Finite Group Databasebeta

← C3×#24198 #10642805

Subgroups — C3×#24198 #10642805

Order 768.

1180 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#559C211yesmaximal, characteristic, complemented
2C2×C3×#459C211yesmaximal, characteristic
3C2×C3×#440C211yesmaximal, characteristic, complemented
4C3×#2580C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2012C212yesmaximal, complemented
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1961C211yesmaximal, characteristic, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1805C211yesmaximal, characteristic, complemented
12#24198C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×D8C2212yes—
15C3×#559—24no—
16C3×#559—24no—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#124—21no—
19C2×C3×#124C2212yes—
20C2×C3×#123—22no—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2212yes—
25C2×C3×C4×D4C2211yescharacteristic
26C3×#459—24no—
27C3×#440—24no—
28C2×C3×#107C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#102C2212yes—
33C2×C3×#102C2212yes—
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2212yes—
36C2×C3×#97C2211yescharacteristic
37C3×#399—21no—
38C3×#393—21no—
39C3×#390C2212yes—
40C3×#389C2212yes—
41C3×#385—22no—
42C3×#382C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#559C2×C311yescharacteristic, complemented
47C2×#459C2×C311yescharacteristic
48C2×#440C2×C311yescharacteristic, complemented
49#2580C2×C311yescharacteristic, complemented
50#2042C2×C312yescomplemented
51#2012C2×C312yescomplemented
52#1985C2×C311yescharacteristic, complemented
53#1961C2×C311yescharacteristic, complemented
54#1879C2×C312yescomplemented
55#1862C2×C312yescomplemented
56#1805C2×C311yescharacteristic, complemented
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4—21no—
61C22×C3×D4C2312yes—
62C22×C3×D4C2311yescharacteristic
63C23×C3×C4—22no—
64C23×C3×C4—21no—
65C2×C3×SD16—22no—
66C2×C3×SD16—22no—
67C2×C3×SD16—22no—
68C2×C3×D8—24no—
69C2×C3×D8—24no—
70C2×C3×D8—42no—
71C22×C3×C8C2312yes—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×#127—22no—
75C3×#127D414yes—
76C3×#124—28no—
77C3×#124—24no—
78C3×#124—24no—
79C3×#123—216no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8D414yes—
84C3×C4×D4—28no—
85C3×C4×D4—24no—
86C3×C4×D4—24no—
87C3×C4×D4—22no—
88C3×C4×D4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4D412yes—
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—22no—
97C2×C3×C22⋊C4—22no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C42—21no—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105C3×#107—22no—
106C3×C4⋊C8—22no—
107C3×#103—22no—
108C3×#103—42no—
109C3×#102—24no—
110C3×#102—24no—
111C3×#102—22no—
112C3×#102—42no—
113C3×#97—22no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—22no—
117C22×SD16C22×C311yescharacteristic
118C22×D8C22×C312yes—
119#559—24no—
120#559—24no—
121C2×#127C22×C311yescharacteristic
122C2×#124—21no—
123C2×#124C22×C312yes—
124C2×#123—22no—
125C2×C4×Q8C22×C311yescharacteristic
126C2×C4×Q8C22×C311yescharacteristic
127C2×C4×D4—21no—
128C2×C4×D4C22×C311yescharacteristic
129C2×C4×D4C22×C312yes—
130#459—24no—
131#440—24no—
132C2×#107C22×C311yescharacteristic
133C2×C4⋊C8C22×C311yescharacteristic
134C2×#103C22×C311yescharacteristic
135C2×#103C22×C312yes—
136C2×#102C22×C312yes—
137C2×#102C22×C312yes—
138C2×#102C22×C311yescharacteristic
139C2×#102C22×C312yes—
140C2×#97C22×C311yescharacteristic
141#399—21no—
142#393—21no—
143#390C22×C312yes—
