Finite Group Databasebeta

← C3×#29647 #10648254

Subgroups — C3×#29647 #10648254

Order 768.

990 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#544C211yesmaximal, characteristic, complemented
2C2×C3×#481C211yesmaximal, characteristic
3C2×C3×#482C211yesmaximal, characteristic
4C3×#2624C211yesmaximal, characteristic
5C3×#2059C212yesmaximal, complemented
6C3×#2058C212yesmaximal, complemented
7C3×#2025C212yesmaximal, complemented
8C3×#2023C212yesmaximal, complemented
9C3×#1825C211yesmaximal, characteristic, complemented
10C3×#1823C211yesmaximal, characteristic, complemented
11C3×#1818C211yesmaximal, characteristic, complemented
12#29647C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#544—24no—
14C3×#544—24no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×#125—21no—
18C2×C3×#125C2212yes—
19C2×C3×C22⋊Q8—22no—
20C2×C3×#121C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C2×C3×#117C2211yescharacteristic
23C22×C3×C4⋊C4C2211yescharacteristic
24C3×#482—24no—
25C3×#481—24no—
26C2×C3×#107C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×#106C2212yes—
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2212yes—
31C2×C3×#102C2211yescharacteristic
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×C3×C22⋊C8C2211yescharacteristic
34C3×#399—21no—
35C3×#397—21no—
36C3×#394—22no—
37C3×#390C2212yes—
38C3×#389C2212yes—
39C3×#380C2211yescharacteristic
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#544C2×C311yescharacteristic, complemented
47C2×#481C2×C311yescharacteristic
48C2×#482C2×C311yescharacteristic
49#2624C2×C311yescharacteristic
50#2059C2×C312yescomplemented
51#2058C2×C312yescomplemented
52#2025C2×C312yescomplemented
53#2023C2×C312yescomplemented
54#1825C2×C311yescharacteristic, complemented
55#1823C2×C311yescharacteristic, complemented
56#1818C2×C311yescharacteristic, complemented
57C22×C3×Q8—22no—
58C22×C3×D4—21no—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4—21no—
61C23×C3×C4C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2311yescharacteristic
65C3×C4⋊Q8—28no—
66C3×C4⋊Q8—24no—
67C3×C4⋊Q8—24no—
68C3×#125—28no—
69C3×#125—24no—
70C3×#125—24no—
71C3×C22⋊Q8—216no—
72C3×#121—22no—
73C3×#121—22no—
74C3×#117—22no—
75C3×#117—22no—
76C3×#117—22no—
77C3×#117D414yes—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4D411yescharacteristic
86C2×C3×C4⋊C4D414yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4D412yes—
98C2×C3×C42—21no—
99C2×C3×C42C2312yes—
100C3×#107—22no—
101C3×#106—24no—
102C3×#106—22no—
103C3×#102—24no—
104C3×#102—22no—
105C3×#102—22no—
106C3×C22⋊C8—22no—
107C3×C22⋊C8—22no—
108C3×#95—21no—
109C3×#95—22no—
110#544—24no—
111#544—24no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8C22×C312yes—
114C2×#125—21no—
115C2×#125C22×C312yes—
116C2×C22⋊Q8—22no—
117C2×#121C22×C311yescharacteristic
118C2×#117C22×C311yescharacteristic
119C2×#117C22×C311yescharacteristic
120C22×C4⋊C4C22×C311yescharacteristic
121#482—24no—
122#481—24no—
123C2×#107C22×C311yescharacteristic
124C2×#106C22×C311yescharacteristic
125C2×#106C22×C312yes—
126C2×#102C22×C311yescharacteristic
127C2×#102C22×C311yescharacteristic
128C2×#102C22×C312yes—
129C2×C22⋊C8C22×C311yescharacteristic
130C2×C22⋊C8C22×C311yescharacteristic
131#399—21no—
132#397—21no—
133#394—22no—
134#390C22×C312yes—
135#389C22×C312yes—
136#380C22×C311yescharacteristic
137#340C22×C312yes—
138#340C22×C312yes—
139#340C22×C312yes—
140#340C22×C312yes—
141#336C22×C312yes—
142#336C22×C312yes—
143C24×C3—21no—
144C24×C3C2×D411yescharacteristic
145C2×C3×Q8—28no—
146C2×C3×Q8—82no—
147C2×C3×Q8—82no—
148C2×C3×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—42no—
153C2×C3×D4—82no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—81no—
163C22×C3×C4—81no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2411yescharacteristic
170C2×C3×C8—22no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C3×C4⋊C4—28no—
178C3×C4⋊C4—28no—
179C3×C4⋊C4—28no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—82no—
191C3×C4⋊C4C2×D414yes—
192C3×C4⋊C4C2×D412yes—
193C3×C4⋊C4C2×D412yes—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—24no—
196C3×C22⋊C4—42no—
197C3×C22⋊C4—82no—
198C3×C22⋊C4—82no—
199C3×C22⋊C4—81no—
200C3×C42—24no—
201C3×C42—22no—
202C3×C42—22no—
203C22×Q8—22no—
204C22×D4—21no—
205C22×D4C23×C311yescharacteristic
206C23×C4—21no—
207C23×C4C23×C311yescharacteristic
208C22×C8C23×C311yescharacteristic
209C22×C8C23×C312yes—
210C22×C8C23×C311yescharacteristic
211C4⋊Q8—28no—
212C4⋊Q8—24no—
