Finite Group Databasebeta

← C3×#18636 #10637243

Subgroups — C3×#18636 #10637243

Order 768.

960 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic
2C3×#3226C211yesmaximal, characteristic, complemented
3C2×C3×#487C211yesmaximal, characteristic, complemented
4C3×#3039C211yesmaximal, characteristic, complemented
5C3×#1646C214yesmaximal, complemented
6C3×#1644C214yesmaximal, complemented
7C3×#1548C211yesmaximal, characteristic, complemented
8C3×#1492C211yesmaximal, characteristic, complemented
9C3×#1476C211yesmaximal, characteristic, complemented
10#18636C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#137—21no—
12C3×#556—21no—
13C3×#555C2212yes—
14C3×#554C2212yes—
15C3×#553—22no—
16C3×#548—22no—
17C3×#547—21no—
18C3×#542—22no—
19C3×#537—21no—
20C3×#533—21no—
21C3×#532—21no—
22C3×#532C2211yescharacteristic
23C2×C3×C4⋊Q8C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C3×#520—21no—
26C3×C4×#41C2211yescharacteristic
27C3×#500—21no—
28C3×#487—22no—
29C3×#487C2214yes—
30C3×#486—21no—
31C3×#480—22no—
32C3×#474C2214yes—
33C3×#462—21no—
34C3×#462C2212yes—
35C3×#461—21no—
36C3×#461C2212yes—
37C3×#431C2211yescharacteristic
38C3×#423C2211yescharacteristic
39C3×#419—21no—
40C2×C3×#103—22no—
41C2×C3×C4×C8C2211yescharacteristic
42C3×#333C2214yes—
43C3×Q8⋊C8C2212yes—
44C3×D4⋊C8C2214yes—
45C3×D4⋊C8C2212yes—
46#3511C2×C311yescharacteristic
47#3226C2×C311yescharacteristic, complemented
48C2×#487C2×C311yescharacteristic, complemented
49#3039C2×C311yescharacteristic, complemented
50#1646C2×C314yescomplemented
51#1644C2×C314yescomplemented
52#1548C2×C311yescharacteristic, complemented
53#1492C2×C311yescharacteristic, complemented
54#1476C2×C311yescharacteristic, complemented
55C2×C3×#41—22no—
56C2×C3×#41—21no—
57C2×C3×#41D411yescharacteristic
58C22×C3×Q8—21no—
59C22×C3×Q8D412yes—
60C3×#137—44no—
61C2×C3×Q16—22no—
62C2×C3×SD16—22no—
63C2×C3×M16—21no—
64C22×C3×C8—21no—
65C3×C4⋊Q8—21no—
66C3×C4⋊Q8—21no—
67C3×C4⋊Q8—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8D412yes—
71C3×C4⋊Q8D411yescharacteristic
72C3×C4⋊Q8C2312yes—
73C3×C4⋊Q8C2312yes—
74C3×#127D411yescharacteristic
75C3×#126—24no—
76C3×#125—22no—
77C3×#124—22no—
78C3×#124—24no—
79C3×#124—21no—
80C3×#123—24no—
81C3×C22⋊Q8—22no—
82C3×C22⋊Q8—22no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8—24no—
85C3×#121—22no—
86C3×#121—24no—
87C3×#121—22no—
88C3×#121—22no—
89C3×#121—21no—
90C3×#121D412yes—
91C3×C4×Q8—22no—
92C3×C4×Q8—22no—
93C3×C4×Q8D411yescharacteristic
94C3×C4×Q8C2311yescharacteristic
95C3×C4×D4—22no—
96C3×C4×D4—24no—
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—22no—
105C3×#117—22no—
106C3×#117—21no—
107C3×#117D411yescharacteristic
108C2×C3×C4⋊C4—21no—
109C2×C3×C4⋊C4—21no—
110C2×C3×C4⋊C4—22no—
111C2×C3×C4⋊C4—21no—
112C2×C3×C4⋊C4—21no—
113C2×C3×C42—21no—
114C2×C3×C42C2311yescharacteristic
115C3×#107—24no—
116C3×C4⋊C8—21no—
117C3×C4⋊C8C2312yes—
118C3×C4⋊C8C2312yes—
119C3×#103—28no—
120C3×#103—22no—
121C3×#103—22no—
122C3×#102—24no—
123C3×C22⋊C8—21no—
124C3×C4×C8—21no—
125C3×C4×C8—21no—
126C3×C4×C8C2312yes—
127C2×#137—21no—
128#556—21no—
129#555C22×C312yes—
130#554C22×C312yes—
131#553—22no—
132#548—22no—
133#547—21no—
134#542—22no—
135#537—21no—
136#533—21no—
137#532—21no—
138#532C22×C311yescharacteristic
139C2×C4⋊Q8C22×C311yescharacteristic
140C2×C4⋊Q8C22×C311yescharacteristic
141#520—21no—
142C4×#41C22×C311yescharacteristic
143#500—21no—
144#487—22no—
145#487C22×C314yes—
146#486—21no—
147#480—22no—
148#474C22×C314yes—
149#462—21no—
150#462C22×C312yes—
151#461—21no—
152#461C22×C312yes—
153#431C22×C311yescharacteristic
154#423C22×C311yescharacteristic
