Finite Group Databasebeta

← C3×#70970 #11778043

Subgroups — C3×#70970 #11778043

Order 1152.

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

normal subgroups only

This group has 643 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×#5602C211yesmaximal, characteristic, complemented
2C32×#447C211yesmaximal, characteristic
3C3×#5222C211yesmaximal, characteristic, complemented
4C3×#5218C211yesmaximal, characteristic, complemented
5C3×#4537C211yesmaximal, characteristic, complemented
6C3×#4469C211yesmaximal, characteristic, complemented
7C3×#4467C211yesmaximal, characteristic, complemented
8C3×#1597—31nomaximal
9#70970C311yesmaximal, characteristic, direct product factor
10C22×C32×D4C2211yescharacteristic
11C2×C32×D8—21no—
12C32×#121C2211yescharacteristic
13C2×C3×#217—21no—
14C2×C3×#217—21no—
15C3×#935—21no—
16C3×#935—21no—
17C3×D4×Dic3—21no—
18C2×C3×#214—21no—
19C3×#924C2211yescharacteristic
20C2×C3×Dic3⋊C4C2211yescharacteristic
21C3×#889—21no—
22C3×#879—21no—
23C3×#878—21no—
24C32×#102—21no—
25C32×C22⋊C8C2211yescharacteristic
26C3×#832—21no—
27C3×C3⋊#95C2211yescharacteristic
28C3×#830C2211yescharacteristic
29C3×#539—31no—
30#5602C2×C311yescharacteristic, complemented
31C3×#450—31no—
32C3×#447—21no—
33C3×#447D311yescharacteristic
34C3×#447C2×C311yescharacteristic
35C3×#394—31no—
36#5222C2×C311yescharacteristic, complemented
37#5218C2×C311yescharacteristic, complemented
38C3×#328—31no—
39C3×#327—31no—
40C3×#324—31no—
41#4537C2×C311yescharacteristic, complemented
42#4469C2×C311yescharacteristic, complemented
43#4467C2×C311yescharacteristic, complemented
44C24×C32—21no—
45C2×C32×D4—21no—
46C2×C32×D4—21no—
47C2×C32×D4—21no—
48C2×C32×D4—42no—
49C2×C32×D4D411yescharacteristic
50C22×C32×C4C2311yescharacteristic
51C22×C3×Dic3—21no—
52C22×C3×Dic3—21no—
53C22×C3×Dic3D411yescharacteristic
54C22×C3×Dic3D411yescharacteristic
55C2×C3×Dic6D411yescharacteristic
56C32×D8—42no—
57C2×C32×C8—21no—
58C32×C4⋊C4D411yescharacteristic
59C32×C22⋊C4—21no—
60C3×#217—21no—
61C3×#217—21no—
62C3×#217—21no—
63C3×#217—21no—
64C3×#217—42no—
65C3×#217—42no—
66C3×#214—42no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3D411yescharacteristic
69C3×Dic3⋊C4—21no—
70C3×Dic3⋊C4—21no—
71C3×Dic3⋊C4—21no—
72C3×Dic3⋊C4—21no—
73C3×Dic3⋊C4—21no—
74C3×C4×Dic3—21no—
75C2×C3×C3⋊C8—21no—
76#1597—31noSylow
77C22×C3×D4—21no—
78C22×C3×D4C2×D311yescharacteristic
79C22×C3×D4C22×C311yescharacteristic
80C2×C3×SD16—61no—
81C2×C3×D8—22no—
82C2×C3×D8—21no—
83C2×C3×D8—21no—
84C3×#126—61no—
85C3×#125—61no—
86C3×#123—61no—
87C3×#123—61no—
88C3×C22⋊Q8—31no—
89C3×C22⋊Q8—61no—
90C3×#121—21no—
91C3×#121C2×D311yescharacteristic
92C3×#121C22×C311yescharacteristic
93C3×C4×D4—61no—
94C2×C3×C4⋊C4—31no—
95C2×C3×C22⋊C4—61no—
96C2×C3×C22⋊C4—61no—
97C2×#217—21no—
98C2×#217—21no—
99#935—21no—
100#935—21no—
101D4×Dic3—21no—
102C2×#214—21no—
103#924C22×C311yescharacteristic
