Finite Group Databasebeta

← C2×C3×#2679⋊C3 #72276702

Subgroups — C2×C3×#2679⋊C3 #72276702

Order 2304.

2586 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 2586 conjugacy classes of subgroups — too many to draw. Only its 52 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
1C3×#2679⋊C3C211yesmaximal, characteristic, direct product factor
2C2×#2679⋊C3C313yesmaximal, direct product factor
3C2×C3×#2679C311yesmaximal, characteristic, complemented
4C22×C3×C22⋊A4—42nomaximal
5C22×C3×C42⋊C3—41nomaximal
6C2×C3×#95⋊C3—42nomaximal
7C22×C3×#120—32no—
8C3×#2679—32no—
9C3×#2679—32no—
10C3×#2679—31no—
11C3×#2679C2×C311yescharacteristic, complemented
12C2×C3×#394—32no—
13C2×C3×#390—31no—
14#2679⋊C3C2×C313yesdirect product factor
15C2×C3×C22⋊A4—44no—
16C2×C3×C22⋊A4—42no—
17C2×C3×C42⋊C3—42no—
18C2×C3×C42⋊C3—41no—
19C3×#95⋊C3—42no—
20C2×#2679C3211yescharacteristic, complemented, Sylow
21C26×C3A412yescomplemented
22C22×C22⋊A4—46no—
23C23×C3×D4—31no—
24C23×C3×D4—32no—
25C2×C3×#120—34no—
26C2×C3×#120—32no—
27C2×C3×#120—32no—
28C2×C3×#120—34no—
29C2×C3×#120—32no—
30C2×C3×#120—32no—
31C2×C3×#120—616no—
32C22×C3×C4⋊C4—31no—
33C22×C3×C22⋊C4—32no—
34C22×C3×C22⋊C4—34no—
35C22×C3×C42A411yescharacteristic, complemented
36C22×C42⋊C3—43no—
37C3×#394—32no—
38C3×#394—34no—
39C3×#394—34no—
40C3×#394—32no—
41C3×#394—32no—
42C3×#394—32no—
43C3×#390—31no—
44C3×#390—34no—
45C3×#390—31no—
46C3×#390—31no—
47C3×#390—31no—
48C2×C3×#95A412yescomplemented
49C2×#95⋊C3—46no—
50C3×C22⋊A4—42no—
51C22×C3×A4—168no—
52C22×C3×A4—161no—
53C3×C42⋊C3—41no—
54C22×#120—32no—
55#2679—32no—
56#2679—31no—
57#2679—32no—
58#2679C2×C3211yescharacteristic, complemented
59C2×#394—32no—
60C2×#390—31no—
61C25×C3—32no—
62C25×C3—32no—
63C25×C3—44no—
64C25×C3—68no—
65C25×C3—122no—
66C25×C3—122no—
67C25×C3C2×A412yescomplemented
68C2×C22⋊A4—412no—
69C2×C22⋊A4—46no—
70C22×C3×D4—31no—
71C22×C3×D4—31no—
72C22×C3×D4—32no—
73C22×C3×D4—38no—
74C22×C3×D4—32no—
75C22×C3×D4—32no—
76C22×C3×D4—34no—
77C22×C3×D4—616no—
78C22×C3×D4—121no—
79C22×C3×D4—121no—
80C22×C3×D4—122no—
81C22×C3×D4—122no—
82C22×C3×D4—122no—
83C23×C3×C4—31no—
84C23×C3×C4—32no—
85C3×#120—68no—
86C3×#120—68no—
87C3×#120—616no—
88C3×#120—68no—
89C3×#120—68no—
90C3×#120—616no—
91C2×C3×C4⋊C4—31no—
92C2×C3×C4⋊C4—31no—
93C2×C3×C4⋊C4—32no—
94C2×C3×C4⋊C4—121no—
95C2×C3×C4⋊C4—121no—
96C2×C3×C22⋊C4—34no—
97C2×C3×C22⋊C4—34no—
98C2×C3×C22⋊C4—34no—
99C2×C3×C22⋊C4—34no—
100C2×C3×C22⋊C4—32no—
101C2×C3×C22⋊C4—34no—
102C2×C3×C22⋊C4—32no—
103C2×C3×C22⋊C4—616no—
104C2×C3×C22⋊C4—68no—
105C2×C3×C42—31no—
106C2×C3×C42—42no—
107C2×C3×C42C2×A411yescharacteristic, complemented
108C2×C42⋊C3—46no—
109C2×C42⋊C3—43no—
110C3×#95—32no—
111C3×#95C2×A412yescomplemented
112#95⋊C3—46no—
113C2×C3×A4—1616no—
114C2×C3×A4—168no—
