Finite Group Databasebeta

← C3×#29780 #10648387

Subgroups — C3×#29780 #10648387

Order 768.

796 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#465C211yesmaximal, characteristic
2C2×C3×#424C211yesmaximal, characteristic, complemented
3C3×#2625C211yesmaximal, characteristic, complemented
4C3×#2624C211yesmaximal, characteristic
5C3×#2088C212yesmaximal, complemented
6C3×#2068C212yesmaximal, complemented
7C3×#2067C212yesmaximal, complemented
8C3×#2055C212yesmaximal, complemented
9C3×#1895C211yesmaximal, characteristic
10C3×#1894C211yesmaximal, characteristic, complemented
11C3×#1808C211yesmaximal, characteristic, complemented
12#29780C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C23×C3×C8C2211yescharacteristic
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×C3×#121C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C3×#465—24no—
20C2×C3×#107C2212yes—
21C2×C3×#106C2211yescharacteristic
22C3×#424—24no—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2212yes—
29C3×#400—21no—
30C3×#399—21no—
31C3×#398—21no—
32C3×#397—21no—
33C3×#397—22no—
34C3×#394—22no—
35C3×#391C2212yes—
36C3×#390C2212yes—
37C3×#389C2212yes—
38C3×#389C2212yes—
39C3×#380C2211yescharacteristic
40C3×#380C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C2×#465C2×C311yescharacteristic
45C2×#424C2×C311yescharacteristic, complemented
46#2625C2×C311yescharacteristic, complemented
47#2624C2×C311yescharacteristic
48#2088C2×C312yescomplemented
49#2068C2×C312yescomplemented
50#2067C2×C312yescomplemented
51#2055C2×C312yescomplemented
52#1895C2×C311yescharacteristic
53#1894C2×C311yescharacteristic, complemented
54#1808C2×C311yescharacteristic, complemented
55C22×C3×Q8C2311yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C2×C3×SD16—22no—
60C2×C3×SD16—42no—
61C2×C3×M16—22no—
62C2×C3×M16—42no—
63C22×C3×C8—42no—
64C22×C3×C8D414yes—
65C22×C3×C8C2312yes—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8—22no—
70C3×#121—22no—
71C3×#121—22no—
72C3×#117—22no—
73C3×#117—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4D411yescharacteristic
78C2×C3×C4⋊C4D411yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4D412yes—
90C2×C3×C42C2312yes—
91C3×#107—42no—
92C3×#106—22no—
93C3×C4⋊C8—24no—
94C3×C4⋊C8—24no—
95C3×#103—22no—
96C3×#103—42no—
97C3×#102—22no—
98C3×#102—42no—
99C3×#95—21no—
100C3×#95—21no—
101C3×#95—22no—
102C3×#95—22no—
103C3×#95—22no—
104C3×#95—22no—
105C22×SD16C22×C311yescharacteristic
106C22×M16C22×C311yescharacteristic
107C23×C8C22×C311yescharacteristic
108C2×C22⋊Q8C22×C311yescharacteristic
109C2×#121C22×C311yescharacteristic
110C2×#117C22×C311yescharacteristic
111#465—24no—
112C2×#107C22×C312yes—
113C2×#106C22×C311yescharacteristic
114#424—24no—
115C2×C4⋊C8C22×C312yes—
116C2×C4⋊C8C22×C312yes—
117C2×#103C22×C311yescharacteristic
118C2×#103C22×C312yes—
119C2×#102C22×C311yescharacteristic
120C2×#102C22×C312yes—
121#400—21no—
122#399—21no—
123#398—21no—
124#397—21no—
125#397—22no—
126#394—22no—
127#391C22×C312yes—
128#390C22×C312yes—
129#389C22×C312yes—
130#389C22×C312yes—
131#380C22×C311yescharacteristic
132#380C22×C311yescharacteristic
133#340C22×C312yes—
134#340C22×C312yes—
135#340C22×C312yes—
136C24×C3—21no—
137C24×C3C2×D411yescharacteristic
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—42no—
147C2×C3×D4—82no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D412yes—
159C22×C3×C4#4112yes—
160C22×C3×C4C2411yescharacteristic
161C3×SD16—44no—
162C3×M16—44no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—44no—
169C2×C3×C8C2×D412yes—
170C2×C3×C8C2×D412yes—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—82no—
183C3×C4⋊C4—82no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—42no—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—82no—
188C3×C22⋊C4—82no—
189C3×C22⋊C4—81no—
190C3×C22⋊C4—81no—
191C3×C42—22no—
192C3×C42—22no—
193C22×Q8C23×C311yescharacteristic
194C22×D4—21no—
195C22×D4C23×C311yescharacteristic
196C23×C4C23×C311yescharacteristic
197C2×SD16—22no—
198C2×SD16—42no—
199C2×M16—22no—
200C2×M16—42no—
201C22×C8—42no—
202C22×C8C3×D414yes—
203C22×C8C23×C312yes—
204C22×C8C23×C311yescharacteristic
205C22×C8C23×C311yescharacteristic
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208#121—22no—
209#121—22no—
210#117—22no—
