Finite Group Databasebeta

← C3×#16696 #10635303

Subgroups — C3×#16696 #10635303

Order 768.

1122 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#391C211yesmaximal, characteristic
2C2×C3×#383C211yesmaximal, characteristic
3C3×#1446C211yesmaximal, characteristic
4C3×#1446C211yesmaximal, characteristic
5C3×#1445C212yesmaximal
6C3×#1444C211yesmaximal, characteristic
7C3×#1443C212yesmaximal
8C3×#1443C212yesmaximal
9C3×#1442C212yesmaximal
10C3×#1442C211yesmaximal, characteristic
11C3×#1434C211yesmaximal, characteristic
12#16696C311yesmaximal, characteristic, direct product factor, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C42C2211yescharacteristic
17C3×#391—24no—
18C3×#383—24no—
19C2×C3×#95C2211yescharacteristic
20C2×C3×#95C2211yescharacteristic
21C2×C3×#95C2211yescharacteristic
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2211yescharacteristic
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3×#95C2212yes—
32C2×C3×#95C2212yes—
33C2×C3×#95C2212yes—
34C2×C3×#95C2212yes—
35C2×C3×#95C2212yes—
36C2×C3×#95C2212yes—
37C2×C3×#95C2212yes—
38C2×#391C2×C311yescharacteristic
39C2×#383C2×C311yescharacteristic
40#1446C2×C311yescharacteristic
41#1446C2×C311yescharacteristic
42#1445C2×C312yes—
43#1444C2×C311yescharacteristic
44#1443C2×C312yes—
45#1443C2×C312yes—
46#1442C2×C312yes—
47#1442C2×C311yescharacteristic
48#1434C2×C311yescharacteristic
49C22×C3×Q8—22no—
50C22×C3×Q8—24no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—42no—
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C23×C3×C4C2312yes—
63C23×C3×C4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—42no—
68C2×C3×C4⋊C4—42no—
69C2×C3×C4⋊C4—42no—
70C2×C3×C42—22no—
71C2×C3×C42—22no—
72C2×C3×C42D414yes—
73C3×#95—22no—
74C3×#95—22no—
75C3×#95—22no—
76C3×#95—22no—
77C3×#95—22no—
78C3×#95—22no—
79C3×#95—24no—
80C3×#95—24no—
81C3×#95—22no—
82C3×#95—24no—
83C3×#95—24no—
84C3×#95—24no—
85C3×#95—24no—
86C3×#95—22no—
87C3×#95—42no—
88C3×#95—42no—
89C3×#95—42no—
90C3×#95—42no—
91C3×#95—42no—
92C23×Q8C22×C311yescharacteristic
93C22×C4⋊C4C22×C312yes—
94C22×C4⋊C4C22×C311yescharacteristic
95C22×C42C22×C311yescharacteristic
96#391—24no—
97#383—24no—
98C2×#95C22×C312yes—
99C2×#95C22×C311yescharacteristic
100C2×#95C22×C311yescharacteristic
101C2×#95C22×C311yescharacteristic
102C2×#95C22×C312yes—
103C2×#95C22×C312yes—
104C2×#95C22×C311yescharacteristic
105C2×#95C22×C312yes—
106C2×#95C22×C312yes—
107C2×#95C22×C311yescharacteristic
108C2×#95C22×C311yescharacteristic
109C2×#95C22×C312yes—
110C2×#95C22×C311yescharacteristic
111C2×#95C22×C311yescharacteristic
112C2×#95C22×C312yes—
113C2×#95C22×C312yes—
114C2×#95C22×C312yes—
115C2×#95C22×C312yes—
116C2×#95C22×C312yes—
117C24×C3C2411yescharacteristic
118C2×C3×Q8—28no—
119C2×C3×Q8—48no—
120C2×C3×Q8—48no—
121C2×C3×Q8—44no—
122C2×C3×Q8—44no—
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—21no—
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—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
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—21no—
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—21no—
170C22×C3×C4—22no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4#4114yes—
185C22×C3×C4#4114yes—
186C22×C3×C4#4112yes—
187C22×C3×C4#4112yes—
188C3×C4⋊C4—44no—
189C3×C4⋊C4—48no—
190C3×C42—28no—
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—42no—
196C23×C4C23×C311yescharacteristic
197C23×C4C23×C312yes—
198C23×C4C23×C311yescharacteristic
199C23×C4C23×C311yescharacteristic
200C23×C4C23×C311yescharacteristic
201C23×C4C23×C312yes—
202C23×C4C23×C311yescharacteristic
203C23×C4C23×C311yescharacteristic
204C23×C4C23×C312yes—
205C23×C4C23×C312yes—
206C23×C4C23×C311yescharacteristic
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42C3×D414yes—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—24no—
222#95—24no—
223#95—22no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—22no—
229#95—42no—
230#95—42no—
231#95—42no—
232#95—42no—
233#95—42no—
234C23×C3C22×D411yescharacteristic
235C23×C3C2×#4111yescharacteristic
