Finite Group Databasebeta

← C3×#33015 #10651622

Subgroups — C3×#33015 #10651622

Order 768.

1258 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#519C211yesmaximal, characteristic, complemented
2C2×C3×#482C211yesmaximal, characteristic
3C2×C3×#448C211yesmaximal, characteristic
4C3×#2624C211yesmaximal, characteristic
5C3×#1880C212yesmaximal, complemented
6C3×#1880C212yesmaximal, complemented
7C3×#1865C212yesmaximal, complemented
8C3×#1863C212yesmaximal, complemented
9C3×#1825C211yesmaximal, characteristic, complemented
10C3×#1818C211yesmaximal, characteristic, complemented
11C3×#1787C211yesmaximal, characteristic, complemented
12#33015C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×#121C2211yescharacteristic
16C3×#519—24no—
17C3×#519—24no—
18C2×C3×C4×Q8—21no—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2212yes—
22C2×C3×#117—22no—
23C2×C3×#117C2211yescharacteristic
24C2×C3×#117C2211yescharacteristic
25C3×#482—24no—
26C3×#448—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2211yescharacteristic
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C3×#399—21no—
37C3×#397—21no—
38C3×#394—22no—
39C3×#390C2212yes—
40C3×#389C2212yes—
41C3×#380C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#519C2×C311yescharacteristic, complemented
47C2×#482C2×C311yescharacteristic
48C2×#448C2×C311yescharacteristic
49#2624C2×C311yescharacteristic
50#1880C2×C312yescomplemented
51#1880C2×C312yescomplemented
52#1865C2×C312yescomplemented
53#1863C2×C312yescomplemented
54#1825C2×C311yescharacteristic, complemented
55#1818C2×C311yescharacteristic, complemented
56#1787C2×C311yescharacteristic, complemented
57C22×C3×Q8—24no—
58C22×C3×Q8—24no—
59C22×C3×Q8—21no—
60C22×C3×Q8—22no—
61C22×C3×Q8D414yes—
62C22×C3×Q8C2312yes—
63C22×C3×D4—21no—
64C22×C3×D4C2311yescharacteristic
65C23×C3×C4—21no—
66C23×C3×C4C2311yescharacteristic
67C2×C3×SD16—24no—
68C2×C3×SD16—24no—
69C2×C3×SD16—24no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2311yescharacteristic
72C22×C3×C8C2311yescharacteristic
73C3×#121—22no—
74C3×#121—22no—
75C3×C4×Q8—28no—
76C3×C4×Q8—28no—
77C3×C4×Q8—24no—
78C3×C4×Q8—24no—
79C3×C4×Q8—24no—
80C3×C4×Q8—24no—
81C3×#117—216no—
82C3×#117—22no—
83C3×#117—22no—
84C3×#117—22no—
85C3×#117D414yes—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4D412yes—
101C2×C3×C42—22no—
102C2×C3×C42—22no—
103C2×C3×C42—21no—
104C2×C3×C42C2312yes—
105C3×#107—22no—
106C3×#106—22no—
107C3×#103—24no—
108C3×#103—24no—
109C3×#103—24no—
110C3×#102—22no—
111C3×#102—22no—
112C3×C22⋊C8—22no—
113C3×C22⋊C8—22no—
114C3×#95—21no—
115C3×#95—22no—
116C23×Q8C22×C311yescharacteristic
117C22×SD16C22×C312yes—
118C2×#121C22×C311yescharacteristic
119#519—24no—
120#519—24no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C22×C312yes—
124C2×C4×Q8C22×C312yes—
125C2×#117—22no—
126C2×#117C22×C311yescharacteristic
127C2×#117C22×C311yescharacteristic
128#482—24no—
129#448—24no—
130C2×#107C22×C311yescharacteristic
131C2×#106C22×C311yescharacteristic
132C2×#103C22×C312yes—
133C2×#103C22×C312yes—
134C2×#103C22×C312yes—
135C2×#102C22×C311yescharacteristic
136C2×#102C22×C311yescharacteristic
137C2×C22⋊C8C22×C311yescharacteristic
