Finite Group Databasebeta

← C3×#29763 #10648370

Subgroups — C3×#29763 #10648370

Order 768.

866 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#481C211yesmaximal, characteristic, complemented
2C2×C3×#480C211yesmaximal, characteristic, complemented
3C3×#2619C211yesmaximal, characteristic
4C3×#2316C211yesmaximal, characteristic, complemented
5C3×#2085C212yesmaximal, complemented
6C3×#2067C212yesmaximal, complemented
7C3×#2065C212yesmaximal, complemented
8C3×#2041C212yesmaximal, complemented
9C3×#1875C211yesmaximal, characteristic, complemented
10C3×#1875C211yesmaximal, characteristic, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#29763C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×#121C2211yescharacteristic
15C2×C3×#121C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C3×#481—24no—
20C3×#480—24no—
21C2×C3×#107C2212yes—
22C2×C3×#106C2212yes—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#394—22no—
31C3×#392—22no—
32C3×#390C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#389—22no—
35C3×#389C2212yes—
36C3×#389C2212yes—
37C3×#382C2211yescharacteristic
38C3×#380—22no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2212yes—
42C2×#481C2×C311yescharacteristic, complemented
43C2×#480C2×C311yescharacteristic, complemented
44#2619C2×C311yescharacteristic
45#2316C2×C311yescharacteristic, complemented
46#2085C2×C312yescomplemented
47#2067C2×C312yescomplemented
48#2065C2×C312yescomplemented
49#2041C2×C312yescomplemented
50#1875C2×C311yescharacteristic, complemented
51#1875C2×C311yescharacteristic, complemented
52#1798C2×C311yescharacteristic, complemented
53C22×C3×D4—21no—
54C22×C3×D4—21no—
55C22×C3×D4—22no—
56C22×C3×D4C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4C2311yescharacteristic
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×#127—42no—
63C3×#121—22no—
64C3×#121—22no—
65C3×#121—22no—
66C3×#121—22no—
67C3×#117—42no—
68C3×#117—42no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4D412yes—
74C2×C3×C4⋊C4D414yescomplemented
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4D412yes—
85C2×C3×C42—21no—
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C3×#107—24no—
89C3×#106—24no—
90C3×C4⋊C8—42no—
91C3×#102—24no—
92C3×#102—24no—
93C3×#102—42no—
94C3×#102—42no—
95C3×C22⋊C8—22no—
96C3×C22⋊C8—22no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C2×#127C22×C311yescharacteristic
101C2×#121C22×C311yescharacteristic
102C2×#121C22×C311yescharacteristic
103C2×#117C22×C311yescharacteristic
104C2×#117C22×C311yescharacteristic
105C22×C4⋊C4C22×C311yescharacteristic
106#481—24no—
107#480—24no—
108C2×#107C22×C312yes—
109C2×#106C22×C312yes—
110C2×C4⋊C8C22×C312yes—
111C2×#102C22×C312yes—
112C2×#102C22×C312yes—
113C2×#102C22×C312yes—
114C2×#102C22×C312yes—
115C2×C22⋊C8C22×C311yescharacteristic
116C2×C22⋊C8C22×C311yescharacteristic
117#394—22no—
118#392—22no—
119#390C22×C311yescharacteristic
120#390C22×C311yescharacteristic
121#389—22no—
122#389C22×C312yes—
123#389C22×C312yes—
124#382C22×C311yescharacteristic
125#380—22no—
126#340C22×C312yes—
127#340C22×C312yes—
128#336C22×C312yes—
129C24×C3—21no—
130C24×C3—21no—
131C24×C3C2×D411yescharacteristic
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4—22no—
137C2×C3×D4—22no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—44no—
141C2×C3×D4—42no—
142C2×C3×D4—42no—
143C2×C3×D4—82no—
144C2×C3×D4—82no—
145C2×C3×D4—82no—
146C2×C3×D4—82no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—24no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—82no—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D412yes—
159C22×C3×C4#4112yes—
160C22×C3×C4C2411yescharacteristic
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—82no—
177C3×C4⋊C4—82no—
178C3×C4⋊C4C2×D414yes—
179C3×C22⋊C4—42no—
180C3×C22⋊C4—42no—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—82no—
184C3×C22⋊C4—82no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—42no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4—22no—
191C22×D4C23×C311yescharacteristic
192C22×D4C23×C311yescharacteristic
193C23×C4—21no—
194C23×C4C23×C311yescharacteristic
195C22×C8C23×C312yes—
196C22×C8C23×C312yes—
197#127—42no—
198#121—22no—
199#121—22no—
200#121—22no—
201#121—22no—
202#117—42no—
