Finite Group Databasebeta

← C3×#24679 #10643286

Subgroups — C3×#24679 #10643286

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#477C211yesmaximal, characteristic, complemented
2C2×C3×C4×SD16C211yesmaximal, characteristic
3C3×#2563C211yesmaximal, characteristic
4C3×#2355C211yesmaximal, characteristic, complemented
5C3×#2074C212yesmaximal, complemented
6C3×#2055C212yesmaximal
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1949C212yesmaximal, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1925C212yesmaximal
11C3×#1802C211yesmaximal, characteristic
12#24679C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C3×#477—24no—
18C3×C4×SD16—24no—
19C2×C3×#107C2212yes—
20C2×C3×#106C2211yescharacteristic
21C2×C3×C4⋊C8C2212yes—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2212yes—
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2212yes—
27C2×C3×C4×C8C2211yescharacteristic
28C3×#393—22no—
29C3×#391C2212yes—
30C3×#390C2212yes—
31C3×#387—22no—
32C3×#384C2211yescharacteristic
33C3×#382—21no—
34C3×#382C2212yes—
35C3×#380—21no—
36C3×#380—21no—
37C3×C4×C4⋊C4—21no—
38C3×C4×C4⋊C4C2212yes—
39C3×C4×C4⋊C4C2211yescharacteristic
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#477C2×C311yescharacteristic, complemented
45C2×C4×SD16C2×C311yescharacteristic
46#2563C2×C311yescharacteristic
47#2355C2×C311yescharacteristic, complemented
48#2074C2×C312yescomplemented
49#2055C2×C312yes—
50#1990C2×C311yescharacteristic, complemented
51#1949C2×C312yescomplemented
52#1939C2×C311yescharacteristic, complemented
53#1925C2×C312yes—
54#1802C2×C311yescharacteristic
55C22×C3×Q8—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C2×C3×SD16—22no—
60C2×C3×SD16—22no—
61C2×C3×SD16—22no—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2311yescharacteristic
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8D414yes—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4D411yescharacteristic
76C2×C3×C4⋊C4D411yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42—21no—
85C2×C3×C42—22no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C3×#107—24no—
89C3×#106—22no—
90C3×C4⋊C8—24no—
91C3×C4⋊C8—22no—
92C3×#103—22no—
93C3×#103—24no—
94C3×#102—24no—
95C3×#102—22no—
96C3×C4×C8—22no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95D412yes—
103C22×SD16C22×C311yescharacteristic
104C2×C4⋊Q8C22×C311yescharacteristic
105C2×C4×Q8C22×C311yescharacteristic
106C2×C4×D4C22×C311yescharacteristic
107#477—24no—
108C4×SD16—24no—
109C2×#107C22×C312yes—
110C2×#106C22×C311yescharacteristic
111C2×C4⋊C8C22×C311yescharacteristic
112C2×C4⋊C8C22×C312yes—
113C2×#103C22×C312yes—
114C2×#103C22×C311yescharacteristic
115C2×#102C22×C312yes—
116C2×#102C22×C311yescharacteristic
117C2×C4×C8C22×C311yescharacteristic
118#393—22no—
119#391C22×C312yes—
120#390C22×C312yes—
121#387—22no—
122#384C22×C311yescharacteristic
123#382—21no—
124#382C22×C312yes—
125#380—21no—
126#380—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C22×C312yes—
129C4×C4⋊C4C22×C311yescharacteristic
130#340C22×C312yes—
131#340C22×C312yes—
132#340C22×C312yes—
133#336C22×C312yes—
134C24×C3—21no—
135C2×C3×Q8—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—42no—
141C2×C3×Q8—81no—
142C2×C3×Q8—81no—
143C2×C3×Q8C2×D412yes—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4#4112yes—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4—42no—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C3×SD16—28no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8#4112yes—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4—81no—
186C3×C4⋊C4C2×D412yes—
187C3×C22⋊C4—42no—
188C3×C22⋊C4—82no—
189C3×C22⋊C4—82no—
190C3×C42—24no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—82no—
194C3×C42C2×D412yes—
195C3×C42#4112yes—
196C22×Q8—21no—
197C22×Q8C23×C311yescharacteristic
198C22×D4C23×C311yescharacteristic
199C23×C4—21no—
200C2×SD16—22no—
201C2×SD16—22no—
202C2×SD16—22no—
203C22×C8C23×C311yescharacteristic
204C22×C8C23×C312yes—
205C22×C8C23×C311yescharacteristic
206C4⋊Q8—22no—
207C4⋊Q8—22no—
208C4×Q8—22no—
209C4×Q8C3×D414yes—
210C4×D4—22no—
211C4×D4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C3×D411yescharacteristic
217C2×C4⋊C4C3×D411yescharacteristic
218C2×C4⋊C4C23×C311yescharacteristic
219C2×C4⋊C4C23×C312yes—
220C2×C4⋊C4C23×C312yes—
221C2×C4⋊C4C23×C311yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42C23×C312yes—
228C2×C42C23×C311yescharacteristic
229#107—24no—
230#106—22no—
