Finite Group Databasebeta

← C3×#33541 #10652148

Subgroups — C3×#33541 #10652148

Order 768.

1104 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3584C211yesmaximal, characteristic, complemented
2C2×C3×#509C211yesmaximal, characteristic, complemented
3C2×C3×#509C211yesmaximal, characteristic, complemented
4C2×C3×#472C211yesmaximal, characteristic, complemented
5C3×#2193C214yesmaximal, complemented
6C3×#2192C214yesmaximal, complemented
7C3×#2148C211yesmaximal, characteristic, complemented
8C3×#2143C211yesmaximal, characteristic, complemented
9C3×#2117C211yesmaximal, characteristic, complemented
10#33541C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#585—21no—
12C3×#578—21no—
13C3×#578C2212yescomplemented
14C3×#577—22no—
15C3×#577—22no—
16C3×#576—21no—
17C3×#576C2212yescomplemented
18C3×#575—22no—
19C3×#575—21no—
20C2×C3×AGL(1,ℤ/8)—21no—
21C2×C3×#135—22no—
22C2×C3×#135C2211yescharacteristic
23C2×C3×#135C2211yescharacteristic
24C22×C3×D8C2211yescharacteristic, complemented
25C2×C3×#131C2211yescharacteristic
26C3×#509—22no—
27C3×#509—22no—
28C3×#509—22no—
29C3×#509C2214yescomplemented
30C2×C3×SD32—21no—
31C2×C3×SD32C2212yescomplemented
32C2×C3×D16—21no—
33C2×C3×D16C2212yescomplemented
34C2×C3×M32C2211yescharacteristic, complemented
35C2×C3×M32C2211yescharacteristic, complemented
36C3×#472—22no—
37C3×#472C2214yescomplemented
38C2×C3×#108C2211yescharacteristic, complemented
39C2×C3×#100—21no—
40C3×#362C2212yescomplemented
41C3×#361C2212yescomplemented
42C3×#359C2214yescomplemented
43C3×#350C2214yescomplemented
44#3584C2×C311yescharacteristic, complemented
45C2×#509C2×C311yescharacteristic, complemented
46C2×#509C2×C311yescharacteristic, complemented
47C2×#472C2×C311yescharacteristic, complemented
48#2193C2×C314yescomplemented
49#2192C2×C314yescomplemented
50#2148C2×C311yescharacteristic, complemented
51#2143C2×C311yescharacteristic, complemented
52#2117C2×C311yescharacteristic, complemented
53C3×2−1+4—22no—
54C3×2−1+4—22no—
55C3×2−1+4—41no—
56C3×H(5,𝔽2)—22no—
57C3×H(5,𝔽2)—22no—
58C3×H(5,𝔽2)—42no—
59C3×H(5,𝔽2)—41no—
60C2×C3×#41—24no—
61C2×C3×#41—21no—
62C2×C3×#41—21no—
63C2×C3×#41—42no—
64C2×C3×#41—42no—
65C2×C3×#41D411yescharacteristic
66C22×C3×D4—21no—
67C3×#137—22no—
68C3×#137—24no—
69C3×#137—24no—
70C3×#137—22no—
71C3×AGL(1,ℤ/8)—22no—
72C3×AGL(1,ℤ/8)—24no—
73C3×AGL(1,ℤ/8)—24no—
74C3×AGL(1,ℤ/8)—24no—
75C3×AGL(1,ℤ/8)—22no—
76C3×AGL(1,ℤ/8)—42no—
77C3×#135—24no—
78C3×#135—24no—
79C3×#135—22no—
80C3×#135—22no—
81C3×#135—21no—
82C3×#135—21no—
83C3×#135—21no—
84C3×#135—21no—
85C3×#135D412yes—
86C3×#135D412yes—
87C3×#135D412yes—
88C3×#135C2312yes—
89C2×C3×Q16—22no—
90C2×C3×Q16D411yescharacteristic
91C2×C3×Q16C2311yescharacteristic
92C2×C3×SD16—24no—
93C2×C3×SD16—21no—
94C2×C3×SD16—21no—
