Finite Group Databasebeta

← C3×#68107 #11778171

Subgroups — C3×#68107 #11778171

Order 1152.

689 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 689 conjugacy classes of subgroups — too many to draw. Only its 98 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×#5494C211yesmaximal, characteristic, complemented
2C3×#4960C211yesmaximal, characteristic, complemented
3C3×#4732C211yesmaximal, characteristic, complemented
4C3×#4731C211yesmaximal, characteristic, complemented
5C32×#323C211yesmaximal, characteristic, complemented
6C3×#4470C211yesmaximal, characteristic, complemented
7C3×#4467C211yesmaximal, characteristic, complemented
8C3×#1600—31nomaximal
9#68107C311yesmaximal, characteristic, direct product factor
10C22×C3×#67—21no—
11C22×C3×D12C2211yescharacteristic
12C2×C32×C22⋊C4C2211yescharacteristic
13C3×#953—21no—
14C3×#930C2211yescharacteristic
15C3×C4×#67—21no—
16C2×C3×(C2×D3)⋊C4—21no—
17C3×#924C2211yescharacteristic
18C2×C3×D24—21no—
19C2×C3×#204—21no—
20C3×#885—21no—
21C3×#884—21no—
22C3×#883—21no—
23C3×#879—21no—
24C32×C22⋊C8C2211yescharacteristic
25C32×C22⋊C8C2211yescharacteristic
26C3×C3⋊#95C2211yescharacteristic
27C3×#821—21no—
28C3×#821—21no—
29C3×#538—31no—
30#5494C2×C311yescharacteristic, complemented
31C3×#450—31no—
32C3×#447—31no—
33C3×#392—31no—
34#4960C2×C311yescharacteristic, complemented
35#4732C2×C311yescharacteristic, complemented
36#4731C2×C311yescharacteristic, complemented
37C3×#328—31no—
38C3×#327—31no—
39C3×#323—21no—
40C3×#323D311yescharacteristic, complemented
41C3×#323C2×C311yescharacteristic, complemented
42#4470C2×C311yescharacteristic, complemented
43#4467C2×C311yescharacteristic, complemented
44C24×C32D411yescharacteristic
45C23×C3×D3—21no—
46C22×C32×C4D411yescharacteristic
47C22×C32×C4C2311yescharacteristic
48C2×C3×#67—21no—
49C2×C3×#67—21no—
50C2×C3×#67—21no—
51C2×C3×#67—42no—
52C2×C3×#67—42no—
53C2×C3×#67—42no—
54C22×C3×Dic3—21no—
55C2×C3×D12—21no—
56C2×C3×D12—21no—
57C2×C3×D12—42no—
58C2×C3×D12D411yescharacteristic, complemented
59C2×C3×C4×D3—21no—
60C2×C3×Dic6D411yescharacteristic, complemented
61C2×C32×C8—21no—
62C2×C32×C8—21no—
63C32×C22⋊C4—21no—
64C32×C22⋊C4—21no—
65C3×#217—21no—
66C3×#217—21no—
67C3×#217—21no—
68C3×(C2×D3)⋊C4—21no—
69C3×(C2×D3)⋊C4—21no—
70C3×(C2×D3)⋊C4—42no—
71C3×C4⋊Dic3D411yescharacteristic, complemented
72C3×C4⋊Dic3D411yescharacteristic, complemented
73C3×Dic3⋊C4—21no—
74C3×Dic3⋊C4—21no—
75C3×C4×Dic3—21no—
76C3×D24—42no—
77C3×#204—42no—
78#1600—31noSylow
79C22×C3×D4—31no—
80C22×C3×D4—61no—
81C22×#67—21no—
82C22×D12C22×C311yescharacteristic
83C2×C3×SD16—61no—
84C2×C3×D8—61no—
85C3×#126—61no—
86C3×#124—61no—
87C3×#123—61no—
88C3×C22⋊Q8—31no—
89C3×#121—31no—
90C3×#121—61no—
91C3×#120—61no—
92C3×C4×D4—61no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—61no—
95C2×C3×C22⋊C4C2×D311yescharacteristic
96C2×C3×C22⋊C4C22×C311yescharacteristic
97#953—21no—
98#930C22×C311yescharacteristic
99C4×#67—21no—
100C2×(C2×D3)⋊C4—21no—
101#924C22×C311yescharacteristic
102C2×D24—21no—
103C2×#204—21no—
104#885—21no—
105#884—21no—
106#883—21no—
107#879—21no—
108C3×#102—61no—
109C3×#102—61no—
110C3×C22⋊C8—21no—
111C3×C22⋊C8—21no—
112C3×C22⋊C8C2×D311yescharacteristic
113C3×C22⋊C8C2×D311yescharacteristic
114C3×C22⋊C8C22×C311yescharacteristic
115C3×C22⋊C8C22×C311yescharacteristic
116C3×#95—31no—
117C3⋊#95C22×C311yescharacteristic
118#821—21no—
119#821—21no—
120C23×C32—21no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32—41no—
124C23×C32—41no—
125C23×C32C2×D411yescharacteristic
126C22×C3×D3—22no—
127C22×C3×D3—21no—
128C22×C3×D3—41no—
129C22×C3×D3—41no—
130C22×C3×D3—41no—
131C22×C3×D3—41no—
132C22×C3×D3—41no—
133C22×C3×D3—41no—
134C2×C32×C4—21no—
135C2×C32×C4—41no—
136C2×C32×C4—41no—
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C3×#67—48no—
140C3×#67—41no—
141C3×#67—41no—
142C3×#67—81no—
143C3×#67—81no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—41no—
149C2×C3×Dic3—41no—
