Finite Group Databasebeta

← C3×#68288 #11778439

Subgroups — C3×#68288 #11778439

Order 1152.

605 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5605C211yesmaximal, characteristic
2C3×#5033C211yesmaximal, characteristic, complemented
3C3×#4736C211yesmaximal, characteristic, complemented
4C3×#4674C211yesmaximal, characteristic, complemented
5C32×#327C211yesmaximal, characteristic, complemented
6C3×#4468C211yesmaximal, characteristic, complemented
7C3×#4451C211yesmaximal, characteristic, complemented
8C3×#1607—31nomaximal
9#68288C311yesmaximal, characteristic, direct product factor
10C2×C3×D3×D4—21no—
11C32×#121C2211yescharacteristic
12C3×#939—21no—
13C3×#937—21no—
14C3×#930C2211yescharacteristic
15C2×C3×(C2×D3)⋊C4—21no—
16C2×C3×(C2×D3)⋊C4C2211yescharacteristic
17C2×C3×C4⋊Dic3C2211yescharacteristic, complemented
18C3×#894—21no—
19C3×#883—21no—
20C3×#882—21no—
21C32×C22⋊C8C2211yescharacteristic, complemented
22C32×#95C2211yescharacteristic, complemented
23C3×#832—21no—
24C3×C3⋊#95—21no—
25C3×#830C2211yescharacteristic, complemented
26C3×#821—21no—
27C3×#818—21no—
28C3×#807—21no—
29C3×#534—31no—
30#5605C2×C311yescharacteristic
31C3×#480—31no—
32C3×#480—31no—
33C3×#397—31no—
34#5033C2×C311yescharacteristic, complemented
35#4736C2×C311yescharacteristic, complemented
36#4674C2×C311yescharacteristic, complemented
37C3×#327—21no—
38C3×#327—31no—
39C3×#327D311yescharacteristic, complemented
40C3×#327C2×C311yescharacteristic, complemented
41C3×#323—31no—
42#4468C2×C311yescharacteristic, complemented
43#4451C2×C311yescharacteristic, complemented
44C23×C3×D3—21no—
45C23×C3×D3D411yescharacteristic
46C2×C32×D4—21no—
47C2×C32×D4D411yescharacteristic
48C22×C32×C4—21no—
49C22×C32×C4C2311yescharacteristic
50C2×C3×#67—21no—
51C2×C3×#67—21no—
52C2×C3×#67—21no—
53C22×C3×Dic3—21no—
54C22×C3×Dic3D411yescharacteristic
55C3×D3×D4—42no—
56C3×D3×D4—42no—
57C2×C3×D12—21no—
58C2×C3×D12D411yescharacteristic
59C2×C3×C4×D3—21no—
60C2×C32×C8—21no—
61C32×C4⋊C4D411yescharacteristic
62C32×C22⋊C4—21no—
63C3×#217—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—21no—
66C3×(C2×D3)⋊C4—21no—
67C3×(C2×D3)⋊C4—21no—
68C3×(C2×D3)⋊C4—42no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3D411yescharacteristic
72C3×Dic3⋊C4—21no—
73C2×C3×C3⋊C8—21no—
74#1607—31noSylow
75C22×C3×D4—61no—
76C2×D3×D4—21no—
77C3×#123—61no—
78C3×#123—61no—
79C3×#121—21no—
80C3×#121—31no—
81C3×#121—61no—
82C3×#121C2×D311yescharacteristic
83C3×#121C22×C311yescharacteristic
84C3×#120—61no—
85C3×#120—61no—
86C2×C3×C4⋊C4—31no—
87C2×C3×C22⋊C4—31no—
88C2×C3×C22⋊C4—61no—
89#939—21no—
90#937—21no—
91#930C22×C311yescharacteristic
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4C22×C311yescharacteristic
94C2×C4⋊Dic3C22×C311yescharacteristic, complemented
95#894—21no—
96#883—21no—
97#882—21no—
98C3×#107—61no—
99C3×#107—61no—
100C3×#102—61no—
101C3×#102—61no—
102C3×C22⋊C8—21no—
103C3×C22⋊C8—31no—
104C3×C22⋊C8C2×D311yescharacteristic, complemented