95C2×C3×SD16—41no—
96C2×C3×D8—22no—
97C2×C3×D8—21no—
98C2×C3×D8—21no—
99C2×C3×D8—21no—
100C2×C3×D8—21no—
101C2×C3×D8—21no—
102C2×C3×D8—41no—
103C2×C3×D8D411yescharacteristic
104C2×C3×D8C2312yes—
105C2×C3×D8C2311yescharacteristic
106C3×#131—21no—
107C3×#131—21no—
108C3×#131D412yes—
109C3×#131C2312yes—
110C2×C3×M16—21no—
111C2×C3×M16—21no—
112C2×C3×M16D411yescharacteristic
113C22×C3×C8—21no—
114C22×C3×C8C2311yescharacteristic
115C3×SD32—24no—
116C3×SD32—24no—
117C3×D16—24no—
118C3×D16—24no—
119C3×M32—21no—
120C3×M32—21no—
121C3×M32—21no—
122C3×M32C2312yes—
123C2×C3×C16—21no—
124C2×C3×C16C2312yes—
125C3×#108—21no—
126C3×#108C2312yes—
127C3×#100—24no—
128#585—21no—
129#578—21no—
130#578C22×C312yescomplemented
131#577—22no—
132#577—22no—
133#576—21no—
134#576C22×C312yescomplemented
135#575—22no—
136#575—21no—
137C2×AGL(1,ℤ/8)—21no—
138C2×#135—22no—
139C2×#135C22×C311yescharacteristic
140C2×#135C22×C311yescharacteristic
141C22×D8C22×C311yescharacteristic, complemented
142C2×#131C22×C311yescharacteristic
143#509—22no—
144#509—22no—
145#509—22no—
146#509C22×C314yescomplemented
147C2×SD32—21no—
148C2×SD32C22×C312yescomplemented
149C2×D16—21no—
150C2×D16C22×C312yescomplemented
151C2×M32C22×C311yescharacteristic, complemented
152C2×M32C22×C311yescharacteristic, complemented
153#472—22no—
154#472C22×C314yescomplemented
155C2×#108C22×C311yescharacteristic, complemented
156C2×#100—21no—
157#362C22×C312yescomplemented
158#361C22×C312yescomplemented
159#359C22×C314yescomplemented
160#350C22×C314yescomplemented
161C24×C3—41no—
162C3×#41—21no—
163C3×#41—21no—
164C3×#41—24no—
165C3×#41—24no—
166C3×#41—22no—
167C3×#41—22no—
168C3×#41—22no—
169C3×#41—22no—
170C3×#41—44no—
171C3×#41—44no—
172C3×#41—44no—
173C3×#41—44no—
174C3×#41—42no—
175C3×#41—42no—
176C3×#41—42no—
177C3×#41—42no—
178C3×#41—41no—
179C3×#41—41no—
180C3×#41—41no—
181C3×#41—41no—
182C3×#41D812yes—
183C3×#41C2×D412yes—
184C2×C3×Q8—24no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—42no—
188C2×C3×Q8—42no—
189C2×C3×Q8D811yescharacteristic
190C2×C3×D4—21no—
191C2×C3×D4—24no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—22no—
195C2×C3×D4—44no—
196C2×C3×D4—44no—
197C2×C3×D4—42no—
198C2×C3×D4—44no—
199C2×C3×D4—41no—
200C2×C3×D4—41no—
201C2×C3×D4—42no—
202C2×C3×D4—41no—
203C2×C3×D4—41no—
204C2×C3×D4—41no—
205C2×C3×D4—41no—
206C2×C3×D4—41no—
207C2×C3×D4D811yescharacteristic
208C22×C3×C4—21no—
209C22×C3×C4—44no—
210C22×C3×C4—41no—
211C22×C3×C4—41no—
212C22×C3×C4C2×D411yescharacteristic
213C3×Q16—24no—
214C3×Q16—21no—
215C3×Q16—21no—
216C3×Q16C2×D412yes—
217C3×Q16C2×D412yes—
218C3×SD16—24no—
219C3×SD16—24no—
220C3×SD16—22no—
221C3×SD16—22no—
222C3×SD16—22no—
223C3×SD16—22no—
224C3×SD16—44no—
225C3×D8—24no—
226C3×D8—21no—
227C3×D8—21no—
228C3×D8—22no—