211#117—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4C3×D411yescharacteristic
216C2×C4⋊C4C3×D411yescharacteristic
217C2×C4⋊C4C23×C311yescharacteristic
218C2×C4⋊C4C23×C311yescharacteristic
219C2×C4⋊C4C23×C312yes—
220C2×C4⋊C4C23×C312yes—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4C3×D412yes—
228C2×C42C23×C312yes—
229#107—42no—
230#106—22no—
231C4⋊C8—24no—
232C4⋊C8—24no—
233#103—22no—
234#103—42no—
235#102—22no—
236#102—42no—
237#95—21no—
238#95—21no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243C23×C3—22no—
244C23×C3—21no—
245C23×C3—22no—
246C23×C3—21no—
247C23×C3—41no—
248C23×C3—42no—
249C23×C3—41no—
250C23×C3—42no—
251C23×C3—41no—
252C23×C3#12112yes—
253C23×C3C22×D411yescharacteristic
254C3×Q8—41no—
255C3×Q8—41no—
256C3×Q8—42no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—42no—
260C3×D4—84no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—24no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4#12112yes—
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4H(5,𝔽2)12yes—
288C2×C3×C42−1+412yes—
289C3×C8—22no—
290C3×C8—22no—
291C3×C8—42no—
292C24—21no—
293C24C2×C3×D411yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—42no—
303C2×D4—82no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4C2×C3×D412yes—
312C22×C4C2×C3×D412yes—
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D412yes—
315C22×C4C3×#4112yes—
316C22×C4C24×C311yescharacteristic
317SD16—44no—
318M16—44no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—44no—
325C2×C8C2×C3×D412yes—
326C2×C8C2×C3×D412yes—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—42no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C22⋊C4—24no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—81no—
346C22⋊C4—81no—
347C42—22no—
348C42—22no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—42no—
353C22×C3—42no—
354C22×C3—81no—
355C22×C3—82no—
356C22×C3—82no—
357C22×C3—81no—
358C22×C3—81no—
359C22×C3C2×#12111yescharacteristic
360C22×C3#53712yes—
361C22×C3#57611yescharacteristic
362C22×C3#57712yes—
363C22×C3#57811yescharacteristic
364C3×C4—22no—
365C3×C4—42no—
366C3×C4—42no—
367C3×C4—81no—
368C3×C4—82no—
369C3×C4—82no—
370C3×C4—81no—
371C3×C4—81no—
372C3×C4#53411yescharacteristic
373C3×C4#53611yescharacteristic
374C3×C4#53912yes—
375C23—21no—
376C23—22no—
377C23—21no—
378C23—22no—
379C23—41no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23C3×#12112yes—
385C23C22×C3×D411yescharacteristic
386Q8—41no—
387Q8—41no—
388Q8—42no—
389D4—41no—
390D4—41no—
391D4—42no—
392D4—84no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4C3×#12112yes—
417C2×C4C22×C3×D411yescharacteristic
418C2×C4C2×C3×#4111yescharacteristic
419C2×C4C3×H(5,𝔽2)12yes—
420C2×C4C3×2−1+412yes—
421C8—22no—
422C8—22no—
423C8—42no—
424C2×C3—42no—
425C2×C3—161no—
426C2×C3#261811yescharacteristic
427C2×C3#305711yescharacteristic
428C2×C3#306111yescharacteristic
429C2×C3#325712yes—
430C2×C3#325812yes—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—42no—
435C22—42no—
436C22—81no—
437C22—82no—
438C22—82no—
439C22—81no—
440C22—81no—
441C22C2×C3×#12111yescharacteristic
442C22C3×#53712yes—
443C22C3×#57611yescharacteristic
444C22C3×#57712yes—
445C22C3×#57811yescharacteristic
446C4—22no—
447C4—42no—
448C4—42no—
449C4—81no—
450C4—82no—
451C4—82no—
452C4—81no—
453C4—81no—
454C4C3×#53411yescharacteristic
455C4C3×#53611yescharacteristic
456C4C3×#53912yes—
457C3#2978011yescharacteristic, direct product factor, Sylow
458C2—42no—
459C2—161no—
460C2C3×#261811yescharacteristic
461C2C3×#305711yescharacteristic
462C2C3×#306111yescharacteristic
463C2C3×#325712yes—
464C2C3×#325812yes—
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
2. C2×C3×#424C2semidirect
3. C3×#2625C2semidirect
5. C3×#2088C2semidirect
6. C3×#2068C2semidirect
7. C3×#2067C2semidirect
8. C3×#2055C2semidirect
10. C3×#1894C2semidirect
11. C3×#1808C2semidirect
12. #29780C3direct (N × H)
45. C2×#424C2×C3semidirect
46. #2625C2×C3semidirect
48. #2088C2×C3semidirect
49. #2068C2×C3semidirect
50. #2067C2×C3semidirect
51. #2055C2×C3semidirect
53. #1894C2×C3semidirect
54. #1808C2×C3semidirect
457. C3#29780direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4652
C2×C3×#4242
C3×#26252
C3×#26242
C3×#20882
C3×#20682
C3×#20672
C3×#20552
C3×#18952
C3×#18942
C3×#18082
#297803