Finite Group Databasebeta

← C3×#68857 #11789891

Subgroups — C3×#68857 #11789891

Order 1152.

625 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#821C211yesmaximal, characteristic
2C2×C3×#815C211yesmaximal, characteristic, complemented
3C3×#4988C211yesmaximal, characteristic
4C3×#4984C211yesmaximal, characteristic, complemented
5C2×C3×#809C211yesmaximal, characteristic
6C32×#340C211yesmaximal, characteristic, complemented
7C3×#4529C211yesmaximal, characteristic, complemented
8C3×#1936—31nomaximal
9#68857C311yesmaximal, characteristic, direct product factor
10C22×C3×D12C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C2×C32×C4⋊C4C2211yescharacteristic
13C2×C32×C4⋊C4C2211yescharacteristic
14C2×C3×(C2×D3)⋊C4—21no—
15C2×C3×(C2×D3)⋊C4—21no—
16C2×C3×C4⋊Dic3—21no—
17C2×C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C4×Dic3C2211yescharacteristic
19C22×C3×C3⋊C8C2211yescharacteristic
20C3×C3⋊#95—21no—
21C3×#821—22no—
22C3×#815—22no—
23C3×#809C414yescomplemented
24C2×C3×#102—31no—
25C2×C3×#102—31no—
26C2×C3×#97—31no—
27C3×#390—31no—
28C3×#389—31no—
29C2×#821C2×C311yescharacteristic
30C2×#815C2×C311yescharacteristic, complemented
31#4988C2×C311yescharacteristic
32#4984C2×C311yescharacteristic, complemented
33C2×#809C2×C311yescharacteristic
34C3×#340—21no—
35C3×#340—31no—
36C3×#340D311yescharacteristic, complemented
37C3×#340C2×C311yescharacteristic, complemented
38#4529C2×C311yescharacteristic, complemented
39C23×C3×D3—21no—
40C22×C32×C4—21no—
41C22×C32×C4—21no—
42C22×C32×C4C2311yescharacteristic
43C22×C3×Dic3—21no—
44C22×C3×Dic3D411yescharacteristic
45C22×C3×Dic3D411yescharacteristic
46C2×C3×D12—21no—
47C2×C3×D12—21no—
48C2×C3×D12—21no—
49C2×C3×D12—21no—
50C2×C3×D12—21no—
51C2×C3×D12C2×C412yes—
52C2×C32×C8—21no—
53C2×C32×C8—21no—
54C2×C32×C8D412yes—
55C32×C4⋊C4—21no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4C2×C412yes—
59C3×(C2×D3)⋊C4—42no—
60C3×(C2×D3)⋊C4—42no—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3—42no—
64C3×C4×Dic3—21no—
65C3×C4×Dic3—21no—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8—21no—
68C2×C3×C3⋊C8C2×C412yes—
69#1936—31noSylow
70C22×C3×D4—31no—
71C22×D12C22×C311yescharacteristic
72C22×C3×C8—21no—
73C22×C3×C8—31no—
74C22×C3×C8C2×D311yescharacteristic
75C22×C3×C8C22×C311yescharacteristic
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—31no—
79C2×C3×C4⋊C4—61no—
80C2×C3×C4⋊C4C2×D311yescharacteristic
81C2×C3×C4⋊C4C2×D311yescharacteristic
82C2×C3×C4⋊C4C22×C311yescharacteristic
83C2×C3×C4⋊C4C22×C311yescharacteristic
84C2×C3×C22⋊C4—61no—
85C2×C3×C22⋊C4—61no—
86C2×C3×C42—31no—
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×C4⋊Dic3—21no—
90C2×C4⋊Dic3C22×C311yescharacteristic
91C2×C4×Dic3C22×C311yescharacteristic
92C22×C3⋊C8C22×C311yescharacteristic
93C3×#102—34no—
94C3×#102—62no—
95C3×#97—62no—
96C3×#95—61no—
97C3⋊#95—21no—
98#821—22no—
99#815—22no—
100#809C3×C414yescomplemented
101C23×C32C2×D411yescharacteristic
102C22×C3×D3—22no—
