Finite Group Databasebeta

← C3×#69768 #11789795

Subgroups — C3×#69768 #11789795

Order 1152.

665 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×C8C211yesmaximal, characteristic, complemented
2C2×C3×#821C211yesmaximal, characteristic
3C2×C3×#821C211yesmaximal, characteristic
4C3×#4939C211yesmaximal, characteristic
5C3×#4928C211yesmaximal, characteristic, complemented
6C3×#4551C211yesmaximal, characteristic, complemented
7C3×#4551C211yesmaximal, characteristic, complemented
8C3×#1934—31nomaximal
9#69768C311yesmaximal, characteristic, direct product factor
10C22×C3×D12C2211yescharacteristic
11C22×C32×C8C2211yescharacteristic
12C22×C32×C8C2211yescharacteristic
13C2×C32×C42C2211yescharacteristic
14C2×C3×(C2×D3)⋊C4—21no—
15C2×C3×(C2×D3)⋊C4—21no—
16C2×C3×C4⋊Dic3C2211yescharacteristic
17C2×C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C4⋊Dic3C2211yescharacteristic
19C2×C3×Dic3⋊C4—21no—
20C32×C4×C8—22no—
21C3×C3⋊#95—21no—
22C3×#821—22no—
23C3×#821C414yescomplemented
24C2×C3×#102—31no—
25C2×C3×#102—31no—
26C2×C3×C4×C8—21no—
27C2×C3×C4×C8D311yescharacteristic, complemented
28C2×C3×C4×C8C2×C311yescharacteristic, complemented
29C3×#390—31no—
30C3×#389—31no—
31C2×#821C2×C311yescharacteristic
32C2×#821C2×C311yescharacteristic
33#4939C2×C311yescharacteristic
34#4928C2×C311yescharacteristic, complemented
35C3×#340—31no—
36C3×#340—31no—
37#4551C2×C311yescharacteristic, complemented
38#4551C2×C311yescharacteristic, complemented
39C23×C3×D3—21no—
40C22×C32×C4D411yescharacteristic
41C22×C32×C4D411yescharacteristic
42C22×C32×C4C2311yescharacteristic
43C22×C3×Dic3—21no—
44C22×C3×Dic3—21no—
45C22×C3×Dic3—21no—
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×C8—21no—
55C2×C32×C8—21no—
56C2×C32×C8C2×C412yes—
57C2×C32×C8D412yes—
58C32×C42—21no—
59C32×C42—21no—
60C3×(C2×D3)⋊C4—42no—
61C3×(C2×D3)⋊C4—42no—
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3C2×C412yes—
68C3×Dic3⋊C4—42no—
69#1934—31noSylow
70C22×C3×D4—31no—
71C22×D12C22×C311yescharacteristic
72C22×C3×C8—21no—
73C22×C3×C8—21no—
74C22×C3×C8C2×D311yescharacteristic
75C22×C3×C8C2×D311yescharacteristic
76C22×C3×C8C22×C311yescharacteristic
77C22×C3×C8C22×C311yescharacteristic
78C2×C3×C4⋊C4—31no—
79C2×C3×C4⋊C4—31no—
80C2×C3×C4⋊C4—31no—
81C2×C3×C4⋊C4—61no—
82C2×C3×C22⋊C4—61no—
83C2×C3×C22⋊C4—61no—
84C2×C3×C42—21no—
85C2×C3×C42C2×D311yescharacteristic
86C2×C3×C42C22×C311yescharacteristic
87C2×(C2×D3)⋊C4—21no—
88C2×(C2×D3)⋊C4—21no—
89C2×C4⋊Dic3C22×C311yescharacteristic
90C2×C4⋊Dic3C22×C311yescharacteristic
91C2×C4⋊Dic3C22×C311yescharacteristic
92C2×Dic3⋊C4—21no—
93C3×#102—34no—
94C3×#102—62no—
95C3×C4×C8—24no—
96C3×C4×C8—22no—
97C3×C4×C8—22no—
98C3×#95—61no—
99C3⋊#95—21no—
100#821—22no—
101#821C3×C414yescomplemented
102C23×C32C2×D411yescharacteristic
103C22×C3×D3—22no—
104C22×C3×D3—41no—
105C22×C3×D3—41no—
106C22×C3×D3—41no—
107C22×C3×D3—41no—
108C22×C3×D3—41no—
109C22×C3×D3—41no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4D812yes—
115C2×C32×C4SD1612yes—