236C23×C3C2×#4111yescharacteristic
237C23×C3C2×#4112yes—
238C23×C3C2×#4112yes—
239C23×C3H(5,𝔽2)12yes—
240C23×C3H(5,𝔽2)11yescharacteristic
241C23×C32−1+411yescharacteristic
242C23×C32−1+412yes—
243C23×C32−1+411yescharacteristic
244C23×C32−1+411yescharacteristic
245C3×Q8—416no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—24no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
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—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
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×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4#12314yes—
311C24C24×C311yescharacteristic
312C2×Q8—28no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—44no—
316C2×Q8—44no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
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—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4C2×C3×D412yes—
376C22×C4C2×C3×D412yes—
377C22×C4C2×C3×D412yes—
378C22×C4C3×#4114yes—
379C22×C4C3×#4114yes—
380C22×C4C3×#4112yes—
381C22×C4C3×#4112yes—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C42—28no—
385C22×C3C2×#12311yescharacteristic
386C22×C3#52511yescharacteristic
387C22×C3#53211yescharacteristic
388C22×C3#53511yescharacteristic
389C22×C3#53611yescharacteristic
390C22×C3#53611yescharacteristic
391C22×C3#53711yescharacteristic
392C22×C3#53812yes—
393C22×C3#54111yescharacteristic
394C22×C3#54111yescharacteristic
395C22×C3#54112yes—
396C22×C3#54211yescharacteristic
397C22×C3#54211yescharacteristic
398C22×C3#54212yes—
399C22×C3#54212yes—
400C22×C3#55112yes—
401C22×C3#55112yes—
402C22×C3#55212yes—
403C22×C3#55212yes—
404C22×C3#55612yes—
405C22×C3#55612yes—
406C22×C3#56312yes—
407C22×C3#56312yes—
408C3×C4—24no—
409C3×C4—42no—
410C3×C4—44no—
411C3×C4—42no—
412C3×C4—44no—
413C3×C4—81no—
414C3×C4—81no—
415C3×C4—81no—
416C3×C4—82no—
417C3×C4—81no—
418C3×C4—82no—
419C23C22×C3×D411yescharacteristic
420C23C2×C3×#4111yescharacteristic
421C23C2×C3×#4111yescharacteristic
422C23C2×C3×#4112yes—
423C23C2×C3×#4112yes—
424C23C3×H(5,𝔽2)12yes—
425C23C3×H(5,𝔽2)11yescharacteristic
426C23C3×2−1+411yescharacteristic
427C23C3×2−1+412yes—
428C23C3×2−1+411yescharacteristic
429C23C3×2−1+411yescharacteristic
430Q8—416no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
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—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4C3×#12314yes—
496C2×C3#248211yescharacteristic
497C2×C3#249911yescharacteristic
498C2×C3#260411yescharacteristic
499C2×C3#262511yescharacteristic
500C2×C3#263711yescharacteristic
501C2×C3#266211yescharacteristic
502C2×C3#275812yes—
503C2×C3#275912yes—
504C2×C3#276212yes—
505C2×C3#276412yes—
506C2×C3#282111yescharacteristic
507C22C2×C3×#12311yescharacteristic
508C22C3×#52511yescharacteristic
509C22C3×#53211yescharacteristic
510C22C3×#53511yescharacteristic
511C22C3×#53611yescharacteristic
512C22C3×#53611yescharacteristic
513C22C3×#53711yescharacteristic
514C22C3×#53812yes—
515C22C3×#54111yescharacteristic
516C22C3×#54111yescharacteristic
517C22C3×#54112yes—
518C22C3×#54211yescharacteristic
519C22C3×#54211yescharacteristic
520C22C3×#54212yes—
521C22C3×#54212yes—
522C22C3×#55112yes—
523C22C3×#55112yes—
524C22C3×#55212yes—
525C22C3×#55212yes—
526C22C3×#55612yes—
527C22C3×#55612yes—
528C22C3×#56312yes—
529C22C3×#56312yes—
530C4—24no—
531C4—42no—
532C4—44no—
533C4—42no—
534C4—44no—
535C4—81no—
536C4—82no—
537C4—81no—
538C4—82no—
539C4—81no—
540C4—81no—
541C3#1669611yescharacteristic, direct product factor, Sylow
542C2C3×#248211yescharacteristic
543C2C3×#249911yescharacteristic
544C2C3×#260411yescharacteristic
545C2C3×#262511yescharacteristic
546C2C3×#263711yescharacteristic
547C2C3×#266211yescharacteristic
548C2C3×#275812yes—
549C2C3×#275912yes—
550C2C3×#276212yes—
551C2C3×#276412yes—
552C2C3×#282111yescharacteristic
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
12. #16696C3direct (N × H)
541. C3#16696direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3912
C2×C3×#3832
C3×#14462
C3×#14452
C3×#14442
C3×#14432
C3×#14422
C3×#14342
#166963