103C22×C3×D3—41no—
104C22×C3×D3—41no—
105C22×C3×D3—41no—
106C22×C3×D3—41no—
107C22×C3×D3—41no—
108C22×C3×D3—41no—
109C2×C32×C4—22no—
110C2×C32×C4—41no—
111C2×C32×C4—41no—
112C2×C32×C4—41no—
113C2×C32×C4—41no—
114C2×C32×C4—41no—
115C2×C32×C4C22×C411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4#4111yescharacteristic
118C2×C32×C4#4111yescharacteristic
119C2×C32×C4#4111yescharacteristic
120C2×C32×C4#4111yescharacteristic
121C2×C3×Dic3—21no—
122C2×C3×Dic3—21no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—21no—
126C2×C3×Dic3—21no—
127C2×C3×Dic3—41no—
128C2×C3×Dic3—41no—
129C2×C3×Dic3—41no—
130C3×D12—22no—
131C3×D12—22no—
132C3×D12—41no—
133C3×D12—41no—
134C32×C8—22no—
135C3×C3⋊C8—22no—
136C2×#102—31no—
137C2×#102—31no—
138C2×#97—31no—
139#390—31no—
140#389—31no—
141#340—31no—
142#340C3×D311yescharacteristic, complemented
143C24×C3—61no—
144C23×D3—21no—
145C2×C3×D4—32no—
146C2×C3×D4—61no—
147C2×C3×D4—61no—
148C2×C3×D4—61no—
149C2×C3×D4—61no—
150C2×C3×D4—61no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—31no—
157C22×C3×C4—31no—
158C22×C3×C4—41no—
159C22×C3×C4—41no—
160C22×C3×C4—61no—
161C22×C3×C4C22×D311yescharacteristic
162C22×C3×C4C23×C311yescharacteristic
163C22×Dic3—21no—
164C22×Dic3C3×D411yescharacteristic
165C22×Dic3C3×D411yescharacteristic
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12C2×C3×C412yes—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—22no—
177C2×C3×C8—32no—
178C2×C3×C8—41no—
179C2×C3×C8—41no—
180C2×C3×C8—61no—
181C2×C3×C8—61no—
182C2×C3×C8D1212yes—
183C2×C3×C8C3×D412yes—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—41no—
193C3×C4⋊C4—41no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—122no—
197C3×C4⋊C4C4×D312yes—
198C3×C4⋊C4C2×C3×C412yes—
199C3×C22⋊C4—122no—
200C3×C22⋊C4—122no—
201C3×C42—61no—
202C3×C42—61no—
203(C2×D3)⋊C4—42no—
204(C2×D3)⋊C4—42no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—42no—
208C4×Dic3—21no—
209C4×Dic3—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8C2×C3×C412yes—
213C22×C32C4×D411yescharacteristic
214C22×C32#12111yescharacteristic
215C22×C32#12411yescharacteristic
216C22×C32AGL(1,ℤ/8)11yescharacteristic
217C22×C32AGL(1,ℤ/8)11yescharacteristic
218C22×C32AGL(1,ℤ/8)11yescharacteristic
219C22×C32#13711yescharacteristic
220C2×C3×D3—42no—
221C2×C3×D3—42no—
222C2×C3×D3—42no—
223C2×C3×D3—81no—
224C2×C3×D3—81no—
225C2×C3×D3—81no—
226C2×C3×D3—81no—
227C32×C4—42no—
228C32×C4—81no—
229C32×C4#11711yescharacteristic
230C32×C4C4×D411yescharacteristic
231C32×C4#12311yescharacteristic
232C32×C4#12611yescharacteristic
233C3×Dic3—41no—
234C3×Dic3—41no—
235C3×Dic3—81no—
236C22×D4—31no—
237C22×C8—31no—
238C22×C8C2×C3×D311yescharacteristic
239C2×C4⋊C4—31no—
240C2×C4⋊C4—61no—
241C2×C4⋊C4C2×C3×D311yescharacteristic
242C2×C4⋊C4C2×C3×D311yescharacteristic
243C2×C22⋊C4—61no—
244C2×C22⋊C4—61no—
245C2×C42—31no—
246#102—34no—
247#102—62no—
248#97—62no—
249#95—61no—
250C23×C3—21no—
251C23×C3—62no—
252C23×C3—121no—
