Finite Group Databasebeta

← C3×#16689 #10635296

Subgroups — C3×#16689 #10635296

Order 768.

1094 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#400C211yesmaximal, characteristic
2C2×C3×#382C211yesmaximal, characteristic
3C3×#1445C212yesmaximal
4C3×#1444C212yesmaximal
5C3×#1444C212yesmaximal
6C3×#1443C212yesmaximal
7C3×#1441C212yesmaximal
8C3×#1441C211yesmaximal, characteristic
9C3×#1434C212yesmaximal
10#16689C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×C3×C4⋊C4C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C3×#400—24no—
19C3×#382—24no—
20C2×C3×#95C2212yes—
21C2×C3×#95C2212yes—
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2211yescharacteristic
34C2×#400C2×C311yescharacteristic
35C2×#382C2×C311yescharacteristic
36#1445C2×C312yes—
37#1444C2×C312yes—
38#1444C2×C312yes—
39#1443C2×C312yes—
40#1441C2×C312yes—
41#1441C2×C311yescharacteristic
42#1434C2×C312yes—
43C23×C3×C4C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2312yes—
46C23×C3×C4C2311yescharacteristic
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4—24no—
56C2×C3×C4⋊C4—41no—
57C2×C3×C4⋊C4—41no—
58C2×C3×C4⋊C4—41no—
59C2×C3×C4⋊C4—41no—
60C2×C3×C4⋊C4—41no—
61C2×C3×C4⋊C4—42no—
62C2×C3×C4⋊C4—42no—
63C2×C3×C4⋊C4—41no—
64C2×C3×C4⋊C4—41no—
65C2×C3×C4⋊C4—42no—
66C2×C3×C4⋊C4—42no—
67C2×C3×C4⋊C4—42no—
68C2×C3×C4⋊C4D414yes—
69C2×C3×C42—22no—
70C2×C3×C42—22no—
71C2×C3×C42Q814yes—
72C3×#95—24no—
73C3×#95—24no—
74C3×#95—22no—
75C3×#95—24no—
76C3×#95—24no—
77C3×#95—24no—
78C3×#95—24no—
79C3×#95—24no—
80C3×#95—24no—
81C3×#95—24no—
82C3×#95—42no—
83C3×#95—42no—
84C3×#95—41no—
85C3×#95—42no—
86C22×C4⋊C4C22×C311yescharacteristic
87C22×C4⋊C4C22×C311yescharacteristic
88C22×C4⋊C4C22×C312yes—
89C22×C4⋊C4C22×C311yescharacteristic
90C22×C4⋊C4C22×C312yes—
91C22×C4⋊C4C22×C311yescharacteristic
92C22×C42C22×C311yescharacteristic
93#400—24no—
94#382—24no—
95C2×#95C22×C312yes—
96C2×#95C22×C312yes—
97C2×#95C22×C311yescharacteristic
98C2×#95C22×C312yes—
99C2×#95C22×C312yes—
100C2×#95C22×C312yes—
101C2×#95C22×C312yes—
102C2×#95C22×C312yes—
103C2×#95C22×C312yes—
104C2×#95C22×C312yes—
105C2×#95C22×C312yes—
106C2×#95C22×C312yes—
107C2×#95C22×C312yes—
108C2×#95C22×C311yescharacteristic
109C24×C3C2411yescharacteristic
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
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—21no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
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×C4C2×D412yes—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×Q814yes—
165C22×C3×C4C2×Q812yes—
166C22×C3×C4#4114yes—
167C22×C3×C4#4114yes—
168C22×C3×C4#4112yes—
169C3×C4⋊C4—44no—
170C3×C4⋊C4—44no—
171C3×C4⋊C4—44no—
172C3×C4⋊C4—44no—
173C3×C4⋊C4—48no—
174C3×C4⋊C4—48no—
175C3×C42—28no—
176C23×C4C23×C312yes—
177C23×C4C23×C312yes—
178C23×C4C23×C312yes—
179C23×C4C23×C311yescharacteristic
180C23×C4C23×C311yescharacteristic
181C23×C4C23×C312yes—
182C23×C4C23×C312yes—
183C23×C4C23×C312yes—
184C23×C4C23×C311yescharacteristic
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4C3×D414yes—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42C3×Q814yes—
205#95—24no—
206#95—24no—
207#95—22no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—42no—
216#95—42no—
217#95—41no—
218#95—42no—
219C23×C3C22×D411yescharacteristic
220C23×C3C22×Q811yescharacteristic
221C23×C3C2×#4111yescharacteristic
222C23×C3C2×#4112yes—
223C23×C3C2×#4112yes—
224C23×C3H(5,𝔽2)11yescharacteristic
225C23×C3H(5,𝔽2)11yescharacteristic
226C23×C3H(5,𝔽2)12yes—
227C23×C32−1+411yescharacteristic
