Finite Group Databasebeta

← C3×#72140 #11784857

Subgroups — C3×#72140 #11784857

Order 1152.

1689 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×H(5,𝔽2)C211yesmaximal, characteristic
2C2×C3×#837C212yesmaximal, complemented
3C3×#5232C212yesmaximal, complemented
4C3×#5221C211yesmaximal, characteristic, complemented
5C2×C3×#833C211yesmaximal, characteristic, complemented
6C3×#1792—31nomaximal
7#72140C311yesmaximal, characteristic, direct product factor
8C32×H(5,𝔽2)—24no—
9C32×H(5,𝔽2)—22no—
10C32×H(5,𝔽2)C414yescomplemented
11C2×C32×#41—22no—
12C2×C32×#41C2212yes—
13C22×C32×D4—22no—
14C22×C32×D4—22no—
15C22×C32×D4C2211yescharacteristic
16C2×C3×#217—22no—
17C2×C3×C4⋊Dic3C2211yescharacteristic, complemented
18C2×C3×C4×Dic3C2211yescharacteristic, complemented
19C2×C3×#208C2212yescomplemented
20C3×#837—24no—
21C3×#835—22no—
22C3×#833—22no—
23C3×#832—22no—
24C2×C3×H(5,𝔽2)—21no—
25C2×C3×H(5,𝔽2)D311yescharacteristic
26C2×C3×H(5,𝔽2)C2×C311yescharacteristic
27C2×C3×#104—32no—
28C3×#417—32no—
29C2×C3×#100—31no—
30C3×#390—31no—
31C2×#837C2×C312yescomplemented
32#5232C2×C312yescomplemented
33#5221C2×C311yescharacteristic, complemented
34C2×#833C2×C311yescharacteristic, complemented
35C24×C32—22no—
36C24×C32D412yes—
37C32×#41—28no—
38C32×#41—28no—
39C32×#41—22no—
40C32×#41—22no—
41C32×#41C2×C414yes—
42C32×#41D414yes—
43C2×C32×Q8D412yes—
44C2×C32×D4—24no—
45C2×C32×D4—28no—
46C2×C32×D4—28no—
47C2×C32×D4—28no—
48C2×C32×D4—22no—
49C2×C32×D4—28no—
50C2×C32×D4—22no—
51C2×C32×D4—22no—
52C2×C32×D4C2×C412yes—
53C2×C32×D4D412yes—
54C2×C32×D4D412yes—
55C22×C32×C4—22no—
56C22×C32×C4—22no—
57C22×C32×C4C2311yescharacteristic
58C22×C3×Dic3—21no—
59C22×C3×Dic3—21no—
60C3×#217—44no—
61C3×C4⋊Dic3—22no—
62C3×C4×Dic3—22no—
63C3×#208—22no—
64C3×#208—22no—
65C2×C3×C3⋊C8—22no—
66#1792—31noSylow
67C3×H(5,𝔽2)—28no—
68C3×H(5,𝔽2)—24no—
69C3×H(5,𝔽2)—24no—
70C3×H(5,𝔽2)—24no—
71C3×H(5,𝔽2)—22no—
72C3×H(5,𝔽2)—22no—
73C3×H(5,𝔽2)—24no—
74C3×H(5,𝔽2)Dic314yescomplemented
75C3×H(5,𝔽2)C3×C414yescomplemented
76C2×C3×#41—24no—
77C2×C3×#41—22no—
78C2×C3×#41—22no—
79C2×C3×#41—22no—
80C2×C3×#41C2×D312yes—
81C2×C3×#41C22×C312yes—
82C22×C3×D4—24no—
83C22×C3×D4—24no—
84C22×C3×D4—22no—
85C22×C3×D4—22no—
86C22×C3×D4—22no—
87C22×C3×D4—22no—
88C22×C3×D4—21no—
89C22×C3×D4C2×D311yescharacteristic
90C22×C3×D4C22×C311yescharacteristic
91C2×C3×M16—32no—
92C2×C3×C4⋊C4—31no—
93C2×C3×C22⋊C4—62no—
94C2×C3×C42—31no—
95C2×#217—22no—
96C2×C4⋊Dic3C22×C311yescharacteristic, complemented
97C2×C4×Dic3C22×C311yescharacteristic, complemented
98C2×#208C22×C312yescomplemented
99C3×#104—64no—
100C3×#103—62no—
101C3×#102—62no—
102C3×#100—62no—
103#837—24no—
104#835—22no—
105#833—22no—
106#832—22no—
107C23×C32—28no—
108C23×C32—22no—
109C23×C32—22no—
110C23×C32—22no—
111C23×C32—42no—
112C23×C32—42no—
113C23×C32—82no—
114C23×C32—82no—
115C23×C32C22⋊C414yes—
116C23×C32C2×D411yescharacteristic
117C32×Q8—22no—
118C32×Q8C22⋊C414yescomplemented
119C32×D4—216no—