229C3×D8—22no—
230C3×D8—44no—
231C3×D8—41no—
232C3×D8—41no—
233C3×D8C2×D412yes—
234C3×D8C2×D412yes—
235C3×M16—21no—
236C3×M16—22no—
237C3×M16—22no—
238C3×M16—22no—
239C3×M16—41no—
240C3×M16C2×D412yes—
241C2×C3×C8—21no—
242C2×C3×C8—22no—
243C2×C3×C8—22no—
244C2×C3×C8—41no—
245C2×C3×C8C2×D411yescharacteristic
246C2×C3×C8C2×D411yescharacteristic
247C2×C3×C8C2×D411yescharacteristic
248C2×C3×C8C2×D411yescharacteristic
249C2×C3×C8C2×D411yescharacteristic
250C2×C3×C8C2411yescharacteristic
251C3×C16—22no—
252C3×C16—22no—
2532−1+4—22no—
2542−1+4—22no—
2552−1+4—41no—
256H(5,𝔽2)—22no—
257H(5,𝔽2)—22no—
258H(5,𝔽2)—42no—
259H(5,𝔽2)—41no—
260C2×#41—24no—
261C2×#41—21no—
262C2×#41—21no—
263C2×#41—42no—
264C2×#41—42no—
265C2×#41C3×D411yescharacteristic
266C22×D4—21no—
267#137—22no—
268#137—24no—
269#137—24no—
270#137—22no—
271AGL(1,ℤ/8)—22no—
272AGL(1,ℤ/8)—24no—
273AGL(1,ℤ/8)—24no—
274AGL(1,ℤ/8)—24no—
275AGL(1,ℤ/8)—22no—
276AGL(1,ℤ/8)—42no—
277#135—24no—
278#135—24no—
279#135—22no—
280#135—22no—
281#135—21no—
282#135—21no—
283#135—21no—
284#135—21no—
285#135C3×D412yes—
286#135C3×D412yes—
287#135C3×D412yes—
288#135C23×C312yes—
289C2×Q16—22no—
290C2×Q16C3×D411yescharacteristic
291C2×Q16C23×C311yescharacteristic
292C2×SD16—24no—
293C2×SD16—21no—
294C2×SD16—21no—
295C2×SD16—41no—
296C2×D8—22no—
297C2×D8—21no—
298C2×D8—21no—
299C2×D8—21no—
300C2×D8—21no—
301C2×D8—21no—
302C2×D8—41no—
303C2×D8C3×D411yescharacteristic
304C2×D8C23×C312yes—
305C2×D8C23×C311yescharacteristic
306#131—21no—
307#131—21no—
308#131C3×D412yes—
309#131C23×C312yes—
310C2×M16—21no—
311C2×M16—21no—
312C2×M16C3×D411yescharacteristic
313C22×C8—21no—
314C22×C8C23×C311yescharacteristic
315SD32—24no—
316SD32—24no—
317D16—24no—
318D16—24no—
319M32—21no—
320M32—21no—
321M32—21no—
322M32C23×C312yes—
323C2×C16—21no—
324C2×C16C23×C312yes—
325#108—21no—
326#108C23×C312yes—
327#100—24no—
328C23×C3—21no—
329C23×C3—44no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—42no—
333C23×C3—81no—
334C23×C3—81no—
335C23×C3—162no—
336C23×C3AGL(1,ℤ/8)11yescharacteristic
337C3×Q8—21no—
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—44no—
341C3×Q8—41no—
342C3×Q8—41no—
343C3×Q8C2×D812yes—
344C3×D4—22no—
345C3×D4—22no—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—21no—
349C3×D4—44no—
350C3×D4—44no—
351C3×D4—44no—
352C3×D4—42no—
353C3×D4—42no—
354C3×D4—42no—
355C3×D4—42no—
356C3×D4—42no—
357C3×D4—41no—
358C3×D4—42no—
359C3×D4—41no—
360C3×D4—81no—
361C3×D4—81no—
362C3×D4C2×D812yes—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—44no—
367C2×C3×C4—44no—
368C2×C3×C4—44no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4#12011yescharacteristic
376C2×C3×C4#12011yescharacteristic
377C2×C3×C4C2×D811yescharacteristic
378C2×C3×C4C2×D811yescharacteristic
