Finite Group Databasebeta

← C3×#68292 #11778055

Subgroups — C3×#68292 #11778055

Order 1152.

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

normal subgroups only

This group has 589 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
2C3×#5036C211yesmaximal, characteristic
3C3×#4731C211yesmaximal, characteristic, complemented
4C3×#4672C211yesmaximal, characteristic, complemented
5C32×#327C211yesmaximal, characteristic, complemented
6C3×#4469C211yesmaximal, characteristic, complemented
7C3×#4452C211yesmaximal, characteristic, complemented
8C3×#1597—31nomaximal
9#68292C311yesmaximal, characteristic, direct product factor
10C22×C3×D12C2211yescharacteristic
11C32×#121C2211yescharacteristic
12C3×#935—21no—
13C3×#935—21no—
14C3×D4×Dic3—21no—
15C2×C3×#213—21no—
16C2×C3×(C2×D3)⋊C4—21no—
17C2×C3×(C2×D3)⋊C4—21no—
18C3×#924C2211yescharacteristic
19C2×C3×Dic3⋊C4C2211yescharacteristic
20C2×C3×#204—21no—
21C3×#889—21no—
22C3×#879—21no—
23C3×#878—21no—
24C32×C22⋊C8C2211yescharacteristic
25C32×#95C2211yescharacteristic
26C3×#830C2211yescharacteristic
27C3×#821—21no—
28C3×#809—21no—
29C3×#539—31no—
30#5602C2×C311yescharacteristic, complemented
31C3×#450—31no—
32C3×#447—31no—
33C3×#394—31no—
34#5036C2×C311yescharacteristic
35#4731C2×C311yescharacteristic, complemented
36#4672C2×C311yescharacteristic, complemented
37C3×#328—31no—
38C3×#327—21no—
39C3×#327D311yescharacteristic, complemented
40C3×#327C2×C311yescharacteristic, complemented
41C3×#324—31no—
42#4469C2×C311yescharacteristic, complemented
43#4452C2×C311yescharacteristic, complemented
44C23×C3×D3—21no—
45C2×C32×D4—21no—
46C2×C32×D4D411yescharacteristic
47C22×C32×C4—21no—
48C22×C32×C4C2311yescharacteristic
49C22×C3×Dic3—21no—
50C22×C3×Dic3D411yescharacteristic
51C22×C3×Dic3D411yescharacteristic
52C2×C3×D12—21no—
53C2×C3×D12—21no—
54C2×C3×D12—42no—
55C2×C3×Dic6D411yescharacteristic
56C2×C32×C8—21no—
57C32×C4⋊C4D411yescharacteristic
58C32×C22⋊C4—21no—
59C3×#217—21no—
60C3×#217—21no—
61C3×#217—21no—
62C3×#217—21no—
63C3×#213—42no—
64C3×(C2×D3)⋊C4—42no—
65C3×(C2×D3)⋊C4—42no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3D411yescharacteristic
68C3×Dic3⋊C4—21no—
69C3×Dic3⋊C4—21no—
70C3×Dic3⋊C4—21no—
71C3×Dic3⋊C4—21no—
72C3×Dic3⋊C4—21no—
73C3×C4×Dic3—21no—
74C2×C3×C3⋊C8—21no—
75C3×#204—42no—
76#1597—31noSylow
77C22×C3×D4—31no—
78C22×D12C22×C311yescharacteristic
79C2×C3×SD16—61no—
80C2×C3×D8—61no—
81C3×#126—61no—
82C3×#125—61no—
83C3×#123—61no—
84C3×#123—61no—
85C3×C22⋊Q8—31no—
86C3×C22⋊Q8—61no—
87C3×#121—21no—
88C3×#121C2×D311yescharacteristic
89C3×#121C22×C311yescharacteristic
90C3×C4×D4—61no—
91C2×C3×C4⋊C4—31no—
92C2×C3×C22⋊C4—61no—
93C2×C3×C22⋊C4—61no—
94#935—21no—
95#935—21no—
96D4×Dic3—21no—
97C2×#213—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100#924C22×C311yescharacteristic