144#389C22×C312yes—
145#385—22no—
146#382C22×C311yescharacteristic
147#340C22×C312yes—
148#336C22×C312yes—
149#336C22×C312yes—
150C24×C3—21no—
151C24×C3—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—42no—
159C2×C3×Q8C2×D412yes—
160C2×C3×D4—24no—
161C2×C3×D4—24no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—22no—
165C2×C3×D4—22no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—21no—
172C2×C3×D4—44no—
173C2×C3×D4—82no—
174C2×C3×D4C2×D414yes—
175C2×C3×D4#4112yes—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—28no—
179C22×C3×C4—22no—
180C22×C3×C4—21no—
181C22×C3×C4—42no—
182C22×C3×C4—82no—
183C22×C3×C4—82no—
184C22×C3×C4—82no—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2411yescharacteristic
190C3×SD16—28no—
191C3×D8—48no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—21no—
198C2×C3×C8—22no—
199C2×C3×C8—21no—
200C2×C3×C8#4112yes—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—24no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—22no—
209C3×C4⋊C4—82no—
210C3×C4⋊C4C2×D412yes—
211C3×C22⋊C4—28no—
212C3×C22⋊C4—28no—
213C3×C22⋊C4—28no—
214C3×C22⋊C4—44no—
215C3×C22⋊C4—42no—
216C3×C22⋊C4—82no—
217C3×C42—24no—
218C3×C42—22no—
219C3×C42—22no—
220C3×C42C2×D412yes—
221C3×C42C2×D412yes—
222C22×Q8C3×D411yescharacteristic
223C22×Q8C23×C311yescharacteristic
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C23×C312yes—
227C22×D4C23×C311yescharacteristic
228C23×C4—22no—
229C23×C4—21no—
230C2×SD16—22no—
231C2×SD16—22no—
232C2×SD16—22no—
233C2×D8—24no—
234C2×D8—24no—
235C2×D8—42no—
236C22×C8C23×C312yes—
237C22×C8C23×C311yescharacteristic
238C22×C8C23×C311yescharacteristic
239#127—22no—
240#127C3×D414yes—
241#124—28no—
242#124—24no—
243#124—24no—
244#123—216no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8C3×D414yes—
249C4×D4—28no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—22no—
253C4×D4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4C3×D411yescharacteristic
257C2×C4⋊C4C3×D412yes—
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C4⋊C4C23×C312yes—
260C2×C4⋊C4C23×C311yescharacteristic
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C42—21no—
268C2×C42C23×C312yes—
269C2×C42C23×C311yescharacteristic
270#107—22no—
271C4⋊C8—22no—
272#103—22no—
273#103—42no—
274#102—24no—
275#102—24no—
276#102—22no—
277#102—42no—
278#97—22no—
279#95—21no—
280#95—21no—
281#95—22no—
282C23×C3—24no—
283C23×C3—22no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—41no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3C22×D411yescharacteristic
295C3×Q8—22no—
296C3×Q8—22no—
297C3×Q8—44no—
298C3×Q8—41no—
299C3×Q8—41no—
300C3×D4—22no—
301C3×D4—22no—
302C3×D4—24no—
303C3×D4—24no—
304C3×D4—48no—
305C3×D4—42no—
306C3×D4—41no—
307C3×D4—42no—
308C3×D4—41no—
309C3×D4—84no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—84no—
329C2×C3×C4—88no—
330C2×C3×C4#12012yes—
331C2×C3×C4#12012yes—
332C2×C3×C4C22×D411yescharacteristic
333C2×C3×C4C22×D411yescharacteristic
334C2×C3×C4C2×#4111yescharacteristic
335C2×C3×C4C2×#4111yescharacteristic
336C2×C3×C4H(5,𝔽2)11yescharacteristic
337C2×C3×C42−1+411yescharacteristic
338C3×C8—22no—
339C3×C8—41no—
340C3×C8—42no—
341C24—21no—
342C24—22no—
343C2×Q8—22no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—42no—
350C2×Q8C2×C3×D412yes—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—44no—
364C2×D4—82no—
365C2×D4C2×C3×D414yes—
