Finite Group Databasebeta

← C3×#24693 #10643300

Subgroups — C3×#24693 #10643300

Order 768.

828 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#474C211yesmaximal, characteristic, complemented
2C2×C3×#442C211yesmaximal, characteristic
3C3×#2563C211yesmaximal, characteristic, complemented
4C3×#2357C211yesmaximal, characteristic, complemented
5C3×#2073C212yesmaximal, complemented
6C3×#2054C212yesmaximal
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1949C212yesmaximal, complemented
9C3×#1940C211yesmaximal, characteristic, complemented
10C3×#1925C212yesmaximal, complemented
11C3×#1804C211yesmaximal, characteristic, complemented
12#24693C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C3×#474—24no—
18C3×#442—24no—
19C2×C3×#107C2212yes—
20C2×C3×#106C2211yescharacteristic
21C2×C3×C4⋊C8C2212yes—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C3×#393—21no—
29C3×#393—21no—
30C3×#390C2212yes—
31C3×#390C2212yes—
32C3×#388—22no—
33C3×#382C2211yescharacteristic
34C3×#382C2212yes—
35C3×#381—21no—
36C3×#380—22no—
37C3×C4×C4⋊C4C2212yes—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×C4×C22⋊C4—21no—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#474C2×C311yescharacteristic, complemented
45C2×#442C2×C311yescharacteristic
46#2563C2×C311yescharacteristic, complemented
47#2357C2×C311yescharacteristic, complemented
48#2073C2×C312yescomplemented
49#2054C2×C312yes—
50#1993C2×C311yescharacteristic, complemented
51#1949C2×C312yescomplemented
52#1940C2×C311yescharacteristic, complemented
53#1925C2×C312yescomplemented
54#1804C2×C311yescharacteristic, complemented
55C22×C3×Q8—21no—
56C22×C3×D4C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C2×C3×D8—22no—
61C2×C3×D8—22no—
62C2×C3×D8—22no—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2312yes—
65C22×C3×C8C2311yescharacteristic
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4D414yes—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4D412yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42—22no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C3×#107—24no—
88C3×#106—22no—
89C3×C4⋊C8—24no—
90C3×C4⋊C8—22no—
91C3×#102—22no—
92C3×#102—24no—
93C3×#102—24no—
94C3×#102—22no—
95C3×#97—22no—
96C3×#95—21no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—21no—
101C3×#95D411yescharacteristic
102C3×#95D411yescharacteristic
103C22×D8C22×C311yescharacteristic
104C2×C4⋊Q8C22×C311yescharacteristic
105C2×C4×D4C22×C311yescharacteristic
106C2×C4×D4C22×C311yescharacteristic
107#474—24no—
108#442—24no—
109C2×#107C22×C312yes—
110C2×#106C22×C311yescharacteristic
111C2×C4⋊C8C22×C312yes—
112C2×C4⋊C8C22×C311yescharacteristic
113C2×#102C22×C312yes—
114C2×#102C22×C312yes—
115C2×#102C22×C311yescharacteristic
116C2×#102C22×C311yescharacteristic
117C2×#97C22×C311yescharacteristic
118#393—21no—
119#393—21no—
120#390C22×C312yes—
121#390C22×C312yes—
122#388—22no—
123#382C22×C312yes—
124#382C22×C311yescharacteristic
125#381—21no—
126#380—22no—
127C4×C4⋊C4C22×C312yes—
128C4×C4⋊C4C22×C311yescharacteristic
129C4×C22⋊C4—21no—
130#340C22×C312yes—
131#340C22×C312yes—
132#340C22×C312yes—
133#336C22×C312yes—
134C24×C3—21no—
135C24×C3—21no—
136C2×C3×Q8—42no—
137C2×C3×Q8—82no—
138C2×C3×D4—21no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4C2×D412yes—
149C2×C3×D4#4112yes—
150C22×C3×C4—21no—
151C22×C3×C4—24no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—44no—
156C22×C3×C4—42no—
157C22×C3×C4—82no—
158C22×C3×C4C2×D412yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C3×D8—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—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4C2×D412yes—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—44no—
185C3×C22⋊C4—42no—
186C3×C22⋊C4—82no—
187C3×C22⋊C4—82no—
188C3×C22⋊C4—82no—
189C3×C42—22no—
190C3×C42—22no—
191C3×C42—82no—
192C3×C42C2×D412yes—
193C3×C42#4112yes—
194C22×Q8—21no—
195C22×D4C23×C311yescharacteristic
196C22×D4C23×C311yescharacteristic
197C23×C4—21no—
198C23×C4—21no—
199C2×D8—22no—
200C2×D8—22no—
201C2×D8—22no—
202C22×C8C23×C311yescharacteristic
203C22×C8C23×C312yes—
204C22×C8C23×C311yescharacteristic
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4C3×D414yes—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C3×D412yes—
213C2×C4⋊C4C23×C311yescharacteristic
214C2×C4⋊C4C23×C312yes—
215C2×C4⋊C4C23×C312yes—
216C2×C4⋊C4C23×C311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C42—22no—
224C2×C42C23×C312yes—
225C2×C42C23×C311yescharacteristic