228C23×C32−1+411yescharacteristic
229C23×C32−1+412yes—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—24no—
237C2×C3×C4—24no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
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×C4C22⋊Q814yes—
289C24C24×C311yescharacteristic
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
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×C4C2×C3×D412yes—
342C22×C4C2×C3×D412yes—
343C22×C4C2×C3×D412yes—
344C22×C4C2×C3×Q814yes—
345C22×C4C2×C3×Q812yes—
346C22×C4C3×#4114yes—
347C22×C4C3×#4114yes—
348C22×C4C3×#4112yes—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—48no—
354C4⋊C4—48no—
355C42—28no—
356C22×C3C2×C22⋊Q811yescharacteristic
357C22×C3#53312yes—
358C22×C3#53411yescharacteristic
359C22×C3#53611yescharacteristic
360C22×C3#53812yes—
361C22×C3#53911yescharacteristic
362C22×C3#53911yescharacteristic
363C22×C3#53912yes—
364C22×C3#54112yes—
365C22×C3#54212yes—
366C22×C3#54411yescharacteristic
367C22×C3#54412yes—
368C22×C3#54411yescharacteristic
369C22×C3#54712yes—
370C22×C3#55012yes—
371C22×C3#55111yescharacteristic
372C22×C3#55212yes—
373C22×C3#55212yes—
374C22×C3#55312yes—
375C22×C3#55511yescharacteristic
376C22×C3H(3,𝔽4)11yescharacteristic
377C22×C3#56311yescharacteristic
378C22×C3#56312yes—
379C3×C4—24no—
380C3×C4—44no—
381C3×C4—42no—
382C3×C4—42no—
383C3×C4—44no—
384C3×C4—44no—
385C3×C4—82no—
386C3×C4—82no—
387C3×C4—82no—
388C23C22×C3×D411yescharacteristic
389C23C22×C3×Q811yescharacteristic
390C23C2×C3×#4111yescharacteristic
391C23C2×C3×#4112yes—
392C23C2×C3×#4112yes—
393C23C3×H(5,𝔽2)11yescharacteristic
394C23C3×H(5,𝔽2)12yes—
395C23C3×H(5,𝔽2)11yescharacteristic
396C23C3×2−1+411yescharacteristic
397C23C3×2−1+412yes—
398C23C3×2−1+411yescharacteristic
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
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—42no—
441C2×C4—42no—
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×C4C3×C22⋊Q814yes—
458C2×C3#244112yes—
459C2×C3#261211yescharacteristic
460C2×C3#262711yescharacteristic
461C2×C3#266612yes—
462C2×C3#271711yescharacteristic
463C2×C3#272112yes—
464C2×C3#273311yescharacteristic
465C2×C3#276512yes—
466C2×C3#276911yescharacteristic
467C2×C3#278211yescharacteristic
468C2×C3#284411yescharacteristic
469C22C2×C3×C22⋊Q811yescharacteristic
470C22C3×#53312yes—
471C22C3×#53411yescharacteristic
472C22C3×#53611yescharacteristic
473C22C3×#53812yes—
474C22C3×#53911yescharacteristic
475C22C3×#53911yescharacteristic
476C22C3×#53912yes—
477C22C3×#54112yes—
478C22C3×#54212yes—
479C22C3×#54411yescharacteristic
480C22C3×#54411yescharacteristic
481C22C3×#54412yes—
482C22C3×#54712yes—
483C22C3×#55012yes—
484C22C3×#55111yescharacteristic
485C22C3×#55212yes—
486C22C3×#55212yes—
487C22C3×#55312yes—
488C22C3×#55511yescharacteristic
489C22C3×H(3,𝔽4)11yescharacteristic
490C22C3×#56311yescharacteristic
491C22C3×#56312yes—
492C4—24no—
493C4—44no—
494C4—42no—
495C4—42no—
496C4—44no—
497C4—44no—
498C4—82no—
499C4—82no—
500C4—82no—
501C3#1668911yescharacteristic, direct product factor, Sylow
502C2C3×#244112yes—
503C2C3×#261211yescharacteristic
504C2C3×#262711yescharacteristic
505C2C3×#266612yes—
506C2C3×#271711yescharacteristic
507C2C3×#272112yes—
508C2C3×#273311yescharacteristic
509C2C3×#276512yes—
510C2C3×#276911yescharacteristic
511C2C3×#278211yescharacteristic
512C2C3×#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. #16689C3direct (N × H)
501. C3#16689direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4002
C2×C3×#3822
C3×#14452
C3×#14442
C3×#14432
C3×#14412
C3×#14342
#166893