366C2×D4C3×#4112yes—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—28no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—42no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4C2×C3×D412yes—
377C22×C4C2×C3×D412yes—
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C24×C311yescharacteristic
381SD16—28no—
382D8—48no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—21no—
389C2×C8—22no—
390C2×C8—21no—
391C2×C8C3×#4112yes—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—82no—
401C4⋊C4C2×C3×D412yes—
402C22⋊C4—28no—
403C22⋊C4—28no—
404C22⋊C4—28no—
405C22⋊C4—44no—
406C22⋊C4—42no—
407C22⋊C4—82no—
408C42—24no—
409C42—22no—
410C42—22no—
411C42C2×C3×D412yes—
412C42C2×C3×D412yes—
413C22×C3—44no—
414C22×C3—42no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—44no—
418C22×C3—82no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3C2×#12011yescharacteristic
426C22×C3#54811yescharacteristic
427C22×C3#55011yescharacteristic
428C22×C3#57512yes—
429C22×C3#57612yes—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—82no—
437C3×C4—81no—
438C3×C4#52511yescharacteristic
439C3×C4#53711yescharacteristic
440C3×C4#54611yescharacteristic
441C3×C4#54711yescharacteristic
442C23—24no—
443C23—22no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—41no—
454C23C22×C3×D411yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—44no—
458Q8—41no—
459Q8—41no—
460D4—22no—
461D4—22no—
462D4—24no—
463D4—24no—
464D4—48no—
465D4—42no—
466D4—41no—
467D4—42no—
468D4—41no—
469D4—84no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—84no—
489C2×C4—88no—
490C2×C4C3×#12012yes—
491C2×C4C3×#12012yes—
492C2×C4C22×C3×D411yescharacteristic
493C2×C4C22×C3×D411yescharacteristic
494C2×C4C2×C3×#4111yescharacteristic
495C2×C4C2×C3×#4111yescharacteristic
496C2×C4C3×H(5,𝔽2)11yescharacteristic
497C2×C4C3×2−1+411yescharacteristic
498C8—22no—
499C8—41no—
500C8—42no—
501C2×C3—82no—
502C2×C3—84no—
503C2×C3#240711yescharacteristic
504C2×C3#300511yescharacteristic
505C2×C3#300811yescharacteristic
506C2×C3#337012yes—
507C2×C3#340312yes—
508C22—42no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—81no—
517C22—81no—
518C22—82no—
519C22—82no—
520C22C2×C3×#12011yescharacteristic
521C22C3×#54811yescharacteristic
522C22C3×#55011yescharacteristic
523C22C3×#57512yes—
524C22C3×#57612yes—
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—82no—
532C4—81no—
533C4C3×#52511yescharacteristic
534C4C3×#53711yescharacteristic
535C4C3×#54611yescharacteristic
536C4C3×#54711yescharacteristic
537C3#2419811yescharacteristic, direct product factor, Sylow
538C2—84no—
539C2—82no—
540C2C3×#240711yescharacteristic
541C2C3×#300511yescharacteristic
542C2C3×#300811yescharacteristic
543C2C3×#337012yes—
544C2C3×#340312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#559C2semidirect
3. C2×C3×#440C2semidirect
4. C3×#2580C2semidirect
5. C3×#2042C2semidirect
6. C3×#2012C2semidirect
7. C3×#1985C2semidirect
8. C3×#1961C2semidirect
9. C3×#1879C2semidirect
10. C3×#1862C2semidirect
11. C3×#1805C2semidirect
12. #24198C3direct (N × H)
46. C2×#559C2×C3semidirect
48. C2×#440C2×C3semidirect
49. #2580C2×C3semidirect
50. #2042C2×C3semidirect
51. #2012C2×C3semidirect
52. #1985C2×C3semidirect
53. #1961C2×C3semidirect
54. #1879C2×C3semidirect
55. #1862C2×C3semidirect
56. #1805C2×C3semidirect
537. C3#24198direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5592
C2×C3×#4592
C2×C3×#4402
C3×#25802
C3×#20422
C3×#20122
C3×#19852
C3×#19612
C3×#18792
C3×#18622
C3×#18052
#241983