155#419—21no—
156C2×#103—22no—
157C2×C4×C8C22×C311yescharacteristic
158#333C22×C314yes—
159Q8⋊C8C22×C312yes—
160D4⋊C8C22×C314yes—
161D4⋊C8C22×C312yes—
162C3×#41—22no—
163C3×#41—22no—
164C3×#41—42no—
165C3×#41—42no—
166C3×#41—84no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×Q8—44no—
171C2×C3×Q8—41no—
172C2×C3×Q8—42no—
173C2×C3×Q8—42no—
174C2×C3×Q8C2×D411yescharacteristic
175C2×C3×Q8C2×D414yes—
176C2×C3×D4—21no—
177C2×C3×D4—22no—
178C2×C3×D4—24no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4C2×D412yes—
182C2×C3×D4C2×D411yescharacteristic
183C22×C3×C4—24no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—41no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C3×Q16—44no—
194C3×SD16—44no—
195C3×M16—42no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—42no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—24no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—21no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—22no—
208C3×C4⋊C4—21no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—41no—
212C3×C4⋊C4—41no—
213C3×C4⋊C4C2×D411yescharacteristic
214C3×C4⋊C4C2×D411yescharacteristic
215C3×C4⋊C4C2×D412yes—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—24no—
219C3×C22⋊C4—21no—
220C3×C22⋊C4—21no—
221C3×C22⋊C4—22no—
222C3×C22⋊C4—21no—
223C3×C22⋊C4—22no—
224C3×C42—24no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42C2×D411yescharacteristic
229C3×C42C2×D411yescharacteristic
230C3×C42C2×D411yescharacteristic
231C3×C42C2411yescharacteristic
232C2×#41—22no—
233C2×#41—21no—
234C2×#41C3×D411yescharacteristic
235C22×Q8—21no—
236C22×Q8C3×D412yes—
237#137—44no—
238C2×Q16—22no—
239C2×SD16—22no—
240C2×M16—21no—
241C22×C8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8C3×D412yes—
248C4⋊Q8C3×D411yescharacteristic
249C4⋊Q8C23×C312yes—
250C4⋊Q8C23×C312yes—
251#127C3×D411yescharacteristic
252#126—24no—
253#125—22no—
254#124—22no—
255#124—24no—
256#124—21no—
257#123—24no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8—24no—
262#121—22no—
263#121—24no—
264#121—22no—
265#121—22no—
266#121—21no—
267#121C3×D412yes—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8C3×D411yescharacteristic
271C4×Q8C23×C311yescharacteristic
272C4×D4—22no—
273C4×D4—24no—
274C4×D4—22no—
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4C3×D411yescharacteristic
279C4×D4C23×C311yescharacteristic
280C4×D4C23×C312yes—
281#117—22no—
282#117—22no—
283#117—21no—
284#117C3×D411yescharacteristic
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C42—21no—
291C2×C42C23×C311yescharacteristic
292#107—24no—
293C4⋊C8—21no—
294C4⋊C8C23×C312yes—
295C4⋊C8C23×C312yes—
296#103—28no—
297#103—22no—
298#103—22no—
299#102—24no—
300C22⋊C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8C23×C312yes—
304C23×C3—21no—
305C23×C3—22no—
306C23×C3#12011yescharacteristic
307C3×Q8—21no—
308C3×Q8—21no—
309C3×Q8—44no—
310C3×Q8—42no—
311C3×Q8—44no—
312C3×Q8—42no—
313C3×Q8—42no—
314C3×Q8—82no—
315C3×D4—21no—
316C3×D4—21no—
317C3×D4—24no—
318C3×D4—42no—
319C3×D4—44no—
320C3×D4—42no—
321C3×D4—42no—
322C3×D4—84no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—24no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—82no—
337C2×C3×C4—84no—
338C2×C3×C4—82no—
339C2×C3×C4#12011yescharacteristic
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4#12011yescharacteristic
342C2×C3×C4#13512yes—
343C2×C3×C4AGL(1,ℤ/8)12yes—
344C2×C3×C4C22×D411yescharacteristic
345C2×C3×C4C22×D411yescharacteristic
346C2×C3×C4C22×D411yescharacteristic
347C3×C8—42no—
348C3×C8—42no—
349C3×C8—42no—
350#41—22no—
351#41—22no—
352#41—42no—
353#41—42no—
354#41—84no—
