9—22no—
289C22×C9—22no—
290C22×C9—42no—
291C22×C9—42no—
292C22×C9—48no—
293C22×C9—44no—
294C22×C9D3×D412yes—
295C22×C9#23712yes—
296C22×C9C2×#6712yes—
297C22×C9C23×D311yescharacteristic
298C2×D9—22no—
299C2×D9—22no—
300C2×D9—22no—
301C2×D9—34no—
302C2×D9—44no—
303C2×D9—64no—
304C2×D9—68no—
305C2×D9—62no—
306C2×D9—62no—
307C2×D9—122no—
308C2×D9—122no—
309C2×D9—128no—
310C4×C9—62no—
311C4×C9—62no—
312Dic9—64no—
313Dic9—62no—
314Dic9C2×#6714yescomplemented
315C22×D4—91no—
316C22×D4—271no—
317C22×D4—271no—
318C23×C4—271no—
319#121—278no—
320C2×C4⋊C4—271no—
321C2×C22⋊C4—31no—
322C2×C22⋊C4—271no—
323C3×C9C2×#12111yescharacteristic, complemented, Sylow
324C23×C3—24no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—21no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3—92no—
336C23×C3—182no—
337C23×C3—182no—
338C23×C3—182no—
339C23×C3C22×D912yes—
340C23×C3C22×D911yescharacteristic
341C23×C3C2×D3212yes—
342C23×C3C2×D3211yescharacteristic
343C22×D3—32no—
344C22×D3—62no—
345C22×D3—62no—
346C22×D3—62no—
347C22×D3—92no—
348C22×D3—94no—
349C22×D3—92no—
350C22×D3—94no—
351C22×D3—182no—
352C22×D3—184no—
353C22×D3—182no—
354C22×D3—182no—
355C22×D3—182no—
356C22×D3—182no—
357C22×D3—362no—
358C22×D3—362no—
359C3×D4—188no—
360C2×C3×C4—32no—
361C2×C3×C4—32no—
362C2×C3×C4—62no—
363C2×C3×C4—62no—
364C2×C3×C4—94no—
365C2×C3×C4—92no—
366#67—68no—
367#67—188no—
368#67—188no—
369#67—188no—
370#67—1816no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—32no—
374C2×Dic3—34no—
375C2×Dic3—92no—
376C2×Dic3—92no—
377C2×Dic3—182no—
378C2×Dic3—184no—
379C2×Dic3—182no—
380C2×Dic3—182no—
381C2×Dic3C22×D912yes—
382C2×Dic3C22×D912yes—
383D12—188no—
384C4×D3—188no—
385C4×D3—188no—
386C2×C32—42no—
387C2×C32#88414yes—
388C2×C32C2×#23511yescharacteristic
389C2×C32C2×D3×D411yescharacteristic
390C2×C32C2×D3×D411yescharacteristic
391C3⋊D3—64no—
392C3×D3—122no—
393C2×C9—21no—
394C2×C9—21no—
395C2×C9—21no—
396C2×C9—24no—
397C2×C9—44no—
398C2×C9—42no—
399C2×C9#93814yescomplemented
400C2×C9C2×D3×D411yescharacteristic
401C2×C9C2×#23711yescharacteristic
402C2×C9C22×#6711yescharacteristic
403D9—42no—
404D9—64no—
405D9—124no—
406C24—91no—
407C24—271no—
408C24D3×D911yescharacteristic
409C2×D4—94no—
410C2×D4—184no—
411C2×D4—274no—
412C2×D4—274no—
413C2×D4—544no—
414C2×D4—544no—
415C22×C4—31no—
416C22×C4—31no—
417C22×C4—91no—
418C22×C4—274no—
419C22×C4—271no—
420C22×C4—544no—
421C4⋊C4—274no—
422C22⋊C4—34no—
423C22⋊C4—274no—
424C22×C3—24no—
425C22×C3—22no—
426C22×C3—22no—
427C22×C3—22no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—22no—
431C22×C3—21no—
432C22×C3—22no—
433C22×C3—22no—
434C22×C3—22no—
435C22×C3—42no—
436C22×C3—42no—
437C22×C3—48no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—182no—
441C22×C3—182no—
442C22×C3—182no—
443C22×C3—364no—
444C22×C3#369712yes—
445C22×C3D4×D912yes—
446C22×C3D4×D912yes—
447C22×C3C23×D911yescharacteristic
448C22×C3#380412yes—
449C22×C3D3×#6712yes—
450C22×C3#381112yes—
451C22×C3C22×D3211yescharacteristic
452C2×D3—62no—
453C2×D3—62no—
454C2×D3—62no—
