Finite Group Databasebeta

← C3×#29816 #10648423

Subgroups — C3×#29816 #10648423

Order 768.

966 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#496C211yesmaximal, characteristic, complemented
2C2×C3×#433C211yesmaximal, characteristic, complemented
3C3×#2620C211yesmaximal, characteristic, complemented
4C3×#2619C211yesmaximal, characteristic, complemented
5C3×#2055C212yesmaximal, complemented
6C3×#2054C212yesmaximal, complemented
7C3×#2000C212yesmaximal, complemented
8C3×#1999C212yesmaximal, complemented
9C3×#1958C211yesmaximal, characteristic, complemented
10C3×#1957C211yesmaximal, characteristic, complemented
11C3×#1836C211yesmaximal, characteristic, complemented
12#29816C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2212yes—
19C2×C3×#121C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C3×#496—24no—
22C3×#433—24no—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×#97C2211yescharacteristic
30C2×C3×C4×C8C2211yescharacteristic
31C3×#396—22no—
32C3×#394—22no—
33C3×#394—22no—
34C3×#392—22no—
35C3×#391C2211yescharacteristic
36C3×#390C2212yes—
37C3×#390C2211yescharacteristic
38C3×#384C2211yescharacteristic
39C3×#382C2211yescharacteristic
40C3×#340C2212yes—
41C3×#340C2212yes—
42C2×#496C2×C311yescharacteristic, complemented
43C2×#433C2×C311yescharacteristic, complemented
44#2620C2×C311yescharacteristic, complemented
45#2619C2×C311yescharacteristic, complemented
46#2055C2×C312yescomplemented
47#2054C2×C312yescomplemented
48#2000C2×C312yescomplemented
49#1999C2×C312yescomplemented
50#1958C2×C311yescharacteristic, complemented
51#1957C2×C311yescharacteristic, complemented
52#1836C2×C311yescharacteristic, complemented
53C22×C3×Q8C2311yescharacteristic
54C22×C3×D4—22no—
55C22×C3×D4—21no—
56C22×C3×D4—22no—
57C22×C3×D4C2311yescharacteristic
58C22×C3×D4C2312yes—
59C23×C3×C4C2311yescharacteristic
60C2×C3×SD16—24no—
61C2×C3×SD16—42no—
62C2×C3×SD16—42no—
63C2×C3×D8—24no—
64C2×C3×D8—42no—
65C2×C3×D8—42no—
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C3×#127—22no—
69C3×#127—22no—
70C3×#124—22no—
71C3×#124—22no—
72C3×C22⋊Q8—42no—
73C3×#121—42no—
74C3×#121—42no—
75C3×#121—42no—
76C3×#117—22no—
77C3×#117—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4D412yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4D412yes—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C3×C4⋊C8—24no—
95C3×#103—42no—
96C3×#102—42no—
97C3×#102—42no—
98C3×#102—42no—
99C3×C22⋊C8—24no—
100C3×#97D414yescomplemented
101C3×C4×C8—22no—
102C3×#95—22no—
103C3×#95—22no—
104C22×SD16C22×C312yes—
105C22×D8C22×C312yes—
106C2×#127C22×C311yescharacteristic
107C2×#124C22×C311yescharacteristic
108C2×C22⋊Q8C22×C311yescharacteristic
109C2×#121C22×C312yes—
110C2×#121C22×C311yescharacteristic
111C2×#117C22×C311yescharacteristic
112#496—24no—
113#433—24no—
114C2×C4⋊C8C22×C312yes—
115C2×#103C22×C312yes—
116C2×#102C22×C312yes—
117C2×#102C22×C312yes—
118C2×#102C22×C312yes—
119C2×C22⋊C8C22×C312yes—
120C2×#97C22×C311yescharacteristic
121C2×C4×C8C22×C311yescharacteristic
122#396—22no—
123#394—22no—
124#394—22no—
125#392—22no—
126#391C22×C311yescharacteristic
127#390C22×C311yescharacteristic
128#390C22×C312yes—
129#384C22×C311yescharacteristic
130#382C22×C311yescharacteristic
131#340C22×C312yes—
132#340C22×C312yes—
133C24×C3—21no—
134C24×C3—22no—
135C24×C3C2×D411yescharacteristic
136C2×C3×Q8—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×D4—21no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—22no—
150C2×C3×D4—42no—
151C2×C3×D4—44no—
152C2×C3×D4—44no—
153C2×C3×D4—82no—
154C2×C3×D4—82no—
155C2×C3×D4—82no—
156C2×C3×D4—82no—
157C2×C3×D4—82no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2411yescharacteristic
170C3×SD16—48no—
171C3×D8—48no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8C2×D414yes—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—82no—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—44no—
187C3×C22⋊C4—42no—
188C3×C22⋊C4—42no—
189C3×C22⋊C4—44no—
190C3×C22⋊C4—82no—
191C3×C22⋊C4—82no—
192C3×C22⋊C4—82no—
193C3×C42—21no—
194C3×C42—21no—
195C3×C42C2×D412yes—
196C3×C42#4112yes—
197C22×Q8C23×C311yescharacteristic
198C22×D4—22no—
199C22×D4—21no—
200C22×D4—22no—
201C22×D4C23×C311yescharacteristic
202C22×D4C23×C312yes—
203C23×C4C23×C311yescharacteristic
204C2×SD16—24no—
205C2×SD16—42no—
206C2×SD16—42no—
207C2×D8—24no—
208C2×D8—42no—
209C2×D8—42no—
210C22×C8C23×C312yes—
211C22×C8C23×C312yes—
212#127—22no—
213#127—22no—
214#124—22no—
215#124—22no—
216C22⋊Q8—42no—
217#121—42no—
218#121—42no—
219#121—42no—
220#117—22no—
221#117—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4C3×D412yes—
225C2×C4⋊C4C23×C312yes—