116C2×C32×C4C22×C411yescharacteristic
117C2×C32×C4C2×D411yescharacteristic
118C2×C32×C4#4111yescharacteristic
119C2×C32×C4#4111yescharacteristic
120C2×C32×C4#4111yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C3×Dic3—22no—
123C2×C3×Dic3—41no—
124C2×C3×Dic3—41no—
125C2×C3×Dic3—41no—
126C2×C3×Dic3—41no—
127C2×C3×Dic3—41no—
128C2×C3×Dic3—41no—
129C2×C3×Dic3—41no—
130C2×C3×Dic3—41no—
131C3×D12—22no—
132C3×D12—22no—
133C3×D12—41no—
134C3×D12—41no—
135C32×C8—22no—
136C32×C8—22no—
137C2×#102—31no—
138C2×#102—31no—
139C2×C4×C8C3×D311yescharacteristic, complemented
140#390—31no—
141#389—31no—
142#340—31no—
143#340—31no—
144C24×C3—61no—
145C23×D3—21no—
146C2×C3×D4—32no—
147C2×C3×D4—61no—
148C2×C3×D4—61no—
149C2×C3×D4—61no—
150C2×C3×D4—61no—
151C2×C3×D4—61no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—61no—
156C22×C3×C4—61no—
157C22×C3×C4—61no—
158C22×C3×C4D1211yescharacteristic
159C22×C3×C4D1211yescharacteristic
160C22×C3×C4C3×D411yescharacteristic
161C22×C3×C4C3×D411yescharacteristic
162C22×C3×C4C22×D311yescharacteristic
163C22×C3×C4C23×C311yescharacteristic
164C22×Dic3—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12C2×C3×C412yes—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C2×C3×C8—21no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8C4×D312yes—
188C2×C3×C8#6712yes—
189C2×C3×C8C2×C3×C412yes—
190C2×C3×C8C3×D412yes—
191C3×C4⋊C4—32no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4—122no—
198C3×C22⋊C4—122no—
199C3×C22⋊C4—122no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—21no—
203C3×C42—21no—
204C3×C42—21no—
205C3×C42—21no—
206(C2×D3)⋊C4—42no—
207(C2×D3)⋊C4—42no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3C2×C3×C412yes—
214Dic3⋊C4—42no—
215C22×C32C4×D411yescharacteristic
216C22×C32#12111yescharacteristic
217C22×C32#12411yescharacteristic
218C22×C32C2×D811yescharacteristic
219C22×C32C2×SD1611yescharacteristic
220C22×C32#13511yescharacteristic
221C22×C32#13511yescharacteristic
222C2×C3×D3—42no—
223C2×C3×D3—42no—
224C2×C3×D3—42no—
225C2×C3×D3—81no—
226C2×C3×D3—81no—
227C2×C3×D3—81no—
228C2×C3×D3—81no—
229C32×C4—22no—
230C32×C4—22no—
231C32×C4#11711yescharacteristic
232C32×C4C4×D411yescharacteristic
233C32×C4#12311yescharacteristic
234C32×C4#12611yescharacteristic
235C3×Dic3—42no—
236C3×Dic3—81no—
237C3×Dic3—81no—
238C22×D4—31no—
239C22×C8C2×C3×D311yescharacteristic
240C22×C8C2×C3×D311yescharacteristic
241C2×C4⋊C4—31no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—31no—
244C2×C4⋊C4—61no—
245C2×C22⋊C4—61no—
246C2×C22⋊C4—61no—
247C2×C42C2×C3×D311yescharacteristic
248#102—34no—
249#102—62no—
250C4×C8—22no—
251#95—61no—
252C23×C3—21no—
253C23×C3—62no—
254C23×C3—121no—
255C23×C3—121no—
256C23×C3—121no—
257C23×C3—121no—
258C23×C3—121no—
259C23×C3—121no—
260C23×C3C2×D1211yescharacteristic
261C23×C3C2×C3×D411yescharacteristic
262C22×D3—22no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3—41no—
267C22×D3—41no—
268C22×D3—41no—
269C3×D4—62no—
270C3×D4—62no—
271C3×D4—121no—