213C4⋊Q8—24no—
214#125—28no—
215#125—24no—
216#125—24no—
217C22⋊Q8—216no—
218#121—22no—
219#121—22no—
220#117—22no—
221#117—22no—
222#117—22no—
223#117C3×D414yes—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4C3×D411yescharacteristic
231C2×C4⋊C4C3×D411yescharacteristic
232C2×C4⋊C4C3×D414yes—
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C4⋊C4C23×C312yes—
235C2×C4⋊C4C23×C312yes—
236C2×C4⋊C4C23×C311yescharacteristic
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4C3×D412yes—
244C2×C42—21no—
245C2×C42C23×C312yes—
246#107—22no—
247#106—24no—
248#106—22no—
249#102—24no—
250#102—22no—
251#102—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254#95—21no—
255#95—22no—
256C23×C3—22no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—22no—
260C23×C3—41no—
261C23×C3—42no—
262C23×C3—41no—
263C23×C3—42no—
264C23×C3—41no—
265C23×C3#12012yes—
266C23×C3C22×D411yescharacteristic
267C3×Q8—88no—
268C3×D4—42no—
269C3×D4—41no—
270C3×D4—41no—
271C3×D4—84no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—84no—
296C2×C3×C4#12012yes—
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C42−1+412yes—
300C2×C3×C42−1+412yes—
301C3×C8—42no—
302C3×C8—41no—
303C3×C8—41no—
304C24—21no—
305C24C2×C3×D411yescharacteristic
306C2×Q8—28no—
307C2×Q8—82no—
308C2×Q8—82no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—42no—
314C2×D4—82no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4C2×C3×D412yes—
326C22×C4C2×C3×D412yes—
327C22×C4C2×C3×D411yescharacteristic
328C22×C4C2×C3×D412yes—
329C22×C4C2×C3×D412yes—
330C22×C4C24×C311yescharacteristic
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C4⋊C4—28no—
339C4⋊C4—28no—
340C4⋊C4—28no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—82no—
352C4⋊C4C2×C3×D414yes—
353C4⋊C4C2×C3×D412yes—
354C4⋊C4C2×C3×D412yes—
355C22⋊C4—24no—
356C22⋊C4—24no—
357C22⋊C4—42no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—81no—
361C42—24no—
362C42—22no—
363C42—22no—
364C22×C3—22no—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3—81no—
370C22×C3—82no—
371C22×C3—82no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3C2×#12011yescharacteristic
375C22×C3#53612yes—
376C22×C3#57611yescharacteristic
377C22×C3#57712yes—
378C22×C3#57711yescharacteristic
379C3×C4—22no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—42no—
383C3×C4—44no—
384C3×C4—42no—
385C3×C4—82no—
386C3×C4—81no—
387C3×C4#53611yescharacteristic
388C3×C4#53612yes—
389C3×C4#53611yescharacteristic
390C23—21no—
391C23—21no—
392C23—22no—
393C23—22no—
394C23—41no—
395C23—42no—
396C23—41no—
397C23—41no—
398C23—42no—
399C23C3×#12012yes—
400C23C22×C3×D411yescharacteristic
401Q8—88no—
402D4—42no—
403D4—41no—
404D4—41no—
405D4—84no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—84no—
430C2×C4C3×#12012yes—
431C2×C4C22×C3×D411yescharacteristic
432C2×C4C22×C3×D411yescharacteristic
433C2×C4C3×2−1+412yes—
434C2×C4C3×2−1+412yes—
435C8—42no—
436C8—41no—
437C8—41no—
438C2×C3—42no—
439C2×C3—161no—
440C2×C3#261411yescharacteristic
441C2×C3#301011yescharacteristic
442C2×C3#301211yescharacteristic
443C2×C3#324112yes—
444C2×C3#324412yes—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—42no—
449C22—42no—
450C22—82no—
451C22—82no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22C2×C3×#12011yescharacteristic
456C22C3×#53612yes—
457C22C3×#57611yescharacteristic
458C22C3×#57712yes—
459C22C3×#57711yescharacteristic
460C4—22no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—44no—
465C4—42no—
466C4—82no—
467C4—81no—
468C4C3×#53611yescharacteristic
469C4C3×#53612yes—
470C4C3×#53611yescharacteristic
471C3#2964711yescharacteristic, direct product factor, Sylow
472C2—42no—
473C2—161no—
474C2C3×#261411yescharacteristic
475C2C3×#301011yescharacteristic
476C2C3×#301211yescharacteristic
477C2C3×#324112yes—
478C2C3×#324412yes—
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×#544C2semidirect
5. C3×#2059C2semidirect
6. C3×#2058C2semidirect
7. C3×#2025C2semidirect
8. C3×#2023C2semidirect
9. C3×#1825C2semidirect
10. C3×#1823C2semidirect
11. C3×#1818C2semidirect
12. #29647C3direct (N × H)
46. C2×#544C2×C3semidirect
50. #2059C2×C3semidirect
51. #2058C2×C3semidirect
52. #2025C2×C3semidirect
53. #2023C2×C3semidirect
54. #1825C2×C3semidirect
55. #1823C2×C3semidirect
56. #1818C2×C3semidirect
471. C3#29647direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5442
C2×C3×#4812
C2×C3×#4822
C3×#26242
C3×#20592
C3×#20582
C3×#20252
C3×#20232
C3×#18252
C3×#18232
C3×#18182
#296473