Finite Group Databasebeta

← C3×#16653 #10635260

Subgroups — C3×#16653 #10635260

Order 768.

1158 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#400C211yesmaximal, characteristic
2C2×C3×#391C211yesmaximal, characteristic
3C3×#1446C212yesmaximal
4C3×#1445C212yesmaximal
5C3×#1443C212yesmaximal
6C3×#1442C212yesmaximal
7C3×#1441C212yesmaximal
8C3×#1441C211yesmaximal, characteristic
9C3×#1436C212yesmaximal
10#16653C311yesmaximal, 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×#400—24no—
18C3×#391—24no—
19C2×C3×#95C2212yes—
20C2×C3×#95C2212yes—
21C2×C3×#95C2211yescharacteristic
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×#400C2×C311yescharacteristic
35C2×#391C2×C311yescharacteristic
36#1446C2×C312yes—
37#1445C2×C312yes—
38#1443C2×C312yes—
39#1442C2×C312yes—
40#1441C2×C312yes—
41#1441C2×C311yescharacteristic
42#1436C2×C312yes—
43C22×C3×Q8—22no—
44C22×C3×Q8—22no—
45C22×C3×Q8—41no—
46C22×C3×Q8—41no—
47C22×C3×Q8—41no—
48C22×C3×Q8—41no—
49C22×C3×Q8Q814yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2312yes—
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4—24no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—41no—
64C2×C3×C4⋊C4—41no—
65C2×C3×C4⋊C4—41no—
66C2×C3×C4⋊C4—42no—
67C2×C3×C4⋊C4—41no—
68C2×C3×C4⋊C4—41no—
69C2×C3×C4⋊C4D414yes—
70C2×C3×C42—22no—
71C2×C3×C42—41no—
72C2×C3×C42—41no—
73C3×#95—24no—
74C3×#95—24no—
75C3×#95—22no—
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—42no—
85C3×#95—41no—
86C3×#95—42no—
87C3×#95—42no—
88C23×Q8C22×C311yescharacteristic
89C22×C4⋊C4C22×C312yes—
90C22×C4⋊C4C22×C311yescharacteristic
91C22×C4⋊C4C22×C311yescharacteristic
92C22×C4⋊C4C22×C311yescharacteristic
93C22×C42C22×C311yescharacteristic
94#400—24no—
95#391—24no—
96C2×#95C22×C312yes—
97C2×#95C22×C312yes—
98C2×#95C22×C311yescharacteristic
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×C311yescharacteristic
108C2×#95C22×C312yes—
109C2×#95C22×C312yes—
110C2×#95C22×C312yes—
111C24×C3C2411yescharacteristic
112C2×C3×Q8—28no—
113C2×C3×Q8—44no—
114C2×C3×Q8—44no—
115C2×C3×Q8—44no—
116C2×C3×Q8—44no—
117C2×C3×Q8—44no—
118C2×C3×Q8—44no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
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—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
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—21no—
143C22×C3×C4—21no—
144C22×C3×C4—21no—
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×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×Q814yes—
174C22×C3×C4C2×Q812yes—
175C22×C3×C4#4114yes—
176C22×C3×C4#4114yes—
177C22×C3×C4#4112yes—
178C3×C4⋊C4—44no—
179C3×C4⋊C4—44no—
180C3×C4⋊C4—48no—
181C3×C4⋊C4—44no—
182C3×C42—44no—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—41no—
189C22×Q8C3×Q814yes—
190C23×C4C23×C312yes—
191C23×C4C23×C312yes—
192C23×C4C23×C312yes—
193C23×C4C23×C311yescharacteristic
194C23×C4C23×C311yescharacteristic
195C23×C4C23×C312yes—
196C23×C4C23×C312yes—
197C23×C4C23×C312yes—
198C23×C4C23×C311yescharacteristic
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4C3×D414yes—
210C2×C42—22no—
211C2×C42—41no—
212C2×C42—41no—
213#95—24no—
214#95—24no—
215#95—22no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—42no—
223#95—42no—
224#95—42no—
225#95—41no—
226#95—42no—
227#95—42no—
228C23×C3C22×D411yescharacteristic
229C23×C3C22×Q811yescharacteristic
230C23×C3C2×#4111yescharacteristic
231C23×C3C2×#4112yes—
232C23×C3C2×#4112yes—
233C23×C3H(5,𝔽2)11yescharacteristic
234C23×C3H(5,𝔽2)11yescharacteristic
235C23×C3H(5,𝔽2)11yescharacteristic
236C23×C3H(5,𝔽2)11yescharacteristic
237C23×C32−1+412yes—
238C23×C32−1+412yes—
239C3×Q8—416no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—24no—
247C2×C3×C4—24no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—24no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
