Finite Group Databasebeta

← C3×#16686 #10635293

Subgroups — C3×#16686 #10635293

Order 768.

1158 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#395C211yesmaximal, characteristic
2C2×C3×#382C211yesmaximal, characteristic
3C3×#1446C212yesmaximal
4C3×#1443C212yesmaximal
5C3×#1442C212yesmaximal
6C3×#1442C212yesmaximal
7C3×#1441C212yesmaximal
8C3×#1441C211yesmaximal, characteristic
9C3×#1434C212yesmaximal
10#16686C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C3×#395—24no—
18C3×#382—24no—
19C2×C3×#95C2212yes—
20C2×C3×#95C2211yescharacteristic
21C2×C3×#95C2212yes—
22C2×C3×#95C2212yes—
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×#395C2×C311yescharacteristic
35C2×#382C2×C311yescharacteristic
36#1446C2×C312yes—
37#1443C2×C312yes—
38#1442C2×C312yes—
39#1442C2×C312yes—
40#1441C2×C312yes—
41#1441C2×C311yescharacteristic
42#1434C2×C312yes—
43C22×C3×Q8—22no—
44C22×C3×Q8—24no—
45C22×C3×Q8—42no—
46C22×C3×Q8—41no—
47C22×C3×Q8—41no—
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2312yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C2×C3×C4⋊C4—22no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C4⋊C4—41no—
61C2×C3×C4⋊C4—41no—
62C2×C3×C4⋊C4—42no—
63C2×C3×C4⋊C4—41no—
64C2×C3×C4⋊C4—41no—
65C2×C3×C4⋊C4—41no—
66C2×C3×C4⋊C4—42no—
67C2×C3×C4⋊C4D414yes—
68C2×C3×C42—22no—
69C2×C3×C42—22no—
70C2×C3×C42Q814yescomplemented
71C3×#95—24no—
72C3×#95—24no—
73C3×#95—22no—
74C3×#95—24no—
75C3×#95—24no—
76C3×#95—24no—
77C3×#95—24no—
78C3×#95—24no—
79C3×#95—24no—
80C3×#95—42no—
81C3×#95—42no—
82C3×#95—41no—
83C3×#95—42no—
84C3×#95—42no—
85C3×#95—42no—
86C23×Q8C22×C311yescharacteristic
87C22×C4⋊C4C22×C312yes—
88C22×C4⋊C4C22×C311yescharacteristic
89C22×C4⋊C4C22×C311yescharacteristic
90C22×C4⋊C4C22×C311yescharacteristic
91C22×C42C22×C311yescharacteristic
92#395—24no—
93#382—24no—
94C2×#95C22×C312yes—
95C2×#95C22×C311yescharacteristic
96C2×#95C22×C312yes—
97C2×#95C22×C312yes—
98C2×#95C22×C312yes—
99C2×#95C22×C312yes—
100C2×#95C22×C312yes—
101C2×#95C22×C312yes—
102C2×#95C22×C312yes—
103C2×#95C22×C312yes—
104C2×#95C22×C312yes—
105C2×#95C22×C311yescharacteristic
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C24×C3C2411yescharacteristic
110C2×C3×Q8—48no—
111C2×C3×Q8—44no—
112C2×C3×Q8—44no—
113C2×C3×Q8—48no—
114C2×C3×Q8—44no—
115C22×C3×C4—21no—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4—21no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×Q814yes—
170C22×C3×C4C2×Q812yes—
171C22×C3×C4#4114yes—
172C22×C3×C4#4114yes—
173C22×C3×C4#4112yes—
174C3×C4⋊C4—44no—
175C3×C4⋊C4—48no—
176C3×C4⋊C4—44no—
177C3×C4⋊C4—44no—
178C3×C42—28no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—42no—
182C22×Q8—41no—
183C22×Q8—41no—
184C23×C4C23×C312yes—
185C23×C4C23×C312yes—
186C23×C4C23×C312yes—
187C23×C4C23×C311yescharacteristic
188C23×C4C23×C311yescharacteristic
189C23×C4C23×C312yes—
190C23×C4C23×C312yes—
191C23×C4C23×C312yes—
192C23×C4C23×C311yescharacteristic
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C3×D414yes—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42C3×Q814yescomplemented
207#95—24no—
208#95—24no—
209#95—22no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—42no—
217#95—42no—
218#95—41no—
219#95—42no—
220#95—42no—
221#95—42no—
222C23×C3C22×D411yescharacteristic
223C23×C3C22×Q811yescharacteristic
224C23×C3C2×#4111yescharacteristic
225C23×C3C2×#4112yes—
226C23×C3C2×#4112yes—
227C23×C3H(5,𝔽2)11yescharacteristic
228C23×C3H(5,𝔽2)11yescharacteristic
229C23×C3H(5,𝔽2)12yes—
230C23×C32−1+411yescharacteristic