150C3×D12—22no—
151C3×D12—22no—
152C3×D12—41no—
153C3×D12—41no—
154C3×D12—41no—
155C3×C4×D3—81no—
156C3×Dic6—41no—
157C32×C8—41no—
158C32×C8—41no—
159#538—31no—
160#450—31no—
161#447—31no—
162#392—31no—
163#328—31no—
164#327—31no—
165#323C3×D311yescharacteristic, complemented
166C24×C3—21no—
167C24×C3—61no—
168C24×C3D1211yescharacteristic
169C24×C3C3×D411yescharacteristic
170C23×D3—21no—
171C2×C3×Q8—31no—
172C2×C3×D4—31no—
173C2×C3×D4—61no—
174C2×C3×D4—61no—
175C2×C3×D4—61no—
176C2×C3×D4—61no—
177C2×C3×D4—61no—
178C2×C3×D4—122no—
179C2×C3×D4—122no—
180C2×C3×D4—122no—
181C2×C3×D4—122no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—61no—
185C22×C3×C4—61no—
186C22×C3×C4D1211yescharacteristic
187C22×C3×C4C3×D411yescharacteristic
188C22×C3×C4C22×D311yescharacteristic
189C22×C3×C4C23×C311yescharacteristic
190C2×#67—21no—
191C2×#67—21no—
192C2×#67—21no—
193C2×#67—42no—
194C2×#67—42no—
195C2×#67—42no—
196C22×Dic3—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—42no—
200C2×D12C3×D411yescharacteristic, complemented
201C2×C4×D3—21no—
202C2×Dic6C3×D411yescharacteristic, complemented
203C3×SD16—122no—
204C3×D8—122no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—41no—
210C2×C3×C8—41no—
211C3×C4⋊C4—31no—
212C3×C4⋊C4—31no—
213C3×C4⋊C4—61no—
214C3×C4⋊C4—61no—
215C3×C22⋊C4—21no—
216C3×C22⋊C4—21no—
217C3×C22⋊C4—21no—
218C3×C22⋊C4—21no—
219C3×C22⋊C4—41no—
220C3×C22⋊C4—41no—
221C3×C22⋊C4—61no—
222C3×C22⋊C4—61no—
223C3×C22⋊C4—61no—
224C3×C22⋊C4—61no—
225C3×C22⋊C4—61no—
226C3×C22⋊C4—122no—
227C3×C42—61no—
228#217—21no—
229#217—21no—
230#217—21no—
231(C2×D3)⋊C4—21no—
232(C2×D3)⋊C4—21no—
233(C2×D3)⋊C4—42no—
234C4⋊Dic3C3×D411yescharacteristic, complemented
235C4⋊Dic3C3×D411yescharacteristic, complemented
236Dic3⋊C4—21no—
237Dic3⋊C4—21no—
238C4×Dic3—21no—
239D24—42no—
240#204—42no—
241C22×C32—21no—
242C22×C32—21no—
243C22×C32—41no—
244C22×C32—42no—
245C22×C32—41no—
246C22×C32—41no—
247C22×C32—81no—
248C22×C32#12011yescharacteristic
249C22×C32#13511yescharacteristic
250C22×C32AGL(1,ℤ/8)11yescharacteristic
251C2×C3×D3—42no—
252C2×C3×D3—42no—
253C2×C3×D3—41no—
254C2×C3×D3—41no—
255C2×C3×D3—42no—
256C2×C3×D3—41no—
257C2×C3×D3—81no—
258C2×C3×D3—81no—
259C2×C3×D3—81no—
260C2×C3×D3—81no—
261C32×C4—21no—
262C32×C4—21no—
263C32×C4—41no—
264C3×Dic3—42no—
265C3×Dic3—41no—
266C3×Dic3—41no—
267C3×Dic3—41no—
268C22×D4—31no—
269C22×D4—61no—
270C2×SD16—61no—
271C2×D8—61no—
272#126—61no—
273#124—61no—
274#123—61no—
275C22⋊Q8—31no—
276#121—31no—
277#121—61no—
278#120—61no—
279C4×D4—61no—
280C2×C22⋊C4—61no—
281C2×C22⋊C4C2×C3×D311yescharacteristic
282#102—61no—
283#102—61no—
284C22⋊C8C2×C3×D311yescharacteristic
285C22⋊C8C2×C3×D311yescharacteristic
286#95—31no—
287C23×C3—22no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—21no—
291C23×C3—21no—
292C23×C3—21no—
293C23×C3—21no—
294C23×C3—21no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—41no—
300C23×C3—41no—
301C23×C3—62no—
302C23×C3—61no—
303C23×C3—81no—
304C23×C3—81no—
305C23×C3—121no—
306C23×C3—121no—
307C23×C3—121no—
308C23×C3—121no—
309C23×C3—121no—
310C23×C3—121no—
311C23×C3C2×D1211yescharacteristic
312C23×C3C2×C3×D411yescharacteristic
313C22×D3—22no—
314C22×D3—21no—
315C22×D3—41no—
316C22×D3—41no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—41no—
320C22×D3—41no—
321C3×Q8—121no—
322C3×D4—62no—
323C3×D4—62no—
324C3×D4—121no—
325C3×D4—121no—
326C3×D4—121no—
327C3×D4—128no—
328C3×D4—121no—
329C3×D4—121no—
330C3×D4—241no—
331C3×D4—241no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
341C2×C3×C4—62no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—61no—
345C2×C3×C4—81no—
346C2×C3×C4—81no—
347C2×C3×C4—121no—
348C2×C3×C4—121no—
349C2×C3×C4—241no—
350C2×C3×C4D3×D411yescharacteristic
351C2×C3×C4D3×D411yescharacteristic
352C2×C3×C4C2×C3×D411yescharacteristic
353C2×C3×C4C2×C3×D411yescharacteristic
354#67—48no—
355#67—41no—
356#67—41no—
357#67—81no—
358#67—81no—
