Finite Group Databasebeta

← C3×#25735 #10644342

Subgroups — C3×#25735 #10644342

Order 768.

1466 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#536C211yesmaximal, characteristic
2C2×C3×#497C211yesmaximal, characteristic, complemented
3C2×C3×#494C211yesmaximal, characteristic, complemented
4C2×C3×#452C212yesmaximal
5C2×C3×#451C212yesmaximal, complemented
6C3×#2621C211yesmaximal, characteristic, complemented
7C3×#2016C212yesmaximal, complemented
8C3×#2015C212yesmaximal, complemented
9C3×#1960C211yesmaximal, characteristic, complemented
10C3×#1958C211yesmaximal, characteristic, complemented
11C3×#1750C211yesmaximal, characteristic, complemented
12#25735C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C22×C3×Q16C2212yes—
16C22×C3×Q16C2212yes—
17C22×C3×Q16C2212yes—
18C22×C3×SD16C2212yes—
19C3×#536—24no—
20C3×#536—24no—
21C2×C3×C4⋊Q8C2211yescharacteristic
22C2×C3×C4⋊Q8C2211yescharacteristic
23C2×C3×C4⋊Q8C2211yescharacteristic
24C2×C3×#124C2211yescharacteristic
25C2×C3×#124C2211yescharacteristic
26C2×C3×#123—22no—
27C2×C3×C22⋊Q8—22no—
28C2×C3×C22⋊Q8C2212yes—
29C2×C3×#117C2211yescharacteristic
30C3×#497—24no—
31C3×#494—24no—
32C3×#452—28no—
33C3×#451—28no—
34C2×C3×#103C2212yes—
35C2×C3×#103C2212yes—
36C2×C3×#103C2212yes—
37C2×C3×#102C2212yes—
38C2×C3×C22⋊C8C2212yes—
39C2×C3×#97C2211yescharacteristic
40C2×C3×C4×C8C2211yescharacteristic
41C3×#398—22no—
42C3×#397—22no—
43C3×#391C2212yes—
44C3×#382C2211yescharacteristic
45C3×#336C2212yes—
46C2×#536C2×C311yescharacteristic
47C2×#497C2×C311yescharacteristic, complemented
48C2×#494C2×C311yescharacteristic, complemented
49C2×#452C2×C312yes—
50C2×#451C2×C312yescomplemented
51#2621C2×C311yescharacteristic, complemented
52#2016C2×C312yescomplemented
53#2015C2×C312yescomplemented
54#1960C2×C311yescharacteristic, complemented
55#1958C2×C311yescharacteristic, complemented
56#1750C2×C311yescharacteristic, complemented
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—24no—
60C2×C3×#41—42no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—21no—
64C22×C3×Q8—24no—
65C22×C3×Q8—42no—
66C22×C3×Q8C2312yes—
67C22×C3×Q8C2311yescharacteristic
68C22×C3×Q8C2311yescharacteristic
69C22×C3×Q8C2311yescharacteristic
70C22×C3×D4—21no—
71C22×C3×D4C2311yescharacteristic
72C23×C3×C4—21no—
73C23×C3×C4—21no—
74C23×C3×C4C2311yescharacteristic
75C2×C3×Q16—24no—
76C2×C3×Q16—24no—
77C2×C3×Q16—24no—
78C2×C3×Q16—24no—
79C2×C3×Q16—24no—
80C2×C3×Q16—24no—
81C2×C3×Q16—42no—
82C2×C3×Q16—42no—
83C2×C3×Q16—42no—
84C2×C3×SD16—24no—
85C2×C3×SD16—24no—
86C2×C3×SD16—42no—
87C22×C3×C8C2312yes—
88C22×C3×C8C2312yes—
89C3×C4⋊Q8—22no—
90C3×C4⋊Q8—22no—
91C3×C4⋊Q8—22no—
92C3×C4⋊Q8—22no—
93C3×C4⋊Q8—22no—
94C3×C4⋊Q8D414yes—
95C3×#124—22no—
96C3×#124—22no—
97C3×#124—22no—
98C3×#124D414yes—
99C3×#123—216no—
100C3×C22⋊Q8—216no—
101C3×C22⋊Q8—24no—
102C3×C22⋊Q8—24no—
103C3×#117—22no—
104C3×#117—22no—
105C2×C3×C4⋊C4—22no—
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—22no—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4D412yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C22⋊C4—22no—
112C2×C3×C22⋊C4—22no—
113C2×C3×C22⋊C4—22no—
114C2×C3×C22⋊C4D412yes—
115C2×C3×C42C2311yescharacteristic
116C2×C3×C42C2311yescharacteristic
117C3×#103—24no—
118C3×#103—24no—
119C3×#103—24no—
120C3×#102—24no—
121C3×C22⋊C8D418yes—
122C3×#97—22no—
123C3×C4×C8—22no—
124C3×#95—22no—
125C3×#95—24no—
126C22×#41C22×C311yescharacteristic
127C23×Q8C22×C311yescharacteristic
128C22×Q16C22×C312yes—
129C22×Q16C22×C312yes—
130C22×Q16C22×C312yes—
131C22×SD16C22×C312yes—
132#536—24no—
133#536—24no—
134C2×C4⋊Q8C22×C311yescharacteristic
135C2×C4⋊Q8C22×C311yescharacteristic
136C2×C4⋊Q8C22×C311yescharacteristic
