Finite Group Databasebeta

← C2×#11721048 #13150521

Subgroups — C2×#11721048 #13150521

Order 1728.

2258 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D3×D9C211yesmaximal, characteristic
2C22×C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C22×(C3⋊D9)⋊C4C211yesmaximal, characteristic, complemented
4C2×C9×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
5C2×C3×(C2×D9)⋊C4C211yesmaximal, characteristic, complemented
6C2×(C2×D3)⋊Dic9C211yesmaximal, characteristic, complemented
7C2×(C2×D9)⋊Dic3C211yesmaximal, characteristic, complemented
8#11721048C218yesmaximal, direct product factor
9C2×#63921—31nomaximal
10C2×C22⋊C4×D9—31nomaximal
11C23×C3⋊D9C2211yescharacteristic
12C23×D3×C9C2211yescharacteristic
13C23×C3×D9C2211yescharacteristic
14C22×D3×D9—24no—
15C22×D3×D9—24no—
16C22×D3×D9C418yescomplemented
17C22×D3×D9C2214yes—
18C22×C3×C4×C9C2211yescharacteristic, complemented
19C22×C3⋊Dic9C2211yescharacteristic, complemented
20C2×C4×C3⋊D9—24no—
21C2×C4×C3⋊D9C2214yescomplemented
22C22×C9×Dic3C2211yescharacteristic, complemented
23C22×C3×Dic9C2211yescharacteristic, complemented
24C2×(C3⋊D9)⋊C4—24no—
25C2×(C3⋊D9)⋊C4C2214yescomplemented
26C9×(C2×D3)⋊C4C2214yescomplemented
27C3×(C2×D9)⋊C4C2214yescomplemented
28(C2×D3)⋊Dic9C2214yescomplemented
29(C2×D9)⋊Dic3C2214yescomplemented
30C23×D32—31no—
31C22×C4×C3⋊D3—31no—
32C22×#624—31no—
33C24×D9—31no—
34C2×C3×(C2×D3)⋊C4—31no—
35C2×C3×(C2×D3)⋊C4D311yescharacteristic
36C2×(C2×D3)⋊Dic3—31no—
37C2×(C2×D3)⋊Dic3—31no—
38#63921—38no—
39C22×C4×D9—31no—
40C22×C4×D9—31no—
41C22×C4×D9—61no—
42C2×C9×C22⋊C4—31no—
43C2×#3676—31no—
44C2×(C2×D9)⋊C4—31no—
45C2×(C2×D9)⋊C4D311yescharacteristic, complemented
46C22⋊C4×D9—38no—
47C23×C3×C9C2311yescharacteristic
48C22×C3⋊D9C2×C414yes—
49C22×C3⋊D9D414yes—
50C22×C3⋊D9D414yes—
51C22×C3⋊D9C2312yes—
52C22×D3×C9—22no—
53C22×D3×C9—22no—
54C22×D3×C9C2×C414yescomplemented
55C22×D3×C9C2312yes—
56C22×C3×D9—22no—
57C22×C3×D9—22no—
58C22×C3×D9C2×C414yescomplemented
59C22×C3×D9C2312yes—
60C2×D3×D9—232no—
61C2×D3×D9—28no—
62C2×D3×D9—28no—
63C2×D3×D9C2×C4116yescomplemented
64C2×C3×C4×C9—22no—
65C2×C3×C4×C9C2312yescomplemented
66C2×C3⋊Dic9—22no—
67C2×C3⋊Dic9C2312yescomplemented
68C4×C3⋊D9—28no—
69C2×C9×Dic3—22no—
70C2×C9×Dic3C2312yescomplemented
71C2×C3×Dic9—22no—
72C2×C3×Dic9C2312yescomplemented
73(C3⋊D9)⋊C4—28no—
74C2×D3×C22⋊C4—91no—
75C2×D3×C22⋊C4—91no—