359C2×Dic3—22no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—41no—
364C2×Dic3—41no—
365D12—22no—
366D12—22no—
367D12—41no—
368D12—41no—
369D12—41no—
370C4×D3—81no—
371Dic6—41no—
372C3×C8—41no—
373C3×C8—41no—
374C3×C8—41no—
375C3×C8—41no—
376C3×C8—81no—
377C3×C8—81no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—81no—
381C2×C32#44911yescharacteristic
382C2×C32#45311yescharacteristic
383C2×C32B(4,𝔽2)11yescharacteristic
384C3×D3—81no—
385C3×D3—82no—
386C24—61no—
387C24C3×D1211yescharacteristic
388C2×Q8—31no—
389C2×D4—31no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—122no—
396C2×D4—122no—
397C2×D4—122no—
398C2×D4—122no—
399C22×C4—61no—
400C22×C4—61no—
401C22×C4C3×D1211yescharacteristic
402C22×C4C22×C3×D311yescharacteristic
403SD16—122no—
404D8—122no—
405C2×C8—21no—
406C2×C8—21no—
407C4⋊C4—31no—
408C4⋊C4—31no—
409C4⋊C4—61no—
410C4⋊C4—61no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—61no—
414C22⋊C4—61no—
415C22⋊C4—61no—
416C22⋊C4—61no—
417C22⋊C4—61no—
418C22⋊C4—122no—
419C42—61no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3—42no—
432C22×C3—41no—
433C22×C3—41no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—81no—
439C22×C3—81no—
440C22×C3—82no—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3—82no—
444C22×C3—122no—
445C22×C3—122no—
446C22×C3—121no—
447C22×C3—122no—
448C22×C3—121no—
449C22×C3—121no—
450C22×C3—241no—
451C22×C3—241no—
452C22×C3—241no—
453C22×C3—241no—
454C22×C3#88211yescharacteristic
455C22×C3#90411yescharacteristic
456C22×C3#90811yescharacteristic
457C22×C3C3×#12011yescharacteristic
458C22×C3C3×#13511yescharacteristic
459C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
460C2×D3—42no—
461C2×D3—42no—
462C2×D3—41no—
463C2×D3—41no—
464C2×D3—42no—
465C2×D3—41no—
466C2×D3—81no—
467C2×D3—81no—
468C2×D3—81no—
469C2×D3—81no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—41no—
475C3×C4—41no—
476C3×C4—41no—
477C3×C4—41no—
478C3×C4—81no—
479C3×C4—122no—
480C3×C4—121no—
481C3×C4—121no—
482C3×C4—121no—
483Dic3—42no—
484Dic3—41no—
485Dic3—41no—
486Dic3—41no—
487C32#160011yescharacteristic, complemented, Sylow
488C23—21no—
489C23—21no—
490C23—21no—
491C23—41no—
492C23—41no—
493C23—62no—
494C23—61no—
495C23—121no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23—121no—
501C23C2×C3×D1211yescharacteristic
502Q8—121no—
503D4—62no—
504D4—62no—
505D4—121no—
506D4—121no—
507D4—121no—
508D4—128no—
509D4—121no—
510D4—121no—
511D4—241no—
512D4—241no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—62no—
517C2×C4—61no—
518C2×C4—61no—
519C2×C4—61no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4—241no—
523C2×C4C3×D3×D411yescharacteristic
524C2×C4C3×D3×D411yescharacteristic
525C8—41no—
526C8—41no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—41no—
535C2×C3—41no—
536C2×C3—82no—
537C2×C3—81no—
538C2×C3—81no—
539C2×C3—242no—
540C2×C3—241no—
541C2×C3#473411yescharacteristic
542C2×C3#474111yescharacteristic
543C2×C3#474711yescharacteristic
544C2×C3C3×#44911yescharacteristic
545C2×C3C3×#45311yescharacteristic
546C2×C3C3×B(4,𝔽2)11yescharacteristic
547D3—82no—
548D3—81no—
549C22—21no—
550C22—21no—
551C22—42no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—81no—
556C22—122no—
557C22—122no—
558C22—121no—
559C22—122no—
560C22—121no—
561C22—121no—
562C22—241no—
563C22—241no—
564C22—241no—
565C22—241no—
566C22C3×#88211yescharacteristic
567C22C3×#90411yescharacteristic
568C22C3×#90811yescharacteristic
569C4—21no—
570C4—21no—
571C4—41no—
572C4—122no—
573C4—121no—
574C4—121no—
575C4—121no—
576C3—21no—
577C3#6810711yescharacteristic, direct product factor
578C3C3×#160011yescharacteristic, complemented
579C2—21no—
580C2—21no—
581C2—81no—
582C2—242no—
583C2—241no—
584C2C3×#473411yescharacteristic
585C2C3×#474111yescharacteristic
586C2C3×#474711yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#323C2
socleC22×C32#120
hypercenterC3×#323D3
hypocenterC3C3×#1600
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#323