253C23×C3—121no—
254C23×C3—121no—
255C23×C3—121no—
256C23×C3—121no—
257C23×C3—121no—
258C23×C3D3×D411yescharacteristic
259C23×C3C2×C3×D411yescharacteristic
260C22×D3—22no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—41no—
267C3×D4—62no—
268C3×D4—62no—
269C3×D4—121no—
270C3×D4—121no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—81no—
298C2×C3×C4—81no—
299C2×C3×C4—81no—
300C2×C3×C4—81no—
301C2×C3×C4—121no—
302C2×C3×C4—121no—
303C2×C3×C4—121no—
304C2×C3×C4C2×C4×D311yescharacteristic
305C2×C3×C4C2×D1211yescharacteristic
306C2×C3×C4#23511yescharacteristic
307C2×C3×C4#23711yescharacteristic
308C2×C3×C4#23711yescharacteristic
309C2×C3×C4#23911yescharacteristic
310C2×C3×C4C22×C3×C411yescharacteristic
311C2×C3×C4C2×C3×D411yescharacteristic
312C2×C3×C4C3×#4111yescharacteristic
313C2×C3×C4C3×#4111yescharacteristic
314C2×C3×C4C3×#4111yescharacteristic
315C2×C3×C4C3×#4111yescharacteristic
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325D12—22no—
326D12—22no—
327D12—41no—
328D12—41no—
329C3×C8—22no—
330C3×C8—22no—
331C3×C8—42no—
332C3×C8—62no—
333C3⋊C8—22no—
334C2×C32#38811yescharacteristic
335C2×C32#44011yescharacteristic
336C2×C32#44211yescharacteristic
337C2×C32#46811yescharacteristic
338C2×C32#46911yescharacteristic
339C2×C32#48911yescharacteristic
340C2×C32#49011yescharacteristic
341C3×D3—82no—
342C24—61no—
343C2×D4—32no—
344C2×D4—61no—
345C2×D4—61no—
346C2×D4—61no—
347C2×D4—61no—
348C2×D4—61no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—31no—
352C22×C4—31no—
353C22×C4—61no—
354C22×C4C22×C3×D311yescharacteristic
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—32no—
358C2×C8—61no—
359C2×C8—61no—
360C2×C8C3×D1212yes—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—61no—
365C4⋊C4—61no—
366C4⋊C4—122no—
367C4⋊C4C3×C4×D312yes—
368C22⋊C4—122no—
369C22⋊C4—122no—
370C42—61no—
371C42—61no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—122no—
380C22×C3—122no—
381C22×C3—122no—
382C22×C3—241no—
383C22×C3—241no—
384C22×C3—241no—
385C22×C3—241no—
386C22×C3#67⋊C411yescharacteristic
387C22×C3#88511yescharacteristic
388C22×C3#89511yescharacteristic
389C22×C3#91111yescharacteristic
390C22×C3#91411yescharacteristic
391C22×C3#91411yescharacteristic
392C22×C3#91811yescharacteristic
393C22×C3C3×C4×D411yescharacteristic
394C22×C3C3×#12111yescharacteristic
395C22×C3C3×#12411yescharacteristic
396C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
397C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
398C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
399C22×C3C3×#13711yescharacteristic
400C2×D3—42no—
401C2×D3—42no—
402C2×D3—42no—
403C2×D3—81no—
404C2×D3—81no—
405C2×D3—81no—
406C2×D3—81no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—82no—
414C3×C4—82no—
415C3×C4—81no—
416C3×C4—81no—
417C3×C4—121no—
418C3×C4—121no—
419C3×C4—241no—
420C3×C4C4×D1211yescharacteristic
421C3×C4#88611yescharacteristic
422C3×C4#89211yescharacteristic
423C3×C4#89811yescharacteristic