76C23×C3⋊D3—31no—
77C23×C3×D3—31no—
78C23×C3×D3C2×D311yescharacteristic
79C22×D32—38no—
80C22×D32—34no—
81C22×D32—64no—
82C22×D32—64no—
83C22×C32×C4C2×D311yescharacteristic
84C22×C3⋊Dic3—31no—
85C2×C4×C3⋊D3—34no—
86C2×C4×C3⋊D3—64no—
87C22×C3×Dic3—31no—
88C22×C3×Dic3C2×D311yescharacteristic
89C2×#624—34no—
90C2×#624—64no—
91C24×C9—31no—
92C23×D9—31no—
93C23×D9—38no—
94C23×D9—34no—
95C23×D9—64no—
96C23×D9—64no—
97C23×D9—61no—
98C23×D9C2×D311yescharacteristic
99C3×(C2×D3)⋊C4—34no—
100C3×(C2×D3)⋊C4C2×D314yes—
101(C2×D3)⋊Dic3—34no—
102(C2×D3)⋊Dic3—34no—
103C22×C4×C9—21no—
104C22×C4×C9—31no—
105C22×C4×C9C2×D311yescharacteristic, complemented
106C22×Dic9—31no—
107C22×Dic9—61no—
108C22×Dic9C2×D311yescharacteristic, complemented
109C2×C4×D9—34no—
110C2×C4×D9—34no—
111C2×C4×D9—64no—
112C2×C4×D9—64no—
113C2×C4×D9—64no—
114C2×C4×D9—68no—
115C9×C22⋊C4—34no—
116#3676—34no—
117(C2×D9)⋊C4—34no—
118(C2×D9)⋊C4C2×D314yescomplemented
119C22×C3×C9C22×C412yes—
120C22×C3×C9C2×D412yes—
121C22×C3×C9C2×D412yes—
122C22×C3×C9C2411yescharacteristic
123C2×C3⋊D9C22⋊C4116yescomplemented
124C2×C3⋊D9C22×C414yes—
125C2×C3⋊D9C2×D414yes—
126C2×C3⋊D9C2×D414yes—
127C2×D3×C9—28no—
128C2×D3×C9—22no—
129C2×D3×C9—22no—
130C2×D3×C9C22×C414yescomplemented
131C2×C3×D9—28no—
132C2×C3×D9—22no—
133C2×C3×D9—22no—
134C2×C3×D9C22×C414yescomplemented
135D3×D9—232no—
136C3×C4×C9—22no—
137C3⋊Dic9—22no—
138C9×Dic3—22no—
139C3×Dic9—22no—
140C24×D3—91no—
141C24×D3—91no—
142C22×C4×D3—91no—
143C22×C4×D3—91no—
144C22×C4×D3—91no—
145C22×C4×D3—91no—
146C22×C4×D3—181no—
147C2×C3×C22⋊C4—31no—
148C2×C3×C22⋊C4—91no—
149C2×#217—91no—
150C2×#217—91no—
151C2×(C2×D3)⋊C4—31no—
152C2×(C2×D3)⋊C4—91no—
153C2×(C2×D3)⋊C4—91no—
154C2×(C2×D3)⋊C4D911yescharacteristic, complemented
155D3×C22⋊C4—98no—
156D3×C22⋊C4—98no—
157C23×C32C22×D311yescharacteristic
158C22×C3⋊D3—34no—
159C22×C3⋊D3—34no—
160C22×C3⋊D3—34no—
161C22×C3⋊D3—32no—
162C22×C3×D3—22no—
163C22×C3×D3—22no—
164C22×C3×D3—34no—
165C22×C3×D3—32no—
166C22×C3×D3—62no—
167C22×C3×D3—62no—
168C22×C3×D3C4×D314yes—
169C22×C3×D3C22×D312yes—
170C2×D32—316no—
171C2×D32—632no—
172C2×D32—68no—
173C2×D32—68no—
174C2×C32×C4—22no—
175C2×C32×C4C22×D312yes—
176C2×C3⋊Dic3—32no—
177C2×C3⋊Dic3—62no—
178C4×C3⋊D3—68no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—62no—
182C2×C3×Dic3C22×D312yes—
183#624—68no—
184C23×C9—21no—
185C23×C9—34no—
