Finite Group Databasebeta

← C3×#33570 #10652177

Subgroups — C3×#33570 #10652177

Order 768.

1212 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3549C211yesmaximal, characteristic, complemented
2C3×#3021C212yesmaximal, complemented
3C3×#3018C211yesmaximal, characteristic, complemented
4C3×#2204C214yesmaximal, complemented
5C3×#2202C214yesmaximal, complemented
6C3×#2134C212yesmaximal, complemented
7C3×#2132C211yesmaximal, characteristic, complemented
8#33570C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#585C2211yescharacteristic, complemented
10C3×#575—21no—
11C2×C3×AGL(1,ℤ/8)—22no—
12C3×H(3,𝔽4)—21no—
13C3×H(3,𝔽4)C2212yescomplemented
14C3×#555—22no—
15C3×#555C2212yescomplemented
16C3×D42—22no—
17C3×D42—21no—
18C3×#535—21no—
19C3×#535—24no—
20C3×#535C2212yes—
21C3×#532—21no—
22C3×#532C2211yescharacteristic
23C3×#456C2212yescomplemented
24C3×#455—21no—
25C3×#454C2212yescomplemented
26C3×#454C2218yescomplemented
27C3×#453—21no—
28C3×#453—24no—
29C3×#421—22no—
30C2×C3×#104—21no—
31C3×#413C2212yescomplemented
32C3×#413C2211yescharacteristic, complemented
33C3×#356C2212yescomplemented
34C3×#355C2212yescomplemented
35C3×#355C2218yescomplemented
36#3549C2×C311yescharacteristic, complemented
37#3021C2×C312yescomplemented
38#3018C2×C311yescharacteristic, complemented
39#2204C2×C314yescomplemented
40#2202C2×C314yescomplemented
41#2134C2×C312yescomplemented
42#2132C2×C311yescharacteristic, complemented
43C3×2−1+4—41no—
44C3×2−1+4C2312yes—
45C3×H(5,𝔽2)—22no—
46C3×H(5,𝔽2)—21no—
47C3×H(5,𝔽2)—41no—
48C2×C3×#41—22no—
49C2×C3×#41—22no—
50C2×C3×#41—21no—
51C2×C3×#41—42no—
52C2×C3×#41D411yescharacteristic
53C2×C3×#41C2311yescharacteristic
54C22×C3×D4—24no—
55C22×C3×D4—21no—
56C22×C3×D4D412yes—
57C3×#137—22no—
58C3×#137—41no—
59C3×AGL(1,ℤ/8)—28no—
60C3×AGL(1,ℤ/8)—22no—
61C3×AGL(1,ℤ/8)—41no—
62C3×AGL(1,ℤ/8)—44no—
63C3×#135—42no—
64C2×C3×SD16—42no—
65C2×C3×D8—42no—
66C2×C3×M16—22no—
67C2×C3×M16—21no—
68C3×C4⋊Q8—21no—
69C3×C4⋊Q8C2311yescharacteristic
70C3×#127—21no—
71C3×#127—22no—
72C3×#127D411yescharacteristic
73C3×#127D414yes—
74C3×#124—24no—
75C3×#124—22no—
76C3×#124—22no—
77C3×#124—24no—
78C3×#124D411yescharacteristic
79C3×#124C2311yescharacteristic
80C3×#124C2314yes—
81C3×#121—22no—
82C3×#121—22no—
83C3×#121—24no—
84C3×#121—22no—
85C3×#121—24no—
86C3×#121—22no—
87C3×#121—22no—
88C3×#120—24no—
89C3×#120—22no—
90C3×#120—22no—
91C3×#120—22no—
92C3×#120—24no—
93C3×#120—22no—
94C3×C4×D4—22no—
95C3×C4×D4—22no—
96C3×C4×D4—22no—
97C3×#117—21no—
98C3×#117—24no—
99C3×#117D412yes—
100C2×C3×C42D411yescharacteristic
101C3×#104—22no—
102C3×#104—22no—
103C3×#104—28no—
104C3×#101C2314yes—
105C3×#101C2312yes—
106C3×#100—22no—
107C3×#100—21no—
108#585C22×C311yescharacteristic, complemented
109#575—21no—
110C2×AGL(1,ℤ/8)—22no—
111H(3,𝔽4)—21no—
112H(3,𝔽4)C22×C312yescomplemented
113#555—22no—