120C32×D4—24no—
121C32×D4—24no—
122C32×D4—24no—
123C32×D4—24no—
124C32×D4—22no—
125C32×D4C22⋊C414yescomplemented
126C2×C32×C4—28no—
127C2×C32×C4—22no—
128C2×C32×C4—24no—
129C2×C32×C4—24no—
130C2×C32×C4C22×C411yescharacteristic
131C2×C32×C4C2×D412yes—
132C2×C32×C4C2×D412yes—
133C2×C32×C4C2×D411yescharacteristic
134C2×C3×Dic3—22no—
135C2×C3×Dic3—41no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—42no—
138C2×C3×Dic3—41no—
139C3×C3⋊C8—42no—
140C2×H(5,𝔽2)C3×D311yescharacteristic
141C2×#104—32no—
142#417—32no—
143C2×#100—31no—
144#390—31no—
145C24×C3—24no—
146C24×C3—22no—
147C24×C3—22no—
148C24×C3—22no—
149C24×C3#6712yes—
150C24×C3C3×D412yes—
151C3×#41—216no—
152C3×#41—28no—
153C3×#41—28no—
154C3×#41—216no—
155C3×#41—24no—
156C3×#41—24no—
157C3×#41—28no—
158C3×#41—28no—
159C3×#41—22no—
160C3×#41—22no—
161C3×#41—22no—
162C3×#41—22no—
163C3×#41—24no—
164C3×#41—24no—
165C3×#41C2×Dic314yes—
166C3×#41#6714yes—
167C3×#41C2×C3×C414yes—
168C3×#41C3×D414yes—
169C2×C3×Q8—22no—
170C2×C3×Q8#6712yes—
171C2×C3×Q8C3×D412yes—
172C2×C3×D4—28no—
173C2×C3×D4—24no—
174C2×C3×D4—24no—
175C2×C3×D4—24no—
176C2×C3×D4—24no—
177C2×C3×D4—216no—
178C2×C3×D4—216no—
179C2×C3×D4—28no—
180C2×C3×D4—28no—
181C2×C3×D4—28no—
182C2×C3×D4—28no—
183C2×C3×D4—216no—
184C2×C3×D4—216no—
185C2×C3×D4—24no—
186C2×C3×D4—28no—
187C2×C3×D4—28no—
188C2×C3×D4—22no—
189C2×C3×D4—22no—
190C2×C3×D4—28no—
191C2×C3×D4—28no—
192C2×C3×D4—22no—
193C2×C3×D4—22no—
194C2×C3×D4—22no—
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4C2×Dic312yes—
200C2×C3×D4#6712yes—
201C2×C3×D4#6712yes—
202C2×C3×D4C2×C3×C412yes—
203C2×C3×D4C3×D412yes—
204C2×C3×D4C3×D412yes—
205C22×C3×C4—24no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—24no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—21no—
212C22×C3×C4—61no—
213C22×C3×C4—61no—
214C22×C3×C4C22×D311yescharacteristic
215C22×C3×C4C23×C311yescharacteristic
216C22×Dic3—21no—
217C22×Dic3—21no—
218C3×M16—62no—
219C3×M16—62no—
220C2×C3×C8—62no—
221C3×C4⋊C4—62no—
222C3×C22⋊C4—124no—
223C3×C42—62no—
224#217—44no—
225C4⋊Dic3—22no—
226C4×Dic3—22no—
227#208—22no—
228#208—22no—
229C2×C3⋊C8—22no—
230C22×C32—24no—
231C22×C32—24no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—42no—
235C22×C32—42no—
236C22×C32—42no—
237C22×C32—42no—
238C22×C32—44no—
239C22×C32—82no—
240C22×C32—88no—
241C22×C32C2×C22⋊C411yescharacteristic
242C22×C32#12011yescharacteristic
243C22×C32#12011yescharacteristic
244C32×C4—24no—
245C32×C4C2×C22⋊C412yes—
246C32×C4#12012yes—
247C3×Dic3—42no—
248C3×Dic3—81no—
249H(5,𝔽2)—24no—
250H(5,𝔽2)—22no—
251H(5,𝔽2)C3×Dic314yescomplemented
252C2×#41—22no—
253C2×#41C2×C3×D312yes—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4C2×C3×D311yescharacteristic
257C2×M16—32no—
258C2×C4⋊C4—31no—
259C2×C22⋊C4—62no—
260C2×C42—31no—
261#104—64no—
262#103—62no—
263#102—62no—
264#100—62no—
265C23×C3—24no—
266C23×C3—24no—
267C23×C3—216no—
268C23×C3—28no—
269C23×C3—28no—