186C23×C9—32no—
187C23×C9—62no—
188C23×C9—62no—
189C23×C9C22×D311yescharacteristic
190C22×D9—22no—
191C22×D9—22no—
192C22×D9—34no—
193C22×D9—34no—
194C22×D9—34no—
195C22×D9—32no—
196C22×D9—316no—
197C22×D9—64no—
198C22×D9—62no—
199C22×D9—64no—
200C22×D9—64no—
201C22×D9—632no—
202C22×D9—68no—
203C22×D9—68no—
204C22×D9C4×D314yescomplemented
205C22×D9C22×D312yes—
206C2×C4×C9—24no—
207C2×C4×C9—22no—
208C2×C4×C9—22no—
209C2×C4×C9—32no—
210C2×C4×C9—62no—
211C2×C4×C9C22×D312yescomplemented
212C2×Dic9—22no—
213C2×Dic9—32no—
214C2×Dic9—62no—
215C2×Dic9—64no—
216C2×Dic9—62no—
217C2×Dic9C22×D312yescomplemented
218C4×D9—68no—
219C4×D9—68no—
220C4×D9—616no—
221C22×C22⋊C4—271noSylow
222C2×C3×C9C2×C22⋊C414yescomplemented
223C2×C3×C9C23×C411yescharacteristic
224C2×C3×C9C22×D411yescharacteristic
225C2×C3×C9C22×D411yescharacteristic
226C3⋊D9C2×C22⋊C418yescomplemented
227D3×C9—24no—
228C3×D9—24no—
229C24×C3—31no—
230C24×C3—91no—
231C23×D3—31no—
232C23×D3—94no—
233C23×D3—98no—
234C23×D3—91no—
235C23×D3—91no—
236C23×D3—98no—
237C23×D3—94no—
238C23×D3—181no—
239C23×D3—184no—
240C23×D3—184no—
241C23×D3—184no—
242C23×D3—184no—
243C23×D3C2×D911yescharacteristic
244C22×C3×C4—21no—
245C22×C3×C4—31no—
246C22×C3×C4—91no—
247C22×C3×C4C2×D911yescharacteristic, complemented
248C22×C3×C4D3211yescharacteristic
249C22×Dic3—31no—
250C22×Dic3—91no—
251C22×Dic3—91no—
252C22×Dic3—181no—
253C22×Dic3C2×D911yescharacteristic, complemented
254C2×C4×D3—94no—
255C2×C4×D3—94no—
256C2×C4×D3—94no—
257C2×C4×D3—94no—
258C2×C4×D3—184no—
259C2×C4×D3—184no—
260C2×C4×D3—184no—
261C2×C4×D3—184no—
262C2×C4×D3—184no—
263C2×C4×D3—188no—
264C3×C22⋊C4—34no—
265C3×C22⋊C4—94no—
266#217—94no—
267#217—94no—
268(C2×D3)⋊C4—34no—
269(C2×D3)⋊C4—94no—
270(C2×D3)⋊C4—94no—
271(C2×D3)⋊C4C2×D914yescomplemented
272C22×C32C2×C4×D312yes—
273C22×C32D3×D412yes—
274C22×C32D3×D412yes—
275C22×C32C23×D311yescharacteristic
276C2×C3⋊D3—316no—
277C2×C3⋊D3—34no—
278C2×C3⋊D3—34no—
279C2×C3⋊D3—34no—
280C2×C3×D3—28no—
281C2×C3×D3—22no—
282C2×C3×D3—22no—
283C2×C3×D3—34no—
284C2×C3×D3—68no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3C2×C4×D314yes—
288D32—632no—
289C32×C4—22no—
290C3⋊Dic3—62no—
291C3×Dic3—22no—
292C3×Dic3—62no—
293C22×C9—21no—
294C22×C9—22no—
295C22×C9—22no—
296C22×C9—22no—
297C22×C9—34no—
298C22×C9—68no—
299C22×C9—62no—
300C22×C9—62no—
301C22×C9C2×C4×D312yes—