226#107—24no—
227#106—22no—
228C4⋊C8—24no—
229C4⋊C8—22no—
230#102—22no—
231#102—24no—
232#102—24no—
233#102—22no—
234#97—22no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95C3×D411yescharacteristic
241#95C3×D411yescharacteristic
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—42no—
251C23×C3—42no—
252C23×C3C22×D411yescharacteristic
253C3×Q8—84no—
254C3×D4—22no—
255C3×D4—22no—
256C3×D4—22no—
257C3×D4—22no—
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—41no—
261C3×D4—41no—
262C2×C3×C4—22no—
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—42no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—84no—
281C2×C3×C4—84no—
282C2×C3×C4#12112yes—
283C2×C3×C4#12112yes—
284C2×C3×C4C22×D411yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4C2×#4111yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C42−1+411yescharacteristic
289C2×C3×C42−1+411yescharacteristic
290C3×C8—22no—
291C3×C8—42no—
292C3×C8—41no—
293C24—21no—
294C24—21no—
295C2×Q8—42no—
296C2×Q8—82no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4C2×C3×D412yes—
308C2×D4C3×#4112yes—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—44no—
315C22×C4—42no—
316C22×C4—82no—
317C22×C4C2×C3×D412yes—
318C22×C4C2×C3×D412yes—
319C22×C4C2×C3×D411yescharacteristic
320C22×C4C2×C3×D411yescharacteristic
321C22×C4C24×C311yescharacteristic
322D8—28no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8C3×#4112yes—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4C2×C3×D412yes—
342C22⋊C4—24no—
343C22⋊C4—44no—
344C22⋊C4—42no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C42—22no—
349C42—22no—
350C42—82no—
351C42C2×C3×D412yes—
352C42C3×#4112yes—
353C22×C3—44no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3—42no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—82no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3C2×#12111yescharacteristic
364C22×C3#54711yescharacteristic
365C22×C3#54811yescharacteristic
366C22×C3#57512yes—
367C22×C3#57612yes—
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—42no—
371C3×C4—42no—
372C3×C4—42no—
373C3×C4—82no—
374C3×C4—81no—
375C3×C4—82no—
376C3×C4#52511yescharacteristic
377C3×C4#53611yescharacteristic
378C3×C4#55211yescharacteristic
379C3×C4#55611yescharacteristic
380C23—22no—
381C23—22no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23C22×C3×D411yescharacteristic
391Q8—84no—
392D4—22no—
393D4—22no—
394D4—22no—
395D4—22no—
396D4—41no—
397D4—41no—
398D4—41no—
399D4—41no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4C3×#12112yes—
421C2×C4C3×#12112yes—
422C2×C4C22×C3×D411yescharacteristic
423C2×C4C2×C3×#4111yescharacteristic
424C2×C4C2×C3×#4111yescharacteristic
425C2×C4C2×C3×#4111yescharacteristic
426C2×C4C3×2−1+411yescharacteristic
427C2×C4C3×2−1+411yescharacteristic
428C8—22no—
429C8—42no—
430C8—41no—
431C2×C3—82no—
432C2×C3—82no—
433C2×C3#241511yescharacteristic
434C2×C3#304011yescharacteristic
435C2×C3#304111yescharacteristic
436C2×C3#334712yes—
437C2×C3#337012yes—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—44no—
443C22—81no—
444C22—81no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22C2×C3×#12111yescharacteristic
449C22C3×#54711yescharacteristic
450C22C3×#54811yescharacteristic
451C22C3×#57512yes—
452C22C3×#57612yes—
453C4—22no—
454C4—22no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—81no—
459C4—82no—
460C4—82no—
461C4C3×#52511yescharacteristic
462C4C3×#53611yescharacteristic
463C4C3×#55211yescharacteristic
464C4C3×#55611yescharacteristic
465C3#2469311yescharacteristic, direct product factor, Sylow
466C2—82no—
467C2—82no—
468C2C3×#241511yescharacteristic
469C2C3×#304011yescharacteristic
470C2C3×#304111yescharacteristic
471C2C3×#334712yes—
472C2C3×#337012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
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×#474C2semidirect
3. C3×#2563C2semidirect
4. C3×#2357C2semidirect
5. C3×#2073C2semidirect
7. C3×#1993C2semidirect
8. C3×#1949C2semidirect
9. C3×#1940C2semidirect
10. C3×#1925C2semidirect
11. C3×#1804C2semidirect
12. #24693C3direct (N × H)
44. C2×#474C2×C3semidirect
46. #2563C2×C3semidirect
47. #2357C2×C3semidirect
48. #2073C2×C3semidirect
50. #1993C2×C3semidirect
51. #1949C2×C3semidirect
52. #1940C2×C3semidirect
53. #1925C2×C3semidirect
54. #1804C2×C3semidirect
465. C3#24693direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4742
C2×C3×#4422
C3×#25632
C3×#23572
C3×#20732
C3×#20542
C3×#19932
C3×#19492
C3×#19402
C3×#19252
C3×#18042
#246933