270C23×C3—24no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—22no—
275C23×C3—22no—
276C23×C3—22no—
277C23×C3—24no—
278C23×C3—21no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3—82no—
284C23×C3—82no—
285C23×C3—84no—
286C23×C3—84no—
287C23×C3—82no—
288C23×C3—82no—
289C23×C3—82no—
290C23×C3—82no—
291C23×C3#21714yes—
292C23×C3C3×C22⋊C414yes—
293C23×C3C2×#6711yescharacteristic
294C23×C3C2×C3×D411yescharacteristic
295C3×Q8—24no—
296C3×Q8—22no—
297C3×Q8—22no—
298C3×Q8—24no—
299C3×Q8#21714yescomplemented
300C3×Q8C3×C22⋊C414yescomplemented
301C3×D4—232no—
302C3×D4—216no—
303C3×D4—216no—
304C3×D4—28no—
305C3×D4—28no—
306C3×D4—28no—
307C3×D4—28no—
308C3×D4—24no—
309C3×D4—24no—
310C3×D4—24no—
311C3×D4—24no—
312C3×D4—24no—
313C3×D4—24no—
314C3×D4—24no—
315C3×D4—22no—
316C3×D4—24no—
317C3×D4—24no—
318C3×D4—22no—
319C3×D4—24no—
320C3×D4#21714yescomplemented
321C3×D4C3×C22⋊C414yescomplemented
322C2×C3×C4—24no—
323C2×C3×C4—216no—
324C2×C3×C4—28no—
325C2×C3×C4—28no—
326C2×C3×C4—28no—
327C2×C3×C4—28no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—62no—
339C2×C3×C4—121no—
340C2×C3×C4—121no—
341C2×C3×C4—122no—
342C2×C3×C4—121no—
343C2×C3×C4C22×Dic311yescharacteristic
344C2×C3×C4C2×#6712yes—
345C2×C3×C4C2×#6712yes—
346C2×C3×C4C2×#6711yescharacteristic
347C2×C3×C4C22×C3×C411yescharacteristic
348C2×C3×C4C2×C3×D412yes—
349C2×C3×C4C2×C3×D412yes—
350C2×C3×C4C2×C3×D411yescharacteristic
351C2×Dic3—22no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354C2×Dic3—42no—
355C2×Dic3—41no—
356C3×C8—122no—
357C3⋊C8—42no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—44no—
361C2×C32—44no—
362C2×C32#37911yescharacteristic
363C2×C32#45311yescharacteristic
364C2×C32#45511yescharacteristic
365C24—22no—
366C24C3×#6712yes—
367#41—28no—
368#41—28no—
369#41—22no—
370#41—22no—
371#41C2×C3×Dic314yes—
372#41C3×#6714yes—
373C2×Q8C3×#6712yes—
374C2×D4—24no—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—28no—
378C2×D4—22no—
379C2×D4—28no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4C2×C3×Dic312yes—
383C2×D4C3×#6712yes—
384C2×D4C3×#6712yes—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—61no—
388C22×C4—61no—
389C22×C4C22×C3×D311yescharacteristic
390M16—62no—
391M16—62no—
392C2×C8—62no—
393C4⋊C4—62no—
394C22⋊C4—124no—
395C42—62no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—28no—
401C22×C3—28no—
402C22×C3—24no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—44no—
412C22×C3—44no—
413C22×C3—44no—
414C22×C3—44no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—84no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—84no—
429C22×C3—816no—
430C22×C3—88no—
431C22×C3—88no—
432C22×C3C2×#21711yescharacteristic
433C22×C3#95311yescharacteristic
434C22×C3#95311yescharacteristic
435C22×C3C2×C3×C22⋊C411yescharacteristic
436C22×C3C3×#12011yescharacteristic
437C22×C3C3×#12011yescharacteristic
438C3×C4—28no—
439C3×C4—24no—
440C3×C4—24no—
441C3×C4—22no—
442C3×C4—22no—
443C3×C4—122no—
444C3×C4—241no—