302C22×C9D3×D412yes—
303C22×C9D3×D412yes—
304C22×C9C23×D311yescharacteristic
305C2×D9—28no—
306C2×D9—22no—
307C2×D9—22no—
308C2×D9—316no—
309C2×D9—34no—
310C2×D9—34no—
311C2×D9—34no—
312C2×D9—616no—
313C2×D9—64no—
314C2×D9—64no—
315C2×D9—64no—
316C2×D9—632no—
317C2×D9C2×C4×D314yescomplemented
318C4×C9—24no—
319C4×C9—22no—
320C4×C9—62no—
321Dic9—22no—
322Dic9—64no—
323Dic9—62no—
324C25—271no—
325C23×C4—271no—
326C23×C4—271no—
327C2×C22⋊C4—31no—
328C2×C22⋊C4—91no—
329C2×C22⋊C4—278no—
330C2×C22⋊C4—271no—
331C2×C22⋊C4—271no—
332C3×C9C22×C22⋊C411yescharacteristic, complemented, Sylow
333C23×C3—21no—
334C23×C3—32no—
335C23×C3—34no—
336C23×C3—62no—
337C23×C3—62no—
338C23×C3—94no—
339C23×C3—92no—
340C23×C3—182no—
341C23×C3—182no—
342C23×C3C22×D911yescharacteristic
343C23×C3C2×D3211yescharacteristic
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—32no—
347C22×D3—34no—
348C22×D3—62no—
349C22×D3—62no—
350C22×D3—916no—
351C22×D3—92no—
352C22×D3—94no—
353C22×D3—94no—
354C22×D3—94no—
355C22×D3—92no—
356C22×D3—94no—
357C22×D3—94no—
358C22×D3—94no—
359C22×D3—916no—
360C22×D3—182no—
361C22×D3—184no—
362C22×D3—184no—
363C22×D3—184no—
364C22×D3—1832no—
365C22×D3—188no—
366C22×D3—188no—
367C22×D3—188no—
368C22×D3—188no—
369C22×D3—1832no—
370C22×D3C4×D914yescomplemented
371C22×D3C22×D912yes—
372C2×C3×C4—24no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—32no—
377C2×C3×C4—62no—
378C2×C3×C4—92no—
379C2×C3×C4—182no—
380C2×C3×C4C22×D912yescomplemented
381C2×C3×C4C2×D3212yes—
382C2×Dic3—22no—
383C2×Dic3—32no—
384C2×Dic3—62no—
385C2×Dic3—92no—
386C2×Dic3—92no—
387C2×Dic3—182no—
388C2×Dic3—184no—
389C2×Dic3—182no—
390C2×Dic3—182no—
391C2×Dic3C22×D912yescomplemented
392C4×D3—188no—
393C4×D3—188no—
394C4×D3—188no—
395C4×D3—188no—
396C4×D3—1816no—
397C2×C32D3×C22⋊C414yes—
398C2×C32C22×C4×D311yescharacteristic
399C2×C32C2×D3×D411yescharacteristic
400C2×C32C2×D3×D411yescharacteristic
401C3⋊D3—38no—
402C3×D3—24no—
403C3×D3—64no—
404C2×C9—21no—
405C2×C9—21no—
406C2×C9—21no—
407C2×C9—24no—
408C2×C9—64no—
409C2×C9D3×C22⋊C414yescomplemented
410C2×C9C22×C4×D311yescharacteristic
411C2×C9C2×D3×D411yescharacteristic
412C2×C9C2×D3×D411yescharacteristic
413D9—24no—
414D9—38no—
415D9—68no—
416C24—31no—
417C24—91no—
418C24—274no—
419C24—278no—
420C24—271no—
421C24—544no—
422C24—544no—
423C22×C4—31no—
424C22×C4—91no—
425C22×C4—274no—
426C22×C4—271no—