231C23×C32−1+411yescharacteristic
232C23×C32−1+412yes—
233C3×Q8—416no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
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—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4C22⋊Q814yes—
293C24C24×C311yescharacteristic
294C2×Q8—48no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—48no—
298C2×Q8—44no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4C2×C3×D412yes—
351C22×C4C2×C3×D412yes—
352C22×C4C2×C3×D412yes—
353C22×C4C2×C3×Q814yes—
354C22×C4C2×C3×Q812yes—
355C22×C4C3×#4114yes—
356C22×C4C3×#4114yes—
357C22×C4C3×#4112yes—
358C4⋊C4—44no—
359C4⋊C4—48no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C42—28no—
363C22×C3C2×C22⋊Q811yescharacteristic
364C22×C3#53312yes—
365C22×C3#53411yescharacteristic
366C22×C3#53611yescharacteristic
367C22×C3#53812yes—
368C22×C3#53911yescharacteristic
369C22×C3#53911yescharacteristic
370C22×C3#53912yes—
371C22×C3#54112yes—
372C22×C3#54212yes—
373C22×C3#54411yescharacteristic
374C22×C3#54412yes—
375C22×C3#54411yescharacteristic
376C22×C3#54612yes—
377C22×C3#54812yes—
378C22×C3#55111yescharacteristic
379C22×C3#55212yes—
380C22×C3#55511yescharacteristic
381C22×C3#55512yes—
382C22×C3#55612yes—
383C22×C3H(3,𝔽4)11yescharacteristic
384C22×C3#56312yes—
385C22×C3#56311yescharacteristic
386C3×C4—24no—
387C3×C4—42no—
388C3×C4—42no—
389C3×C4—44no—
390C3×C4—44no—
391C3×C4—44no—
392C3×C4—82no—
393C3×C4—82no—
394C3×C4—82no—
395C23C22×C3×D411yescharacteristic
396C23C22×C3×Q811yescharacteristic
397C23C2×C3×#4111yescharacteristic
398C23C2×C3×#4112yes—
399C23C2×C3×#4112yes—
400C23C3×H(5,𝔽2)11yescharacteristic
401C23C3×H(5,𝔽2)12yes—
402C23C3×H(5,𝔽2)11yescharacteristic
403C23C3×2−1+411yescharacteristic
404C23C3×2−1+412yes—
405C23C3×2−1+411yescharacteristic
406Q8—416no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4C3×C22⋊Q814yes—
466C2×C3#243912yes—
467C2×C3#261211yescharacteristic
468C2×C3#262711yescharacteristic
469C2×C3#266612yes—
470C2×C3#268411yescharacteristic
471C2×C3#269012yes—
472C2×C3#273111yescharacteristic
473C2×C3#277512yes—
474C2×C3#278511yescharacteristic
475C2×C3#278911yescharacteristic
476C2×C3#284411yescharacteristic
477C22C2×C3×C22⋊Q811yescharacteristic
478C22C3×#53312yes—
479C22C3×#53411yescharacteristic
480C22C3×#53611yescharacteristic
481C22C3×#53812yes—
482C22C3×#53911yescharacteristic
483C22C3×#53911yescharacteristic
484C22C3×#53912yes—
485C22C3×#54112yes—
486C22C3×#54212yes—
487C22C3×#54411yescharacteristic
488C22C3×#54411yescharacteristic
489C22C3×#54412yes—
490C22C3×#54612yes—
491C22C3×#54812yes—
492C22C3×#55111yescharacteristic
493C22C3×#55212yes—
494C22C3×#55512yes—
495C22C3×#55511yescharacteristic
496C22C3×#55612yes—
497C22C3×H(3,𝔽4)11yescharacteristic
498C22C3×#56312yes—
499C22C3×#56311yescharacteristic
500C4—24no—
501C4—42no—
502C4—44no—
503C4—42no—
504C4—44no—
505C4—44no—
506C4—82no—
507C4—82no—
508C4—82no—
509C3#1668611yescharacteristic, direct product factor, Sylow
510C2C3×#243912yes—
511C2C3×#261211yescharacteristic
512C2C3×#262711yescharacteristic
513C2C3×#266612yes—
514C2C3×#268411yescharacteristic
515C2C3×#269012yes—
516C2C3×#273111yescharacteristic
517C2C3×#277512yes—
518C2C3×#278511yescharacteristic
519C2C3×#278911yescharacteristic
520C2C3×#284411yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
10. #16686C3direct (N × H)
70. C2×C3×C42Q8semidirect
206. C2×C42C3×Q8semidirect
509. C3#16686direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3952
C2×C3×#3822
C3×#14462
C3×#14432
C3×#14422
C3×#14412
C3×#14342
#166863