137C2×#124C22×C311yescharacteristic
138C2×#124C22×C311yescharacteristic
139C2×#123—22no—
140C2×C22⋊Q8—22no—
141C2×C22⋊Q8C22×C312yes—
142C2×#117C22×C311yescharacteristic
143#497—24no—
144#494—24no—
145#452—28no—
146#451—28no—
147C2×#103C22×C312yes—
148C2×#103C22×C312yes—
149C2×#103C22×C312yes—
150C2×#102C22×C312yes—
151C2×C22⋊C8C22×C312yes—
152C2×#97C22×C311yescharacteristic
153C2×C4×C8C22×C311yescharacteristic
154#398—22no—
155#397—22no—
156#391C22×C312yes—
157#382C22×C311yescharacteristic
158#336C22×C312yes—
159C24×C3—21no—
160C24×C3C2×D411yescharacteristic
161C3×#41—48no—
162C3×#41—48no—
163C2×C3×Q8—24no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8—21no—
167C2×C3×Q8—22no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—22no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8—48no—
180C2×C3×Q8—48no—
181C2×C3×Q8—42no—
182C2×C3×Q8—42no—
183C2×C3×Q8C2×D412yes—
184C2×C3×Q8C2×D412yes—
185C2×C3×Q8C2×D412yes—
186C2×C3×D4—24no—
187C2×C3×D4—22no—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4—42no—
191C2×C3×D4—42no—
192C2×C3×D4C2×D412yes—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—22no—
196C22×C3×C4—24no—
197C22×C3×C4—24no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—22no—
202C22×C3×C4—22no—
203C22×C3×C4—42no—
204C22×C3×C4—42no—
205C22×C3×C4—42no—
206C22×C3×C4—42no—
207C22×C3×C4C2×D412yes—
208C22×C3×C4C2×D412yes—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D414yes—
211C22×C3×C4C2411yescharacteristic
212C3×Q16—48no—
213C3×Q16—48no—
214C3×Q16—48no—
215C3×SD16—48no—
216C2×C3×C8—22no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8C2×D414yes—
221C2×C3×C8C2×D414yes—
222C3×C4⋊C4—28no—
223C3×C4⋊C4—28no—
224C3×C4⋊C4—24no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—44no—
228C3×C4⋊C4—44no—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—28no—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—44no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42C2×D412yes—
236C3×C42C2×D412yes—
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—24no—
240C2×#41—42no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8—21no—
244C22×Q8—24no—
245C22×Q8—42no—
246C22×Q8C23×C312yes—
247C22×Q8C23×C311yescharacteristic
248C22×Q8C23×C311yescharacteristic
249C22×Q8C23×C311yescharacteristic
250C22×D4—21no—
251C22×D4C23×C311yescharacteristic
252C23×C4—21no—
253C23×C4—21no—
254C23×C4C23×C311yescharacteristic
255C2×Q16—24no—
256C2×Q16—24no—
257C2×Q16—24no—
258C2×Q16—24no—
259C2×Q16—24no—
260C2×Q16—24no—
261C2×Q16—42no—
262C2×Q16—42no—
263C2×Q16—42no—
264C2×SD16—24no—
265C2×SD16—24no—
266C2×SD16—42no—
267C22×C8C23×C312yes—
268C22×C8C23×C312yes—
269C4⋊Q8—22no—
270C4⋊Q8—22no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8—22no—
274C4⋊Q8C3×D414yes—
275#124—22no—
276#124—22no—
277#124—22no—
278#124C3×D414yes—
279#123—216no—
280C22⋊Q8—216no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283#117—22no—
284#117—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4C3×D412yes—
290C2×C4⋊C4C23×C312yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4C3×D412yes—
295C2×C42C23×C311yescharacteristic
296C2×C42C23×C311yescharacteristic
297#103—24no—
298#103—24no—
299#103—24no—
300#102—24no—
301C22⋊C8C3×D418yes—
302#97—22no—
303C4×C8—22no—
304#95—22no—
305#95—24no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—24no—
309C23×C3—22no—
310C23×C3—41no—
311C23×C3—41no—
312C23×C3—44no—
313C23×C3#12712yes—
314C23×C3C22×D411yescharacteristic
315C3×Q8—24no—
316C3×Q8—24no—
317C3×Q8—24no—
318C3×Q8—48no—
319C3×Q8—42no—
320C3×Q8—42no—
321C3×Q8—42no—
322C3×Q8—42no—
323C3×Q8—42no—
324C3×Q8—42no—
325C3×Q8—42no—
326C3×D4—24no—
327C3×D4—48no—