355C2×Q8—21no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—44no—
359C2×Q8—41no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8C2×C3×D411yescharacteristic
363C2×Q8C2×C3×D414yes—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4C2×C3×D412yes—
370C2×D4C2×C3×D411yescharacteristic
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—41no—
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C2×C3×D411yescharacteristic
380C22×C4C2×C3×D411yescharacteristic
381Q16—44no—
382SD16—44no—
383M16—42no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—42no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4C2×C3×D411yescharacteristic
402C4⋊C4C2×C3×D411yescharacteristic
403C4⋊C4C2×C3×D412yes—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—21no—
408C22⋊C4—21no—
409C22⋊C4—22no—
410C22⋊C4—21no—
411C22⋊C4—22no—
412C42—24no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42C2×C3×D411yescharacteristic
417C42C2×C3×D411yescharacteristic
418C42C2×C3×D411yescharacteristic
419C42C24×C311yescharacteristic
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—41no—
423C22×C3—42no—
424C22×C3—41no—
425C22×C3—42no—
426C22×C3—41no—
427C22×C3—42no—
428C22×C3#44912yes—
429C22×C3C2×#12011yescharacteristic
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—41no—
433C3×C4—42no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—44no—
437C3×C4—82no—
438C3×C4#44912yes—
439C3×C4#45512yes—
440C3×C4C2×#13511yescharacteristic
441C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
442C3×C4#57812yes—
443C23—21no—
444C23—22no—
445C23C3×#12011yescharacteristic
446Q8—21no—
447Q8—21no—
448Q8—44no—
449Q8—42no—
450Q8—44no—
451Q8—42no—
452Q8—42no—
453Q8—82no—
454D4—21no—
455D4—21no—
456D4—24no—
457D4—42no—
458D4—44no—
459D4—42no—
460D4—42no—
461D4—84no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—82no—
476C2×C4—84no—
477C2×C4—82no—
478C2×C4C3×#12011yescharacteristic
479C2×C4C3×#12011yescharacteristic
480C2×C4C3×#12011yescharacteristic
481C2×C4C3×#13512yes—
482C2×C4C3×AGL(1,ℤ/8)12yes—
483C2×C4C22×C3×D411yescharacteristic
484C2×C4C22×C3×D411yescharacteristic
485C2×C4C22×C3×D411yescharacteristic
486C8—42no—
487C8—42no—
488C8—42no—
489C2×C3—22no—
490C2×C3—81no—
491C2×C3—82no—
492C2×C3C2×#44911yescharacteristic
493C2×C3#301111yescharacteristic
494C2×C3C2×#45511yescharacteristic
495C22—21no—
496C22—21no—
497C22—41no—
498C22—42no—
499C22—41no—
500C22—42no—
501C22—41no—
502C22—42no—
503C22C3×#44912yes—
504C22C2×C3×#12011yescharacteristic
505C4—21no—
506C4—21no—
507C4—41no—
508C4—42no—
509C4—41no—
510C4—41no—
511C4—44no—
512C4—82no—
513C4C3×#44912yes—
514C4C3×#45512yes—
515C4C2×C3×#13511yescharacteristic
516C4C2×C3×AGL(1,ℤ/8)11yescharacteristic
517C4C3×#57812yes—
518C3#1863611yescharacteristic, direct product factor, Sylow
519C2—22no—
520C2—81no—
521C2—82no—
522C2C2×C3×#44911yescharacteristic
523C2C3×#301111yescharacteristic
524C2C2×C3×#45511yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC42C24×C3
prime radicalC3×#3511C2
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×#3226C2semidirect
3. C2×C3×#487C2semidirect
4. C3×#3039C2semidirect
5. C3×#1646C2semidirect
6. C3×#1644C2semidirect
7. C3×#1548C2semidirect
8. C3×#1492C2semidirect
9. C3×#1476C2semidirect
10. #18636C3direct (N × H)
47. #3226C2×C3semidirect
48. C2×#487C2×C3semidirect
49. #3039C2×C3semidirect
50. #1646C2×C3semidirect
51. #1644C2×C3semidirect
52. #1548C2×C3semidirect
53. #1492C2×C3semidirect
54. #1476C2×C3semidirect
518. C3#18636direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35112
C3×#32262
C2×C3×#4872
C3×#30392
C3×#16462
C3×#16442
C3×#15482
C3×#14922
C3×#14762
#186363