105C3×C22⋊C8C22×C311yescharacteristic, complemented
106C3×#95—21no—
107C3×#95—61no—
108C3×#95C2×D311yescharacteristic, complemented
109C3×#95C22×C311yescharacteristic, complemented
110#832—21no—
111C3⋊#95—21no—
112#830C22×C311yescharacteristic, complemented
113#821—21no—
114#818—21no—
115#807—21no—
116C23×C32—21no—
117C23×C32C2×D411yescharacteristic
118C22×C3×D3—21no—
119C22×C3×D3—21no—
120C22×C3×D3—21no—
121C22×C3×D3—42no—
122C22×C3×D3—41no—
123C22×C3×D3—41no—
124C22×C3×D3—81no—
125C22×C3×D3—81no—
126C22×C3×D3D812yes—
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
137C3×#67—42no—
138C3×#67—41no—
139C3×#67—41no—
140C3×#67—81no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—41no—
145C2×C3×Dic3—41no—
146C2×C3×Dic3—41no—
147C3×D12—41no—
148C3×D12—42no—
149C3×C4×D3—42no—
150C32×C8—41no—
151C3×C3⋊C8—41no—
152#534—31no—
153#480—31no—
154#480—31no—
155#397—31no—
156#327—31no—
157#327C3×D311yescharacteristic, complemented
158#323—31no—
159C24×C3—31no—
160C24×C3—61no—
161C23×D3—21no—
162C23×D3C3×D411yescharacteristic
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—31no—
167C2×C3×D4—41no—
168C2×C3×D4—61no—
169C2×C3×D4—61no—
170C2×C3×D4—61no—
171C2×C3×D4—61no—
172C2×C3×D4—122no—
173C2×C3×D4—122no—
174C2×C3×D4D1211yescharacteristic
175C2×C3×D4C3×D411yescharacteristic
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—31no—
180C22×C3×C4—41no—
181C22×C3×C4—61no—
182C22×C3×C4—61no—
183C22×C3×C4C22×D311yescharacteristic
184C22×C3×C4C23×C311yescharacteristic
185C2×#67—21no—
186C2×#67—21no—
187C2×#67—21no—
188C22×Dic3—21no—
189C22×Dic3C3×D411yescharacteristic
190D3×D4—42no—
191D3×D4—42no—
192C2×D12—21no—
193C2×D12C3×D411yescharacteristic
194C2×C4×D3—21no—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8—41no—
198C2×C3×C8—61no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—31no—
201C3×C4⋊C4—61no—
202C3×C4⋊C4—61no—
203C3×C4⋊C4—61no—
204C3×C4⋊C4D1211yescharacteristic
205C3×C4⋊C4C3×D411yescharacteristic
206C3×C22⋊C4—21no—
207C3×C22⋊C4—21no—
208C3×C22⋊C4—41no—
209C3×C22⋊C4—61no—
210C3×C22⋊C4—61no—
211C3×C22⋊C4—61no—
212C3×C22⋊C4—61no—
213C3×C22⋊C4—61no—
214C3×C22⋊C4—122no—
215#217—21no—
216(C2×D3)⋊C4—21no—
217(C2×D3)⋊C4—21no—
218(C2×D3)⋊C4—21no—
219(C2×D3)⋊C4—21no—
220(C2×D3)⋊C4—42no—
221C4⋊Dic3—21no—
222C4⋊Dic3—21no—
223C4⋊Dic3C3×D411yescharacteristic
224Dic3⋊C4—21no—
225C2×C3⋊C8—21no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—41no—
229C22×C32—41no—
230C22×C32—41no—
231C22×C32#12011yescharacteristic
232C22×C32C2×D811yescharacteristic
233C22×C32AGL(1,ℤ/8)11yescharacteristic
234C2×C3×D3—22no—
235C2×C3×D3—41no—
236C2×C3×D3—41no—
237C2×C3×D3—42no—
238C2×C3×D3—41no—
239C2×C3×D3—41no—
240C2×C3×D3—41no—
241C2×C3×D3—81no—
242C2×C3×D3—84no—
243C32×C4—21no—
244C32×C4—21no—
245C32×C4—41no—
246C32×C4—81no—
247C3×Dic3—41no—
248C3×Dic3—41no—
249C3×Dic3—81no—
250C22×D4—61no—
251#123—61no—
252#123—61no—
253#121—31no—
254#121—61no—
255#121C2×C3×D311yescharacteristic
256#120—61no—