424C3×C4C3×#11711yescharacteristic
425C3×C4C3×C4×D411yescharacteristic
426C3×C4C3×#12311yescharacteristic
427C3×C4C3×#12611yescharacteristic
428Dic3—41no—
429Dic3—41no—
430Dic3—81no—
431C32#193611yescharacteristic, complemented, Sylow
432C23—62no—
433C23—121no—
434C23—121no—
435C23—121no—
436C23—121no—
437C23—121no—
438C23—121no—
439C23C3×D3×D411yescharacteristic
440D4—62no—
441D4—62no—
442D4—121no—
443D4—121no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4C2×C3×C4×D311yescharacteristic
460C2×C4C2×C3×D1211yescharacteristic
461C2×C4C3×#23511yescharacteristic
462C2×C4C3×#23711yescharacteristic
463C2×C4C3×#23711yescharacteristic
464C2×C4C3×#23911yescharacteristic
465C8—22no—
466C8—62no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—242no—
475C2×C3#467011yescharacteristic
476C2×C3#476411yescharacteristic
477C2×C3#478511yescharacteristic
478C2×C3#479411yescharacteristic
479C2×C3#481711yescharacteristic
480C2×C3#486511yescharacteristic
481C2×C3#488611yescharacteristic
482C2×C3C3×#38811yescharacteristic
483C2×C3C3×#44011yescharacteristic
484C2×C3C3×#44211yescharacteristic
485C2×C3C3×#46811yescharacteristic
486C2×C3C3×#46911yescharacteristic
487C2×C3C3×#48911yescharacteristic
488C2×C3C3×#49011yescharacteristic
489D3—82no—
490C22—122no—
491C22—122no—
492C22—122no—
493C22—241no—
494C22—241no—
495C22—241no—
496C22—241no—
497C22C3×#67⋊C411yescharacteristic
498C22C3×#88511yescharacteristic
499C22C3×#89511yescharacteristic
500C22C3×#91111yescharacteristic
501C22C3×#91411yescharacteristic
502C22C3×#91411yescharacteristic
503C22C3×#91811yescharacteristic
504C4—42no—
505C4—81no—
506C4—121no—
507C4—121no—
508C4—241no—
509C4C3×C4×D1211yescharacteristic
510C4C3×#88611yescharacteristic
511C4C3×#89211yescharacteristic
512C4C3×#89811yescharacteristic
513C3—21no—
514C3#6885711yescharacteristic, direct product factor
515C3C3×#193611yescharacteristic, complemented
516C2—242no—
517C2C3×#467011yescharacteristic
518C2C3×#476411yescharacteristic
519C2C3×#478511yescharacteristic
520C2C3×#479411yescharacteristic
521C2C3×#481711yescharacteristic
522C2C3×#486511yescharacteristic
523C2C3×#488611yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#340C2
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#340D3
hypocenterC3C3×#1936
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C3×#340

Complements

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

NH ≅ G/Ntype
2. C2×C3×#815C2semidirect
4. C3×#4984C2semidirect
6. C32×#340C2semidirect
7. C3×#4529C2semidirect
9. #68857C3direct (N × H)
23. C3×#809C4semidirect
30. C2×#815C2×C3semidirect
32. #4984C2×C3semidirect
36. C3×#340D3semidirect
37. C3×#340C2×C3semidirect
38. #4529C2×C3semidirect
100. #809C3×C4semidirect
142. #340C3×D3semidirect
431. C32#1936semidirect
514. C3#68857direct (N × H)
515. C3C3×#1936semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8212
C2×C3×#8152
C3×#49882
C3×#49842
C2×C3×#8092
C32×#3402
C3×#45292
C3×#19363
#688573