138C2×C22⋊C8C22×C311yescharacteristic
139#399—21no—
140#397—21no—
141#394—22no—
142#390C22×C312yes—
143#389C22×C312yes—
144#380C22×C311yescharacteristic
145#340C22×C312yes—
146#340C22×C312yes—
147#336C22×C312yes—
148#336C22×C312yes—
149C24×C3—21no—
150C24×C3C2×D411yescharacteristic
151C2×C3×Q8—28no—
152C2×C3×Q8—28no—
153C2×C3×Q8—28no—
154C2×C3×Q8—216no—
155C2×C3×Q8—24no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—22no—
159C2×C3×Q8C2×D414yes—
160C2×C3×Q8#4114yes—
161C2×C3×Q8#4114yes—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—42no—
167C2×C3×D4—82no—
168C22×C3×C4—28no—
169C22×C3×C4—24no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2411yescharacteristic
183C3×SD16—48no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C3×C4⋊C4—28no—
192C3×C4⋊C4—28no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—82no—
202C3×C4⋊C4C2×D412yes—
203C3×C4⋊C4C2×D412yes—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—24no—
206C3×C22⋊C4—42no—
207C3×C22⋊C4—82no—
208C3×C22⋊C4—82no—
209C3×C22⋊C4—81no—
210C3×C42—28no—
211C3×C42—28no—
212C3×C42—24no—
213C3×C42—22no—
214C3×C42—22no—
215C22×Q8—24no—
216C22×Q8—24no—
217C22×Q8—21no—
218C22×Q8—22no—
219C22×Q8C3×D414yes—
220C22×Q8C23×C312yes—
221C22×D4—21no—
222C22×D4C23×C311yescharacteristic
223C23×C4—21no—
224C23×C4C23×C311yescharacteristic
225C2×SD16—24no—
226C2×SD16—24no—
227C2×SD16—24no—
228C22×C8C23×C312yes—
229C22×C8C23×C311yescharacteristic
230C22×C8C23×C311yescharacteristic
231#121—22no—
232#121—22no—
233C4×Q8—28no—
234C4×Q8—28no—
235C4×Q8—24no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—24no—
239#117—216no—
240#117—22no—
241#117—22no—
242#117—22no—
243#117C3×D414yes—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4C3×D411yescharacteristic
248C2×C4⋊C4C3×D411yescharacteristic
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C312yes—
251C2×C4⋊C4C23×C311yescharacteristic
252C2×C4⋊C4C23×C311yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4C3×D412yes—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—21no—
262C2×C42C23×C312yes—
263#107—22no—
264#106—22no—
265#103—24no—
266#103—24no—
267#103—24no—
268#102—22no—
269#102—22no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272#95—21no—
273#95—22no—
274C23×C3—22no—
275C23×C3—21no—
276C23×C3—22no—
277C23×C3—21no—
278C23×C3—41no—
279C23×C3—42no—
280C23×C3—41no—
281C23×C3—42no—
282C23×C3—41no—
283C23×C3#12012yes—
284C23×C3C22×D411yescharacteristic
285C3×Q8—216no—
286C3×Q8—24no—
287C3×Q8—24no—
288C3×Q8—24no—
289C3×Q8#12318yes—
290C3×D4—42no—
291C3×D4—41no—
292C3×D4—41no—
293C3×D4—84no—
294C2×C3×C4—216no—
295C2×C3×C4—24no—
296C2×C3×C4—24no—
297C2×C3×C4—24no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—42no—
318C2×C3×C4#12012yes—
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321C2×C3×C4C2×#4112yes—
322C2×C3×C4C2×#4112yes—
323C3×C8—42no—
324C3×C8—41no—
325C3×C8—41no—
326C24—21no—
327C24C2×C3×D411yescharacteristic
328C2×Q8—28no—
329C2×Q8—28no—
330C2×Q8—28no—
331C2×Q8—216no—
332C2×Q8—24no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8C2×C3×D414yes—
337C2×Q8C3×#4114yes—
338C2×Q8C3×#4114yes—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—42no—