272C3×D4—121no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—62no—
294C2×C3×C4—121no—
295C2×C3×C4—121no—
296C2×C3×C4—121no—
297C2×C3×C4—121no—
298C2×C3×C4—121no—
299C2×C3×C4—121no—
300C2×C3×C4—121no—
301C2×C3×C4—121no—
302C2×C3×C4#20412yes—
303C2×C3×C4D2412yes—
304C2×C3×C4C3×D812yes—
305C2×C3×C4C3×SD1612yes—
306C2×C3×C4C2×C4×D311yescharacteristic
307C2×C3×C4#23511yescharacteristic
308C2×C3×C4#23511yescharacteristic
309C2×C3×C4#23511yescharacteristic
310C2×C3×C4#23511yescharacteristic
311C2×C3×C4C2×#6711yescharacteristic
312C2×C3×C4C22×C3×C411yescharacteristic
313C2×C3×C4C2×C3×D411yescharacteristic
314C2×C3×C4C3×#4111yescharacteristic
315C2×C3×C4C3×#4111yescharacteristic
316C2×C3×C4C3×#4111yescharacteristic
317C2×C3×C4C3×#4111yescharacteristic
318C2×Dic3—22no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327D12—22no—
328D12—22no—
329D12—41no—
330D12—41no—
331C3×C8—24no—
332C3×C8—24no—
333C3×C8—22no—
334C3×C8—22no—
335C3×C8—22no—
336C3×C8—22no—
337C2×C32#38811yescharacteristic
338C2×C32C4×D811yescharacteristic
339C2×C32C4×SD1611yescharacteristic
340C2×C32#46511yescharacteristic
341C2×C32#46611yescharacteristic
342C2×C32#48611yescharacteristic
343C2×C32#48811yescharacteristic
344C3×D3—82no—
345C24—61no—
346C2×D4—32no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C2×D4—61no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4C3×D1211yescharacteristic
356C22×C4C3×D1211yescharacteristic
357C22×C4C22×C3×D311yescharacteristic
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8C3×C4×D312yes—
363C2×C8C3×#6712yes—
364C4⋊C4—32no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—61no—
370C4⋊C4—122no—
371C22⋊C4—122no—
372C22⋊C4—122no—
373C42—21no—
374C42—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—122no—
383C22×C3—122no—
384C22×C3—122no—
385C22×C3—241no—
386C22×C3—241no—
387C22×C3—241no—
388C22×C3—241no—
389C22×C3C4×D1211yescharacteristic
390C22×C3#87511yescharacteristic
391C22×C3C2×#20411yescharacteristic
392C22×C3C2×D2411yescharacteristic
393C22×C3#90411yescharacteristic
394C22×C3#90411yescharacteristic
395C22×C3#93011yescharacteristic
396C22×C3C3×C4×D411yescharacteristic
397C22×C3C3×#12111yescharacteristic
398C22×C3C3×#12411yescharacteristic
399C22×C3C2×C3×D811yescharacteristic
400C22×C3C2×C3×SD1611yescharacteristic
401C22×C3C3×#13511yescharacteristic
402C22×C3C3×#13511yescharacteristic
403C2×D3—42no—
404C2×D3—42no—
405C2×D3—42no—
406C2×D3—81no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—81no—
410C3×C4—24no—
411C3×C4—24no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—22no—
415C3×C4—22no—
416C3×C4—21no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—21no—
420C3×C4—122no—
421C3×C4—241no—
422C3×C4—241no—
423C3×C4#87211yescharacteristic
424C3×C4#87611yescharacteristic
425C3×C4C4×#6711yescharacteristic
426C3×C4#92911yescharacteristic
427C3×C4C3×#11711yescharacteristic
428C3×C4C3×C4×D411yescharacteristic
429C3×C4C3×#12311yescharacteristic
430C3×C4C3×#12611yescharacteristic
431Dic3—42no—
432Dic3—81no—
433Dic3—81no—
