Finite Group Databasebeta

← C3×#24676 #10643283

Subgroups — C3×#24676 #10643283

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#474C211yesmaximal, characteristic, complemented
2C2×C3×C4×D8C211yesmaximal, characteristic
3C3×#2542C211yesmaximal, characteristic, complemented
4C3×#2354C211yesmaximal, characteristic, complemented
5C3×#2071C212yesmaximal, complemented
6C3×#2054C212yesmaximal
7C3×#1990C211yesmaximal, characteristic, complemented
8C3×#1947C212yesmaximal, complemented
9C3×#1939C211yesmaximal, characteristic, complemented
10C3×#1925C212yesmaximal, complemented
11C3×#1802C211yesmaximal, characteristic, complemented
12#24676C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117—21no—
20C3×#474—24no—
21C3×C4×D8—24no—
22C2×C3×#107C2211yescharacteristic
23C2×C3×#107C2212yes—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#102C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×C4×C8C2211yescharacteristic
31C3×#399—22no—
32C3×#390C2212yes—
33C3×#390C2212yes—
34C3×#388—22no—
35C3×#384C2211yescharacteristic
36C3×#384C2212yes—
37C3×#380—21no—
38C3×C4×C4⋊C4C2212yes—
39C3×C4×C4⋊C4C2211yescharacteristic
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#474C2×C311yescharacteristic, complemented
45C2×C4×D8C2×C311yescharacteristic
46#2542C2×C311yescharacteristic, complemented
47#2354C2×C311yescharacteristic, complemented
48#2071C2×C312yescomplemented
49#2054C2×C312yes—
50#1990C2×C311yescharacteristic, complemented
51#1947C2×C312yescomplemented
52#1939C2×C311yescharacteristic, complemented
53#1925C2×C312yescomplemented
54#1802C2×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×C22⋊Q8—44no—
69C3×C22⋊Q8—44no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C3×C4×D4D414yes—
74C3×#117—44no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4D411yescharacteristic
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—22no—
92C2×C3×C42—22no—
93C2×C3×C42C2312yes—
94C2×C3×C42C2311yescharacteristic
95C3×#107—22no—
96C3×#107—24no—
97C3×C4⋊C8—24no—
98C3×C4⋊C8—22no—
99C3×#102—22no—
100C3×#102—24no—
101C3×#102—24no—
102C3×#102—22no—
103C3×C4×C8—22no—
104C3×#95—22no—
105C3×#95D412yes—
106C22×D8C22×C311yescharacteristic
107C2×C4⋊Q8C22×C311yescharacteristic
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C4×D4C22×C311yescharacteristic
111C2×C4×D4C22×C311yescharacteristic
112C2×#117—21no—
113#474—24no—
114C4×D8—24no—
115C2×#107C22×C311yescharacteristic
116C2×#107C22×C312yes—
117C2×C4⋊C8C22×C312yes—
118C2×C4⋊C8C22×C311yescharacteristic
119C2×#102C22×C312yes—
120C2×#102C22×C312yes—
121C2×#102C22×C311yescharacteristic
122C2×#102C22×C311yescharacteristic
123C2×C4×C8C22×C311yescharacteristic
124#399—22no—
125#390C22×C312yes—
126#390C22×C312yes—
127#388—22no—
128#384C22×C311yescharacteristic
129#384C22×C312yes—
130#380—21no—
131C4×C4⋊C4C22×C312yes—
132C4×C4⋊C4C22×C311yescharacteristic
133#340C22×C312yes—
134#340C22×C312yes—
135#340C22×C312yes—
136#336C22×C312yes—
137C24×C3—21no—
138C24×C3—21no—
139C2×C3×Q8—42no—
140C2×C3×Q8—42no—
141C2×C3×Q8—42no—
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×D4—21no—
149C2×C3×D4—21no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4C2×D412yes—
153C2×C3×D4#4112yes—
154C22×C3×C4—21no—
155C22×C3×C4—24no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—42no—
160C22×C3×C4—42no—
161C22×C3×C4—42no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2411yescharacteristic
169C3×D8—28no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8#4112yes—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—44no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4—44no—
191C3×C4⋊C4—81no—
192C3×C4⋊C4D814yescomplemented
193C3×C4⋊C4C2×D412yes—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—44no—
196C3×C22⋊C4—42no—
197C3×C22⋊C4—44no—
198C3×C22⋊C4—44no—
199C3×C22⋊C4—82no—
200C3×C42—22no—
201C3×C42—22no—
202C3×C42—44no—
203C3×C42C2×D412yes—
204C3×C42#4112yes—
205C22×Q8—21no—
206C22×D4C23×C311yescharacteristic
207C22×D4C23×C311yescharacteristic
208C23×C4—21no—
209C23×C4—21no—
210C2×D8—22no—
211C2×D8—22no—
212C2×D8—22no—
213C22×C8C23×C311yescharacteristic
214C22×C8C23×C312yes—
215C22×C8C23×C311yescharacteristic
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C22⋊Q8—44no—
219C22⋊Q8—44no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4C3×D414yes—
224#117—44no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C3×D411yescharacteristic
231C2×C4⋊C4C3×D411yescharacteristic
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C4⋊C4C23×C312yes—
234C2×C4⋊C4C23×C312yes—
235C2×C4⋊C4C23×C311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C42—22no—
243C2×C42C23×C312yes—
244C2×C42C23×C311yescharacteristic
245#107—22no—
246#107—24no—
247C4⋊C8—24no—
248C4⋊C8—22no—
249#102—22no—
250#102—24no—
