Finite Group Databasebeta

← C2×C3×#63694 #13164445

Subgroups — C2×C3×#63694 #13164445

Order 1728.

1154 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C23×C3×C22⋊C9C211yesmaximal, characteristic, complemented
2C22×C3×(C22⋊C9)⋊C2C211yesmaximal, characteristic, complemented
3C3×#63694C214yesmaximal, direct product factor
4C2×C3×(C22⋊C9)⋊C4C211yesmaximal, characteristic, complemented
5C2×C3×#953—31nomaximal
6C2×#63694C311yesmaximal, characteristic, direct product factor
7C22×C3×C22⋊C9—22no—
8C22×C3×C22⋊C9C2211yescharacteristic
9C22×C3×C22⋊C9C2212yescomplemented
10C2×C3×(C22⋊C9)⋊C2—22no—
11C2×C3×(C22⋊C9)⋊C2C2212yescomplemented
12C2×C3×#611—41nomaximal
13C3×(C22⋊C9)⋊C4C2212yescomplemented
14C25×C32D311yescharacteristic
15C22×C3×#67—31no—
16C22×C3×#67—31no—
17C22×C3×#67—31no—
18C23×C22⋊C9—21no—
19C23×C22⋊C9C2×C311yescharacteristic, complemented
20C22×(C22⋊C9)⋊C2C2×C311yescharacteristic, complemented
21C3×#953—34no—
22C3×#953—34no—
23C2×C3×#217—31no—
24C2×C3×#217—31no—
25C2×C3×#217—31no—
26#63694C2×C314yesdirect product factor
27C2×(C22⋊C9)⋊C4C2×C311yescharacteristic, complemented
28C23×C3×C9—41no—
29C22×C3×D9—41no—
30C2×C3×C22⋊C9—22no—
31C2×C3×C22⋊C9D412yescomplemented
32C2×C3×C22⋊C9C2311yescharacteristic
33C3×(C22⋊C9)⋊C2—22no—
34C3×#611—44no—
35C2×C3×Dic9—41no—
36C2×C3×#120—91no—
37C2×#953—31no—
38C24×C32—22no—
39C24×C32—31no—
40C24×C32—31no—
41C24×C32—32no—
42C24×C32—62no—
43C24×C32C2×D312yes—
44C24×C32C2×D311yescharacteristic
45C23×C3×D3—31no—
46C2×C3×#67—32no—
47C2×C3×#67—34no—
48C2×C3×#67—34no—
49C2×C3×#67—32no—
50C2×C3×#67—62no—
51C2×C3×#67—64no—
52C2×C3×#67—62no—
53C2×C3×#67—61no—
54C2×C3×#67—62no—
55C2×C3×#67—61no—
56C22×C3×Dic3—31no—
57C22×C3×Dic3—31no—
58C22×C3×Dic3—31no—
59C22×C22⋊C9—24no—
60C22×C22⋊C9—21no—
61C22×C22⋊C9—22no—
62C22×C22⋊C9—22no—
63C22×C22⋊C9C22×C311yescharacteristic
64C22×C22⋊C9C22×C312yescomplemented
65C2×(C22⋊C9)⋊C2—22no—
66C2×(C22⋊C9)⋊C2C22×C312yescomplemented
67C3×#217—34no—
68C3×#217—32no—
69C3×#217—34no—
70C3×#217—32no—
71C2×#611—41no—
72(C22⋊C9)⋊C4C22×C312yescomplemented
73C22×C3×C9—41no—
74C22×C3×C9—42no—
75C22×C3×C9—82no—
76C2×C3×D9—42no—
77C2×C3×D9—82no—
78C3×C22⋊C9C2×D411yescharacteristic, complemented
79C3×Dic9—42no—
80C25×C3—21no—
81C25×C3D911yescharacteristic, complemented
82C25×C3C3×D311yescharacteristic
83C22×C3×D4—91no—
84C22×C3×D4—91no—
85C22×C3×D4—91no—
86C22×#67—31no—
87C22×#67—31no—
88C22×#67—31no—
89C3×#120—94no—
90C3×#120—94no—
91C2×C3×C22⋊C4—91no—
92C2×C3×C22⋊C4—91no—