434C32#193411yescharacteristic, complemented, Sylow
435C23—62no—
436C23—121no—
437C23—121no—
438C23—121no—
439C23—121no—
440C23—121no—
441C23—121no—
442C23C2×C3×D1211yescharacteristic
443D4—62no—
444D4—62no—
445D4—121no—
446D4—121no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—62no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—121no—
455C2×C4—121no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4C3×#20412yes—
461C2×C4C3×D2412yes—
462C2×C4C2×C3×C4×D311yescharacteristic
463C2×C4C3×#23511yescharacteristic
464C2×C4C3×#23511yescharacteristic
465C2×C4C3×#23511yescharacteristic
466C2×C4C3×#23511yescharacteristic
467C2×C4C2×C3×#6711yescharacteristic
468C8—22no—
469C8—22no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—242no—
478C2×C3C4×#20411yescharacteristic
479C2×C3C4×D2411yescharacteristic
480C2×C3#469411yescharacteristic
481C2×C3#469711yescharacteristic
482C2×C3#494011yescharacteristic
483C2×C3#511411yescharacteristic
484C2×C3#511511yescharacteristic
485C2×C3C3×#38811yescharacteristic
486C2×C3C3×C4×D811yescharacteristic
487C2×C3C3×C4×SD1611yescharacteristic
488C2×C3C3×#46511yescharacteristic
489C2×C3C3×#46611yescharacteristic
490C2×C3C3×#48611yescharacteristic
491C2×C3C3×#48811yescharacteristic
492D3—82no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22—241no—
497C22—241no—
498C22—241no—
499C22—241no—
500C22C3×C4×D1211yescharacteristic
501C22C3×#87511yescharacteristic
502C22C2×C3×#20411yescharacteristic
503C22C2×C3×D2411yescharacteristic
504C22C3×#90411yescharacteristic
505C22C3×#90411yescharacteristic
506C22C3×#93011yescharacteristic
507C4—22no—
508C4—22no—
509C4—122no—
510C4—241no—
511C4—241no—
512C4C3×#87211yescharacteristic
513C4C3×#87611yescharacteristic
514C4C3×C4×#6711yescharacteristic
515C4C3×#92911yescharacteristic
516C3—21no—
517C3#6976811yescharacteristic, direct product factor
518C3C3×#193411yescharacteristic, complemented
519C2—242no—
520C2C3×C4×#20411yescharacteristic
521C2C3×C4×D2411yescharacteristic
522C2C3×#469411yescharacteristic
523C2C3×#469711yescharacteristic
524C2C3×#494011yescharacteristic
525C2C3×#511411yescharacteristic
526C2C3×#511511yescharacteristic
subgroupG/H
centerC23×C3C2×D12
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×C4×C8C2
socleC23×C32C2×D4
prime radicalC22×C3×D12C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×C4×C8D3
hypocenterC3C3×#1934
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×C8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×C8C2semidirect
5. C3×#4928C2semidirect
6. C3×#4551C2semidirect
7. C3×#4551C2semidirect
9. #69768C3direct (N × H)
23. C3×#821C4semidirect
27. C2×C3×C4×C8D3semidirect
28. C2×C3×C4×C8C2×C3semidirect
34. #4928C2×C3semidirect
37. #4551C2×C3semidirect
38. #4551C2×C3semidirect
101. #821C3×C4semidirect
139. C2×C4×C8C3×D3semidirect
434. C32#1934semidirect
517. C3#69768direct (N × H)
518. C3C3×#1934semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×C82
C2×C3×#8212
C3×#49392
C3×#49282
C3×#45512
C3×#19343
#697683