115C2×C3×A4—162no—
116C2×C3×A4—161no—
117C26C3×A412yescomplemented
118C23×D4—31no—
119C23×D4—32no—
120C2×#120—32no—
121C2×#120—34no—
122C2×#120—34no—
123C2×#120—32no—
124C2×#120—32no—
125C2×#120—32no—
126C2×#120—616no—
127C22×C4⋊C4—31no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—34no—
130C22×C42C3×A411yescharacteristic, complemented
131#394—32no—
132#394—34no—
133#394—34no—
134#394—32no—
135#394—32no—
136#394—32no—
137#390—31no—
138#390—34no—
139#390—31no—
140#390—31no—
141#390—31no—
142C2×#95C3×A412yescomplemented
143C24×C3—38no—
144C24×C3—32no—
145C24×C3—32no—
146C24×C3—48no—
147C24×C3—42no—
148C24×C3—68no—
149C24×C3—62no—
150C24×C3—64no—
151C24×C3—62no—
152C24×C3—64no—
153C24×C3—64no—
154C24×C3—64no—
155C24×C3—68no—
156C24×C3—64no—
157C24×C3—122no—
158C24×C3—128no—
159C24×C3—1216no—
160C24×C3—128no—
161C24×C3—122no—
162C24×C3—124no—
163C24×C3—122no—
164C24×C3—128no—
165C24×C3—128no—
166C24×C3—128no—
167C24×C3—1216no—
168C24×C3C22⋊A411yescharacteristic
169C22⋊A4—46no—
170C22×A4—1624no—
171C22×A4—163no—
172C2×C3×D4—34no—
173C2×C3×D4—34no—
174C2×C3×D4—38no—
175C2×C3×D4—68no—
176C2×C3×D4—68no—
177C2×C3×D4—68no—
178C2×C3×D4—68no—
179C2×C3×D4—121no—
180C2×C3×D4—121no—
181C2×C3×D4—122no—
182C2×C3×D4—128no—
183C2×C3×D4—128no—
184C2×C3×D4—121no—
185C2×C3×D4—121no—
186C2×C3×D4—122no—
187C2×C3×D4—122no—
188C2×C3×D4—122no—
189C2×C3×D4—124no—
190C2×C3×D4—1216no—
191C2×C3×D4—1216no—
192C22×C3×C4—34no—
193C22×C3×C4—31no—
194C22×C3×C4—31no—
195C22×C3×C4—32no—
196C22×C3×C4—32no—
197C22×C3×C4—61no—
198C22×C3×C4—61no—
199C22×C3×C4—62no—
200C22×C3×C4—68no—
201C22×C3×C4—62no—
202C22×C3×C4—62no—
203C3×C4⋊C4—121no—
204C3×C4⋊C4—121no—
205C3×C4⋊C4—122no—
206C3×C22⋊C4—68no—
207C3×C22⋊C4—68no—
208C3×C22⋊C4—68no—
209C3×C22⋊C4—68no—
210C3×C22⋊C4—64no—
211C3×C22⋊C4—68no—
212C3×C22⋊C4—64no—
213C3×C42—41no—
214C3×C42—121no—
215C42⋊C3—43no—
216C22×C32—641no—
217C3×A4—168no—
218C3×A4—161no—
219C25—32no—
220C25—32no—
221C25—44no—
222C25—68no—
223C25—122no—
224C25—122no—
225C25C2×C3×A412yescomplemented
226C22×D4—31no—
227C22×D4—31no—
228C22×D4—32no—
229C22×D4—38no—
230C22×D4—32no—
231C22×D4—32no—
232C22×D4—34no—
233C22×D4—616no—
234C22×D4—121no—
235C22×D4—121no—
236C22×D4—122no—
237C22×D4—122no—
238C22×D4—122no—
239C23×C4—31no—
240C23×C4—32no—
241#120—68no—
242#120—68no—
243#120—68no—
244#120—616no—
245#120—616no—
246#120—68no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—121no—
251C2×C4⋊C4—121no—
252C2×C22⋊C4—34no—
253C2×C22⋊C4—34no—
254C2×C22⋊C4—34no—
255C2×C22⋊C4—34no—
256C2×C22⋊C4—32no—
257C2×C22⋊C4—34no—
258C2×C22⋊C4—32no—
259C2×C22⋊C4—616no—
260C2×C22⋊C4—68no—
261C2×C42—31no—
262C2×C42—42no—
263C2×C42C2×C3×A411yescharacteristic, complemented
264#95—32no—
265#95C2×C3×A412yescomplemented
266C23×C3—32no—
267C23×C3—31no—
