Finite Group Databasebeta

← C3×#18641 #10637248

Subgroups — C3×#18641 #10637248

Order 768.

1114 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3499C211yesmaximal, characteristic
2C3×#3227C211yesmaximal, characteristic, complemented
3C2×C3×#487C211yesmaximal, characteristic, complemented
4C3×#3079C211yesmaximal, characteristic, complemented
5C3×#1637C214yesmaximal, complemented
6C3×#1635C214yesmaximal, complemented
7C3×#1546C211yesmaximal, characteristic, complemented
8C3×#1492C211yesmaximal, characteristic, complemented
9C3×#1476C211yesmaximal, characteristic, complemented
10#18641C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×2−1+4—21no—
12C3×D4×Q8—21no—
13C3×D4×Q8C2212yes—
14C3×#547—22no—
15C3×#547—21no—
16C3×#547—22no—
17C3×#547C2212yes—
18C3×#536—22no—
19C3×#536—21no—
20C3×#532—21no—
21C3×#532C2211yescharacteristic
22C2×C3×C4⋊Q8—21no—
23C2×C3×C4⋊Q8C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C3×C4×#41C2211yescharacteristic
26C3×#498—21no—
27C3×#487—22no—
28C3×#487C2214yes—
29C3×#486—21no—
30C3×#481—22no—
31C3×#477C2212yes—
32C3×#476C2214yes—
33C3×#476C2212yes—
34C3×#475—21no—
35C3×#474—21no—
36C3×#431C2211yescharacteristic
37C3×#428—21no—
38C3×#423C2211yescharacteristic
39C3×#417—21no—
40C2×C3×#103—22no—
41C2×C3×C4×C8C2211yescharacteristic
42C3×#332C2214yes—
43C3×Q8⋊C8C2212yes—
44C3×D4⋊C8C2214yes—
45C3×D4⋊C8C2212yes—
46#3499C2×C311yescharacteristic
47#3227C2×C311yescharacteristic, complemented
48C2×#487C2×C311yescharacteristic, complemented
49#3079C2×C311yescharacteristic, complemented
50#1637C2×C314yescomplemented
51#1635C2×C314yescomplemented
52#1546C2×C311yescharacteristic, complemented
53#1492C2×C311yescharacteristic, complemented
54#1476C2×C311yescharacteristic, complemented
55C3×2−1+4—44no—
56C3×2−1+4—44no—
57C2×C3×#41—22no—
58C2×C3×#41—24no—
59C2×C3×#41—22no—
60C2×C3×#41—21no—
61C2×C3×#41D411yescharacteristic
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8—21no—
65C22×C3×Q8—21no—
66C2×C3×M16—21no—
67C22×C3×C8—21no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8—21no—
76C3×C4⋊Q8D411yescharacteristic
77C3×C4⋊Q8D412yes—
78C3×C4⋊Q8C2312yes—
79C3×C4⋊Q8C2312yes—
80C3×#127D411yescharacteristic
81C3×#124—22no—
82C3×#124—21no—
83C3×#123—24no—
84C3×#123—24no—
85C3×#123—24no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—24no—
89C3×C22⋊Q8—22no—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121—21no—
93C3×#121D412yes—
94C3×C4×Q8D411yescharacteristic
95C3×C4×Q8C2311yescharacteristic
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C3×C4×D4—21no—
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×C4×D4D411yescharacteristic
102C3×C4×D4C2311yescharacteristic
103C3×C4×D4C2312yes—
104C3×#117—21no—
105C3×#117D411yescharacteristic
106C2×C3×C4⋊C4—22no—
107C2×C3×C4⋊C4—21no—
108C2×C3×C4⋊C4—22no—
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4—21no—
111C2×C3×C4⋊C4D412yes—
112C2×C3×C42—21no—
113C2×C3×C42C2311yescharacteristic
114C3×#107—22no—
115C3×#106—22no—
116C3×#106—24no—
117C3×C4⋊C8—21no—
118C3×C4⋊C8C2312yes—
119C3×C4⋊C8C2312yes—
120C3×#103—22no—
121C3×#103—28no—
122C3×#102—22no—
123C3×#102—24no—
124C3×C22⋊C8—21no—
125C3×C4×C8—21no—
126C3×C4×C8—21no—
127C3×C4×C8C2312yes—
128C2×2−1+4—21no—
129D4×Q8—21no—
130D4×Q8C22×C312yes—
131#547—22no—
132#547—21no—
133#547—22no—
134#547C22×C312yes—
135#536—22no—
136#536—21no—
137#532—21no—
138#532C22×C311yescharacteristic
139C2×C4⋊Q8—21no—
140C2×C4⋊Q8C22×C311yescharacteristic
141C2×C4⋊Q8C22×C311yescharacteristic
142C4×#41C22×C311yescharacteristic
143#498—21no—
144#487—22no—
145#487C22×C314yes—
146#486—21no—
147#481—22no—
148#477C22×C312yes—
149#476C22×C312yes—
150#476C22×C314yes—
151#475—21no—
152#474—21no—
153#431C22×C311yescharacteristic
154#428—21no—
155#423C22×C311yescharacteristic
