Finite Group Databasebeta

← C3×#29817 #10648424

Subgroups — C3×#29817 #10648424

Order 768.

866 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#497C211yesmaximal, characteristic, complemented
2C2×C3×#433C211yesmaximal, characteristic, complemented
3C3×#2621C211yesmaximal, characteristic, complemented
4C3×#2620C211yesmaximal, characteristic
5C3×#2056C212yesmaximal, complemented
6C3×#2055C212yesmaximal, complemented
7C3×#2001C212yesmaximal, complemented
8C3×#2000C212yesmaximal
9C3×#1958C211yesmaximal, characteristic, complemented
10C3×#1956C211yesmaximal, characteristic, complemented
11C3×#1836C211yesmaximal, characteristic, complemented
12#29817C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#124C2211yescharacteristic
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×#121C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C3×#497—24no—
22C3×#433—24no—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×#97C2211yescharacteristic
30C2×C3×C4×C8C2211yescharacteristic
31C3×#398—22no—
32C3×#397—22no—
33C3×#396—22no—
34C3×#394—22no—
35C3×#391C2211yescharacteristic
36C3×#391C2212yes—
37C3×#390C2211yescharacteristic
38C3×#384C2211yescharacteristic
39C3×#382C2211yescharacteristic
40C3×#340C2212yes—
41C3×#340C2212yes—
42C2×#497C2×C311yescharacteristic, complemented
43C2×#433C2×C311yescharacteristic, complemented
44#2621C2×C311yescharacteristic, complemented
45#2620C2×C311yescharacteristic
46#2056C2×C312yescomplemented
47#2055C2×C312yescomplemented
48#2001C2×C312yescomplemented
49#2000C2×C312yes—
50#1958C2×C311yescharacteristic, complemented
51#1956C2×C311yescharacteristic, complemented
52#1836C2×C311yescharacteristic, complemented
53C22×C3×Q8C2312yes—
54C22×C3×Q8C2311yescharacteristic
55C22×C3×D4—21no—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C2×C3×Q16—24no—
59C2×C3×Q16—42no—
60C2×C3×Q16—42no—
61C2×C3×SD16—24no—
62C2×C3×SD16—42no—
63C2×C3×SD16—42no—
64C22×C3×C8C2312yes—
65C22×C3×C8C2312yes—
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×#124—22no—
69C3×#124—22no—
70C3×C22⋊Q8—42no—
71C3×C22⋊Q8—42no—
72C3×C22⋊Q8—42no—
73C3×#121—42no—
74C3×#117—22no—
75C3×#117—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
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⋊C4D412yes—
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×C4⋊C8—24no—
93C3×#103—42no—
94C3×#103—42no—
95C3×#103—42no—
96C3×#102—42no—
97C3×C22⋊C8—24no—
98C3×#97D414yescomplemented
99C3×C4×C8—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—24no—
103C22×Q16C22×C312yes—
104C22×SD16C22×C312yes—
105C2×C4⋊Q8C22×C311yescharacteristic
106C2×#124C22×C311yescharacteristic
107C2×C22⋊Q8C22×C311yescharacteristic
108C2×C22⋊Q8C22×C312yes—
109C2×#121C22×C311yescharacteristic
110C2×#117C22×C311yescharacteristic
111#497—24no—
112#433—24no—
113C2×C4⋊C8C22×C312yes—
114C2×#103C22×C312yes—
115C2×#103C22×C312yes—
116C2×#103C22×C312yes—
117C2×#102C22×C312yes—
118C2×C22⋊C8C22×C312yes—
119C2×#97C22×C311yescharacteristic
120C2×C4×C8C22×C311yescharacteristic
121#398—22no—
122#397—22no—
123#396—22no—
124#394—22no—
125#391C22×C312yes—
126#391C22×C311yescharacteristic
127#390C22×C311yescharacteristic
128#384C22×C311yescharacteristic
129#382C22×C311yescharacteristic
130#340C22×C312yes—
131#340C22×C312yes—
132C24×C3—21no—
133C24×C3C2×D411yescharacteristic
134C2×C3×Q8—21no—
135C2×C3×Q8—22no—
136C2×C3×Q8—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—42no—
149C2×C3×D4—82no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2411yescharacteristic
163C3×Q16—48no—
164C3×SD16—48no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8C2×D414yes—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—44no—
178C3×C4⋊C4—44no—
179C3×C4⋊C4—82no—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—42no—
182C3×C22⋊C4—44no—
183C3×C22⋊C4—42no—
184C3×C22⋊C4—44no—
185C3×C22⋊C4—82no—
186C3×C42—21no—
187C3×C42—21no—
188C3×C42C2×D412yes—
189C3×C42#4112yes—
190C22×Q8C23×C311yescharacteristic
191C22×Q8C23×C312yes—
192C22×D4—21no—
193C22×D4C23×C311yescharacteristic
194C23×C4C23×C311yescharacteristic
195C2×Q16—24no—
196C2×Q16—42no—
197C2×Q16—42no—
198C2×SD16—24no—
199C2×SD16—42no—
200C2×SD16—42no—
201C22×C8C23×C312yes—
202C22×C8C23×C312yes—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205#124—22no—