455C2×D3—94no—
456C2×D3—94no—
457C2×D3—124no—
458C2×D3—184no—
459C2×D3—182no—
460C2×D3—182no—
461C2×D3—188no—
462C2×D3—188no—
463C2×D3—182no—
464C2×D3—182no—
465C2×D3—362no—
466C2×D3—362no—
467C2×D3—368no—
468C3×C4—62no—
469C3×C4—62no—
470C3×C4—94no—
471Dic3—22no—
472Dic3—22no—
473Dic3—34no—
474Dic3—184no—
475Dic3—182no—
476Dic3—182no—
477C32C2×#88411yescharacteristic
478C9—21no—
479C9C2×#93811yescharacteristic, complemented
480C23—22no—
481C23—22no—
482C23—22no—
483C23—92no—
484C23—182no—
485C23—182no—
486C23—182no—
487C23—272no—
488C23—274no—
489C23—542no—
490C23—542no—
491C23C2×D3×D912yes—
492C23C2×D3×D911yescharacteristic
493D4—188no—
494D4—548no—
495D4—548no—
496C2×C4—32no—
497C2×C4—32no—
498C2×C4—62no—
499C2×C4—62no—
500C2×C4—92no—
501C2×C4—94no—
502C2×C4—272no—
503C2×C4—548no—
504C2×C4—542no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—24no—
509C2×C3—44no—
510C2×C3—42no—
511C2×C3—42no—
512C2×C3—362no—
513C2×C3#6344614yescomplemented
514C2×C3C2×#369711yescharacteristic
515C2×C3C2×D4×D911yescharacteristic
516C2×C3C2×D4×D911yescharacteristic
517C2×C3#6395814yes—
518C2×C3C2×#380411yescharacteristic
519C2×C3C2×D3×#6711yescharacteristic
520C2×C3C2×#381111yescharacteristic
521D3—122no—
522D3—184no—
523D3—184no—
524D3—364no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—44no—
529C22—182no—
530C22—182no—
531C22—182no—
532C22—274no—
533C22—364no—
534C22—542no—
535C22—542no—
536C22—548no—
537C22#8865812yes—
538C22D9×#6712yes—
539C22#8866812yes—
540C22C22×D3×D911yescharacteristic
541C4—62no—
542C4—62no—
543C4—94no—
544C4—542no—
545C3—21no—
546C3C2×#6344611yescharacteristic, complemented
547C3C2×#6395811yescharacteristic
548C2—42no—
549C2—362no—
550C2—544no—
551C2#1172125614yesdirect product factor
552C2C2×#8865811yescharacteristic
553C2C2×D9×#6711yescharacteristic
554C2C2×#8866811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC23×C32C22×D3
hypercenterC24D3×D9
hypocenterC3×C9C2×#121
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C22×#6233C2semidirect
2. C22×C3×#611C2semidirect
3. C22×#6192C2semidirect
4. C22×(C3⋊D9)⋊C4C2semidirect
5. C2×C9×#217C2semidirect
6. C2×(C2×D9)⋊Dic3C2semidirect
7. #11721256C2direct (N × H)
8. C2×Dic9⋊Dic3C2semidirect
15. C2×#6233C22semidirect
18. C2×C3×#611C22semidirect
23. C2×#6192C22semidirect
25. C2×(C3⋊D9)⋊C4C22semidirect
26. C9×#217C22semidirect
27. (C2×D9)⋊Dic3C22semidirect
28. Dic9⋊Dic3C22semidirect
39. C22×#611D3semidirect
53. C22×C3⋊D9D4semidirect
67. C2×C3×Dic9D4semidirect
103. C2×#611C2×D3semidirect
125. C2×C3⋊D9C2×D4semidirect
133. C3×Dic9C2×D4semidirect
145. C2×#217D9semidirect
209. C2×Dic9#67semidirect
215. C2×C3×C9#121semidirect
258. #217C2×D9semidirect
314. Dic9C2×#67semidirect
323. C3×C9C2×#121semidirect
399. C2×C9#938semidirect
479. C9C2×#938semidirect
513. C2×C3#63446semidirect
546. C3C2×#63446semidirect
551. C2#11721256direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#62332
C22×C3×#6112
C22×#61922
C22×(C3⋊D9)⋊C42
C2×C9×#2172
C2×(C2×D9)⋊Dic32
#117212562
C2×Dic9⋊Dic32
C2×#639583
C2×#634463