101C2×Dic3⋊C4C22×C311yescharacteristic
102C2×#204—21no—
103#889—21no—
104#879—21no—
105#878—21no—
106C3×#102—61no—
107C3×#102—61no—
108C3×C22⋊C8—21no—
109C3×C22⋊C8—31no—
110C3×C22⋊C8C2×D311yescharacteristic
111C3×C22⋊C8C22×C311yescharacteristic
112C3×#95—21no—
113C3×#95C2×D311yescharacteristic
114C3×#95C22×C311yescharacteristic
115#830C22×C311yescharacteristic
116#821—21no—
117#809—21no—
118C23×C32—21no—
119C23×C32C2×D411yescharacteristic
120C22×C3×D3—22no—
121C22×C3×D3—41no—
122C22×C3×D3—41no—
123C22×C3×D3—41no—
124C22×C3×D3—41no—
125C22×C3×D3—41no—
126C22×C3×D3—41no—
127C32×D4—41no—
128C32×D4—41no—
129C32×D4—41no—
130C2×C32×C4—21no—
131C2×C32×C4—41no—
132C2×C32×C4—41no—
133C2×C32×C4—41no—
134C2×C32×C4—41no—
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4C2×D411yescharacteristic
137C2×C3×Dic3—21no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—81no—
145C3×D12—22no—
146C3×D12—22no—
147C3×D12—41no—
148C3×D12—41no—
149C3×Dic6—41no—
150C32×C8—41no—
151C3×C3⋊C8—41no—
152#539—31no—
153#450—31no—
154#447—31no—
155#394—31no—
156#328—31no—
157#327C3×D311yescharacteristic, complemented
158#324—31no—
159C24×C3—61no—
160C23×D3—21no—
161C2×C3×Q8—31no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—41no—
166C2×C3×D4—61no—
167C2×C3×D4—61no—
168C2×C3×D4—122no—
169C2×C3×D4D1211yescharacteristic
170C2×C3×D4C3×D411yescharacteristic
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—31no—
175C22×C3×C4—31no—
176C22×C3×C4—41no—
177C22×C3×C4—61no—
178C22×C3×C4C22×D311yescharacteristic
179C22×C3×C4C23×C311yescharacteristic
180C22×Dic3—21no—
181C22×Dic3C3×D411yescharacteristic
182C22×Dic3C3×D411yescharacteristic
183C2×D12—21no—
184C2×D12—21no—
185C2×D12—42no—
186C2×Dic6C3×D411yescharacteristic
187C3×SD16—122no—
188C3×D8—122no—
189C2×C3×C8—21no—
190C2×C3×C8—21no—
191C2×C3×C8—41no—
192C2×C3×C8—61no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—31no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4—61no—
198C3×C4⋊C4—61no—
199C3×C4⋊C4—61no—
200C3×C4⋊C4—61no—
201C3×C4⋊C4D1211yescharacteristic
202C3×C4⋊C4C3×D411yescharacteristic
203C3×C22⋊C4—21no—
204C3×C22⋊C4—21no—
205C3×C22⋊C4—41no—
206C3×C22⋊C4—61no—
207C3×C22⋊C4—61no—
208C3×C22⋊C4—61no—
209C3×C22⋊C4—61no—
210C3×C22⋊C4—122no—
211C3×C22⋊C4—122no—
212C3×C42—61no—
213#217—21no—
214#217—21no—
215#217—21no—
216#217—21no—
217#213—42no—
218(C2×D3)⋊C4—42no—
219(C2×D3)⋊C4—42no—
220C4⋊Dic3—21no—
221C4⋊Dic3C3×D411yescharacteristic
222Dic3⋊C4—21no—
223Dic3⋊C4—21no—
224Dic3⋊C4—21no—
225Dic3⋊C4—21no—
226Dic3⋊C4—21no—
227C4×Dic3—21no—
228C2×C3⋊C8—21no—
229#204—42no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—41no—
233C22×C32—41no—
234C22×C32—41no—
235C22×C32#12011yescharacteristic
236C22×C32#13511yescharacteristic
237C22×C32AGL(1,ℤ/8)11yescharacteristic
238C2×C3×D3—42no—
239C2×C3×D3—42no—
240C2×C3×D3—42no—
241C2×C3×D3—81no—