427C22×C4—274no—
428C22×C4—544no—
429C22×C4—544no—
430C22×C4D3×D911yescharacteristic, complemented
431C22⋊C4—34no—
432C22⋊C4—94no—
433C22⋊C4—274no—
434C22⋊C4—274no—
435C22×C3—21no—
436C22×C3—22no—
437C22×C3—22no—
438C22×C3—22no—
439C22×C3—34no—
440C22×C3—62no—
441C22×C3—62no—
442C22×C3—68no—
443C22×C3—94no—
444C22×C3—188no—
445C22×C3—182no—
446C22×C3—182no—
447C22×C3C2×C4×D912yes—
448C22×C3D4×D912yes—
449C22×C3D4×D912yes—
450C22×C3C23×D911yescharacteristic
451C22×C3C4×D3212yes—
452C22×C3#380212yes—
453C22×C3#381112yes—
454C22×C3C22×D3211yescharacteristic
455C2×D3—28no—
456C2×D3—22no—
457C2×D3—22no—
458C2×D3—34no—
459C2×D3—62no—
460C2×D3—62no—
461C2×D3—68no—
462C2×D3—94no—
463C2×D3—94no—
464C2×D3—94no—
465C2×D3—916no—
466C2×D3—916no—
467C2×D3—94no—
468C2×D3—94no—
469C2×D3—94no—
470C2×D3—184no—
471C2×D3—1816no—
472C2×D3—184no—
473C2×D3—184no—
474C2×D3—1832no—
475C2×D3—1832no—
476C2×D3C2×C4×D914yescomplemented
477C3×C4—24no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—62no—
481C3×C4—182no—
482Dic3—22no—
483Dic3—62no—
484Dic3—184no—
485Dic3—182no—
486Dic3—182no—
487C32C2×D3×C22⋊C411yescharacteristic
488C9—21no—
489C9C2×D3×C22⋊C411yescharacteristic, complemented
490C23—32no—
491C23—34no—
492C23—62no—
493C23—62no—
494C23—92no—
495C23—94no—
496C23—182no—
497C23—182no—
498C23—2716no—
499C23—272no—
500C23—274no—
501C23—274no—
502C23—274no—
503C23—548no—
504C23—5432no—
505C23—548no—
506C23C2×D3×D911yescharacteristic
507C2×C4—22no—
508C2×C4—32no—
509C2×C4—62no—
510C2×C4—92no—
511C2×C4—182no—
512C2×C4—272no—
513C2×C4—548no—
514C2×C4—548no—
515C2×C4—542no—
516C2×C4C2×D3×D912yescomplemented
517C2×C3—21no—
518C2×C3—24no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—64no—
522C2×C3—184no—
523C2×C3C22⋊C4×D914yescomplemented
524C2×C3C22×C4×D911yescharacteristic
525C2×C3C2×D4×D911yescharacteristic
526C2×C3C2×D4×D911yescharacteristic
527C2×C3#6392114yes—
528C2×C3C2×C4×D3211yescharacteristic
529C2×C3C2×#380211yescharacteristic
530C2×C3C2×#381111yescharacteristic
531D3—24no—
532D3—64no—
533D3—98no—
534D3—98no—
535D3—188no—
536C22—34no—
537C22—62no—
538C22—68no—
539C22—62no—
540C22—94no—
541C22—182no—
542C22—188no—
543C22—182no—
544C22—274no—
545C22—2716no—
546C22—274no—
547C22—274no—
548C22—5432no—
549C22C4×D3×D912yes—
550C22#8864612yes—
551C22#8866812yes—
552C22C22×D3×D911yescharacteristic
553C4—22no—
554C4—62no—
