Finite Group Databasebeta

← C3×#71947 #11798667

Subgroups — C3×#71947 #11798667

Order 1152.

646 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C32×#560C211yesmaximal, characteristic, complemented
2C3×#5077C211yesmaximal, characteristic, complemented
3C3×#5053C211yesmaximal, characteristic, complemented
4C3×#5032C211yesmaximal, characteristic, complemented
5C3×#4545C211yesmaximal, characteristic, complemented
6C3×#4539C211yesmaximal, characteristic, complemented
7C3×#4536C211yesmaximal, characteristic, complemented
8C3×#2200—31nomaximal
9#71947C311yesmaximal, characteristic, direct product factor
10C3×#1004C2211yescharacteristic
11C32×#127C2211yescharacteristic
12C32×#126—21no—
13C32×#123—21no—
14C32×#123C2211yescharacteristic
15C32×#121—21no—
16C32×#121—21no—
17C32×#121—21no—
18C32×#120—21no—
19C32×#120C2211yescharacteristic
20C3×#953—21no—
21C3×#932—21no—
22C3×#929—21no—
23C3×#834C2211yescharacteristic
24C3×#834C2211yescharacteristic
25C3×#833C2211yescharacteristic
26C3×#806—21no—
27C3×#806—21no—
28C3×#805—21no—
29C3×#560—21no—
30C3×#560D311yescharacteristic, complemented
31C3×#560C2×C311yescharacteristic, complemented
32C3×#458—31no—
33C3×B(4,𝔽2)—31no—
34C3×#453—31no—
35#5077C2×C311yescharacteristic, complemented
36#5053C2×C311yescharacteristic, complemented
37#5032C2×C311yescharacteristic, complemented
38C3×#353—31no—
39C3×#352—31no—
40C3×AΣL(1,𝔽16)—31no—
41#4545C2×C311yescharacteristic, complemented
42#4539C2×C311yescharacteristic, complemented
43#4536C2×C311yescharacteristic, complemented
44C24×C32D411yescharacteristic
45C2×C32×D4—21no—
46C2×C32×D4—21no—
47C2×C32×D4—21no—
48C2×C32×D4—21no—
49C2×C32×D4—21no—
50C2×C32×D4D411yescharacteristic, complemented
51C2×C32×D4C2311yescharacteristic
52C22×C32×C4—21no—
53C22×C32×C4D411yescharacteristic, complemented
54C2×C3×#67—21no—
55C2×C3×#67—21no—
56C2×C3×#67—41no—
57C3×#237—41no—
58C3×D3×D4—41no—
59C3×#235—21no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4—21no—
63C32×C22⋊C4—21no—
64C32×C22⋊C4—21no—
65C32×C22⋊C4—21no—
66C32×C22⋊C4—21no—
67C32×C22⋊C4—21no—
68C32×C22⋊C4D411yescharacteristic
69C32×C22⋊C4D411yescharacteristic, complemented
70C32×C42D411yescharacteristic, complemented
71C3×#217—21no—
72C3×#217—21no—
73C3×#217—41no—
74C3×#214—41no—
75C3×#213—41no—
76C3×(C2×D3)⋊C4—41no—
77C3×Dic3⋊C4—41no—
78C3×#208—21no—
79#2200—31noSylow
80C3×H(5,𝔽2)—31no—
81#1004C22×C311yescharacteristic
82C3×AGL(1,ℤ/8)—61no—
83C3×#127—21no—
84C3×#127C2×D311yescharacteristic
85C3×#127C22×C311yescharacteristic
86C3×#126—22no—
87C3×#126—21no—
88C3×#126—21no—
89C3×#123—22no—
90C3×#123—21no—
91C3×#123—21no—
92C3×#123—21no—
93C3×#123—61no—
94C3×#123C2×D311yescharacteristic
95C3×#123C22×C311yescharacteristic
96C3×#121—22no—
97C3×#121—22no—
98C3×#121—22no—
99C3×#121—21no—
100C3×#121—21no—
101C3×#121—21no—
102C3×#121—21no—
103C3×#121—21no—
104C3×#121—21no—
105C3×#120—22no—
106C3×#120—21no—
107C3×#120—21no—
108C3×#120—21no—
109C3×#120—61no—
110C3×#120C2×D311yescharacteristic
111C3×#120C22×C311yescharacteristic
112#953—21no—
113#932—21no—
114#929—21no—
115C3×#104—61no—
116C3×#100—31no—
117C3×#99—31no—
118C3×#99—31no—
119C3×#99—61no—
120C3×#99—61no—
121#834C22×C311yescharacteristic
122#834C22×C311yescharacteristic
123#833C22×C311yescharacteristic
124#806—21no—
125#806—21no—
126#805—21no—
127C23×C32—21no—
128C23×C32—21no—
129C23×C32—41no—
130C23×C32—81no—
131C23×C32C2×D411yescharacteristic
132C23×C32C2×D411yescharacteristic
133C22×C3×D3—21no—
134C22×C3×D3—21no—
135C32×D4—41no—
136C32×D4—41no—
137C32×D4—81no—
138C32×D4—81no—
139C32×D4—81no—
140C32×D4—81no—
141C32×D4—81no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—41no—
147C2×C32×C4—81no—
148C2×C32×C4C2×D411yescharacteristic