268C23×C3—31no—
269C23×C3—34no—
270C23×C3—34no—
271C23×C3—416no—
272C23×C3—48no—
273C23×C3—64no—
274C23×C3—64no—
275C23×C3—68no—
276C23×C3—616no—
277C23×C3—68no—
278C23×C3—616no—
279C23×C3—62no—
280C23×C3—64no—
281C23×C3—62no—
282C23×C3—64no—
283C23×C3—62no—
284C23×C3—62no—
285C23×C3—128no—
286C23×C3—124no—
287C23×C3—128no—
288C23×C3—124no—
289C23×C3—124no—
290C23×C3—122no—
291C23×C3—124no—
292C23×C3—128no—
293C23×C3—1216no—
294C23×C3—124no—
295C23×C3—124no—
296C23×C3—1216no—
297C23×C3—124no—
298C23×C3—128no—
299C23×C3—1216no—
300C23×C3—1216no—
301C23×C3—128no—
302C23×C3—1216no—
303C23×C3—1232no—
304C23×C3—122no—
305C23×C3#99712yes—
306C23×C3C2×C22⋊A411yescharacteristic
307C2×A4—1648no—
308C2×A4—1624no—
309C2×A4—166no—
310C2×A4—163no—
311C3×D4—124no—
312C3×D4—124no—
313C3×D4—128no—
314C3×D4—128no—
315C3×D4—1216no—
316C3×D4—128no—
317C2×C3×C4—32no—
318C2×C3×C4—32no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—64no—
322C2×C3×C4—64no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—64no—
326C2×C3×C4—64no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—124no—
330C2×C3×C4—124no—
331C2×C32—642no—
332C2×C32—641no—
333C24—38no—
334C24—32no—
335C24—32no—
336C24—48no—
337C24—42no—
338C24—68no—
339C24—62no—
340C24—64no—
341C24—62no—
342C24—64no—
343C24—64no—
344C24—64no—
345C24—68no—
346C24—64no—
347C24—122no—
348C24—128no—
349C24—1216no—
350C24—128no—
351C24—122no—
352C24—124no—
353C24—122no—
354C24—128no—
355C24—128no—
356C24—128no—
357C24—1216no—
358C24C3×C22⋊A411yescharacteristic
359C2×D4—34no—
360C2×D4—34no—
361C2×D4—38no—
362C2×D4—68no—
363C2×D4—68no—
364C2×D4—68no—
365C2×D4—68no—
366C2×D4—121no—
367C2×D4—121no—
368C2×D4—122no—
369C2×D4—128no—
370C2×D4—128no—
371C2×D4—121no—
372C2×D4—121no—
373C2×D4—122no—
374C2×D4—122no—
375C2×D4—122no—
376C2×D4—124no—
377C2×D4—1216no—
378C2×D4—1216no—
379C22×C4—34no—
380C22×C4—31no—
381C22×C4—31no—
382C22×C4—32no—
383C22×C4—32no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—62no—
387C22×C4—68no—
388C22×C4—62no—
389C22×C4—62no—
390C4⋊C4—121no—
391C4⋊C4—121no—
392C4⋊C4—122no—
393C22⋊C4—68no—
394C22⋊C4—68no—
395C22⋊C4—68no—
396C22⋊C4—68no—
397C22⋊C4—64no—
398C22⋊C4—68no—
399C22⋊C4—64no—
400C42—41no—
401C42—121no—
402C22×C3—31no—
403C22×C3—32no—
404C22×C3—32no—
405C22×C3—32no—
406C22×C3—32no—
407C22×C3—31no—
408C22×C3—31no—
409C22×C3—48no—
410C22×C3—64no—
411C22×C3—64no—
412C22×C3—68no—
413C22×C3—68no—
414C22×C3—124no—
415C22×C3—124no—
416C22×C3—122no—
417C22×C3—122no—
418C22×C3—1216no—
419C22×C3—128no—
420C22×C3—128no—
421C22×C3—1216no—
422C22×C3—1216no—
423C22×C3—128no—
424C22×C3—122no—
425C22×C3—122no—
426C22×C3—643no—
427C22×C3#546411yescharacteristic
428C22×C3C2×#99711yescharacteristic
429A4—1624no—
430A4—163no—
431C3×C4—62no—
432C3×C4—62no—