251#102—24no—
252#102—22no—
253C4×C8—22no—
254#95—22no—
255#95C3×D412yes—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—41no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3C22×D411yescharacteristic
271C3×Q8—84no—
272C3×D4—22no—
273C3×D4—22no—
274C3×D4—22no—
275C3×D4—22no—
276C3×D4—41no—
277C3×D4—41no—
278C3×D4—41no—
279C3×D4—41no—
280C2×C3×C4—22no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—48no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—42no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—84no—
301C2×C3×C4#12112yes—
302C2×C3×C4#12112yes—
303C2×C3×C4C2×D814yes—
304C2×C3×C4C2×D812yes—
305C2×C3×C4AGL(1,ℤ/8)12yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C42−1+411yescharacteristic
311C2×C3×C42−1+411yescharacteristic
312C3×C8—22no—
313C3×C8—42no—
314C3×C8—41no—
315C24—21no—
316C24—21no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4C2×C3×D412yes—
331C2×D4C3×#4112yes—
332C22×C4—21no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4C2×C3×D412yes—
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C2×C3×D412yes—
345C22×C4C2×C3×D411yescharacteristic
346C22×C4C24×C311yescharacteristic
347D8—28no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8C3×#4112yes—
357C4⋊C4—22no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—44no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—44no—
369C4⋊C4—81no—
370C4⋊C4C3×D814yescomplemented
371C4⋊C4C2×C3×D412yes—
372C22⋊C4—24no—
373C22⋊C4—44no—
374C22⋊C4—42no—
375C22⋊C4—44no—
376C22⋊C4—44no—
377C22⋊C4—82no—
378C42—22no—
379C42—22no—
380C42—44no—
381C42C2×C3×D412yes—
382C42C3×#4112yes—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—81no—
396C22×C3—81no—
397C22×C3C2×#12111yescharacteristic
398C22×C3#54711yescharacteristic
399C22×C3#54811yescharacteristic
400C22×C3C22×D811yescharacteristic
401C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
402C22×C3#57711yescharacteristic
403C22×C3#57811yescharacteristic
404C3×C4—24no—
405C3×C4—22no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—82no—
409C3×C4—81no—
410C3×C4—82no—
411C3×C4#45914yes—
412C3×C4#52511yescharacteristic
413C3×C4#53611yescharacteristic
414C3×C4#55211yescharacteristic
415C3×C4#55611yescharacteristic
416C23—22no—
417C23—22no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23C22×C3×D411yescharacteristic
431Q8—84no—
432D4—22no—
433D4—22no—
434D4—22no—
435D4—22no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—84no—
461C2×C4C3×#12112yes—
462C2×C4C3×#12112yes—
463C2×C4C2×C3×D814yes—
464C2×C4C2×C3×D812yes—
465C2×C4C3×AGL(1,ℤ/8)12yes—
466C2×C4C22×C3×D411yescharacteristic
467C2×C4C2×C3×#4111yescharacteristic
468C2×C4C2×C3×#4111yescharacteristic
469C2×C4C2×C3×#4111yescharacteristic
470C2×C4C3×2−1+411yescharacteristic
471C2×C4C3×2−1+411yescharacteristic
472C8—22no—
473C8—42no—
474C8—41no—
475C2×C3—82no—
476C2×C3—82no—
477C2×C3#241511yescharacteristic
478C2×C3C2×#45911yescharacteristic
479C2×C3#304611yescharacteristic
480C2×C3#333411yescharacteristic
481C2×C3#335411yescharacteristic
482C2×C3#335811yescharacteristic
483C2×C3#336711yescharacteristic
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—81no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22C2×C3×#12111yescharacteristic
499C22C3×#54711yescharacteristic
500C22C3×#54811yescharacteristic
501C22C22×C3×D811yescharacteristic
502C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
503C22C3×#57711yescharacteristic
504C22C3×#57811yescharacteristic
505C4—22no—
506C4—24no—
507C4—42no—
508C4—42no—
509C4—81no—
510C4—82no—
511C4—82no—
512C4C3×#45914yes—
513C4C3×#52511yescharacteristic
514C4C3×#53611yescharacteristic
515C4C3×#55211yescharacteristic
516C4C3×#55611yescharacteristic
517C3#2467611yescharacteristic, direct product factor, Sylow
518C2—82no—
519C2—82no—
520C2C3×#241511yescharacteristic
521C2C2×C3×#45911yescharacteristic
522C2C3×#304611yescharacteristic
523C2C3×#333411yescharacteristic
524C2C3×#335411yescharacteristic
525C2C3×#335811yescharacteristic
526C2C3×#336711yescharacteristic
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×#2542C2semidirect
4. C3×#2354C2semidirect
5. C3×#2071C2semidirect
7. C3×#1990C2semidirect
8. C3×#1947C2semidirect
9. C3×#1939C2semidirect
10. C3×#1925C2semidirect
11. C3×#1802C2semidirect
12. #24676C3direct (N × H)
44. C2×#474C2×C3semidirect
46. #2542C2×C3semidirect
47. #2354C2×C3semidirect
48. #2071C2×C3semidirect
50. #1990C2×C3semidirect
51. #1947C2×C3semidirect
52. #1939C2×C3semidirect
53. #1925C2×C3semidirect
54. #1802C2×C3semidirect
192. C3×C4⋊C4D8semidirect
370. C4⋊C4C3×D8semidirect
517. C3#24676direct (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×C4×D82
C3×#25422
C3×#23542
C3×#20712
C3×#20542
C3×#19902
C3×#19472
C3×#19392
C3×#19252
C3×#18022
#246763