Finite Group Databasebeta

← C3×#29728 #10648335

Subgroups — C3×#29728 #10648335

Order 768.

964 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#451C211yesmaximal, characteristic, complemented
2C2×C3×#448C211yesmaximal, characteristic
3C3×#2620C211yesmaximal, characteristic
4C3×#2310C211yesmaximal, characteristic, complemented
5C3×#2072C212yesmaximal, complemented
6C3×#2056C212yesmaximal, complemented
7C3×#2055C212yesmaximal, complemented
8C3×#2015C212yesmaximal, complemented
9C3×#1875C211yesmaximal, characteristic
10C3×#1873C211yesmaximal, characteristic, complemented
11C3×#1798C211yesmaximal, characteristic, complemented
12#29728C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C2×C3×#124C2211yescharacteristic
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C3×#451—24no—
22C3×#448—24no—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#396—22no—
31C3×#394—22no—
32C3×#391—22no—
33C3×#391C2211yescharacteristic
34C3×#391C2212yes—
35C3×#391C2212yes—
36C3×#390C2211yescharacteristic
37C3×#384C2211yescharacteristic
38C3×#381—22no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2212yes—
42C2×#451C2×C311yescharacteristic, complemented
43C2×#448C2×C311yescharacteristic
44#2620C2×C311yescharacteristic
45#2310C2×C311yescharacteristic, complemented
46#2072C2×C312yescomplemented
47#2056C2×C312yescomplemented
48#2055C2×C312yescomplemented
49#2015C2×C312yescomplemented
50#1875C2×C311yescharacteristic
51#1873C2×C311yescharacteristic, complemented
52#1798C2×C311yescharacteristic, complemented
53C22×C3×Q8—21no—
54C22×C3×Q8—22no—
55C22×C3×Q8—42no—
56C22×C3×Q8—42no—
57C22×C3×Q8D414yes—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×Q8C2312yes—
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C2×C3×Q16—24no—
65C2×C3×Q16—42no—
66C2×C3×Q16—42no—
67C2×C3×SD16—24no—
68C2×C3×SD16—42no—
69C2×C3×SD16—42no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C3×#124—42no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C3×#121—22no—
76C3×#121—22no—
77C3×#117—42no—
78C3×#117—42no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4D412yes—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4D412yes—
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C3×C4⋊C8—42no—
95C3×#103—24no—
96C3×#103—24no—
97C3×#103—42no—
98C3×#102—42no—
99C3×C22⋊C8—22no—
100C3×C22⋊C8—22no—
101C3×#95—24no—
102C3×#95—24no—
103C3×#95—22no—
104C23×Q8C22×C311yescharacteristic
105C22×Q16C22×C312yes—
106C22×SD16C22×C312yes—
107C2×#124C22×C311yescharacteristic
108C2×C22⋊Q8C22×C311yescharacteristic
109C2×#121C22×C311yescharacteristic
110C2×#117C22×C311yescharacteristic
111C2×#117C22×C311yescharacteristic
112#451—24no—
113#448—24no—
114C2×C4⋊C8C22×C312yes—
115C2×#103C22×C312yes—
116C2×#103C22×C312yes—
117C2×#103C22×C312yes—
118C2×#102C22×C312yes—
119C2×C22⋊C8C22×C311yescharacteristic
120C2×C22⋊C8C22×C311yescharacteristic
121#396—22no—
122#394—22no—
123#391—22no—
124#391C22×C312yes—
125#391C22×C312yes—
126#391C22×C311yescharacteristic
127#390C22×C311yescharacteristic
128#384C22×C311yescharacteristic
129#381—22no—
130#340C22×C312yes—
131#340C22×C312yes—
132#336C22×C312yes—
133C24×C3—21no—
134C24×C3C2×D411yescharacteristic
135C2×C3×Q8—24no—
136C2×C3×Q8—22no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8—48no—
146C2×C3×Q8—42no—
147C2×C3×Q8—42no—
148C2×C3×Q8—48no—
149C2×C3×Q8C2×D414yes—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—42no—
155C2×C3×D4—82no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—24no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D412yes—
170C22×C3×C4#4112yes—
171C22×C3×C4C2411yescharacteristic
172C3×Q16—48no—
173C3×SD16—48no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—82no—
189C3×C22⋊C4—42no—
190C3×C22⋊C4—44no—
191C3×C22⋊C4—42no—
192C3×C22⋊C4—42no—
193C3×C22⋊C4—42no—
194C3×C22⋊C4—82no—
195C3×C42—22no—
196C3×C42—22no—
197C3×C42—42no—
198C22×Q8—21no—
199C22×Q8—22no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8C3×D414yes—
203C22×Q8C23×C311yescharacteristic
204C22×Q8C23×C312yes—
205C22×D4—21no—
206C22×D4C23×C311yescharacteristic
207C23×C4—21no—
208C23×C4C23×C311yescharacteristic
209C2×Q16—24no—
210C2×Q16—42no—
211C2×Q16—42no—
212C2×SD16—24no—
213C2×SD16—42no—
214C2×SD16—42no—
215C22×C8C23×C312yes—
216C22×C8C23×C312yes—