242C2×C3×D3—81no—
243C2×C3×D3—81no—
244C2×C3×D3—81no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—41no—
248C32×C4—81no—
249C3×Dic3—41no—
250C3×Dic3—41no—
251C3×Dic3—41no—
252C3×Dic3—41no—
253C22×D4—31no—
254C2×SD16—61no—
255C2×D8—61no—
256#126—61no—
257#125—61no—
258#123—61no—
259#123—61no—
260C22⋊Q8—31no—
261C22⋊Q8—61no—
262#121C2×C3×D311yescharacteristic
263C4×D4—61no—
264C2×C4⋊C4—31no—
265C2×C22⋊C4—61no—
266C2×C22⋊C4—61no—
267#102—61no—
268#102—61no—
269C22⋊C8—31no—
270C22⋊C8C2×C3×D311yescharacteristic
271#95C2×C3×D311yescharacteristic
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—41no—
276C23×C3—62no—
277C23×C3—121no—
278C23×C3—121no—
279C23×C3—121no—
280C23×C3—121no—
281C23×C3—121no—
282C23×C3—121no—
283C23×C3D3×D411yescharacteristic
284C23×C3C2×C3×D411yescharacteristic
285C22×D3—22no—
286C22×D3—41no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—41no—
291C22×D3—41no—
292C3×Q8—121no—
293C3×D4—41no—
294C3×D4—41no—
295C3×D4—41no—
296C3×D4—41no—
297C3×D4—41no—
298C3×D4—41no—
299C3×D4—62no—
300C3×D4—62no—
301C3×D4—81no—
302C3×D4—81no—
303C3×D4—81no—
304C3×D4—121no—
305C3×D4—121no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—42no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—81no—
327C2×C3×C4—81no—
328C2×C3×C4—81no—
329C2×C3×C4—121no—
330C2×C3×C4—241no—
331C2×C3×C4C2×D1211yescharacteristic
332C2×C3×C4D3×D411yescharacteristic
333C2×C3×C4C2×C3×D411yescharacteristic
334C2×C3×C4C2×C3×D411yescharacteristic
335C2×Dic3—21no—
336C2×Dic3—21no—
337C2×Dic3—21no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—41no—
342C2×Dic3—81no—
343D12—22no—
344D12—22no—
345D12—41no—
346D12—41no—
347Dic6—41no—
348C3×C8—41no—
349C3×C8—41no—
350C3×C8—81no—
351C3×C8—121no—
352C3⋊C8—41no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—81no—
356C2×C32#44911yescharacteristic
357C2×C32#45411yescharacteristic
358C2×C32B(4,𝔽2)11yescharacteristic
359C3×D3—82no—
360C24—61no—
361C2×Q8—31no—
362C2×D4—21no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—122no—
366C2×D4C3×D1211yescharacteristic
367C22×C4—21no—
368C22×C4—31no—
369C22×C4—31no—
370C22×C4—61no—
371C22×C4C22×C3×D311yescharacteristic
372SD16—122no—
373D8—122no—
374C2×C8—21no—
375C2×C8—61no—
376C4⋊C4—31no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—61no—
380C4⋊C4—61no—
381C4⋊C4—61no—
382C4⋊C4—61no—
383C4⋊C4C3×D1211yescharacteristic
384C22⋊C4—21no—
385C22⋊C4—61no—
386C22⋊C4—61no—
387C22⋊C4—61no—
388C22⋊C4—61no—
389C22⋊C4—122no—
390C22⋊C4—122no—
391C42—61no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—81no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—122no—
411C22×C3—122no—
412C22×C3—122no—
413C22×C3—241no—
414C22×C3—241no—
415C22×C3—241no—
416C22×C3—241no—
417C22×C3#88211yescharacteristic
418C22×C3#91111yescharacteristic
419C22×C3#91611yescharacteristic