114#555C22×C312yescomplemented
115D42—22no—
116D42—21no—
117#535—21no—
118#535—24no—
119#535C22×C312yes—
120#532—21no—
121#532C22×C311yescharacteristic
122#456C22×C312yescomplemented
123#455—21no—
124#454C22×C312yescomplemented
125#454C22×C318yescomplemented
126#453—21no—
127#453—24no—
128#421—22no—
129C2×#104—21no—
130#413C22×C312yescomplemented
131#413C22×C311yescharacteristic, complemented
132#356C22×C312yescomplemented
133#355C22×C312yescomplemented
134#355C22×C318yescomplemented
135C24×C3—22no—
136C24×C3—22no—
137C3×#41—22no—
138C3×#41—24no—
139C3×#41—24no—
140C3×#41—22no—
141C3×#41—41no—
142C3×#41—41no—
143C3×#41—42no—
144C3×#41—41no—
145C3×#41—41no—
146C3×#41—41no—
147C3×#41—42no—
148C3×#41—44no—
149C3×#41—42no—
150C3×#41—41no—
151C3×#41—84no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8—42no—
156C2×C3×Q8C2×D411yescharacteristic
157C2×C3×Q8C2411yescharacteristic
158C2×C3×D4—22no—
159C2×C3×D4—21no—
160C2×C3×D4—22no—
161C2×C3×D4—22no—
162C2×C3×D4—22no—
163C2×C3×D4—22no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—21no—
167C2×C3×D4—24no—
168C2×C3×D4—22no—
169C2×C3×D4—42no—
170C2×C3×D4—42no—
171C2×C3×D4—44no—
172C2×C3×D4—42no—
173C2×C3×D4—42no—
174C2×C3×D4—41no—
175C2×C3×D4—88no—
176C2×C3×D4—81no—
177C2×C3×D4—81no—
178C2×C3×D4C2×D411yescharacteristic
179C2×C3×D4C2×D414yes—
180C2×C3×D4C2×D411yescharacteristic
181C2×C3×D4C2×D412yes—
182C22×C3×C4—24no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4—41no—
186C22×C3×C4—42no—
187C22×C3×C4C2×D411yescharacteristic
188C22×C3×C4C2×D412yes—
189C3×Q16—42no—
190C3×SD16—42no—
191C3×SD16—48no—
192C3×SD16—42no—
193C3×D8—48no—
194C3×D8—42no—
195C3×M16—24no—
196C3×M16—22no—
197C3×M16—42no—
198C3×M16—41no—
199C2×C3×C8—42no—
200C2×C3×C8—41no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C22⋊C4—24no—
205C3×C22⋊C4—22no—
206C3×C22⋊C4—24no—
207C3×C22⋊C4—22no—
208C3×C22⋊C4—22no—
209C3×C22⋊C4—24no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—24no—
212C3×C42—22no—
213C3×C42—21no—
214C3×C42C2×D411yescharacteristic
215C3×C42C2×D411yescharacteristic
216C3×C42C2×D414yes—
2172−1+4—41no—
2182−1+4C23×C312yes—
219H(5,𝔽2)—22no—
220H(5,𝔽2)—21no—
221H(5,𝔽2)—41no—
222C2×#41—22no—
223C2×#41—22no—
224C2×#41—21no—
225C2×#41—42no—
226C2×#41C3×D411yescharacteristic
227C2×#41C23×C311yescharacteristic
228C22×D4—24no—
229C22×D4—21no—
230C22×D4C3×D412yes—
231#137—22no—
232#137—41no—
233AGL(1,ℤ/8)—28no—
234AGL(1,ℤ/8)—22no—
235AGL(1,ℤ/8)—41no—
236AGL(1,ℤ/8)—44no—
237#135—42no—
238C2×SD16—42no—
239C2×D8—42no—
240C2×M16—22no—
241C2×M16—21no—
242C4⋊Q8—21no—
243C4⋊Q8C23×C311yescharacteristic
244#127—21no—
245#127—22no—
246#127C3×D411yescharacteristic
247#127C3×D414yes—
248#124—24no—
249#124—22no—
250#124—22no—
251#124—24no—
252#124C3×D411yescharacteristic
253#124C23×C311yescharacteristic
254#124C23×C314yes—
255#121—22no—
256#121—22no—