104C2×Dic3⋊C4C22×C311yescharacteristic
105#889—21no—
106#879—21no—
107#878—21no—
108C3×#102—22no—
109C3×#102—21no—
110C3×#102—21no—
111C3×#102—61no—
112C3×C22⋊C8—21no—
113C3×C22⋊C8—31no—
114C3×C22⋊C8C2×D311yescharacteristic
115C3×C22⋊C8C22×C311yescharacteristic
116C3×#95—31no—
117#832—21no—
118C3⋊#95C22×C311yescharacteristic
119#830C22×C311yescharacteristic
120C23×C32—22no—
121C23×C32—21no—
122C23×C32—41no—
123C23×C32—41no—
124C23×C32—41no—
125C23×C32—41no—
126C23×C32—41no—
127C23×C32—41no—
128C23×C32C2×D411yescharacteristic
129C32×D4—22no—
130C32×D4—22no—
131C32×D4—41no—
132C32×D4—41no—
133C32×D4—41no—
134C32×D4—41no—
135C32×D4—41no—
136C2×C32×C4—21no—
137C2×C32×C4—41no—
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C3×Dic3—21no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—21no—
146C2×C3×Dic3—41no—
147C2×C3×Dic3—41no—
148C2×C3×Dic3—41no—
149C2×C3×Dic3—41no—
150C2×C3×Dic3—81no—
151C3×Dic6—41no—
152C32×C8—41no—
153C3×C3⋊C8—41no—
154#539—31no—
155#450—31no—
156#447C3×D311yescharacteristic
157#394—31no—
158#328—31no—
159#327—31no—
160#324—31no—
161C24×C3—21no—
162C24×C3—21no—
163C24×C3—41no—
164C2×C3×Q8—31no—
165C2×C3×D4—22no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—41no—
174C2×C3×D4—41no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C2×C3×D4—44no—
178C2×C3×D4#6711yescharacteristic
179C2×C3×D4C3×D411yescharacteristic
180C22×C3×C4—21no—
181C22×C3×C4—31no—
182C22×C3×C4—31no—
183C22×C3×C4—61no—
184C22×C3×C4—61no—
185C22×C3×C4C22×D311yescharacteristic
186C22×C3×C4C23×C311yescharacteristic
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3C3×D411yescharacteristic
190C22×Dic3C3×D411yescharacteristic
191C2×Dic6C3×D411yescharacteristic
192C3×SD16—122no—
193C3×D8—42no—
194C3×D8—42no—
195C3×D8—44no—
196C2×C3×C8—22no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8—61no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—31no—
202C3×C4⋊C4—61no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4—61no—
205C3×C4⋊C4—61no—
206C3×C4⋊C4—61no—
207C3×C4⋊C4—61no—
208C3×C4⋊C4#6711yescharacteristic
209C3×C4⋊C4C3×D411yescharacteristic
210C3×C22⋊C4—21no—
211C3×C22⋊C4—21no—
212C3×C22⋊C4—41no—
213C3×C22⋊C4—61no—
214C3×C22⋊C4—61no—
215C3×C22⋊C4—61no—
216C3×C22⋊C4—61no—
217C3×C22⋊C4—122no—
218C3×C22⋊C4—122no—
219C3×C42—61no—
220#217—21no—
221#217—21no—
222#217—21no—
223#217—21no—
224#217—42no—
225#217—42no—
226#214—42no—
227C4⋊Dic3—21no—
228C4⋊Dic3C3×D411yescharacteristic
229Dic3⋊C4—21no—
230Dic3⋊C4—21no—
231Dic3⋊C4—21no—
232Dic3⋊C4—21no—
233Dic3⋊C4—21no—
234C4×Dic3—21no—
235C2×C3⋊C8—21no—
236C22×C32—21no—
237C22×C32—21no—
238C22×C32—42no—
239C22×C32—42no—
240C22×C32—42no—
241C22×C32—41no—
242C22×C32—41no—
243C22×C32—41no—
244C22×C32—81no—
245C22×C32—81no—
246C22×C32—81no—
247C22×C32—81no—
248C22×C32#12011yescharacteristic
249C22×C32#13511yescharacteristic
250C22×C32AGL(1,ℤ/8)11yescharacteristic
251C32×C4—21no—
