Finite Group Databasebeta

← C3×#32891 #10651498

Subgroups — C3×#32891 #10651498

Order 768.

3940 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×H(5,𝔽2)C211yesmaximal, characteristic
2C2×C3×#455C214yesmaximal, complemented
3C3×#3007C212yesmaximal, complemented
4C3×#2435C211yesmaximal, characteristic, complemented
5C3×#1792C214yesmaximal, complemented
6C3×#1789C212yesmaximal, complemented
7C3×#1783C211yesmaximal, characteristic, complemented
8#32891C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×H(5,𝔽2)—216no—
10C2×C3×H(5,𝔽2)—24no—
11C2×C3×H(5,𝔽2)—22no—
12C2×C3×H(5,𝔽2)C2214yes—
13C22×C3×#41—22no—
14C22×C3×#41C2212yes—
15C23×C3×D4—22no—
16C23×C3×D4—22no—
17C23×C3×D4C2211yescharacteristic
18C2×C3×#137C2214yescomplemented
19C2×C3×#124C2212yescomplemented
20C2×C3×C22⋊Q8C2211yescharacteristic, complemented
21C3×#455—216no—
22C3×#452—24no—
23C3×#451—22no—
24C3×#450—22no—
25C2×C3×#104C2218yescomplemented
26C3×#417C2214yescomplemented
27C2×C3×#100C2214yescomplemented
28C3×#407C2212yescomplemented
29C3×#394—22no—
30C3×#390C2212yescomplemented
31C3×#379—22no—
32C3×C4×C22⋊C4C2211yescharacteristic, complemented
33C22×H(5,𝔽2)C2×C311yescharacteristic
34C2×#455C2×C314yescomplemented
35#3007C2×C312yescomplemented
36#2435C2×C311yescharacteristic, complemented
37#1792C2×C314yescomplemented
38#1789C2×C312yescomplemented
39#1783C2×C311yescharacteristic, complemented
40C25×C3—22no—
41C25×C3D412yes—
42C3×H(5,𝔽2)—232no—
43C3×H(5,𝔽2)—232no—
44C3×H(5,𝔽2)—232no—
45C3×H(5,𝔽2)—216no—
46C3×H(5,𝔽2)—28no—
47C3×H(5,𝔽2)D4116yes—
48C2×C3×#41—232no—
49C2×C3×#41—28no—
50C2×C3×#41—216no—
51C2×C3×#41—28no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41D414yes—
55C2×C3×#41C2314yes—
56C22×C3×Q8D412yes—
57C22×C3×Q8C2311yescharacteristic, complemented
58C22×C3×D4—24no—
59C22×C3×D4—232no—
60C22×C3×D4—232no—
61C22×C3×D4—28no—
62C22×C3×D4—28no—
63C22×C3×D4—22no—
64C22×C3×D4—28no—
65C22×C3×D4—28no—
66C22×C3×D4—28no—
67C22×C3×D4—22no—
68C22×C3×D4—22no—
69C22×C3×D4D412yes—
70C22×C3×D4D412yes—
71C22×C3×D4C2312yes—
72C23×C3×C4—22no—
73C23×C3×C4—22no—
74C23×C3×C4C2311yescharacteristic
75C3×#137—216no—
76C2×C3×Q16—24no—
77C2×C3×SD16—24no—
78C2×C3×M16C2314yescomplemented
79C3×#124—28no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic, complemented
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—24no—
87C2×C3×C22⋊C4—24no—
88C2×C3×C42C2312yescomplemented
89C3×#104—216no—
90C3×#103—24no—
91C3×#102—24no—
92C3×#100—28no—
93C3×C22⋊C8—22no—
94C3×C22⋊C8—22no—
95C3×#95—21no—
96C2×H(5,𝔽2)—216no—
97C2×H(5,𝔽2)—24no—
98C2×H(5,𝔽2)—22no—
99C2×H(5,𝔽2)C22×C314yes—
100C22×#41—22no—
101C22×#41C22×C312yes—
102C23×D4—22no—
103C23×D4—22no—
104C23×D4C22×C311yescharacteristic
105C2×#137C22×C314yescomplemented
106C2×#124C22×C312yescomplemented
107C2×C22⋊Q8C22×C311yescharacteristic, complemented
108#455—216no—
109#452—24no—
110#451—22no—
111#450—22no—
112C2×#104C22×C318yescomplemented
113#417C22×C314yescomplemented
114C2×#100C22×C314yescomplemented
115#407C22×C312yescomplemented
116#394—22no—
117#390C22×C312yescomplemented
118#379—22no—
119C4×C22⋊C4C22×C311yescharacteristic, complemented
120C24×C3—216no—
121C24×C3—28no—
122C24×C3—22no—
123C24×C3—22no—
124C24×C3—28no—
125C24×C3—22no—
126C24×C3—44no—
127C24×C3—82no—
128C24×C3—84no—
129C24×C3—84no—
130C24×C3C2×D411yescharacteristic
131C24×C3C2×D414yes—
132C3×#41—264no—
133C3×#41—216no—
134C3×#41—232no—
135C3×#41—216no—
136C3×#41—232no—
137C3×#41—28no—
138C3×#41—28no—
139C3×#41—28no—
140C3×#41C2×D4116yes—
141C2×C3×Q8—28no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—24no—
145C2×C3×Q8C2×D414yes—