231C4⋊C8—24no—
232C4⋊C8—22no—
233#103—22no—
234#103—24no—
235#102—24no—
236#102—22no—
237C4×C8—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95C3×D412yes—
244C23×C3—22no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3C22×D411yescharacteristic
252C3×Q8—22no—
253C3×Q8—22no—
254C3×Q8—41no—
255C3×Q8—41no—
256C3×Q8—84no—
257C3×D4—22no—
258C3×D4—22no—
259C3×D4—41no—
260C3×D4—41no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
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—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—84no—
285C2×C3×C4#12112yes—
286C2×C3×C4#12112yes—
287C2×C3×C4#13512yes—
288C2×C3×C4#13712yes—
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×C3×C42−1+411yescharacteristic
294C2×C3×C42−1+411yescharacteristic
295C3×C8—22no—
296C3×C8—42no—
297C3×C8—41no—
298C24—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—42no—
305C2×Q8—81no—
306C2×Q8—81no—
307C2×Q8C2×C3×D412yes—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4C3×#4112yes—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4C2×C3×D412yes—
323C22×C4C2×C3×D412yes—
324C22×C4C2×C3×D411yescharacteristic
325C22×C4C2×C3×D411yescharacteristic
326C22×C4C24×C311yescharacteristic
327SD16—28no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8C3×#4112yes—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—81no—
350C4⋊C4C2×C3×D412yes—
351C22⋊C4—42no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C42—24no—
355C42—22no—
356C42—22no—
357C42—82no—
358C42C2×C3×D412yes—
359C42C3×#4112yes—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—81no—
364C22×C3—81no—
365C22×C3—81no—
366C22×C3—81no—
367C22×C3C2×#12111yescharacteristic
368C22×C3#54711yescharacteristic
369C22×C3#54811yescharacteristic
370C22×C3C2×#13511yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C22×C3#57611yescharacteristic
373C22×C3#57711yescharacteristic
374C3×C4—22no—
375C3×C4—42no—
376C3×C4—42no—
377C3×C4—42no—
378C3×C4—42no—
379C3×C4—82no—
380C3×C4—81no—
381C3×C4—82no—
382C3×C4#46414yes—
383C3×C4#52511yescharacteristic
384C3×C4#53611yescharacteristic
385C3×C4#55211yescharacteristic
386C3×C4#55611yescharacteristic
387C23—22no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23C22×C3×D411yescharacteristic
395Q8—22no—
396Q8—22no—
397Q8—41no—
398Q8—41no—
399Q8—84no—
400D4—22no—
401D4—22no—
402D4—41no—
403D4—41no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—84no—
428C2×C4C3×#12112yes—
429C2×C4C3×#12112yes—
430C2×C4C3×#13512yes—
431C2×C4C3×#13712yes—
432C2×C4C22×C3×D411yescharacteristic
433C2×C4C2×C3×#4111yescharacteristic
434C2×C4C2×C3×#4111yescharacteristic
435C2×C4C2×C3×#4111yescharacteristic
436C2×C4C3×2−1+411yescharacteristic
437C2×C4C3×2−1+411yescharacteristic
438C8—22no—
439C8—42no—
440C8—41no—
441C2×C3—82no—
442C2×C3#241511yescharacteristic
443C2×C3C2×#46411yescharacteristic
444C2×C3#304511yescharacteristic
445C2×C3#334011yescharacteristic
446C2×C3#335711yescharacteristic
447C2×C3#336111yescharacteristic
448C2×C3#337211yescharacteristic
449C22—42no—
450C22—42no—
451C22—42no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22C2×C3×#12111yescharacteristic
457C22C3×#54711yescharacteristic
458C22C3×#54811yescharacteristic
459C22C2×C3×#13511yescharacteristic
460C22C2×C3×#13711yescharacteristic
461C22C3×#57611yescharacteristic
462C22C3×#57711yescharacteristic
463C4—22no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—42no—
468C4—81no—
469C4—82no—
470C4—82no—
471C4C3×#46414yes—
472C4C3×#52511yescharacteristic
473C4C3×#53611yescharacteristic
474C4C3×#55211yescharacteristic
475C4C3×#55611yescharacteristic
476C3#2467911yescharacteristic, direct product factor, Sylow
477C2—82no—
478C2C3×#241511yescharacteristic
479C2C2×C3×#46411yescharacteristic
480C2C3×#304511yescharacteristic
481C2C3×#334011yescharacteristic
482C2C3×#335711yescharacteristic
483C2C3×#336111yescharacteristic
484C2C3×#337211yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C3×#477C2semidirect
4. C3×#2355C2semidirect
5. C3×#2074C2semidirect
7. C3×#1990C2semidirect
8. C3×#1949C2semidirect
9. C3×#1939C2semidirect
12. #24679C3direct (N × H)
44. C2×#477C2×C3semidirect
47. #2355C2×C3semidirect
48. #2074C2×C3semidirect
50. #1990C2×C3semidirect
51. #1949C2×C3semidirect
52. #1939C2×C3semidirect
476. C3#24679direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4772
C2×C3×C4×SD162
C3×#25632
C3×#23552
C3×#20742
C3×#20552
C3×#19902
C3×#19492
C3×#19392
C3×#19252
C3×#18022
#246793