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×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4C22⋊Q814yes—
299C24C24×C311yescharacteristic
300C2×Q8—28no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—44no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
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—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
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×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4C2×C3×D412yes—
359C22×C4C2×C3×D412yes—
360C22×C4C2×C3×D412yes—
361C22×C4C2×C3×Q814yes—
362C22×C4C2×C3×Q812yes—
363C22×C4C3×#4114yes—
364C22×C4C3×#4114yes—
365C22×C4C3×#4112yes—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—48no—
369C4⋊C4—44no—
370C42—44no—
371C22×C3C2×C22⋊Q811yescharacteristic
372C22×C3#52912yes—
373C22×C3#53411yescharacteristic
374C22×C3#53511yescharacteristic
375C22×C3#53811yescharacteristic
376C22×C3#53811yescharacteristic
377C22×C3#53912yes—
378C22×C3#54112yes—
379C22×C3#54212yes—
380C22×C3#54212yes—
381C22×C3#54212yes—
382C22×C3#54311yescharacteristic
383C22×C3#54311yescharacteristic
384C22×C3#54712yes—
385C22×C3#54812yes—
386C22×C3#55111yescharacteristic
387C22×C3#55112yes—
388C22×C3#55511yescharacteristic
389C22×C3#55712yes—
390C22×C3#55712yes—
391C22×C3#56311yescharacteristic
392C22×C3#56312yes—
393C22×C3#56311yescharacteristic
394C3×C4—24no—
395C3×C4—44no—
396C3×C4—44no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—44no—
400C3×C4—82no—
401C3×C4—82no—
402C3×C4—82no—
403C23C22×C3×D411yescharacteristic
404C23C22×C3×Q811yescharacteristic
405C23C2×C3×#4111yescharacteristic
406C23C2×C3×#4112yes—
407C23C2×C3×#4112yes—
408C23C3×H(5,𝔽2)11yescharacteristic
409C23C3×H(5,𝔽2)11yescharacteristic
410C23C3×H(5,𝔽2)11yescharacteristic
411C23C3×H(5,𝔽2)11yescharacteristic
412C23C3×2−1+412yes—
413C23C3×2−1+412yes—
414Q8—416no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—41no—
435C2×C4—41no—
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—41no—
449C2×C4—41no—
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×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4C3×C22⋊Q814yes—
474C2×C3#245312yes—
475C2×C3#257211yescharacteristic
476C2×C3#262611yescharacteristic
477C2×C3#265012yes—
478C2×C3#271711yescharacteristic
479C2×C3#272312yes—
480C2×C3#273111yescharacteristic
481C2×C3#279912yes—
482C2×C3#279911yescharacteristic
483C2×C3#280911yescharacteristic
484C2×C3#283311yescharacteristic
485C22C2×C3×C22⋊Q811yescharacteristic
486C22C3×#52912yes—
487C22C3×#53411yescharacteristic
488C22C3×#53511yescharacteristic
489C22C3×#53811yescharacteristic
490C22C3×#53811yescharacteristic
491C22C3×#53912yes—
492C22C3×#54112yes—
493C22C3×#54212yes—
494C22C3×#54212yes—
495C22C3×#54212yes—
496C22C3×#54311yescharacteristic
497C22C3×#54311yescharacteristic
498C22C3×#54712yes—
499C22C3×#54812yes—
500C22C3×#55111yescharacteristic
501C22C3×#55112yes—
502C22C3×#55511yescharacteristic
503C22C3×#55712yes—
504C22C3×#55712yes—
505C22C3×#56311yescharacteristic
506C22C3×#56312yes—
507C22C3×#56311yescharacteristic
508C4—24no—
509C4—44no—
510C4—42no—
511C4—42no—
512C4—44no—
513C4—44no—
514C4—82no—
515C4—82no—
516C4—82no—
517C3#1665311yescharacteristic, direct product factor, Sylow
518C2C3×#245312yes—
519C2C3×#257211yescharacteristic
520C2C3×#262611yescharacteristic
521C2C3×#265012yes—
522C2C3×#271711yescharacteristic
523C2C3×#272312yes—
524C2C3×#273111yescharacteristic
525C2C3×#279912yes—
526C2C3×#279911yescharacteristic
527C2C3×#280911yescharacteristic
528C2C3×#283311yescharacteristic
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. #16653C3direct (N × H)
517. C3#16653direct (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×#3912
C3×#14462
C3×#14452
C3×#14432
C3×#14422
C3×#14412
C3×#14362
#166533