257#121—24no—
258#121—22no—
259#121—24no—
260#121—22no—
261#121—22no—
262#120—22no—
263#120—24no—
264#120—22no—
265#120—22no—
266#120—24no—
267#120—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271#117—21no—
272#117—24no—
273#117C3×D412yes—
274C2×C42C3×D411yescharacteristic
275#104—22no—
276#104—22no—
277#104—28no—
278#101C23×C312yes—
279#101C23×C314yes—
280#100—22no—
281#100—21no—
282C23×C3—24no—
283C23×C3—21no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—82no—
287C23×C3—84no—
288C23×C3—84no—
289C23×C3—82no—
290C23×C3#12012yes—
291C23×C3#12011yescharacteristic
292C3×Q8—24no—
293C3×Q8—22no—
294C3×Q8—41no—
295C3×Q8—41no—
296C3×Q8—41no—
297C3×Q8—82no—
298C3×D4—24no—
299C3×D4—22no—
300C3×D4—42no—
301C3×D4—41no—
302C3×D4—44no—
303C3×D4—41no—
304C3×D4—44no—
305C3×D4—44no—
306C3×D4—42no—
307C3×D4—44no—
308C3×D4—42no—
309C3×D4—41no—
310C3×D4—42no—
311C3×D4—42no—
312C3×D4—81no—
313C3×D4—84no—
314C3×D4—84no—
315C3×D4—82no—
316C3×D4—84no—
317C3×D4—82no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—24no—
326C2×C3×C4—48no—
327C2×C3×C4—42no—
328C2×C3×C4—44no—
329C2×C3×C4—42no—
330C2×C3×C4—41no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4—41no—
334C2×C3×C4—42no—
335C2×C3×C4#12011yescharacteristic
336C2×C3×C4C22×D411yescharacteristic
337C2×C3×C4C22×D412yes—
338C3×C8—44no—
339C3×C8—42no—
340C24—22no—
341C24—22no—
342#41—22no—
343#41—24no—
344#41—24no—
345#41—22no—
346#41—41no—
347#41—41no—
348#41—42no—
349#41—41no—
350#41—41no—
351#41—41no—
352#41—42no—
353#41—44no—
354#41—42no—
355#41—41no—
356#41—84no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—42no—
361C2×Q8C2×C3×D411yescharacteristic
362C2×Q8C24×C311yescharacteristic
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—24no—
370C2×D4—24no—
371C2×D4—21no—
372C2×D4—24no—
373C2×D4—22no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—41no—
380C2×D4—88no—
381C2×D4—81no—
382C2×D4—81no—
383C2×D4C2×C3×D411yescharacteristic
384C2×D4C2×C3×D414yes—
385C2×D4C2×C3×D411yescharacteristic
386C2×D4C2×C3×D412yes—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4C2×C3×D411yescharacteristic
393C22×C4C2×C3×D412yes—
394Q16—42no—
395SD16—42no—
396SD16—48no—
397SD16—42no—
398D8—48no—
399D8—42no—
400M16—24no—
401M16—22no—
402M16—42no—
403M16—41no—
404C2×C8—42no—
405C2×C8—41no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C22⋊C4—24no—
410C22⋊C4—22no—
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—24no—
415C22⋊C4—22no—
416C22⋊C4—24no—
417C42—22no—
418C42—21no—
419C42C2×C3×D411yescharacteristic
420C42C2×C3×D411yescharacteristic
421C42C2×C3×D414yes—
422C22×C3—21no—
423C22×C3—22no—
424C22×C3—41no—
425C22×C3—42no—
426C22×C3—44no—
427C22×C3—44no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—41no—
431C22×C3—44no—
432C22×C3—161no—
433C22×C3—162no—
434C22×C3—164no—
435C22×C3—164no—
