Finite Group Databasebeta

← C3×#24201 #10642808

Subgroups — C3×#24201 #10642808

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#556C211yesmaximal, characteristic
2C2×C3×#462C211yesmaximal, characteristic, complemented
3C2×C3×#440C211yesmaximal, characteristic
4C3×#2581C211yesmaximal, characteristic, complemented
5C3×#2042C212yesmaximal
6C3×#2015C212yesmaximal
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1960C211yesmaximal, characteristic, complemented
9C3×#1878C212yesmaximal, complemented
10C3×#1863C212yesmaximal, complemented
11C3×#1804C211yesmaximal, characteristic
12#24201C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2212yes—
14C22×C3×SD16C2211yescharacteristic
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×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×#117—21no—
26C3×#462—24no—
27C3×#440—24no—
28C2×C3×#107C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2212yes—
31C2×C3×#103C2212yes—
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2212yes—
36C2×C3×#97C2211yescharacteristic
37C3×#400—21no—
38C3×#395—21no—
39C3×#391C2212yes—
40C3×#389C2212yes—
41C3×#387—22no—
42C3×#382C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#556C2×C311yescharacteristic
47C2×#462C2×C311yescharacteristic, complemented
48C2×#440C2×C311yescharacteristic
49#2581C2×C311yescharacteristic, complemented
50#2042C2×C312yes—
51#2015C2×C312yes—
52#1985C2×C311yescharacteristic, complemented
53#1960C2×C311yescharacteristic, complemented
54#1878C2×C312yescomplemented
55#1863C2×C312yescomplemented
56#1804C2×C311yescharacteristic
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C2×C3×Q16—24no—
63C2×C3×Q16—24no—
64C2×C3×Q16—42no—
65C2×C3×SD16—22no—
66C2×C3×SD16—22no—
67C2×C3×SD16—22no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×C4⋊Q8—22no—
72C3×C4⋊Q8D414yes—
73C3×#126—216no—
74C3×#124—24no—
75C3×#124—24no—
76C3×C22⋊Q8—28no—
77C3×C4×Q8—24no—
78C3×C4×Q8—24no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—22no—
84C3×C4×D4D414yes—
85C3×#117—28no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—22no—
90C2×C3×C4⋊C4—21no—
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×C42—22no—
100C2×C3×C42—21no—
101C2×C3×C42C2312yes—
102C2×C3×C42C2311yescharacteristic
103C3×#107—22no—
104C3×C4⋊C8—22no—
105C3×#103—24no—
106C3×#103—24no—
107C3×#103—22no—
108C3×#103—42no—
109C3×#102—22no—
110C3×#102—42no—
111C3×#97—22no—
112C3×#95—21no—
113C3×#95—21no—
114C3×#95—22no—
115C3×#95—22no—
116C3×#95—22no—
117C22×Q16C22×C312yes—
118C22×SD16C22×C311yescharacteristic
119#556—24no—
120#556—24no—
121C2×C4⋊Q8C22×C311yescharacteristic
122C2×#126—22no—
123C2×#124C22×C312yes—
124C2×C22⋊Q8—21no—
125C2×C4×Q8C22×C311yescharacteristic
126C2×C4×Q8C22×C312yes—
127C2×C4×Q8C22×C311yescharacteristic
128C2×C4×D4C22×C311yescharacteristic
129C2×#117—21no—
130#462—24no—
131#440—24no—
132C2×#107C22×C311yescharacteristic
133C2×C4⋊C8C22×C311yescharacteristic
134C2×#103C22×C312yes—
135C2×#103C22×C312yes—
136C2×#103C22×C311yescharacteristic
137C2×#103C22×C312yes—
138C2×#102C22×C311yescharacteristic
139C2×#102C22×C312yes—
140C2×#97C22×C311yescharacteristic
141#400—21no—
142#395—21no—
143#391C22×C312yes—
144#389C22×C312yes—
145#387—22no—
146#382C22×C311yescharacteristic
147#340C22×C312yes—
148#336C22×C312yes—
149#336C22×C312yes—
150C24×C3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—22no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—22no—
157C2×C3×Q8—21no—
158C2×C3×Q8—22no—
159C2×C3×Q8—22no—
160C2×C3×Q8—21no—
161C2×C3×Q8—22no—
162C2×C3×Q8—42no—
163C2×C3×Q8C2×D414yes—
164C2×C3×Q8#4112yes—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4C2×D412yes—
171C22×C3×C4—21no—
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×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D411yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C3×Q16—48no—
184C3×SD16—28no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8#4112yes—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—24no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—44no—
205C3×C4⋊C4—42no—
206C3×C4⋊C4C2×D412yes—
207C3×C22⋊C4—28no—
208C3×C22⋊C4—28no—
209C3×C22⋊C4—24no—
210C3×C42—28no—
211C3×C42—22no—
212C3×C42—22no—
213C3×C42—42no—
214C3×C42C2×D412yes—
215C3×C42C2×D412yes—
216C22×Q8C3×D411yescharacteristic
217C22×Q8C23×C312yes—
218C22×Q8C23×C311yescharacteristic
219C22×D4C23×C311yescharacteristic
220C23×C4—21no—
221C2×Q16—24no—
222C2×Q16—24no—
223C2×Q16—42no—
224C2×SD16—22no—
225C2×SD16—22no—
226C2×SD16—22no—
227C22×C8C23×C312yes—
228C22×C8C23×C311yescharacteristic
229C22×C8C23×C311yescharacteristic
230C4⋊Q8—22no—
231C4⋊Q8C3×D414yes—
232#126—216no—
233#124—24no—
234#124—24no—
235C22⋊Q8—28no—
236C4×Q8—24no—
237C4×Q8—24no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×D4—22no—