Complements

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

NH ≅ G/Ntype
1. C3×#5494C2semidirect
2. C3×#4960C2semidirect
3. C3×#4732C2semidirect
4. C3×#4731C2semidirect
5. C32×#323C2semidirect
6. C3×#4470C2semidirect
7. C3×#4467C2semidirect
9. #68107C3direct (N × H)
30. #5494C2×C3semidirect
34. #4960C2×C3semidirect
35. #4732C2×C3semidirect
36. #4731C2×C3semidirect
40. C3×#323D3semidirect
41. C3×#323C2×C3semidirect
42. #4470C2×C3semidirect
43. #4467C2×C3semidirect
58. C2×C3×D12D4semidirect
60. C2×C3×Dic6D4semidirect
71. C3×C4⋊Dic3D4semidirect
72. C3×C4⋊Dic3D4semidirect
165. #323C3×D3semidirect
200. C2×D12C3×D4semidirect
202. C2×Dic6C3×D4semidirect
234. C4⋊Dic3C3×D4semidirect
235. C4⋊Dic3C3×D4semidirect
487. C32#1600semidirect
577. C3#68107direct (N × H)
578. C3C3×#1600semidirect

Maximal subgroups

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

subgroupindex
C3×#54942
C3×#49602
C3×#47322
C3×#47312
C32×#3232
C3×#44702
C3×#44672
C3×#16003
#681073