436C22×C3C2×#12011yescharacteristic
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—21no—
440C3×C4—44no—
441C3×C4—41no—
442C3×C4—41no—
443C3×C4—42no—
444C3×C4B(4,𝔽2)11yescharacteristic
445C3×C4B(4,𝔽2)11yescharacteristic
446C23—24no—
447C23—21no—
448C23—41no—
449C23—42no—
450C23—82no—
451C23—84no—
452C23—84no—
453C23—82no—
454C23C3×#12012yes—
455C23C3×#12011yescharacteristic
456Q8—24no—
457Q8—22no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461Q8—82no—
462D4—24no—
463D4—22no—
464D4—42no—
465D4—41no—
466D4—44no—
467D4—41no—
468D4—44no—
469D4—44no—
470D4—42no—
471D4—44no—
472D4—42no—
473D4—41no—
474D4—42no—
475D4—42no—
476D4—81no—
477D4—84no—
478D4—84no—
479D4—82no—
480D4—84no—
481D4—82no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—48no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4C3×#12011yescharacteristic
500C2×C4C22×C3×D411yescharacteristic
501C2×C4C22×C3×D412yes—
502C8—44no—
503C8—42no—
504C2×C3—21no—
505C2×C3—41no—
506C2×C3—42no—
507C2×C3—84no—
508C2×C3—81no—
509C2×C3—82no—
510C2×C3C2×B(4,𝔽2)11yescharacteristic
511C22—21no—
512C22—22no—
513C22—41no—
514C22—42no—
515C22—44no—
516C22—44no—
517C22—42no—
518C22—42no—
519C22—41no—
520C22—44no—
521C22—162no—
522C22—161no—
523C22—164no—
524C22—164no—
525C22C2×C3×#12011yescharacteristic
526C4—22no—
527C4—22no—
528C4—21no—
529C4—44no—
530C4—41no—
531C4—41no—
532C4—42no—
533C4C3×B(4,𝔽2)11yescharacteristic
534C4C3×B(4,𝔽2)11yescharacteristic
535C3#3357011yescharacteristic, direct product factor, Sylow
536C2—21no—
537C2—42no—
538C2—41no—
539C2—84no—
540C2—81no—
541C2—82no—
542C2C2×C3×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C3C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×Q8C24×C3
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3549C2semidirect
2. C3×#3021C2semidirect
3. C3×#3018C2semidirect
4. C3×#2204C2semidirect
5. C3×#2202C2semidirect
6. C3×#2134C2semidirect
7. C3×#2132C2semidirect
8. #33570C3direct (N × H)
9. C3×#585C22semidirect
13. C3×H(3,𝔽4)C22semidirect
15. C3×#555C22semidirect
23. C3×#456C22semidirect
25. C3×#454C22semidirect
26. C3×#454C22semidirect
31. C3×#413C22semidirect
32. C3×#413C22semidirect
33. C3×#356C22semidirect
34. C3×#355C22semidirect
35. C3×#355C22semidirect
36. #3549C2×C3semidirect
37. #3021C2×C3semidirect
38. #3018C2×C3semidirect
39. #2204C2×C3semidirect
40. #2202C2×C3semidirect
41. #2134C2×C3semidirect
42. #2132C2×C3semidirect
108. #585C22×C3semidirect
112. H(3,𝔽4)C22×C3semidirect
114. #555C22×C3semidirect
122. #456C22×C3semidirect
124. #454C22×C3semidirect
125. #454C22×C3semidirect
130. #413C22×C3semidirect
131. #413C22×C3semidirect
132. #356C22×C3semidirect
133. #355C22×C3semidirect
134. #355C22×C3semidirect
535. C3#33570direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35492
C3×#30212
C3×#30182
C3×#22042
C3×#22022
C3×#21342
C3×#21322
#335703