93C2×C3×C22⋊C4—91no—
94#953—34no—
95#953—34no—
96C2×#217—31no—
97C2×#217—31no—
98C2×#217—31no—
99C23×C32—22no—
100C23×C32—31no—
101C23×C32—31no—
102C23×C32—32no—
103C23×C32—32no—
104C23×C32—34no—
105C23×C32—32no—
106C23×C32—31no—
107C23×C32—32no—
108C23×C32—62no—
109C23×C32—62no—
110C23×C32—62no—
111C23×C32—61no—
112C23×C32—64no—
113C23×C32—62no—
114C23×C32—64no—
115C23×C32#6712yes—
116C23×C32S411yescharacteristic
117C23×C32C22×D311yescharacteristic
118C22×C3×D3—32no—
119C22×C3×D3—62no—
120C22×C3×D3—62no—
121C22×C3×D3—61no—
122C22×C3×D3—61no—
123C3×#67—64no—
124C3×#67—62no—
125C3×#67—68no—
126C3×#67—64no—
127C3×#67—62no—
128C3×#67—64no—
129C2×C3×Dic3—32no—
130C2×C3×Dic3—32no—
131C2×C3×Dic3—32no—
132C2×C3×Dic3—62no—
133C2×C3×Dic3—62no—
134C2×C3×Dic3—61no—
135C2×C3×Dic3—61no—
136C23×C9—41no—
137C23×C9—81no—
138C22×D9—41no—
139C2×C22⋊C9—24no—
140C2×C22⋊C9—22no—
141C2×C22⋊C9—22no—
142C2×C22⋊C9—21no—
143C2×C22⋊C9C3×D412yescomplemented
144C2×C22⋊C9C23×C311yescharacteristic
145(C22⋊C9)⋊C2—22no—
146#611—44no—
147C2×Dic9—41no—
148C2×#120—91noSylow
149C2×C3×C9—42no—
150C2×C3×C9—41no—
151C2×C3×C9—82no—
152C3×D9—82no—
153C24×C3—24no—
154C24×C3—22no—
155C24×C3—22no—
156C24×C3—21no—
157C24×C3—22no—
158C24×C3—31no—
159C24×C3—31no—
160C24×C3—31no—
161C24×C3—31no—
162C24×C3—32no—
163C24×C3—32no—
164C24×C3—61no—
165C24×C3—61no—
166C24×C3—62no—
167C24×C3—62no—
168C24×C3—62no—
169C24×C3—64no—
170C24×C3—91no—
171C24×C3C2×D911yescharacteristic
172C24×C3C2×D912yescomplemented
173C24×C3C2×C3×D311yescharacteristic
174C24×C3C2×C3×D312yes—
175C23×D3—31no—
176C2×C3×D4—92no—
177C2×C3×D4—94no—
178C2×C3×D4—94no—
179C2×C3×D4—92no—
180C2×C3×D4—182no—
181C2×C3×D4—184no—
182C2×C3×D4—182no—
183C2×C3×D4—181no—
184C2×C3×D4—182no—
185C2×C3×D4—181no—
186C22×C3×C4—91no—
187C22×C3×C4—91no—
188C22×C3×C4—91no—
189C2×#67—32no—
190C2×#67—34no—
191C2×#67—34no—
192C2×#67—32no—
193C2×#67—62no—
194C2×#67—64no—
195C2×#67—62no—
196C2×#67—61no—
197C2×#67—62no—
198C2×#67—61no—
199C22×Dic3—31no—
200C22×Dic3—31no—
201C22×Dic3—31no—
202C3×C22⋊C4—92no—
203C3×C22⋊C4—94no—
204C3×C22⋊C4—94no—
205C3×C22⋊C4—92no—
206#217—34no—
207#217—32no—
208#217—34no—
209#217—32no—
210C22×C32—22no—
211C22×C32—32no—
212C22×C32—31no—
213C22×C32—32no—
214C22×C32—32no—
215C22×C32—31no—
216C22×C32—31no—
217C22×C32—34no—
218C22×C32—32no—
219C22×C32—62no—
220C22×C32—62no—
221C22×C32—64no—
222C22×C32—61no—
223C22×C32—62no—
224C22×C32—64no—
225C22×C32—62no—
226C22×C32C2×#6711yescharacteristic
227C22×C32C2×S411yescharacteristic