257#120—61no—
258C2×C4⋊C4—31no—
259C2×C22⋊C4—31no—
260C2×C22⋊C4—61no—
261#107—61no—
262#107—61no—
263#102—61no—
264#102—61no—
265C22⋊C8—31no—
266C22⋊C8C2×C3×D311yescharacteristic, complemented
267#95—61no—
268#95C2×C3×D311yescharacteristic, complemented
269C23×C3—21no—
270C23×C3—21no—
271C23×C3—21no—
272C23×C3—32no—
273C23×C3—41no—
274C23×C3—61no—
275C23×C3—61no—
276C23×C3—61no—
277C23×C3—121no—
278C23×C3—121no—
279C23×C3—122no—
280C23×C3—241no—
281C23×C3—241no—
282C23×C3D3×D411yescharacteristic
283C23×C3C2×C3×D411yescharacteristic
284C22×D3—21no—
285C22×D3—21no—
286C22×D3—21no—
287C22×D3—42no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—81no—
291C22×D3—81no—
292C22×D3C3×D812yes—
293C3×D4—41no—
294C3×D4—41no—
295C3×D4—41no—
296C3×D4—41no—
297C3×D4—41no—
298C3×D4—41no—
299C3×D4—81no—
300C3×D4—81no—
301C3×D4—81no—
302C3×D4—121no—
303C3×D4—122no—
304C3×D4—122no—
305C3×D4—121no—
306C3×D4—121no—
307C3×D4—241no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—81no—
325C2×C3×C4—81no—
326C2×C3×C4—81no—
327C2×C3×C4—122no—
328C2×C3×C4—121no—
329C2×C3×C4—121no—
330C2×C3×C4—121no—
331C2×C3×C4—121no—
332C2×C3×C4C2×D1211yescharacteristic
333C2×C3×C4D3×D411yescharacteristic
334C2×C3×C4C2×C3×D411yescharacteristic
335C2×C3×C4C2×C3×D411yescharacteristic
336#67—42no—
337#67—41no—
338#67—41no—
339#67—81no—
340C2×Dic3—21no—
341C2×Dic3—21no—
342C2×Dic3—41no—
343C2×Dic3—41no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346D12—41no—
347D12—42no—
348C4×D3—42no—
349C3×C8—41no—
350C3×C8—41no—
351C3×C8—81no—
352C3×C8—121no—
353C3⋊C8—41no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—81no—
357C2×C32#44711yescharacteristic
358C2×C32#45511yescharacteristic
359C2×C32#45811yescharacteristic
360C3×D3—42no—
361C3×D3—81no—
362C24—31no—
363C24—61no—
364C2×D4—21no—
365C2×D4—31no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—61no—
369C2×D4—61no—
370C2×D4—122no—
371C2×D4—122no—
372C2×D4C3×D1211yescharacteristic
373C22×C4—21no—
374C22×C4—31no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4C22×C3×D311yescharacteristic
378C2×C8—21no—
379C2×C8—61no—
380C4⋊C4—31no—
381C4⋊C4—61no—
382C4⋊C4—61no—
383C4⋊C4—61no—
384C4⋊C4C3×D1211yescharacteristic
385C22⋊C4—21no—
386C22⋊C4—61no—
387C22⋊C4—61no—
388C22⋊C4—61no—
389C22⋊C4—61no—
390C22⋊C4—61no—
391C22⋊C4—122no—
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—62no—
408C22×C3—81no—
409C22×C3—81no—
410C22×C3—81no—
411C22×C3—122no—
412C22×C3—121no—
413C22×C3—121no—
414C22×C3—121no—
415C22×C3—121no—
416C22×C3—121no—
417C22×C3—241no—
418C22×C3—244no—
419C22×C3#88211yescharacteristic
420C22×C3D3×D811yescharacteristic
421C22×C3#91411yescharacteristic
422C22×C3C3×#12011yescharacteristic
423C22×C3C2×C3×D811yescharacteristic
424C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
425C2×D3—22no—
426C2×D3—41no—
427C2×D3—41no—
428C2×D3—42no—
429C2×D3—41no—
430C2×D3—41no—
431C2×D3—41no—
432C2×D3—81no—
433C2×D3—84no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—41no—