243C4×D4C3×D414yes—
244#117—28no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C3×D411yescharacteristic
251C2×C4⋊C4C3×D412yes—
252C2×C4⋊C4C23×C311yescharacteristic
253C2×C4⋊C4C23×C312yes—
254C2×C4⋊C4C23×C311yescharacteristic
255C2×C22⋊C4—22no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—22no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42C23×C312yes—
261C2×C42C23×C311yescharacteristic
262#107—22no—
263C4⋊C8—22no—
264#103—24no—
265#103—24no—
266#103—22no—
267#103—42no—
268#102—22no—
269#102—42no—
270#97—22no—
271#95—21no—
272#95—21no—
273#95—22no—
274#95—22no—
275#95—22no—
276C23×C3—22no—
277C23×C3—41no—
278C23×C3—41no—
279C23×C3—42no—
280C23×C3—42no—
281C23×C3C22×D411yescharacteristic
282C3×Q8—22no—
283C3×Q8—22no—
284C3×Q8—24no—
285C3×Q8—24no—
286C3×Q8—44no—
287C3×Q8—42no—
288C3×Q8—42no—
289C3×Q8—41no—
290C3×Q8—41no—
291C3×D4—22no—
292C3×D4—22no—
293C3×D4—41no—
294C3×D4—41no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—24no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—24no—
305C2×C3×C4—48no—
306C2×C3×C4—41no—
307C2×C3×C4—42no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4#12012yes—
317C2×C3×C4#12012yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C42−1+411yescharacteristic
324C3×C8—22no—
325C3×C8—41no—
326C3×C8—42no—
327C24—21no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—21no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—21no—
338C2×Q8—22no—
339C2×Q8—42no—
340C2×Q8C2×C3×D414yes—
341C2×Q8C3×#4112yes—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4C2×C3×D412yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—44no—
355C22×C4C2×C3×D412yes—
356C22×C4C2×C3×D412yes—
357C22×C4C2×C3×D411yescharacteristic
358C22×C4C2×C3×D411yescharacteristic
359C22×C4C24×C311yescharacteristic
360Q16—48no—
361SD16—28no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8C3×#4112yes—
371C4⋊C4—28no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4C2×C3×D412yes—
384C22⋊C4—28no—
385C22⋊C4—28no—
386C22⋊C4—24no—
387C42—28no—
388C42—22no—
389C42—22no—
390C42—42no—
391C42C2×C3×D412yes—
392C42C2×C3×D412yes—
393C22×C3—42no—
394C22×C3—42no—
395C22×C3—42no—
396C22×C3—82no—
397C22×C3—82no—
398C22×C3C2×#12011yescharacteristic
399C22×C3#54811yescharacteristic
400C22×C3#55011yescharacteristic
401C22×C3#57512yes—
402C22×C3#57812yes—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—44no—
406C3×C4—42no—
407C3×C4—42no—
408C3×C4—42no—
409C3×C4—42no—
410C3×C4—82no—
411C3×C4—81no—
412C3×C4#52511yescharacteristic
413C3×C4#53711yescharacteristic
414C3×C4#54611yescharacteristic
415C3×C4#54711yescharacteristic
416C23—22no—
417C23—42no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23C22×C3×D411yescharacteristic
422Q8—22no—
423Q8—22no—
424Q8—24no—
425Q8—24no—
426Q8—44no—
427Q8—42no—
428Q8—42no—
429Q8—41no—
430Q8—41no—
431D4—22no—
432D4—22no—
433D4—41no—
434D4—41no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—48no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4C3×#12012yes—
457C2×C4C3×#12012yes—
458C2×C4C22×C3×D411yescharacteristic
459C2×C4C22×C3×D411yescharacteristic
460C2×C4C2×C3×#4111yescharacteristic
461C2×C4C2×C3×#4111yescharacteristic
462C2×C4C3×H(5,𝔽2)11yescharacteristic
463C2×C4C3×2−1+411yescharacteristic
464C8—22no—
465C8—41no—
466C8—42no—
467C2×C3—82no—
468C2×C3#240711yescharacteristic
469C2×C3#300511yescharacteristic
470C2×C3#301111yescharacteristic
471C2×C3#337712yes—
472C2×C3#340212yes—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—82no—
477C22—82no—
478C22C2×C3×#12011yescharacteristic
479C22C3×#54811yescharacteristic
480C22C3×#55011yescharacteristic
481C22C3×#57512yes—
482C22C3×#57812yes—
483C4—22no—
484C4—22no—
485C4—44no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—82no—
491C4—81no—
492C4C3×#52511yescharacteristic
493C4C3×#53711yescharacteristic
494C4C3×#54611yescharacteristic
495C4C3×#54711yescharacteristic
496C3#2420111yescharacteristic, direct product factor, Sylow
497C2—82no—
498C2C3×#240711yescharacteristic
499C2C3×#300511yescharacteristic
500C2C3×#301111yescharacteristic
501C2C3×#337712yes—
502C2C3×#340212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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
2. C2×C3×#462C2semidirect
4. C3×#2581C2semidirect
7. C3×#1985C2semidirect
8. C3×#1960C2semidirect
9. C3×#1878C2semidirect
10. C3×#1863C2semidirect
12. #24201C3direct (N × H)
47. C2×#462C2×C3semidirect
49. #2581C2×C3semidirect
52. #1985C2×C3semidirect
53. #1960C2×C3semidirect
54. #1878C2×C3semidirect
55. #1863C2×C3semidirect
496. C3#24201direct (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×#4622
C2×C3×#4402
C3×#25812
C3×#20422
C3×#20152
C3×#19852
C3×#19602
C3×#18782
C3×#18632
C3×#18042
#242013