228C22×C32C2×S412yes—
229C2×C3×D3—62no—
230C2×C3×D3—62no—
231C2×C3×D3—62no—
232C2×C3×D3—122no—
233C2×C3×D3—122no—
234C3×Dic3—62no—
235C3×Dic3—62no—
236C3×Dic3—62no—
237C22×C9—41no—
238C22×C9—42no—
239C22×C9—82no—
240C22×C9—84no—
241C22×C9—81no—
242C22×C9—82no—
243C2×D9—42no—
244C2×D9—82no—
245C22⋊C9—21no—
246C22⋊C9C2×C3×D411yescharacteristic, complemented
247Dic9—42no—
248C25C3×D911yescharacteristic, complemented
249C22×D4—91no—
250C22×D4—91no—
251C22×D4—91no—
252#120—94no—
253#120—94no—
254C2×C22⋊C4—91no—
255C2×C22⋊C4—91no—
256C2×C22⋊C4—91no—
257C3×C9—41noSylow
258C23×C3—24no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—22no—
262C23×C3—21no—
263C23×C3—21no—
264C23×C3—31no—
265C23×C3—31no—
266C23×C3—31no—
267C23×C3—32no—
268C23×C3—31no—
269C23×C3—32no—
270C23×C3—32no—
271C23×C3—32no—
272C23×C3—34no—
273C23×C3—34no—
274C23×C3—32no—
275C23×C3—31no—
276C23×C3—32no—
277C23×C3—31no—
278C23×C3—32no—
279C23×C3—32no—
280C23×C3—61no—
281C23×C3—62no—
282C23×C3—61no—
283C23×C3—62no—
284C23×C3—61no—
285C23×C3—62no—
286C23×C3—62no—
287C23×C3—64no—
288C23×C3—62no—
289C23×C3—62no—
290C23×C3—62no—
291C23×C3—62no—
292C23×C3—64no—
293C23×C3—64no—
294C23×C3—62no—
295C23×C3—62no—
296C23×C3—61no—
297C23×C3—61no—
298C23×C3—64no—
299C23×C3—64no—
300C23×C3—62no—
301C23×C3—62no—
302C23×C3—64no—
303C23×C3—64no—
304C23×C3—64no—
305C23×C3—64no—
306C23×C3—68no—
307C23×C3—62no—
308C23×C3—68no—
309C23×C3—92no—
310C23×C3—181no—
311C23×C3—182no—
312C23×C3—182no—
313C23×C3—181no—
314C23×C3#61112yescomplemented
315C23×C3(C22⋊C9)⋊C211yescharacteristic, complemented
316C23×C3C22×D911yescharacteristic
317C23×C3C3×#6712yes—
318C23×C3C3×S411yescharacteristic
319C23×C3C22×C3×D311yescharacteristic
320C22×D3—32no—
321C22×D3—62no—
322C22×D3—62no—
323C22×D3—61no—
324C22×D3—61no—
325C3×D4—184no—
326C3×D4—182no—
327C3×D4—188no—
328C3×D4—184no—
329C3×D4—182no—
330C3×D4—184no—
331C2×C3×C4—92no—
332C2×C3×C4—92no—
333C2×C3×C4—92no—
334C2×C3×C4—182no—
335C2×C3×C4—182no—
336C2×C3×C4—181no—
337C2×C3×C4—181no—
338#67—64no—
339#67—62no—
340#67—68no—
341#67—64no—
342#67—64no—
343#67—62no—
344C2×Dic3—32no—
345C2×Dic3—32no—
346C2×Dic3—32no—
347C2×Dic3—62no—
348C2×Dic3—62no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×C32—22no—
352C2×C32—31no—
353C2×C32—31no—
354C2×C32—32no—
355C2×C32—62no—
356C2×C32GL(2,ℤ/4)12yes—
357C2×C32C22×S411yescharacteristic
358C3×D3—122no—
359C2×C9—42no—
360C2×C9—41no—
361C2×C9—82no—
362C2×C9—84no—
363C2×C9—82no—
364C2×C9—81no—
365D9—82no—
366C24—22no—