439C3×C4—41no—
440C3×C4—41no—
441C3×C4—41no—
442C3×C4—81no—
443C3×C4—81no—
444C3×C4—81no—
445C3×C4—82no—
446C3×C4—121no—
447C3×C4—121no—
448C3×C4—241no—
449Dic3—41no—
450Dic3—41no—
451Dic3—81no—
452C32#160711yescharacteristic, complemented, Sylow
453C23—21no—
454C23—32no—
455C23—61no—
456C23—61no—
457C23—61no—
458C23—121no—
459C23—121no—
460C23—122no—
461C23—241no—
462C23—241no—
463C23C3×D3×D411yescharacteristic
464D4—41no—
465D4—41no—
466D4—41no—
467D4—121no—
468D4—122no—
469D4—122no—
470D4—121no—
471D4—121no—
472D4—241no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—61no—
479C2×C4—61no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4C2×C3×D1211yescharacteristic
486C2×C4C3×D3×D411yescharacteristic
487C8—41no—
488C8—121no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—41no—
497C2×C3—41no—
498C2×C3—81no—
499C2×C3—81no—
500C2×C3—122no—
501C2×C3—161no—
502C2×C3—241no—
503C2×C3#474211yescharacteristic
504C2×C3#477311yescharacteristic
505C2×C3#482511yescharacteristic
506C2×C3C3×#44711yescharacteristic
507C2×C3C3×#45511yescharacteristic
508C2×C3C3×#45811yescharacteristic
509D3—42no—
510D3—81no—
511C22—21no—
512C22—21no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—62no—
517C22—122no—
518C22—121no—
519C22—121no—
520C22—121no—
521C22—121no—
522C22—121no—
523C22—241no—
524C22—244no—
525C22C3×#88211yescharacteristic
526C22C3×D3×D811yescharacteristic
527C22C3×#91411yescharacteristic
528C4—21no—
529C4—21no—
530C4—41no—
531C4—81no—
532C4—121no—
533C4—121no—
534C4—241no—
535C3—21no—
536C3#6828811yescharacteristic, direct product factor
537C3C3×#160711yescharacteristic, complemented
538C2—21no—
539C2—21no—
540C2—81no—
541C2—122no—
542C2—241no—
543C2C3×#474211yescharacteristic
544C2C3×#477311yescharacteristic
545C2C3×#482511yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#327C2
socleC22×C32#120
prime radicalC3×#5605C2
hypercenterC3×#327D3
hypocenterC3C3×#1607
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
2. C3×#5033C2semidirect
3. C3×#4736C2semidirect
4. C3×#4674C2semidirect
5. C32×#327C2semidirect
6. C3×#4468C2semidirect
7. C3×#4451C2semidirect
9. #68288C3direct (N × H)
17. C2×C3×C4⋊Dic3C22semidirect
21. C32×C22⋊C8C22semidirect
22. C32×#95C22semidirect
25. C3×#830C22semidirect
34. #5033C2×C3semidirect
35. #4736C2×C3semidirect
36. #4674C2×C3semidirect
39. C3×#327D3semidirect
40. C3×#327C2×C3semidirect
42. #4468C2×C3semidirect
43. #4451C2×C3semidirect
94. C2×C4⋊Dic3C22×C3semidirect
104. C3×C22⋊C8C2×D3semidirect
105. C3×C22⋊C8C22×C3semidirect
108. C3×#95C2×D3semidirect
109. C3×#95C22×C3semidirect
112. #830C22×C3semidirect
157. #327C3×D3semidirect
266. C22⋊C8C2×C3×D3semidirect
268. #95C2×C3×D3semidirect
452. C32#1607semidirect
536. C3#68288direct (N × H)
537. C3C3×#1607semidirect

Maximal subgroups

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

subgroupindex
C3×#56052
C3×#50332
C3×#47362
C3×#46742
C32×#3272
C3×#44682
C3×#44512
C3×#16073
#682883