156#417—21no—
157C2×#103—22no—
158C2×C4×C8C22×C311yescharacteristic
159#332C22×C314yes—
160Q8⋊C8C22×C312yes—
161D4⋊C8C22×C312yes—
162D4⋊C8C22×C314yes—
163C3×#41—22no—
164C3×#41—48no—
165C3×#41—416no—
166C3×#41—44no—
167C3×#41—44no—
168C3×#41—42no—
169C3×#41—42no—
170C3×#41SD1614yes—
171C2×C3×Q8—24no—
172C2×C3×Q8—22no—
173C2×C3×Q8—21no—
174C2×C3×Q8—22no—
175C2×C3×Q8—22no—
176C2×C3×Q8—48no—
177C2×C3×Q8—44no—
178C2×C3×Q8—44no—
179C2×C3×Q8—44no—
180C2×C3×Q8—41no—
181C2×C3×Q8—42no—
182C2×C3×Q8—42no—
183C2×C3×Q8C2×D411yescharacteristic
184C2×C3×D4—21no—
185C2×C3×D4—24no—
186C2×C3×D4—22no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4C2×D412yes—
190C2×C3×D4C2×D411yescharacteristic
191C22×C3×C4—21no—
192C22×C3×C4—24no—
193C22×C3×C4—24no—
194C22×C3×C4—22no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—22no—
198C22×C3×C4—41no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C3×M16—42no—
203C2×C3×C8—22no—
204C2×C3×C8—22no—
205C2×C3×C8—22no—
206C2×C3×C8—42no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—22no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—41no—
218C3×C4⋊C4—41no—
219C3×C4⋊C4C2×D414yes—
220C3×C4⋊C4C2×D412yes—
221C3×C4⋊C4C2×D411yescharacteristic
222C3×C4⋊C4C2×D411yescharacteristic
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—21no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—22no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—22no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42C2×D411yescharacteristic
234C3×C42C2×D411yescharacteristic
235C3×C42C2×D411yescharacteristic
236C3×C42C2411yescharacteristic
2372−1+4—44no—
2382−1+4—44no—
239C2×#41—22no—
240C2×#41—24no—
241C2×#41—22no—
242C2×#41—21no—
243C2×#41C3×D411yescharacteristic
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—21no—
247C22×Q8—21no—
248C2×M16—21no—
249C22×C8—21no—
250C4⋊Q8—22no—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—21no—
258C4⋊Q8C3×D411yescharacteristic
259C4⋊Q8C3×D412yes—
260C4⋊Q8C23×C312yes—
261C4⋊Q8C23×C312yes—
262#127C3×D411yescharacteristic
263#124—22no—
264#124—21no—
265#123—24no—
266#123—24no—
267#123—24no—
268C22⋊Q8—24no—
269C22⋊Q8—22no—
270C22⋊Q8—24no—
271C22⋊Q8—22no—
272#121—22no—
273#121—22no—
274#121—21no—
275#121C3×D412yes—
276C4×Q8C3×D411yescharacteristic
277C4×Q8C23×C311yescharacteristic
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4C3×D411yescharacteristic
284C4×D4C23×C311yescharacteristic
285C4×D4C23×C312yes—
286#117—21no—
287#117C3×D411yescharacteristic
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4C3×D412yes—
294C2×C42—21no—
295C2×C42C23×C311yescharacteristic
296#107—22no—
297#106—22no—
298#106—24no—
299C4⋊C8—21no—
300C4⋊C8C23×C312yes—
301C4⋊C8C23×C312yes—
302#103—22no—
303#103—28no—
304#102—22no—
305#102—24no—
306C22⋊C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8C23×C312yes—
310C23×C3—21no—
311C23×C3—22no—
312C23×C3#12011yescharacteristic
313C3×Q8—21no—
314C3×Q8—48no—
315C3×Q8—44no—
316C3×Q8—42no—
317C3×Q8—42no—
318C3×Q8—42no—
319C3×Q8—44no—
320C3×Q8—82no—
321C3×Q8C2×SD1612yes—
322C3×D4—24no—
323C3×D4—21no—
324C3×D4—24no—
325C3×D4—48no—
326C3×D4—44no—
327C3×D4—42no—
328C3×D4—42no—
329C3×D4C2×SD1612yes—
330C2×C3×C4—21no—
331C2×C3×C4—22no—
332C2×C3×C4—24no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—24no—
337C2×C3×C4—22no—
338C2×C3×C4—48no—
339C2×C3×C4—48no—
340C2×C3×C4—44no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—82no—
347C2×C3×C4#12011yescharacteristic
348C2×C3×C4#12011yescharacteristic
349C2×C3×C4#12011yescharacteristic
350C2×C3×C4C2×SD1612yes—
351C2×C3×C4AGL(1,ℤ/8)12yes—
352C2×C3×C4C22×D411yescharacteristic
353C2×C3×C4C22×D411yescharacteristic
354C2×C3×C4C22×D411yescharacteristic
355C3×C8—42no—
356C3×C8—42no—
357C3×C8—42no—
358#41—22no—
359#41—48no—