420C22×C3C3×#12011yescharacteristic
421C22×C3C3×#13511yescharacteristic
422C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
423C2×D3—42no—
424C2×D3—42no—
425C2×D3—42no—
426C2×D3—81no—
427C2×D3—81no—
428C2×D3—81no—
429C2×D3—81no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—81no—
439C3×C4—81no—
440C3×C4—81no—
441C3×C4—82no—
442C3×C4—121no—
443C3×C4—121no—
444C3×C4—121no—
445C3×C4—121no—
446Dic3—41no—
447Dic3—41no—
448Dic3—41no—
449Dic3—41no—
450C32#159711yescharacteristic, complemented, Sylow
451C23—21no—
452C23—62no—
453C23—121no—
454C23—121no—
455C23—121no—
456C23—121no—
457C23—121no—
458C23—121no—
459C23C3×D3×D411yescharacteristic
460Q8—121no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—62no—
465D4—62no—
466D4—121no—
467D4—121no—
468C2×C4—21no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—61no—
479C2×C4—121no—
480C2×C4—241no—
481C2×C4C2×C3×D1211yescharacteristic
482C2×C4C3×D3×D411yescharacteristic
483C8—41no—
484C8—121no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—81no—
495C2×C3—81no—
496C2×C3—161no—
497C2×C3—242no—
498C2×C3#474111yescharacteristic
499C2×C3#478111yescharacteristic
500C2×C3#482711yescharacteristic
501C2×C3C3×#44911yescharacteristic
502C2×C3C3×#45411yescharacteristic
503C2×C3C3×B(4,𝔽2)11yescharacteristic
504D3—82no—
505C22—21no—
506C22—21no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—122no—
511C22—122no—
512C22—122no—
513C22—241no—
514C22—241no—
515C22—241no—
516C22—241no—
517C22C3×#88211yescharacteristic
518C22C3×#91111yescharacteristic
519C22C3×#91611yescharacteristic
520C4—21no—
521C4—21no—
522C4—41no—
523C4—81no—
524C4—121no—
525C4—121no—
526C4—121no—
527C4—121no—
528C3—21no—
529C3#6829211yescharacteristic, direct product factor
530C3C3×#159711yescharacteristic, complemented
531C2—21no—
532C2—21no—
533C2—81no—
534C2—242no—
535C2C3×#474111yescharacteristic
536C2C3×#478111yescharacteristic
537C2C3×#482711yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#327C2
socleC22×C32#120
prime radicalC3×#5036C2
hypercenterC3×#327D3
hypocenterC3C3×#1597
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#327

Complements

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

NH ≅ G/Ntype
1. C3×#5602C2semidirect
3. C3×#4731C2semidirect
4. C3×#4672C2semidirect
5. C32×#327C2semidirect
6. C3×#4469C2semidirect
7. C3×#4452C2semidirect
9. #68292C3direct (N × H)
30. #5602C2×C3semidirect
35. #4731C2×C3semidirect
36. #4672C2×C3semidirect
39. C3×#327D3semidirect
40. C3×#327C2×C3semidirect
42. #4469C2×C3semidirect
43. #4452C2×C3semidirect
157. #327C3×D3semidirect
450. C32#1597semidirect
529. C3#68292direct (N × H)
530. C3C3×#1597semidirect

Maximal subgroups

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

subgroupindex
C3×#56022
C3×#50362
C3×#47312
C3×#46722
C32×#3272
C3×#44692
C3×#44522
C3×#15973
#682923