379C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
380C2×C3×C4C22×D411yescharacteristic
381C3×C8—22no—
382C3×C8—42no—
383C3×C8#12011yescharacteristic
384C3×C8#12011yescharacteristic
385C3×C8C22×D411yescharacteristic
386C3×C8C22×D411yescharacteristic
387C24—41no—
388#41—21no—
389#41—21no—
390#41—24no—
391#41—24no—
392#41—22no—
393#41—22no—
394#41—22no—
395#41—22no—
396#41—44no—
397#41—44no—
398#41—44no—
399#41—44no—
400#41—42no—
401#41—42no—
402#41—42no—
403#41—42no—
404#41—41no—
405#41—41no—
406#41—41no—
407#41—41no—
408#41C3×D812yes—
409#41C2×C3×D412yes—
410C2×Q8—24no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8C3×D811yescharacteristic
416C2×D4—21no—
417C2×D4—24no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—22no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—44no—
425C2×D4—41no—
426C2×D4—41no—
427C2×D4—42no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—41no—
433C2×D4C3×D811yescharacteristic
434C22×C4—21no—
435C22×C4—44no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4C2×C3×D411yescharacteristic
439Q16—24no—
440Q16—21no—
441Q16—21no—
442Q16C2×C3×D412yes—
443Q16C2×C3×D412yes—
444SD16—24no—
445SD16—24no—
446SD16—22no—
447SD16—22no—
448SD16—22no—
449SD16—22no—
450SD16—44no—
451D8—24no—
452D8—21no—
453D8—21no—
454D8—22no—
455D8—22no—
456D8—44no—
457D8—41no—
458D8—41no—
459D8C2×C3×D412yes—
460D8C2×C3×D412yes—
461M16—21no—
462M16—22no—
463M16—22no—
464M16—22no—
465M16—41no—
466M16C2×C3×D412yes—
467C2×C8—21no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—41no—
471C2×C8C2×C3×D411yescharacteristic
472C2×C8C2×C3×D411yescharacteristic
473C2×C8C2×C3×D411yescharacteristic
474C2×C8C2×C3×D411yescharacteristic
475C2×C8C2×C3×D411yescharacteristic
476C2×C8C24×C311yescharacteristic
477C16—22no—
478C16—22no—
479C22×C3—22no—
480C22×C3—41no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—81no—
484C22×C3—82no—
485C22×C3—161no—
486C22×C3—161no—
487C22×C3—162no—
488C22×C3—162no—
489C22×C3—162no—
490C22×C3—164no—
491C22×C3#44711yescharacteristic
492C22×C3#44711yescharacteristic
493C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
494C3×C4—22no—
495C3×C4—42no—
496C3×C4—42no—
497C3×C4#44711yescharacteristic
498C3×C4#44711yescharacteristic
499C3×C4C2×#12011yescharacteristic
500C3×C4C22×D811yescharacteristic
501C23—21no—
502C23—44no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—81no—
507C23—81no—
508C23—162no—
509C23C3×AGL(1,ℤ/8)11yescharacteristic
510Q8—21no—
511Q8—22no—
512Q8—22no—
513Q8—44no—
514Q8—41no—
515Q8—41no—
516Q8C2×C3×D812yes—
517D4—22no—
518D4—22no—
519D4—21no—
520D4—22no—
521D4—22no—
522D4—44no—
523D4—44no—
524D4—44no—
525D4—42no—
526D4—42no—
527D4—42no—
528D4—42no—
529D4—41no—
530D4—42no—
531D4—41no—
532D4—42no—
533D4—81no—
534D4—81no—