328C3×D4—42no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—24no—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—24no—
339C2×C3×C4—48no—
340C2×C3×C4—48no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—44no—
352C2×C3×C4#12014yes—
353C2×C3×C4#12712yes—
354C2×C3×C4C22×D411yescharacteristic
355C2×C3×C4C22×D411yescharacteristic
356C2×C3×C4C22×D412yes—
357C2×C3×C4H(5,𝔽2)12yes—
358C3×C8—24no—
359C3×C8—24no—
360C24—21no—
361C24C2×C3×D411yescharacteristic
362#41—48no—
363#41—48no—
364C2×Q8—24no—
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8C2×C3×D412yes—
385C2×Q8C2×C3×D412yes—
386C2×Q8C2×C3×D412yes—
387C2×D4—24no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—42no—
392C2×D4—42no—
393C2×D4C2×C3×D412yes—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4C2×C3×D412yes—
409C22×C4C2×C3×D412yes—
410C22×C4C2×C3×D411yescharacteristic
411C22×C4C2×C3×D414yes—
412C22×C4C24×C311yescharacteristic
413Q16—48no—
414Q16—48no—
415Q16—48no—
416SD16—48no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8C2×C3×D414yes—
422C2×C8C2×C3×D414yes—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C22⋊C4—28no—
431C22⋊C4—28no—
432C22⋊C4—24no—
433C22⋊C4—44no—
434C42—21no—
435C42—21no—
436C42C2×C3×D412yes—
437C42C2×C3×D412yes—
438C22×C3—24no—
439C22×C3—22no—
440C22×C3—42no—
441C22×C3—42no—
442C22×C3—42no—
443C22×C3—44no—
444C22×C3—82no—
445C22×C3—82no—
446C22×C3C2×#12011yescharacteristic
447C22×C3C2×#12711yescharacteristic
448C22×C3#53511yescharacteristic
449C22×C3#57814yes—
450C3×C4—22no—
451C3×C4—24no—
452C3×C4—42no—
453C3×C4—42no—
454C3×C4—42no—
455C3×C4—42no—
456C3×C4—82no—
457C3×C4—82no—
458C3×C4D4212yes—
459C3×C4D4212yes—
460C23—24no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23—44no—
465C23—41no—
466C23—41no—
467C23C3×#12712yes—
468C23C22×C3×D411yescharacteristic
469Q8—24no—
470Q8—24no—
471Q8—24no—
472Q8—48no—
473Q8—42no—
474Q8—42no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480D4—24no—
481D4—48no—
482D4—42no—
483C2×C4—24no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—48no—
494C2×C4—48no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4C3×#12014yes—
507C2×C4C3×#12712yes—
508C2×C4C22×C3×D411yescharacteristic
509C2×C4C22×C3×D411yescharacteristic
510C2×C4C22×C3×D412yes—
511C2×C4C3×H(5,𝔽2)12yes—
512C8—24no—
513C8—24no—
514C2×C3—42no—
515C2×C3—82no—
516C2×C3#243311yescharacteristic
517C2×C3#301112yes—
518C2×C3#315612yes—
519C2×C3#322312yes—
520C22—24no—
521C22—22no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—44no—
526C22—82no—
527C22—82no—
528C22C2×C3×#12011yescharacteristic
529C22C2×C3×#12711yescharacteristic
530C22C3×#53511yescharacteristic
531C22C3×#57814yes—
532C4—22no—
533C4—24no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—42no—
538C4—82no—
539C4—82no—
540C4C3×D4212yes—
541C4C3×D4212yes—
542C3#2573511yescharacteristic, direct product factor, Sylow
543C2—42no—
544C2—82no—
545C2C3×#243311yescharacteristic
546C2C3×#301112yes—
547C2C3×#315612yes—
548C2C3×#322312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#497C2semidirect
3. C2×C3×#494C2semidirect
5. C2×C3×#451C2semidirect
6. C3×#2621C2semidirect
7. C3×#2016C2semidirect
8. C3×#2015C2semidirect
9. C3×#1960C2semidirect
10. C3×#1958C2semidirect
11. C3×#1750C2semidirect
12. #25735C3direct (N × H)
47. C2×#497C2×C3semidirect
48. C2×#494C2×C3semidirect
50. C2×#451C2×C3semidirect
51. #2621C2×C3semidirect
52. #2016C2×C3semidirect
53. #2015C2×C3semidirect
54. #1960C2×C3semidirect
55. #1958C2×C3semidirect
56. #1750C2×C3semidirect
542. C3#25735direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5362
C2×C3×#4972
C2×C3×#4942
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#257353