433C3×C4—122no—
434C3×C4—122no—
435C32—641noSylow
436C23—32no—
437C23—31no—
438C23—31no—
439C23—34no—
440C23—34no—
441C23—416no—
442C23—48no—
443C23—64no—
444C23—64no—
445C23—68no—
446C23—616no—
447C23—68no—
448C23—616no—
449C23—62no—
450C23—64no—
451C23—62no—
452C23—64no—
453C23—62no—
454C23—62no—
455C23—128no—
456C23—124no—
457C23—128no—
458C23—124no—
459C23—124no—
460C23—122no—
461C23—124no—
462C23—128no—
463C23—1216no—
464C23—124no—
465C23—124no—
466C23—1216no—
467C23—124no—
468C23—128no—
469C23—1216no—
470C23—1216no—
471C23—128no—
472C23—1216no—
473C23—1232no—
474C23—122no—
475C23C3×#99712yes—
476C23C2×C3×C22⋊A411yescharacteristic
477D4—124no—
478D4—124no—
479D4—128no—
480D4—128no—
481D4—1216no—
482D4—128no—
483C2×C4—32no—
484C2×C4—32no—
485C2×C4—61no—
486C2×C4—61no—
487C2×C4—64no—
488C2×C4—64no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—64no—
492C2×C4—64no—
493C2×C4—62no—
494C2×C4—62no—
495C2×C4—124no—
496C2×C4—124no—
497C2×C3—31no—
498C2×C3—31no—
499C2×C3—32no—
500C2×C3—124no—
501C2×C3—124no—
502C2×C3—646no—
503C2×C3—643no—
504C2×C3#2679⋊C312yesdirect product factor
505C2×C3C2×#546411yescharacteristic
506C22—31no—
507C22—32no—
508C22—32no—
509C22—32no—
510C22—32no—
511C22—31no—
512C22—31no—
513C22—48no—
514C22—64no—
515C22—64no—
516C22—68no—
517C22—68no—
518C22—124no—
519C22—124no—
520C22—122no—
521C22—122no—
522C22—1216no—
523C22—128no—
524C22—128no—
525C22—1216no—
526C22—1216no—
527C22—128no—
528C22—122no—
529C22—122no—
530C22C3×#546411yescharacteristic
531C22C2×C3×#99711yescharacteristic
532C4—62no—
533C4—62no—
534C4—122no—
535C4—122no—
536C3—643no—
537C3C2×#2679⋊C311yescharacteristic, direct product factor
538C2—31no—
539C2—31no—
540C2—32no—
541C2—124no—
542C2—124no—
543C2C3×#2679⋊C312yesdirect product factor
544C2C2×C3×#546411yescharacteristic
subgroupG/H
center, hypercenterC22×C3#5464
commutator, cosocle, hypocenter#2679C2×C32
FrattiniC23C2×C3×C22⋊A4
FittingC2×C3×#2679C3
socleC24×C3C22⋊A4
LCSG ▷ #2679
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C3×#2679⋊C3C2direct (N × H)
2. C2×#2679⋊C3C3direct (N × H)
3. C2×C3×#2679C3semidirect
11. C3×#2679C2×C3semidirect
14. #2679⋊C3C2×C3direct (N × H)
20. C2×#2679C32semidirect
21. C26×C3A4semidirect
35. C22×C3×C42A4semidirect
48. C2×C3×#95A4semidirect
58. #2679C2×C32semidirect
67. C25×C3C2×A4semidirect
107. C2×C3×C42C2×A4semidirect
111. C3×#95C2×A4semidirect
117. C26C3×A4semidirect
130. C22×C42C3×A4semidirect
142. C2×#95C3×A4semidirect
225. C25C2×C3×A4semidirect
263. C2×C42C2×C3×A4semidirect
265. #95C2×C3×A4semidirect
504. C2×C3#2679⋊C3direct (N × H)
537. C3C2×#2679⋊C3direct (N × H)
543. C2C3×#2679⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#2679⋊C32
C2×#2679⋊C33
C2×C3×#26793
C22×C3×C22⋊A44
C22×C3×C42⋊C34
C2×C3×#95⋊C34