203#117—42no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C3×D412yes—
209C2×C4⋊C4C3×D414yescomplemented
210C2×C4⋊C4C23×C312yes—
211C2×C4⋊C4C23×C311yescharacteristic
212C2×C4⋊C4C23×C311yescharacteristic
213C2×C4⋊C4C23×C312yes—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4C3×D412yes—
220C2×C42—21no—
221C2×C42C23×C311yescharacteristic
222C2×C42C23×C311yescharacteristic
223#107—24no—
224#106—24no—
225C4⋊C8—42no—
226#102—24no—
227#102—24no—
228#102—42no—
229#102—42no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—22no—
238C23×C3—22no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—41no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3#12112yes—
250C23×C3C22×D411yescharacteristic
251C3×D4—42no—
252C3×D4—42no—
253C3×D4—42no—
254C3×D4—42no—
255C3×D4—88no—
256C3×D4—84no—
257C3×D4—84no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—84no—
279C2×C3×C4#12112yes—
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4H(5,𝔽2)12yes—
283C2×C3×C42−1+412yes—
284C3×C8—42no—
285C3×C8—42no—
286C24—21no—
287C24—21no—
288C24C2×C3×D411yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—44no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4—82no—
303C2×D4—82no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—82no—
312C22×C4C2×C3×D412yes—
313C22×C4C2×C3×D412yes—
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C2×C3×D412yes—
316C22×C4C3×#4112yes—
317C22×C4C24×C311yescharacteristic
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4C2×C3×D414yes—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—82no—
341C22⋊C4—82no—
342C42—22no—
343C42—22no—
344C42—42no—
345C22×C3—22no—
346C22×C3—22no—
347C22×C3—22no—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3—82no—
353C22×C3—82no—
354C22×C3—82no—
355C22×C3—82no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3—81no—
360C22×C3C2×#12111yescharacteristic
361C22×C3#53411yescharacteristic
362C22×C3#53611yescharacteristic
363C22×C3#57612yes—
364C22×C3#57712yes—
365C3×C4—22no—
366C3×C4—42no—
367C3×C4—42no—
368C3×C4—44no—
369C3×C4—82no—
370C3×C4—81no—
371C3×C4—81no—
372C3×C4#53712yes—
373C3×C4#53912yes—
374C23—22no—
375C23—22no—
376C23—21no—
377C23—21no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23C3×#12112yes—
389C23C22×C3×D411yescharacteristic
390D4—42no—
391D4—42no—
392D4—42no—
393D4—42no—
394D4—88no—
395D4—84no—
396D4—84no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—84no—
418C2×C4C3×#12112yes—
419C2×C4C22×C3×D411yescharacteristic
420C2×C4C2×C3×#4111yescharacteristic
421C2×C4C3×H(5,𝔽2)12yes—
422C2×C4C3×2−1+412yes—
423C8—42no—
424C8—42no—
425C2×C3—42no—
426C2×C3—161no—
427C2×C3—161no—
428C2×C3#261811yescharacteristic
429C2×C3#304111yescharacteristic
430C2×C3#304311yescharacteristic
431C2×C3#319812yes—
432C2×C3#324112yes—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—42no—
437C22—42no—
438C22—81no—
439C22—81no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22C2×C3×#12111yescharacteristic
449C22C3×#53411yescharacteristic
450C22C3×#53611yescharacteristic
451C22C3×#57612yes—
452C22C3×#57712yes—
453C4—22no—
454C4—42no—
455C4—42no—
456C4—44no—
457C4—82no—
458C4—81no—
459C4—81no—
460C4C3×#53712yes—
461C4C3×#53912yes—
462C3#2976311yescharacteristic, direct product factor, Sylow
463C2—42no—
464C2—161no—
465C2—161no—
466C2C3×#261811yescharacteristic
467C2C3×#304111yescharacteristic
468C2C3×#304311yescharacteristic
469C2C3×#319812yes—
470C2C3×#324112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2619C2
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×#481C2semidirect
2. C2×C3×#480C2semidirect
4. C3×#2316C2semidirect
5. C3×#2085C2semidirect
6. C3×#2067C2semidirect
7. C3×#2065C2semidirect
8. C3×#2041C2semidirect
9. C3×#1875C2semidirect
10. C3×#1875C2semidirect
11. C3×#1798C2semidirect
12. #29763C3direct (N × H)
42. C2×#481C2×C3semidirect
43. C2×#480C2×C3semidirect
45. #2316C2×C3semidirect
46. #2085C2×C3semidirect
47. #2067C2×C3semidirect
48. #2065C2×C3semidirect
49. #2041C2×C3semidirect
50. #1875C2×C3semidirect
51. #1875C2×C3semidirect
52. #1798C2×C3semidirect
74. C2×C3×C4⋊C4D4semidirect
209. C2×C4⋊C4C3×D4semidirect
462. C3#29763direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4812
C2×C3×#4802
C3×#26192
C3×#23162
C3×#20852
C3×#20672
C3×#20652
C3×#20412
C3×#18752
C3×#17982
#297633