149C3×#67—41no—
150C3×#67—41no—
151C3×#67—41no—
152C3×#67—82no—
153C2×C3×Dic3—21no—
154C2×C3×Dic3—21no—
155C2×C3×Dic3—41no—
156C2×C3×Dic3—41no—
157C3×D12—21no—
158C3×C4×D3—41no—
159C3×Dic6—21no—
160C3×C3⋊C8—41no—
161#560C3×D311yescharacteristic, complemented
162#458—31no—
163B(4,𝔽2)—31no—
164#453—31no—
165#353—31no—
166#352—31no—
167AΣL(1,𝔽16)—31no—
168C24×C3—21no—
169C24×C3#6711yescharacteristic
170C24×C3C3×D411yescharacteristic
171C3×#41—61no—
172C3×#41—121no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—22no—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—61no—
191C2×C3×D4—61no—
192C2×C3×D4—121no—
193C2×C3×D4—121no—
194C2×C3×D4#6711yescharacteristic, complemented
195C2×C3×D4C3×D411yescharacteristic, complemented
196C2×C3×D4C22×D311yescharacteristic
197C2×C3×D4C23×C311yescharacteristic
198C22×C3×C4—22no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4#6711yescharacteristic, complemented
203C22×C3×C4C3×D411yescharacteristic, complemented
204C2×#67—21no—
205C2×#67—21no—
206C2×#67—41no—
207#237—41no—
208D3×D4—41no—
209#235—21no—
210C3×SD16—121no—
211C3×D8—121no—
212C3×M16—61no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—22no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—121no—
223C3×C22⋊C4—22no—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—21no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—21no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—21no—
237C3×C22⋊C4—21no—
238C3×C22⋊C4—21no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—61no—
241C3×C22⋊C4—61no—
242C3×C22⋊C4—121no—
243C3×C22⋊C4—121no—
244C3×C22⋊C4#6711yescharacteristic
245C3×C22⋊C4#6711yescharacteristic, complemented
246C3×C22⋊C4C3×D411yescharacteristic
247C3×C22⋊C4C3×D411yescharacteristic, complemented
248C3×C42—21no—
249C3×C42#6711yescharacteristic, complemented
250C3×C42C3×D411yescharacteristic, complemented
251#217—21no—
252#217—21no—
253#217—41no—
254#214—41no—
255#213—41no—
256(C2×D3)⋊C4—41no—
257Dic3⋊C4—41no—
258#208—21no—
259C22×C32—21no—
260C22×C32—21no—
261C22×C32—41no—
262C22×C32—41no—
263C22×C32—41no—
264C22×C32—41no—
265C22×C32—41no—
266C22×C32—41no—
267C22×C32—41no—
268C22×C32—41no—
269C22×C32—42no—
270C22×C32—81no—
271C22×C32#12011yescharacteristic
272C2×C3×D3—41no—
273C2×C3×D3—81no—
274C2×C3×D3—81no—
275C32×C4—21no—
276C32×C4—41no—
277C32×C4—41no—
278C32×C4—41no—
279C32×C4—41no—
280C3×Dic3—41no—
281C3×Dic3—81no—
282C3×Dic3—81no—
283H(5,𝔽2)—31no—
284AGL(1,ℤ/8)—61no—
285#127C2×C3×D311yescharacteristic
286#126—21no—
287#123—21no—
288#123—61no—
289#123C2×C3×D311yescharacteristic
290#121—21no—
291#121—21no—
292#121—21no—
293#120—21no—
294#120—61no—
295#120C2×C3×D311yescharacteristic
296#104—61no—
297#100—31no—
298#99—31no—
299#99—31no—
300#99—61no—
301#99—61no—
302C23×C3—22no—
303C23×C3—22no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3—61no—
313C23×C3—61no—
314C23×C3—81no—
315C23×C3—81no—
316C23×C3—81no—
317C23×C3—82no—
318C23×C3C2×#6711yescharacteristic
319C23×C3C2×#6711yescharacteristic
320C23×C3C2×C3×D411yescharacteristic
321C23×C3C2×C3×D411yescharacteristic
322C22×D3—21no—
323C22×D3—21no—
324C3×Q8—61no—
325C3×D4—41no—
326C3×D4—41no—
327C3×D4—41no—
328C3×D4—41no—
329C3×D4—61no—
330C3×D4—81no—
331C3×D4—81no—
332C3×D4—81no—
333C3×D4—81no—
334C3×D4—82no—
335C3×D4—82no—
336C3×D4—81no—
337C3×D4—81no—
338C3×D4—82no—
339C3×D4—81no—
340C3×D4—81no—
341C3×D4—81no—
342C3×D4—81no—
343C3×D4—82no—
344C3×D4—82no—
345C3×D4—81no—
346C3×D4—81no—
347C3×D4—121no—
348C3×D4—121no—
349C3×D4—121no—
350C3×D4—242no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—81no—
369C2×C3×C4—82no—
370C2×C3×C4—81no—
371C2×C3×C4—81no—
372C2×C3×C4—121no—
373C2×C3×C4—121no—
374C2×C3×C4—121no—