344C2×D4—82no—
345C22×C4—28no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4C2×C3×D412yes—
355C22×C4C2×C3×D412yes—
356C22×C4C2×C3×D411yescharacteristic
357C22×C4C2×C3×D412yes—
358C22×C4C2×C3×D412yes—
359C22×C4C24×C311yescharacteristic
360SD16—48no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C4⋊C4—28no—
369C4⋊C4—28no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—82no—
379C4⋊C4C2×C3×D412yes—
380C4⋊C4C2×C3×D412yes—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—42no—
384C22⋊C4—82no—
385C22⋊C4—82no—
386C22⋊C4—81no—
387C42—28no—
388C42—28no—
389C42—24no—
390C42—22no—
391C42—22no—
392C22×C3—22no—
393C22×C3—22no—
394C22×C3—22no—
395C22×C3—42no—
396C22×C3—42no—
397C22×C3—81no—
398C22×C3—82no—
399C22×C3—82no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3C2×#12011yescharacteristic
403C22×C3#52512yes—
404C22×C3#57712yes—
405C22×C3#57711yescharacteristic
406C22×C3#57811yescharacteristic
407C3×C4—28no—
408C3×C4—22no—
409C3×C4—42no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—82no—
414C3×C4—81no—
415C3×C4C2×#12312yes—
416C3×C4#52511yescharacteristic
417C3×C4#52511yescharacteristic
418C23—21no—
419C23—22no—
420C23—21no—
421C23—22no—
422C23—41no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23C3×#12012yes—
428C23C22×C3×D411yescharacteristic
429Q8—216no—
430Q8—24no—
431Q8—24no—
432Q8—24no—
433Q8C3×#12318yes—
434D4—42no—
435D4—41no—
436D4—41no—
437D4—84no—
438C2×C4—216no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4C3×#12012yes—
463C2×C4C22×C3×D411yescharacteristic
464C2×C4C22×C3×D411yescharacteristic
465C2×C4C2×C3×#4112yes—
466C2×C4C2×C3×#4112yes—
467C8—42no—
468C8—41no—
469C8—41no—
470C2×C3—42no—
471C2×C3—161no—
472C2×C3#284911yescharacteristic
473C2×C3#301011yescharacteristic
474C2×C3#301311yescharacteristic
475C2×C3#312112yes—
476C2×C3#312612yes—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—42no—
481C22—42no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—82no—
486C22—82no—
487C22C2×C3×#12011yescharacteristic
488C22C3×#52512yes—
489C22C3×#57712yes—
490C22C3×#57711yescharacteristic
491C22C3×#57811yescharacteristic
492C4—28no—
493C4—22no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—81no—
499C4—82no—
500C4C2×C3×#12312yes—
501C4C3×#52511yescharacteristic
502C4C3×#52511yescharacteristic
503C3#3301511yescharacteristic, direct product factor, Sylow
504C2—42no—
505C2—161no—
506C2C3×#284911yescharacteristic
507C2C3×#301011yescharacteristic
508C2C3×#301311yescharacteristic
509C2C3×#312112yes—
510C2C3×#312612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#519C2semidirect
5. C3×#1880C2semidirect
6. C3×#1880C2semidirect
7. C3×#1865C2semidirect
8. C3×#1863C2semidirect
9. C3×#1825C2semidirect
10. C3×#1818C2semidirect
11. C3×#1787C2semidirect
12. #33015C3direct (N × H)
46. C2×#519C2×C3semidirect
50. #1880C2×C3semidirect
51. #1880C2×C3semidirect
52. #1865C2×C3semidirect
53. #1863C2×C3semidirect
54. #1825C2×C3semidirect
55. #1818C2×C3semidirect
56. #1787C2×C3semidirect
503. C3#33015direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5192
C2×C3×#4822
C2×C3×#4482
C3×#26242
C3×#18802
C3×#18652
C3×#18632
C3×#18252
C3×#18182
C3×#17872
#330153