252C32×C4—21no—
253C32×C4—41no—
254C3×Dic3—41no—
255C3×Dic3—41no—
256C3×Dic3—41no—
257C3×Dic3—41no—
258C3×Dic3—81no—
259C22×D4C2×C3×D311yescharacteristic
260C2×SD16—61no—
261C2×D8—21no—
262#126—61no—
263#125—61no—
264#123—61no—
265#123—61no—
266C22⋊Q8—31no—
267C22⋊Q8—61no—
268#121C2×C3×D311yescharacteristic
269C4×D4—61no—
270C2×C4⋊C4—31no—
271C2×C22⋊C4—61no—
272C2×C22⋊C4—61no—
273#102—21no—
274#102—61no—
275C22⋊C8—31no—
276C22⋊C8C2×C3×D311yescharacteristic
277#95—31no—
278C23×C3—22no—
279C23×C3—22no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3—21no—
283C23×C3—41no—
284C23×C3—42no—
285C23×C3—42no—
286C23×C3—41no—
287C23×C3—41no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—42no—
293C23×C3—41no—
294C23×C3—41no—
295C23×C3—41no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—81no—
300C23×C3—81no—
301C23×C3—81no—
302C23×C3—81no—
303C23×C3D3×D411yescharacteristic
304C23×C3C2×C3×D411yescharacteristic
305C3×Q8—121no—
306C3×D4—22no—
307C3×D4—24no—
308C3×D4—22no—
309C3×D4—22no—
310C3×D4—22no—
311C3×D4—42no—
312C3×D4—41no—
313C3×D4—41no—
314C3×D4—42no—
315C3×D4—41no—
316C3×D4—41no—
317C3×D4—41no—
318C3×D4—41no—
319C3×D4—42no—
320C3×D4—42no—
321C3×D4—41no—
322C3×D4—41no—
323C3×D4—41no—
324C3×D4—41no—
325C3×D4—81no—
326C3×D4—81no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—121no—
342C2×C3×C4—121no—
343C2×C3×C4—121no—
344C2×C3×C4—121no—
345C2×C3×C4—241no—
346C2×C3×C4D3×D411yescharacteristic
347C2×C3×C4C2×#6711yescharacteristic
348C2×C3×C4C2×C3×D411yescharacteristic
349C2×C3×C4C2×C3×D411yescharacteristic
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—81no—
361Dic6—41no—
362C3×C8—42no—
363C3×C8—41no—
364C3×C8—41no—
365C3×C8—121no—
366C3⋊C8—41no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—82no—
370C2×C32—81no—
371C2×C32#44911yescharacteristic
372C2×C32#45411yescharacteristic
373C2×C32B(4,𝔽2)11yescharacteristic
374C24—21no—
375C2×Q8—31no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—42no—
380C2×D4C3×#6711yescharacteristic
381C22×C4—31no—
382C22×C4—31no—
383C22×C4—61no—
384C22×C4—61no—
385C22×C4C22×C3×D311yescharacteristic
386SD16—122no—
387D8—42no—
388C2×C8—21no—
389C2×C8—61no—
390C4⋊C4—31no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C4⋊C4—61no—
394C4⋊C4—61no—
395C4⋊C4—61no—
396C4⋊C4—61no—
397C4⋊C4C3×#6711yescharacteristic
398C22⋊C4—21no—
399C22⋊C4—61no—
400C22⋊C4—61no—
401C22⋊C4—61no—
402C22⋊C4—61no—
403C22⋊C4—122no—
404C22⋊C4—122no—
405C42—61no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—41no—
414C22×C3—41no—
415C22×C3—44no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—42no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—42no—
428C22×C3—81no—
429C22×C3—81no—
430C22×C3—81no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3—82no—
438C22×C3—82no—
439C22×C3—82no—
440C22×C3—82no—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3—81no—
444C22×C3—81no—