217#124—42no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220#121—22no—
221#121—22no—
222#117—42no—
223#117—42no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4C3×D412yes—
227C2×C4⋊C4C23×C312yes—
228C2×C4⋊C4C23×C311yescharacteristic
229C2×C4⋊C4C23×C311yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4C3×D412yes—
236C2×C42—21no—
237C2×C42C23×C311yescharacteristic
238C2×C42C23×C311yescharacteristic
239C4⋊C8—42no—
240#103—24no—
241#103—24no—
242#103—42no—
243#102—42no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246#95—24no—
247#95—24no—
248#95—22no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—22no—
252C23×C3—22no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3#12112yes—
259C23×C3C22×D411yescharacteristic
260C3×Q8—24no—
261C3×Q8—24no—
262C3×Q8—48no—
263C3×Q8—42no—
264C3×Q8—42no—
265C3×Q8—42no—
266C3×Q8—42no—
267C3×D4—42no—
268C3×D4—42no—
269C3×D4—84no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—48no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4#12112yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4H(5,𝔽2)12yes—
298C2×C3×C4H(5,𝔽2)12yes—
299C3×C8—42no—
300C3×C8—42no—
301C24—21no—
302C24C2×C3×D411yescharacteristic
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—48no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—48no—
317C2×Q8C2×C3×D414yes—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—42no—
323C2×D4—82no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4C2×C3×D412yes—
335C22×C4C2×C3×D412yes—
336C22×C4C2×C3×D411yescharacteristic
337C22×C4C2×C3×D412yes—
338C22×C4C3×#4112yes—
339C22×C4C24×C311yescharacteristic
340Q16—48no—
341SD16—48no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—82no—
357C22⋊C4—42no—
358C22⋊C4—44no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—82no—
363C42—22no—
364C42—22no—
365C42—42no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—42no—
370C22×C3—42no—
371C22×C3—81no—
372C22×C3—82no—
373C22×C3—82no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3C2×#12111yescharacteristic
377C22×C3#53411yescharacteristic
378C22×C3#53511yescharacteristic
379C22×C3#57712yes—
380C22×C3#57812yes—
381C3×C4—22no—
382C3×C4—44no—
383C3×C4—42no—
384C3×C4—42no—
385C3×C4—81no—
386C3×C4—82no—
387C3×C4—81no—
388C3×C4—81no—
389C3×C4#53412yes—
390C3×C4#53812yes—
391C23—22no—
392C23—22no—
393C23—21no—
394C23—21no—
395C23—42no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23C3×#12112yes—
401C23C22×C3×D411yescharacteristic
402Q8—24no—
403Q8—24no—
404Q8—48no—
405Q8—42no—
406Q8—42no—
407Q8—42no—
408Q8—42no—
409D4—42no—
410D4—42no—
411D4—84no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—48no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4C3×#12112yes—
437C2×C4C22×C3×D411yescharacteristic
438C2×C4C2×C3×#4111yescharacteristic
439C2×C4C3×H(5,𝔽2)12yes—
440C2×C4C3×H(5,𝔽2)12yes—
441C8—42no—
442C8—42no—
443C2×C3—42no—
444C2×C3—161no—
445C2×C3#261711yescharacteristic
446C2×C3#304311yescharacteristic
447C2×C3#304411yescharacteristic
448C2×C3#320112yes—
449C2×C3#322512yes—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—42no—
454C22—42no—
455C22—81no—
456C22—82no—
457C22—82no—
458C22—81no—
459C22—81no—
460C22C2×C3×#12111yescharacteristic
461C22C3×#53411yescharacteristic
462C22C3×#53511yescharacteristic
463C22C3×#57712yes—
464C22C3×#57812yes—
465C4—22no—
466C4—44no—
467C4—42no—
468C4—42no—
469C4—81no—
470C4—82no—
471C4—81no—
472C4—81no—
473C4C3×#53412yes—
474C4C3×#53812yes—
475C3#2972811yescharacteristic, direct product factor, Sylow
476C2—42no—
477C2—161no—
478C2C3×#261711yescharacteristic
479C2C3×#304311yescharacteristic
480C2C3×#304411yescharacteristic
481C2C3×#320112yes—
482C2C3×#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×#451C2semidirect
4. C3×#2310C2semidirect
5. C3×#2072C2semidirect
6. C3×#2056C2semidirect
7. C3×#2055C2semidirect
8. C3×#2015C2semidirect
10. C3×#1873C2semidirect
11. C3×#1798C2semidirect
12. #29728C3direct (N × H)
42. C2×#451C2×C3semidirect
45. #2310C2×C3semidirect
46. #2072C2×C3semidirect
47. #2056C2×C3semidirect
48. #2055C2×C3semidirect
49. #2015C2×C3semidirect
51. #1873C2×C3semidirect
52. #1798C2×C3semidirect
475. C3#29728direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4512
C2×C3×#4482
C3×#26202
C3×#23102
C3×#20722
C3×#20562
C3×#20552
C3×#20152
C3×#18752
C3×#18732
C3×#17982
#297283