367C24—31no—
368C24—31no—
369C24—32no—
370C24—62no—
371C24—91no—
372C24C2×C3×D911yescharacteristic
373C24C2×C3×D912yescomplemented
374C2×D4—92no—
375C2×D4—94no—
376C2×D4—94no—
377C2×D4—92no—
378C2×D4—182no—
379C2×D4—184no—
380C2×D4—182no—
381C2×D4—181no—
382C2×D4—182no—
383C2×D4—181no—
384C22×C4—91no—
385C22×C4—91no—
386C22×C4—91no—
387C22⋊C4—92no—
388C22⋊C4—94no—
389C22⋊C4—94no—
390C22⋊C4—92no—
391C22×C3—24no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—21no—
395C22×C3—22no—
396C22×C3—21no—
397C22×C3—32no—
398C22×C3—31no—
399C22×C3—32no—
400C22×C3—31no—
401C22×C3—32no—
402C22×C3—32no—
403C22×C3—31no—
404C22×C3—31no—
405C22×C3—31no—
406C22×C3—31no—
407C22×C3—32no—
408C22×C3—32no—
409C22×C3—34no—
410C22×C3—34no—
411C22×C3—32no—
412C22×C3—32no—
413C22×C3—61no—
414C22×C3—62no—
415C22×C3—61no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3—62no—
419C22×C3—61no—
420C22×C3—64no—
421C22×C3—62no—
422C22×C3—62no—
423C22×C3—62no—
424C22×C3—64no—
425C22×C3—62no—
426C22×C3—64no—
427C22×C3—64no—
428C22×C3—64no—
429C22×C3—68no—
430C22×C3—64no—
431C22×C3—61no—
432C22×C3—61no—
433C22×C3—62no—
434C22×C3—62no—
435C22×C3—64no—
436C22×C3—64no—
437C22×C3—64no—
438C22×C3—62no—
439C22×C3—68no—
440C22×C3—62no—
441C22×C3—62no—
442C22×C3—182no—
443C22×C3—182no—
444C22×C3—182no—
445C22×C3—362no—
446C22×C3—362no—
447C22×C3C2×#61111yescharacteristic, complemented
448C22×C3C2×(C22⋊C9)⋊C211yescharacteristic
449C22×C3C2×(C22⋊C9)⋊C212yescomplemented
450C22×C3C2×C3×#6711yescharacteristic
451C22×C3C2×C3×S411yescharacteristic
452C22×C3C2×C3×S412yes—
453C2×D3—62no—
454C2×D3—62no—
455C2×D3—62no—
456C2×D3—122no—
457C2×D3—122no—
458C3×C4—182no—
459C3×C4—182no—
460C3×C4—182no—
461Dic3—62no—
462Dic3—62no—
463Dic3—62no—
464C32C2×GL(2,ℤ/4)11yescharacteristic
465C9—41no—
466C9—81no—
467C23—22no—
468C23—31no—
469C23—31no—
470C23—32no—
471C23—32no—
472C23—34no—
473C23—31no—
474C23—32no—
475C23—32no—
476C23—62no—
477C23—62no—
478C23—62no—
479C23—61no—
480C23—64no—
481C23—62no—
482C23—64no—
483C23—92no—
484C23—181no—
485C23—182no—
486C23—182no—
487C23—181no—
488C23C3×#61112yescomplemented
489C23C3×(C22⋊C9)⋊C211yescharacteristic, complemented
490C23C22×C3×D911yescharacteristic
491D4—184no—
492D4—182no—
493D4—188no—
494D4—184no—
495D4—184no—
496D4—182no—
497C2×C4—92no—
498C2×C4—92no—
499C2×C4—92no—
500C2×C4—182no—
501C2×C4—182no—
502C2×C4—181no—
503C2×C4—181no—
504C2×C3—24no—
505C2×C3—22no—
506C2×C3—22no—
507C2×C3—21no—
508C2×C3—22no—
509C2×C3—31no—
510C2×C3—31no—