360#41—416no—
361#41—44no—
362#41—44no—
363#41—42no—
364#41—42no—
365#41C3×SD1614yes—
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—21no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—48no—
372C2×Q8—44no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—41no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8C2×C3×D411yescharacteristic
379C2×D4—21no—
380C2×D4—24no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4C2×C3×D412yes—
385C2×D4C2×C3×D411yescharacteristic
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—41no—
394C22×C4C2×C3×D411yescharacteristic
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D411yescharacteristic
397M16—42no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—42no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4C2×C3×D414yes—
415C4⋊C4C2×C3×D412yes—
416C4⋊C4C2×C3×D411yescharacteristic
417C4⋊C4C2×C3×D411yescharacteristic
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—21no—
421C22⋊C4—21no—
422C22⋊C4—22no—
423C22⋊C4—21no—
424C22⋊C4—22no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42C2×C3×D411yescharacteristic
429C42C2×C3×D411yescharacteristic
430C42C2×C3×D411yescharacteristic
431C42C24×C311yescharacteristic
432C22×C3—21no—
433C22×C3—21no—
434C22×C3—22no—
435C22×C3—41no—
436C22×C3—41no—
437C22×C3—42no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3#45012yes—
441C22×C3C2×#12011yescharacteristic
442C3×C4—22no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—44no—
446C3×C4—42no—
447C3×C4—41no—
448C3×C4—41no—
449C3×C4—82no—
450C3×C4#45012yes—
451C3×C4#45412yes—
452C3×C4C22×SD1611yescharacteristic
453C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
454C3×C4#57812yes—
455C23—21no—
456C23—22no—
457C23C3×#12011yescharacteristic
458Q8—21no—
459Q8—48no—
460Q8—44no—
461Q8—42no—
462Q8—42no—
463Q8—42no—
464Q8—44no—
465Q8—82no—
466Q8C2×C3×SD1612yes—
467D4—24no—
468D4—21no—
469D4—24no—
470D4—48no—
471D4—44no—
472D4—42no—
473D4—42no—
474D4C2×C3×SD1612yes—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—82no—
492C2×C4C3×#12011yescharacteristic
493C2×C4C3×#12011yescharacteristic
494C2×C4C3×#12011yescharacteristic
495C2×C4C2×C3×SD1612yes—
496C2×C4C3×AGL(1,ℤ/8)12yes—
497C2×C4C22×C3×D411yescharacteristic
498C2×C4C22×C3×D411yescharacteristic
499C2×C4C22×C3×D411yescharacteristic
500C8—42no—
501C8—42no—
502C8—42no—
503C2×C3—22no—
504C2×C3—42no—
505C2×C3—82no—
506C2×C3C2×#45011yescharacteristic
507C2×C3#301111yescharacteristic
508C2×C3C2×#45411yescharacteristic
509C22—21no—
510C22—21no—
511C22—22no—
512C22—41no—
513C22—41no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22C3×#45012yes—
518C22C2×C3×#12011yescharacteristic
519C4—22no—
520C4—21no—
521C4—21no—
522C4—44no—
523C4—42no—
524C4—41no—
525C4—41no—
526C4—82no—
527C4C3×#45012yes—
528C4C3×#45412yes—
529C4C22×C3×SD1611yescharacteristic
530C4C2×C3×AGL(1,ℤ/8)11yescharacteristic
531C4C3×#57812yes—
532C3#1864111yescharacteristic, direct product factor, Sylow
533C2—22no—
534C2—42no—
535C2—82no—
536C2C2×C3×#45011yescharacteristic
537C2C3×#301111yescharacteristic
538C2C2×C3×#45411yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3499C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3227C2semidirect
3. C2×C3×#487C2semidirect
4. C3×#3079C2semidirect
5. C3×#1637C2semidirect
6. C3×#1635C2semidirect
7. C3×#1546C2semidirect
8. C3×#1492C2semidirect
9. C3×#1476C2semidirect
10. #18641C3direct (N × H)
47. #3227C2×C3semidirect
48. C2×#487C2×C3semidirect
49. #3079C2×C3semidirect
50. #1637C2×C3semidirect
51. #1635C2×C3semidirect
52. #1546C2×C3semidirect
53. #1492C2×C3semidirect
54. #1476C2×C3semidirect
532. C3#18641direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34992
C3×#32272
C2×C3×#4872
C3×#30792
C3×#16372
C3×#16352
C3×#15462
C3×#14922
C3×#14762
#186413