Finite Group Databasebeta

← C3×#24197 #10642804

Subgroups — C3×#24197 #10642804

Order 768.

1116 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×#556C211yesmaximal, characteristic, complemented
2C2×C3×#461C211yesmaximal, characteristic, complemented
3C2×C3×#442C211yesmaximal, characteristic
4C3×#2580C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal, complemented
6C3×#2012C212yesmaximal
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1960C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal, complemented
10C3×#1865C212yesmaximal, complemented
11C3×#1804C211yesmaximal, characteristic, complemented
12#24197C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C22×C3×D8C2211yescharacteristic
15C3×#556—24no—
16C3×#556—24no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#126—22no—
19C2×C3×#124C2212yes—
20C2×C3×C22⋊Q8—21no—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117—21no—
26C3×#461—24no—
27C3×#442—24no—
28C2×C3×#106C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2212yes—
33C2×C3×#102C2211yescharacteristic
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×#556C2×C311yescharacteristic, complemented
47C2×#461C2×C311yescharacteristic, complemented
48C2×#442C2×C311yescharacteristic
49#2580C2×C311yescharacteristic, complemented
50#2042C2×C312yescomplemented
51#2012C2×C312yes—
52#1985C2×C311yescharacteristic, complemented
53#1960C2×C311yescharacteristic, complemented
54#1880C2×C312yescomplemented
55#1865C2×C312yescomplemented
56#1804C2×C311yescharacteristic, complemented
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×D4C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4—21no—
63C2×C3×SD16—24no—
64C2×C3×SD16—24no—
65C2×C3×SD16—42no—
66C2×C3×D8—22no—
67C2×C3×D8—22no—
68C2×C3×D8—22no—
69C22×C3×C8C2312yes—
70C22×C3×C8C2311yescharacteristic
71C22×C3×C8C2311yescharacteristic
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8D414yes—
74C3×#126—216no—
75C3×#124—24no—
76C3×#124—24no—
77C3×C22⋊Q8—28no—
78C3×C4×Q8—24no—
79C3×C4×Q8—24no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C3×C4×D4D414yes—
86C3×#117—28no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—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—21no—
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—22no—
100C2×C3×C22⋊C4—21no—
101C2×C3×C22⋊C4—22no—
102C2×C3×C42—22no—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105C3×#106—22no—
106C3×C4⋊C8—22no—
107C3×#103—24no—
108C3×#103—24no—
109C3×#103—42no—
110C3×#102—22no—
111C3×#102—22no—
112C3×#102—42no—
113C3×#97—22no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—22no—
117C22×SD16C22×C312yes—
118C22×D8C22×C311yescharacteristic
119#556—24no—
120#556—24no—
121C2×C4⋊Q8C22×C311yescharacteristic
122C2×#126—22no—
123C2×#124C22×C312yes—
124C2×C22⋊Q8—21no—
125C2×C4×Q8C22×C312yes—
126C2×C4×Q8C22×C311yescharacteristic
127C2×C4×D4C22×C311yescharacteristic
128C2×C4×D4C22×C311yescharacteristic
129C2×#117—21no—
130#461—24no—
131#442—24no—
132C2×#106C22×C311yescharacteristic
133C2×C4⋊C8C22×C311yescharacteristic
134C2×#103C22×C312yes—
135C2×#103C22×C312yes—
136C2×#103C22×C312yes—
137C2×#102C22×C311yescharacteristic
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—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—22no—
155C2×C3×Q8—22no—
156C2×C3×Q8—22no—
157C2×C3×Q8—22no—
158C2×C3×Q8—42no—
159C2×C3×Q8C2×D414yes—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4—21no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4C2×D412yes—
171C2×C3×D4#4112yes—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—24no—
175C22×C3×C4—22no—
176C22×C3×C4—21no—
177C22×C3×C4—44no—
178C22×C3×C4—42no—
179C22×C3×C4—82no—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D411yescharacteristic
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2411yescharacteristic
185C3×SD16—48no—
186C3×D8—28no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—21no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8#4112yes—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—82no—
207C3×C4⋊C4C2×D412yes—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—28no—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—44no—
212C3×C22⋊C4—42no—
213C3×C22⋊C4—82no—
214C3×C42—28no—
215C3×C42—22no—
216C3×C42—22no—
217C3×C42C2×D412yes—
218C3×C42C2×D412yes—
219C22×Q8C3×D411yescharacteristic
220C22×Q8C23×C312yes—
221C22×D4C23×C311yescharacteristic
222C22×D4C23×C311yescharacteristic
223C23×C4—21no—
224C23×C4—21no—
225C2×SD16—24no—
226C2×SD16—24no—
227C2×SD16—42no—
228C2×D8—22no—
229C2×D8—22no—
230C2×D8—22no—
231C22×C8C23×C312yes—
232C22×C8C23×C311yescharacteristic
233C22×C8C23×C311yescharacteristic
234C4⋊Q8—22no—
235C4⋊Q8C3×D414yes—
236#126—216no—
237#124—24no—
238#124—24no—
239C22⋊Q8—28no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4C3×D414yes—
248#117—28no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4C3×D411yescharacteristic
254C2×C4⋊C4C3×D412yes—
255C2×C4⋊C4C23×C311yescharacteristic