445C3×C4C2×#21712yes—
446C3×C4#95312yes—
447C3×C4C2×C3×C22⋊C412yes—
448C3×C4C3×#12012yes—
449Dic3—42no—
450Dic3—81no—
451C32#179211yescharacteristic, complemented, Sylow
452C23—28no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—42no—
457C23—42no—
458C23—82no—
459C23—82no—
460C23C3×#21714yes—
461C23C2×C3×#6711yescharacteristic
462Q8—22no—
463Q8C3×#21714yescomplemented
464D4—216no—
465D4—24no—
466D4—24no—
467D4—24no—
468D4—24no—
469D4—22no—
470D4C3×#21714yescomplemented
471C2×C4—28no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—62no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—122no—
479C2×C4—121no—
480C2×C4C22×C3×Dic311yescharacteristic
481C2×C4C2×C3×#6712yes—
482C2×C4C2×C3×#6712yes—
483C2×C4C2×C3×#6711yescharacteristic
484C8—122no—
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—48no—
495C2×C3—48no—
496C2×C3—44no—
497C2×C3—44no—
498C2×C3—44no—
499C2×C3—44no—
500C2×C3#524111yescharacteristic
501C2×C3#524211yescharacteristic
502C2×C3#524711yescharacteristic
503C2×C3C3×#37911yescharacteristic
504C2×C3C3×#45311yescharacteristic
505C2×C3C3×#45511yescharacteristic
506C22—21no—
507C22—21no—
508C22—24no—
509C22—24no—
510C22—44no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—82no—
516C22—88no—
517C22C2×C3×#21711yescharacteristic
518C22C3×#95311yescharacteristic
519C22C3×#95311yescharacteristic
520C4—24no—
521C4—122no—
522C4—241no—
523C4C2×C3×#21712yes—
524C4C3×#95312yes—
525C3—21no—
526C3#7214011yescharacteristic, direct product factor
527C3C3×#179211yescharacteristic, complemented
528C2—21no—
529C2—21no—
530C2—44no—
531C2—44no—
532C2C3×#524111yescharacteristic
533C2C3×#524211yescharacteristic
534C2C3×#524711yescharacteristic
subgroupG/H
centerC22×C3#953
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
Fitting, prime radicalC2×C32×H(5,𝔽2)C2
socleC22×C32#120
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×H(5,𝔽2)D3
hypocenterC3C3×#1792
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C2×C3×#837C2semidirect
3. C3×#5232C2semidirect
4. C3×#5221C2semidirect
5. C2×C3×#833C2semidirect
7. #72140C3direct (N × H)
10. C32×H(5,𝔽2)C4semidirect
17. C2×C3×C4⋊Dic3C22semidirect
18. C2×C3×C4×Dic3C22semidirect
19. C2×C3×#208C22semidirect
31. C2×#837C2×C3semidirect
32. #5232C2×C3semidirect
33. #5221C2×C3semidirect
34. C2×#833C2×C3semidirect
74. C3×H(5,𝔽2)Dic3semidirect
75. C3×H(5,𝔽2)C3×C4semidirect
96. C2×C4⋊Dic3C22×C3semidirect
97. C2×C4×Dic3C22×C3semidirect
98. C2×#208C22×C3semidirect
118. C32×Q8C22⋊C4semidirect
125. C32×D4C22⋊C4semidirect
251. H(5,𝔽2)C3×Dic3semidirect
299. C3×Q8#217semidirect
300. C3×Q8C3×C22⋊C4semidirect
320. C3×D4#217semidirect
321. C3×D4C3×C22⋊C4semidirect
451. C32#1792semidirect
463. Q8C3×#217semidirect
470. D4C3×#217semidirect
526. C3#72140direct (N × H)
527. C3C3×#1792semidirect

Maximal subgroups

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

subgroupindex
C2×C32×H(5,𝔽2)2
C2×C3×#8372
C3×#52322
C3×#52212
C2×C3×#8332
C3×#17923
#721403