555C4—182no—
556C4—542no—
557C3—21no—
558C3C2×C22⋊C4×D911yescharacteristic, complemented
559C3C2×#6392111yescharacteristic
560C2—64no—
561C2—184no—
562C2—278no—
563C2#1172104814yesdirect product factor
564C2C2×C4×D3×D911yescharacteristic
565C2C2×#8864611yescharacteristic
566C2C2×#8866811yescharacteristic
subgroupG/H
centerC23C2×D3×D9
commutatorC2×C3×C9C23×C4
FrattiniC22×C3C22×D32
FittingC22×C3×C4×C9C22
socleC23×C32C22×D3
prime radicalC23×D3×D9C2
cosocleC22×C3×C9C24
hypercenterC22×C4D3×D9
hypocenterC3×C9C22×C22⋊C4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×C4×C3⋊D9C2semidirect
3. C22×(C3⋊D9)⋊C4C2semidirect
4. C2×C9×(C2×D3)⋊C4C2semidirect
5. C2×C3×(C2×D9)⋊C4C2semidirect
6. C2×(C2×D3)⋊Dic9C2semidirect
7. C2×(C2×D9)⋊Dic3C2semidirect
8. #11721048C2direct (N × H)
16. C22×D3×D9C4semidirect
18. C22×C3×C4×C9C22semidirect
19. C22×C3⋊Dic9C22semidirect
21. C2×C4×C3⋊D9C22semidirect
22. C22×C9×Dic3C22semidirect
23. C22×C3×Dic9C22semidirect
25. C2×(C3⋊D9)⋊C4C22semidirect
26. C9×(C2×D3)⋊C4C22semidirect
27. C3×(C2×D9)⋊C4C22semidirect
28. (C2×D3)⋊Dic9C22semidirect
29. (C2×D9)⋊Dic3C22semidirect
45. C2×(C2×D9)⋊C4D3semidirect
54. C22×D3×C9C2×C4semidirect
58. C22×C3×D9C2×C4semidirect
63. C2×D3×D9C2×C4semidirect
65. C2×C3×C4×C9C23semidirect
67. C2×C3⋊Dic9C23semidirect
70. C2×C9×Dic3C23semidirect
72. C2×C3×Dic9C23semidirect
105. C22×C4×C9C2×D3semidirect
108. C22×Dic9C2×D3semidirect
118. (C2×D9)⋊C4C2×D3semidirect
123. C2×C3⋊D9C22⋊C4semidirect
130. C2×D3×C9C22×C4semidirect
134. C2×C3×D9C22×C4semidirect
154. C2×(C2×D3)⋊C4D9semidirect
204. C22×D9C4×D3semidirect
211. C2×C4×C9C22×D3semidirect
217. C2×Dic9C22×D3semidirect
222. C2×C3×C9C2×C22⋊C4semidirect
226. C3⋊D9C2×C22⋊C4semidirect
247. C22×C3×C4C2×D9semidirect
253. C22×Dic3C2×D9semidirect
271. (C2×D3)⋊C4C2×D9semidirect
317. C2×D9C2×C4×D3semidirect
332. C3×C9C22×C22⋊C4semidirect
370. C22×D3C4×D9semidirect
380. C2×C3×C4C22×D9semidirect
391. C2×Dic3C22×D9semidirect
409. C2×C9D3×C22⋊C4semidirect
430. C22×C4D3×D9semidirect
476. C2×D3C2×C4×D9semidirect
489. C9C2×D3×C22⋊C4semidirect
516. C2×C4C2×D3×D9semidirect
523. C2×C3C22⋊C4×D9semidirect
558. C3C2×C22⋊C4×D9semidirect
563. C2#11721048direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D3×D92
C22×C4×C3⋊D92
C22×(C3⋊D9)⋊C42
C2×C9×(C2×D3)⋊C42
C2×C3×(C2×D9)⋊C42
C2×(C2×D3)⋊Dic92
C2×(C2×D9)⋊Dic32
#117210482
C2×#639213
C2×C22⋊C4×D93