206#124—22no—
207C22⋊Q8—42no—
208C22⋊Q8—42no—
209C22⋊Q8—42no—
210#121—42no—
211#117—22no—
212#117—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4C3×D412yes—
218C2×C4⋊C4C23×C311yescharacteristic
219C2×C4⋊C4C23×C312yes—
220C2×C4⋊C4C23×C311yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4C3×D412yes—
227C2×C42C23×C311yescharacteristic
228C2×C42C23×C311yescharacteristic
229C4⋊C8—24no—
230#103—42no—
231#103—42no—
232#103—42no—
233#102—42no—
234C22⋊C8—24no—
235#97C3×D414yescomplemented
236C4×C8—22no—
237#95—22no—
238#95—22no—
239#95—24no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—22no—
244C23×C3—22no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3C22×D411yescharacteristic
251C3×Q8—42no—
252C3×Q8—42no—
253C3×Q8—42no—
254C3×Q8—42no—
255C3×Q8—42no—
256C3×Q8—42no—
257C3×D4—42no—
258C3×D4—42no—
259C3×D4—84no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4#12114yes—
284C2×C3×C4C22×D411yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4H(5,𝔽2)12yes—
287C2×C3×C4H(5,𝔽2)12yes—
288C3×C8—24no—
289C3×C8—42no—
290C24—21no—
291C24C2×C3×D411yescharacteristic
292C2×Q8—21no—
293C2×Q8—22no—
294C2×Q8—21no—
295C2×Q8—22no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—42no—
307C2×D4—82no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4C2×C3×D412yes—
318C22×C4C2×C3×D411yescharacteristic
319C22×C4C2×C3×D412yes—
320C22×C4C24×C311yescharacteristic
321Q16—48no—
322SD16—48no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8C2×C3×D414yes—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—82no—
338C22⋊C4—24no—
339C22⋊C4—42no—
340C22⋊C4—44no—
341C22⋊C4—42no—
342C22⋊C4—44no—
343C22⋊C4—82no—
344C42—21no—
345C42—21no—
346C42C2×C3×D412yes—
347C42C3×#4112yes—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3—81no—
356C22×C3—82no—
357C22×C3—82no—
358C22×C3C2×#12111yescharacteristic
359C22×C3#53511yescharacteristic
360C22×C3#53511yescharacteristic
361C22×C3#57712yes—
362C22×C3#57812yes—
363C3×C4—24no—
364C3×C4—41no—
365C3×C4—42no—
366C3×C4—81no—
367C3×C4—82no—
368C3×C4—81no—
369C3×C4—81no—
370C3×C4—82no—
371C3×C4#53512yes—
372C3×C4#54012yes—
373C23—22no—
374C23—22no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—42no—
379C23—42no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23C22×C3×D411yescharacteristic
384Q8—42no—
385Q8—42no—
386Q8—42no—
387Q8—42no—
388Q8—42no—
389Q8—42no—
390D4—42no—
391D4—42no—
392D4—84no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4C3×#12114yes—
417C2×C4C22×C3×D411yescharacteristic
418C2×C4C2×C3×#4111yescharacteristic
419C2×C4C3×H(5,𝔽2)12yes—
420C2×C4C3×H(5,𝔽2)12yes—
421C8—24no—
422C8—42no—
423C2×C3—81no—
424C2×C3—161no—
425C2×C3#261911yescharacteristic
426C2×C3#306312yes—
427C2×C3#322211yescharacteristic
428C2×C3#322311yescharacteristic
429C2×C3#322512yes—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—42no—
434C22—42no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—82no—
439C22—82no—
440C22C2×C3×#12111yescharacteristic
441C22C3×#53511yescharacteristic
442C22C3×#53511yescharacteristic
443C22C3×#57712yes—
444C22C3×#57812yes—
445C4—24no—
446C4—41no—
447C4—42no—
448C4—81no—
449C4—82no—
450C4—81no—
451C4—81no—
452C4—82no—
453C4C3×#53512yes—
454C4C3×#54012yes—
455C3#2981711yescharacteristic, direct product factor, Sylow
456C2—81no—
457C2—161no—
458C2C3×#261911yescharacteristic
459C2C3×#306312yes—
460C2C3×#322211yescharacteristic
461C2C3×#322311yescharacteristic
462C2C3×#322512yes—
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×#497C2semidirect
2. C2×C3×#433C2semidirect
3. C3×#2621C2semidirect
5. C3×#2056C2semidirect
6. C3×#2055C2semidirect
7. C3×#2001C2semidirect
9. C3×#1958C2semidirect
10. C3×#1956C2semidirect
11. C3×#1836C2semidirect
12. #29817C3direct (N × H)
42. C2×#497C2×C3semidirect
43. C2×#433C2×C3semidirect
44. #2621C2×C3semidirect
46. #2056C2×C3semidirect
47. #2055C2×C3semidirect
48. #2001C2×C3semidirect
50. #1958C2×C3semidirect
51. #1956C2×C3semidirect
52. #1836C2×C3semidirect
98. C3×#97D4semidirect
235. #97C3×D4semidirect
455. C3#29817direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4972
C2×C3×#4332
C3×#26212
C3×#26202
C3×#20562
C3×#20552
C3×#20012
C3×#20002
C3×#19582
C3×#19562
C3×#18362
#298173