256C2×C4⋊C4C23×C312yes—
257C2×C4⋊C4C23×C311yescharacteristic
258C2×C22⋊C4—22no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C42—22no—
265C2×C42C23×C312yes—
266C2×C42C23×C311yescharacteristic
267#106—22no—
268C4⋊C8—22no—
269#103—24no—
270#103—24no—
271#103—42no—
272#102—22no—
273#102—22no—
274#102—42no—
275#97—22no—
276#95—21no—
277#95—21no—
278#95—22no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—42no—
289C23×C3C22×D411yescharacteristic
290C3×Q8—24no—
291C3×Q8—24no—
292C3×Q8—44no—
293C3×Q8—42no—
294C3×Q8—42no—
295C3×D4—22no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—22no—
299C3×D4—41no—
300C3×D4—41no—
301C3×D4—41no—
302C3×D4—41no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—24no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—48no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—41no—
318C2×C3×C4—42no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—84no—
323C2×C3×C4#12012yes—
324C2×C3×C4#12012yes—
325C2×C3×C4C22×D411yescharacteristic
326C2×C3×C4C22×D411yescharacteristic
327C2×C3×C4C2×#4111yescharacteristic
328C2×C3×C4C2×#4111yescharacteristic
329C2×C3×C4H(5,𝔽2)11yescharacteristic
330C2×C3×C42−1+411yescharacteristic
331C3×C8—22no—
332C3×C8—41no—
333C3×C8—42no—
334C24—21no—
335C24—21no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—42no—
343C2×Q8C2×C3×D414yes—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4C2×C3×D412yes—
355C2×D4C3×#4112yes—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—82no—
364C22×C4C2×C3×D412yes—
365C22×C4C2×C3×D412yes—
366C22×C4C2×C3×D411yescharacteristic
367C22×C4C2×C3×D411yescharacteristic
368C22×C4C24×C311yescharacteristic
369SD16—48no—
370D8—28no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8C3×#4112yes—
380C4⋊C4—28no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—82no—
391C4⋊C4C2×C3×D412yes—
392C22⋊C4—28no—
393C22⋊C4—28no—
394C22⋊C4—24no—
395C22⋊C4—44no—
396C22⋊C4—42no—
397C22⋊C4—82no—
398C42—28no—
399C42—22no—
400C42—22no—
401C42C2×C3×D412yes—
402C42C2×C3×D412yes—
403C22×C3—44no—
404C22×C3—42no—
405C22×C3—42no—
406C22×C3—42no—
407C22×C3—42no—
408C22×C3—82no—
409C22×C3—82no—
410C22×C3—82no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3C2×#12011yescharacteristic
414C22×C3#54811yescharacteristic
415C22×C3#55011yescharacteristic
416C22×C3#57512yes—
417C22×C3#57712yes—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—44no—
421C3×C4—42no—
422C3×C4—42no—
423C3×C4—42no—
424C3×C4—82no—
425C3×C4—81no—
426C3×C4#52511yescharacteristic
427C3×C4#53711yescharacteristic
428C3×C4#54611yescharacteristic
429C3×C4#54711yescharacteristic
430C23—22no—
431C23—22no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23C22×C3×D411yescharacteristic
441Q8—24no—
442Q8—24no—
443Q8—44no—
444Q8—42no—
445Q8—42no—
446D4—22no—
447D4—22no—
448D4—22no—
449D4—22no—
450D4—41no—
451D4—41no—
452D4—41no—
453D4—41no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—22no—
464C2×C4—48no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—84no—
474C2×C4C3×#12012yes—
475C2×C4C3×#12012yes—
476C2×C4C22×C3×D411yescharacteristic
477C2×C4C22×C3×D411yescharacteristic
478C2×C4C2×C3×#4111yescharacteristic
479C2×C4C2×C3×#4111yescharacteristic
480C2×C4C3×H(5,𝔽2)11yescharacteristic
481C2×C4C3×2−1+411yescharacteristic
482C8—22no—
483C8—41no—
484C8—42no—
485C2×C3—82no—
486C2×C3—82no—
487C2×C3#240711yescharacteristic
488C2×C3#300511yescharacteristic
489C2×C3#301011yescharacteristic
490C2×C3#337612yes—
491C2×C3#340112yes—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22—82no—
498C22—82no—
499C22—81no—
500C22—81no—
501C22—82no—
502C22C2×C3×#12011yescharacteristic
503C22C3×#54811yescharacteristic
504C22C3×#55011yescharacteristic
505C22C3×#57512yes—
506C22C3×#57712yes—
507C4—22no—
508C4—22no—
509C4—44no—
510C4—42no—
511C4—42no—
512C4—42no—
513C4—82no—
514C4—81no—
515C4C3×#52511yescharacteristic
516C4C3×#53711yescharacteristic
517C4C3×#54611yescharacteristic
518C4C3×#54711yescharacteristic
519C3#2419711yescharacteristic, direct product factor, Sylow
520C2—82no—
521C2—82no—
522C2C3×#240711yescharacteristic
523C2C3×#300511yescharacteristic
524C2C3×#301011yescharacteristic
525C2C3×#337612yes—
526C2C3×#340112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
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×#556C2semidirect
2. C2×C3×#461C2semidirect
4. C3×#2580C2semidirect
5. C3×#2042C2semidirect
7. C3×#1985C2semidirect
8. C3×#1960C2semidirect
9. C3×#1880C2semidirect
10. C3×#1865C2semidirect
11. C3×#1804C2semidirect
12. #24197C3direct (N × H)
46. C2×#556C2×C3semidirect
47. C2×#461C2×C3semidirect
49. #2580C2×C3semidirect
50. #2042C2×C3semidirect
52. #1985C2×C3semidirect
53. #1960C2×C3semidirect
54. #1880C2×C3semidirect
55. #1865C2×C3semidirect
56. #1804C2×C3semidirect
519. C3#24197direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5562
C2×C3×#4612
C2×C3×#4422
C3×#25802
C3×#20422
C3×#20122
C3×#19852
C3×#19602
C3×#18802
C3×#18652
C3×#18042
#241973