445C22×C3#91111yescharacteristic
446C22×C3#91211yescharacteristic
447C22×C3#93711yescharacteristic
448C22×C3C3×#12011yescharacteristic
449C22×C3C3×#13511yescharacteristic
450C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
451C3×C4—22no—
452C3×C4—21no—
453C3×C4—21no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—41no—
457C3×C4—41no—
458C3×C4—41no—
459C3×C4—81no—
460C3×C4—121no—
461C3×C4—121no—
462C3×C4—121no—
463C3×C4—121no—
464C3×C4—241no—
465Dic3—41no—
466Dic3—41no—
467Dic3—41no—
468Dic3—41no—
469Dic3—81no—
470C32#159711yescharacteristic, complemented, Sylow
471C23—22no—
472C23—21no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23C3×D3×D411yescharacteristic
480Q8—121no—
481D4—22no—
482D4—22no—
483D4—41no—
484D4—41no—
485D4—41no—
486D4—41no—
487D4—41no—
488C2×C4—21no—
489C2×C4—41no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—61no—
494C2×C4—61no—
495C2×C4—61no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4—121no—
500C2×C4—241no—
501C2×C4C3×D3×D411yescharacteristic
502C2×C4C2×C3×#6711yescharacteristic
503C8—41no—
504C8—121no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—84no—
515C2×C3—82no—
516C2×C3—81no—
517C2×C3—82no—
518C2×C3—81no—
519C2×C3—161no—
520C2×C3#503211yescharacteristic
521C2×C3#515811yescharacteristic
522C2×C3#520111yescharacteristic
523C2×C3C3×#44911yescharacteristic
524C2×C3C3×#45411yescharacteristic
525C2×C3C3×B(4,𝔽2)11yescharacteristic
526C22—21no—
527C22—21no—
528C22—42no—
529C22—42no—
530C22—41no—
531C22—41no—
532C22—42no—
533C22—41no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22C3×#91111yescharacteristic
539C22C3×#91211yescharacteristic
540C22C3×#93711yescharacteristic
541C4—21no—
542C4—21no—
543C4—41no—
544C4—121no—
545C4—121no—
546C4—121no—
547C4—121no—
548C4—241no—
549C3—21no—
550C3#7097011yescharacteristic, direct product factor
551C3C3×#159711yescharacteristic, complemented
552C2—21no—
553C2—21no—
554C2—82no—
555C2—81no—
556C2C3×#503211yescharacteristic
557C2C3×#515811yescharacteristic
558C2C3×#520111yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
Fitting, prime radicalC32×#447C2
socleC22×C32#120
hypercenterC3×#447D3
hypocenterC3C3×#1597
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#447

Complements

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

NH ≅ G/Ntype
1. C3×#5602C2semidirect
3. C3×#5222C2semidirect
4. C3×#5218C2semidirect
5. C3×#4537C2semidirect
6. C3×#4469C2semidirect
7. C3×#4467C2semidirect
9. #70970C3direct (N × H)
30. #5602C2×C3semidirect
36. #5222C2×C3semidirect
37. #5218C2×C3semidirect
41. #4537C2×C3semidirect
42. #4469C2×C3semidirect
43. #4467C2×C3semidirect
470. C32#1597semidirect
550. C3#70970direct (N × H)
551. C3C3×#1597semidirect

Maximal subgroups

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

subgroupindex
C3×#56022
C32×#4472
C3×#52222
C3×#52182
C3×#45372
C3×#44692
C3×#44672
C3×#15973
#709703