535D4C2×C3×D812yes—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4C3×#12011yescharacteristic
549C2×C4C3×#12011yescharacteristic
550C2×C4C2×C3×D811yescharacteristic
551C2×C4C2×C3×D811yescharacteristic
552C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
553C2×C4C22×C3×D411yescharacteristic
554C8—22no—
555C8—42no—
556C8C3×#12011yescharacteristic
557C8C3×#12011yescharacteristic
558C8C22×C3×D411yescharacteristic
559C8C22×C3×D411yescharacteristic
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—21no—
563C2×C3—42no—
564C2×C3—82no—
565C2×C3—82no—
566C2×C3—162no—
567C2×C3C2×#44711yescharacteristic
568C22—22no—
569C22—41no—
570C22—42no—
571C22—42no—
572C22—81no—
573C22—82no—
574C22—161no—
575C22—161no—
576C22—162no—
577C22—162no—
578C22—162no—
579C22—164no—
580C22C3×#44711yescharacteristic
581C22C3×#44711yescharacteristic
582C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
583C4—22no—
584C4—42no—
585C4—42no—
586C4C3×#44711yescharacteristic
587C4C3×#44711yescharacteristic
588C4C2×C3×#12011yescharacteristic
589C4C22×C3×D811yescharacteristic
590C3#3354111yescharacteristic, direct product factor, Sylow
591C2—21no—
592C2—21no—
593C2—21no—
594C2—42no—
595C2—82no—
596C2—82no—
597C2—162no—
598C2C2×C3×#44711yescharacteristic
subgroupG/H
center, socleC2×C3C2×#447
commutator, Frattini, cosocleC2×C8C24×C3
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3584C2semidirect
2. C2×C3×#509C2semidirect
3. C2×C3×#509C2semidirect
4. C2×C3×#472C2semidirect
5. C3×#2193C2semidirect
6. C3×#2192C2semidirect
7. C3×#2148C2semidirect
8. C3×#2143C2semidirect
9. C3×#2117C2semidirect
10. #33541C3direct (N × H)
13. C3×#578C22semidirect
17. C3×#576C22semidirect
24. C22×C3×D8C22semidirect
29. C3×#509C22semidirect
31. C2×C3×SD32C22semidirect
33. C2×C3×D16C22semidirect
34. C2×C3×M32C22semidirect
35. C2×C3×M32C22semidirect
37. C3×#472C22semidirect
38. C2×C3×#108C22semidirect
40. C3×#362C22semidirect
41. C3×#361C22semidirect
42. C3×#359C22semidirect
43. C3×#350C22semidirect
44. #3584C2×C3semidirect
45. C2×#509C2×C3semidirect
46. C2×#509C2×C3semidirect
47. C2×#472C2×C3semidirect
48. #2193C2×C3semidirect
49. #2192C2×C3semidirect
50. #2148C2×C3semidirect
51. #2143C2×C3semidirect
52. #2117C2×C3semidirect
130. #578C22×C3semidirect
134. #576C22×C3semidirect
141. C22×D8C22×C3semidirect
146. #509C22×C3semidirect
148. C2×SD32C22×C3semidirect
150. C2×D16C22×C3semidirect
151. C2×M32C22×C3semidirect
152. C2×M32C22×C3semidirect
154. #472C22×C3semidirect
155. C2×#108C22×C3semidirect
157. #362C22×C3semidirect
158. #361C22×C3semidirect
159. #359C22×C3semidirect
160. #350C22×C3semidirect
590. C3#33541direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35842
C2×C3×#5092
C2×C3×#4722
C3×#21932
C3×#21922
C3×#21482
C3×#21432
C3×#21172
#335413