511C2×C3—32no—
512C2×C3—31no—
513C2×C3—31no—
514C2×C3—32no—
515C2×C3—61no—
516C2×C3—61no—
517C2×C3—62no—
518C2×C3—64no—
519C2×C3—62no—
520C2×C3—62no—
521C2×C3—362no—
522C2×C3#6369412yesdirect product factor
523C2×C3C22×(C22⋊C9)⋊C211yescharacteristic
524C2×C3C3×GL(2,ℤ/4)12yes—
525C2×C3C22×C3×S411yescharacteristic
526D3—122no—
527C22—22no—
528C22—32no—
529C22—31no—
530C22—31no—
531C22—31no—
532C22—32no—
533C22—32no—
534C22—34no—
535C22—32no—
536C22—62no—
537C22—62no—
538C22—64no—
539C22—61no—
540C22—62no—
541C22—64no—
542C22—62no—
543C22—182no—
544C22—182no—
545C22—182no—
546C22—362no—
547C22—362no—
548C22C2×C3×#61111yescharacteristic, complemented
549C22C2×C3×(C22⋊C9)⋊C211yescharacteristic
550C22C2×C3×(C22⋊C9)⋊C212yescomplemented
551C4—182no—
552C4—182no—
553C4—182no—
554C3—21no—
555C3C2×#6369411yescharacteristic, direct product factor
556C3C2×C3×GL(2,ℤ/4)11yescharacteristic
557C2—22no—
558C2—31no—
559C2—31no—
560C2—32no—
561C2—62no—
562C2—362no—
563C2C3×#6369412yesdirect product factor
564C2C22×C3×(C22⋊C9)⋊C211yescharacteristic
subgroupG/H
centerC22×C3C2×(C22⋊C9)⋊C2
commutator, cosocleC2×C22⋊C9C23×C3
FrattiniC2×C3C22×C3×S4
FittingC25×C32D3
socleC24×C32C2×D3
hypercenterC23×C3(C22⋊C9)⋊C2
hypocenterC22⋊C9C2×C3×D4
LCSG ▷ C2×C22⋊C9 ▷ C22⋊C9
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×C22⋊C9C2semidirect
2. C22×C3×(C22⋊C9)⋊C2C2semidirect
3. C3×#63694C2direct (N × H)
4. C2×C3×(C22⋊C9)⋊C4C2semidirect
6. C2×#63694C3direct (N × H)
9. C22×C3×C22⋊C9C22semidirect
11. C2×C3×(C22⋊C9)⋊C2C22semidirect
13. C3×(C22⋊C9)⋊C4C22semidirect
19. C23×C22⋊C9C2×C3semidirect
20. C22×(C22⋊C9)⋊C2C2×C3semidirect
26. #63694C2×C3direct (N × H)
27. C2×(C22⋊C9)⋊C4C2×C3semidirect
31. C2×C3×C22⋊C9D4semidirect
64. C22×C22⋊C9C22×C3semidirect
66. C2×(C22⋊C9)⋊C2C22×C3semidirect
72. (C22⋊C9)⋊C4C22×C3semidirect
78. C3×C22⋊C9C2×D4semidirect
81. C25×C3D9semidirect
143. C2×C22⋊C9C3×D4semidirect
172. C24×C3C2×D9semidirect
246. C22⋊C9C2×C3×D4semidirect
248. C25C3×D9semidirect
314. C23×C3#611semidirect
315. C23×C3(C22⋊C9)⋊C2semidirect
373. C24C2×C3×D9semidirect
447. C22×C3C2×#611semidirect
449. C22×C3C2×(C22⋊C9)⋊C2semidirect
488. C23C3×#611semidirect
489. C23C3×(C22⋊C9)⋊C2semidirect
522. C2×C3#63694direct (N × H)
548. C22C2×C3×#611semidirect
550. C22C2×C3×(C22⋊C9)⋊C2semidirect
555. C3C2×#63694direct (N × H)
563. C2C3×#63694direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C22⋊C92
C22×C3×(C22⋊C9)⋊C22
C3×#636942
C2×C3×(C22⋊C9)⋊C42
C2×C3×#9533
C2×#636943
C2×C3×#6114