226C2×C4⋊C4C23×C311yescharacteristic
227C2×C4⋊C4C23×C311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4C3×D412yes—
236C2×C42C23×C311yescharacteristic
237C2×C42C23×C311yescharacteristic
238C4⋊C8—24no—
239#103—42no—
240#102—42no—
241#102—42no—
242#102—42no—
243C22⋊C8—24no—
244#97C3×D414yescomplemented
245C4×C8—22no—
246#95—22no—
247#95—22no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—41no—
254C23×C3—42no—
255C23×C3—41no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—41no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3C22×D411yescharacteristic
266C3×Q8—42no—
267C3×Q8—42no—
268C3×D4—42no—
269C3×D4—42no—
270C3×D4—42no—
271C3×D4—42no—
272C3×D4—42no—
273C3×D4—42no—
274C3×D4—88no—
275C3×D4—84no—
276C3×D4—88no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4#12114yes—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4H(5,𝔽2)12yes—
301C2×C3×C4H(5,𝔽2)12yes—
302C3×C8—24no—
303C3×C8—42no—
304C24—21no—
305C24—22no—
306C24C2×C3×D411yescharacteristic
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×D4—21no—
312C2×D4—22no—
313C2×D4—21no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—42no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—82no—
328C2×D4—82no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4C2×C3×D412yes—
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×D412yes—
340C22×C4C24×C311yescharacteristic
341SD16—48no—
342D8—48no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8C2×C3×D414yes—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—82no—
356C22⋊C4—24no—
357C22⋊C4—44no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—44no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C42—21no—
365C42—21no—
366C42C2×C3×D412yes—
367C42C3×#4112yes—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—81no—
374C22×C3—82no—
375C22×C3—81no—
376C22×C3—82no—
377C22×C3—81no—
378C22×C3—82no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3—82no—
382C22×C3—82no—
383C22×C3—82no—
384C22×C3—82no—
385C22×C3C2×#12111yescharacteristic
386C22×C3#53511yescharacteristic
387C22×C3#53511yescharacteristic
388C22×C3#57612yes—
389C22×C3#57712yes—
390C3×C4—24no—
391C3×C4—41no—
392C3×C4—42no—
393C3×C4—81no—
394C3×C4—81no—
395C3×C4—81no—
396C3×C4—82no—
397C3×C4#53512yes—
398C3×C4#54012yes—
399C23—22no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—41no—
411C23—42no—
412C23—41no—
413C23—42no—
414C23—42no—
415C23—41no—
416C23C22×C3×D411yescharacteristic
417Q8—42no—
418Q8—42no—
419D4—42no—
420D4—42no—
421D4—42no—
422D4—42no—
423D4—42no—
424D4—42no—
425D4—88no—
426D4—84no—
427D4—88no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4C3×#12114yes—
449C2×C4C22×C3×D411yescharacteristic
450C2×C4C2×C3×#4111yescharacteristic
451C2×C4C3×H(5,𝔽2)12yes—
452C2×C4C3×H(5,𝔽2)12yes—
453C8—24no—
454C8—42no—
455C2×C3—81no—
456C2×C3—161no—
457C2×C3—162no—
458C2×C3#261911yescharacteristic
459C2×C3#306212yes—
460C2×C3#322011yescharacteristic
461C2×C3#322111yescharacteristic
462C2×C3#322412yes—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—42no—
467C22—42no—
468C22—81no—
469C22—82no—
470C22—81no—
471C22—82no—
472C22—82no—
473C22—81no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22C2×C3×#12111yescharacteristic
481C22C3×#53511yescharacteristic
482C22C3×#53511yescharacteristic
483C22C3×#57612yes—
484C22C3×#57712yes—
485C4—24no—
486C4—41no—
487C4—42no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—82no—
492C4C3×#53512yes—
493C4C3×#54012yes—
494C3#2981611yescharacteristic, direct product factor, Sylow
495C2—81no—
496C2—161no—
497C2—162no—
498C2C3×#261911yescharacteristic
499C2C3×#306212yes—
500C2C3×#322011yescharacteristic
501C2C3×#322111yescharacteristic
502C2C3×#322412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
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×#496C2semidirect
2. C2×C3×#433C2semidirect
3. C3×#2620C2semidirect
4. C3×#2619C2semidirect
5. C3×#2055C2semidirect
6. C3×#2054C2semidirect
7. C3×#2000C2semidirect
8. C3×#1999C2semidirect
9. C3×#1958C2semidirect
10. C3×#1957C2semidirect
11. C3×#1836C2semidirect
12. #29816C3direct (N × H)
42. C2×#496C2×C3semidirect
43. C2×#433C2×C3semidirect
44. #2620C2×C3semidirect
45. #2619C2×C3semidirect
46. #2055C2×C3semidirect
47. #2054C2×C3semidirect
48. #2000C2×C3semidirect
49. #1999C2×C3semidirect
50. #1958C2×C3semidirect
51. #1957C2×C3semidirect
52. #1836C2×C3semidirect
100. C3×#97D4semidirect
244. #97C3×D4semidirect
494. C3#29816direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4962
C2×C3×#4332
C3×#26202
C3×#26192
C3×#20552
C3×#20542
C3×#20002
C3×#19992
C3×#19582
C3×#19572
C3×#18362
#298163