375C2×C3×C4C2×#6711yescharacteristic
376C2×C3×C4C2×C3×D411yescharacteristic
377#67—41no—
378#67—41no—
379#67—41no—
380#67—82no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385D12—21no—
386C4×D3—41no—
387Dic6—21no—
388C3×C8—121no—
389C3⋊C8—41no—
390C2×C32—21no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32—81no—
394C2×C32—81no—
395C2×C32B(4,𝔽2)11yescharacteristic
396C3×D3—81no—
397C24C3×#6711yescharacteristic
398#41—61no—
399#41—121no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—121no—
408C2×D4—121no—
409C2×D4C3×#6711yescharacteristic, complemented
410C2×D4C22×C3×D311yescharacteristic
411C22×C4—21no—
412C22×C4C3×#6711yescharacteristic, complemented
413SD16—121no—
414D8—121no—
415M16—61no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—121no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—21no—
423C22⋊C4—21no—
424C22⋊C4—21no—
425C22⋊C4—61no—
426C22⋊C4—61no—
427C22⋊C4—121no—
428C22⋊C4—121no—
429C22⋊C4C3×#6711yescharacteristic
430C22⋊C4C3×#6711yescharacteristic, complemented
431C42C3×#6711yescharacteristic, complemented
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—21no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—41no—
441C22×C3—41no—
442C22×C3—41no—
443C22×C3—42no—
444C22×C3—41no—
445C22×C3—41no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—41no—
449C22×C3—41no—
450C22×C3—42no—
451C22×C3—42no—
452C22×C3—42no—
453C22×C3—42no—
454C22×C3—41no—
455C22×C3—41no—
456C22×C3—41no—
457C22×C3—41no—
458C22×C3—42no—
459C22×C3—42no—
460C22×C3—42no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—41no—
464C22×C3—41no—
465C22×C3—82no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—82no—
469C22×C3—121no—
470C22×C3—241no—
471C22×C3—241no—
472C22×C3#95311yescharacteristic
473C22×C3C3×#12011yescharacteristic
474C2×D3—41no—
475C2×D3—81no—
476C2×D3—81no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—41no—
480C3×C4—42no—
481C3×C4—42no—
482C3×C4—41no—
483C3×C4—41no—
484C3×C4—41no—
485C3×C4—41no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—121no—
493C3×C4—241no—
494C3×C4—241no—
495Dic3—41no—
496Dic3—81no—
497Dic3—81no—
498C32#220011yescharacteristic, complemented, Sylow
499C23—21no—
500C23—21no—
501C23—41no—
502C23—61no—
503C23—61no—
504C23—81no—
505C23C2×C3×#6711yescharacteristic
506C23C2×C3×#6711yescharacteristic
507Q8—61no—
508D4—41no—
509D4—41no—
510D4—61no—
511D4—81no—
512D4—81no—
513D4—81no—
514D4—81no—
515D4—81no—
516D4—121no—
517D4—121no—
518D4—121no—
519D4—242no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—81no—
528C2×C4—121no—
529C2×C4—121no—
530C2×C4—121no—
531C2×C4C2×C3×#6711yescharacteristic
532C8—121no—
533C2×C3—22no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—41no—
540C2×C3—41no—
541C2×C3—81no—
542C2×C3—81no—
543C2×C3—82no—
544C2×C3—82no—
545C2×C3—81no—
546C2×C3—81no—
547C2×C3—81no—
548C2×C3—81no—
549C2×C3—241no—
550C2×C3#524411yescharacteristic
551C2×C3C3×B(4,𝔽2)11yescharacteristic
552D3—81no—
553C22—21no—
554C22—21no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—41no—
560C22—41no—
561C22—42no—
562C22—41no—
563C22—41no—
564C22—81no—
565C22—121no—
566C22—241no—
567C22—241no—
568C22C3×#95311yescharacteristic
569C4—21no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—121no—
575C4—241no—
576C4—241no—
577C3—21no—
578C3#7194711yescharacteristic, direct product factor
579C3C3×#220011yescharacteristic, complemented
580C2—21no—
581C2—41no—
582C2—41no—
583C2—81no—
584C2—81no—
585C2—241no—
586C2C3×#524411yescharacteristic
subgroupG/H
centerC2×C3#5244
commutator, cosocleC2×C3×D4C23×C3
FrattiniC2×D4C22×C3×D3
FittingC32×#560C2
socleC2×C32B(4,𝔽2)
hypercenterC3×#560D3
hypocenterC3C3×#2200
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#560

Complements

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

NH ≅ G/Ntype
1. C32×#560C2semidirect
2. C3×#5077C2semidirect
3. C3×#5053C2semidirect
4. C3×#5032C2semidirect
5. C3×#4545C2semidirect
6. C3×#4539C2semidirect
7. C3×#4536C2semidirect
9. #71947C3direct (N × H)
30. C3×#560D3semidirect
31. C3×#560C2×C3semidirect
35. #5077C2×C3semidirect
36. #5053C2×C3semidirect
37. #5032C2×C3semidirect
41. #4545C2×C3semidirect
42. #4539C2×C3semidirect
43. #4536C2×C3semidirect
50. C2×C32×D4D4semidirect
53. C22×C32×C4D4semidirect
69. C32×C22⋊C4D4semidirect
70. C32×C42D4semidirect
161. #560C3×D3semidirect
194. C2×C3×D4#67semidirect
195. C2×C3×D4C3×D4semidirect
202. C22×C3×C4#67semidirect
203. C22×C3×C4C3×D4semidirect
245. C3×C22⋊C4#67semidirect
247. C3×C22⋊C4C3×D4semidirect
249. C3×C42#67semidirect
250. C3×C42C3×D4semidirect
409. C2×D4C3×#67semidirect
412. C22×C4C3×#67semidirect
430. C22⋊C4C3×#67semidirect
431. C42C3×#67semidirect
498. C32#2200semidirect
578. C3#71947direct (N × H)
579. C3C3×#2200semidirect

Maximal subgroups

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

subgroupindex
C32×#5602
C3×#50772
C3×#50532
C3×#50322
C3×#45452
C3×#45392
C3×#45362
C3×#22003
#719473