146C2×C3×D4—232no—
147C2×C3×D4—216no—
148C2×C3×D4—216no—
149C2×C3×D4—216no—
150C2×C3×D4—264no—
151C2×C3×D4—264no—
152C2×C3×D4—232no—
153C2×C3×D4—232no—
154C2×C3×D4—216no—
155C2×C3×D4—24no—
156C2×C3×D4—232no—
157C2×C3×D4—28no—
158C2×C3×D4—24no—
159C2×C3×D4—28no—
160C2×C3×D4—232no—
161C2×C3×D4—24no—
162C2×C3×D4—22no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4C2×D418yes—
166C2×C3×D4C2×D414yes—
167C22×C3×C4—216no—
168C22×C3×C4—216no—
169C22×C3×C4—28no—
170C22×C3×C4—22no—
171C22×C3×C4—24no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—21no—
177C22×C3×C4C2×D411yescharacteristic
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2411yescharacteristic
181C3×Q16—48no—
182C3×SD16—48no—
183C3×M16—28no—
184C2×C3×C8—24no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—42no—
187C3×C22⋊C4—42no—
188C3×C22⋊C4—41no—
189C3×C22⋊C4—41no—
190C3×C22⋊C4—48no—
191C3×C22⋊C4—84no—
192C3×C42—24no—
193C25—22no—
194C25C3×D412yes—
195H(5,𝔽2)—232no—
196H(5,𝔽2)—232no—
197H(5,𝔽2)—232no—
198H(5,𝔽2)—216no—
199H(5,𝔽2)—28no—
200H(5,𝔽2)C3×D4116yes—
201C2×#41—232no—
202C2×#41—28no—
203C2×#41—216no—
204C2×#41—28no—
205C2×#41—22no—
206C2×#41—22no—
207C2×#41C3×D414yes—
208C2×#41C23×C314yes—
209C22×Q8C3×D412yes—
210C22×Q8C23×C311yescharacteristic, complemented
211C22×D4—24no—
212C22×D4—232no—
213C22×D4—232no—
214C22×D4—28no—
215C22×D4—28no—
216C22×D4—22no—
217C22×D4—28no—
218C22×D4—28no—
219C22×D4—28no—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4C3×D412yes—
223C22×D4C3×D412yes—
224C22×D4C23×C312yes—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4C23×C311yescharacteristic
228#137—216no—
229C2×Q16—24no—
230C2×SD16—24no—
231C2×M16C23×C314yescomplemented
232#124—28no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C23×C311yescharacteristic, complemented
237C2×C22⋊C4—21no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—24no—
241C2×C42C23×C312yescomplemented
242#104—216no—
243#103—24no—
244#102—24no—
245#100—28no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248#95—21no—
249C23×C3—28no—
250C23×C3—28no—
251C23×C3—28no—
252C23×C3—232no—
253C23×C3—22no—
254C23×C3—24no—
255C23×C3—24no—
256C23×C3—22no—
257C23×C3—24no—
258C23×C3—41no—
259C23×C3—44no—
260C23×C3—44no—
261C23×C3—44no—
262C23×C3—44no—
263C23×C3—44no—
264C23×C3—44no—
265C23×C3—82no—
266C23×C3—82no—
267C23×C3—816no—
268C23×C3—832no—
269C23×C3—88no—
270C23×C3—88no—
271C23×C3#12011yescharacteristic
272C23×C3#12011yescharacteristic
273C23×C3#12018yes—
274C23×C3C22×D411yescharacteristic
275C3×Q8—28no—
276C3×Q8—24no—
277C3×Q8—44no—
278C3×Q8#12018yes—
279C3×D4—264no—
280C3×D4—216no—
281C3×D4—216no—
282C3×D4—28no—
283C3×D4—216no—
284C3×D4—216no—
285C3×D4—24no—
286C3×D4#12018yes—
287C2×C3×C4—28no—
288C2×C3×C4—232no—
289C2×C3×C4—216no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—24no—
293C2×C3×C4—216no—
294C2×C3×C4—24no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4—41no—
301C2×C3×C4—41no—
302C2×C3×C4#12012yes—
303C2×C3×C4C22×D414yes—
304C2×C3×C4C22×D412yes—
305C3×C8—44no—
306C24—216no—
307C24—28no—
308C24—22no—
309C24—22no—
310C24—28no—
311C24—22no—
312C24—44no—
313C24—82no—
314C24—84no—
315C24—84no—
316C24C2×C3×D411yescharacteristic
317C24C2×C3×D414yes—
318#41—264no—
319#41—216no—
320#41—232no—
321#41—216no—
322#41—232no—
323#41—28no—
324#41—28no—
325#41—28no—
326#41C2×C3×D4116yes—
327C2×Q8—28no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8C2×C3×D414yes—
332C2×D4—232no—
333C2×D4—216no—
334C2×D4—216no—
335C2×D4—216no—
336C2×D4—264no—
337C2×D4—264no—
338C2×D4—232no—
339C2×D4—232no—
340C2×D4—216no—
341C2×D4—24no—
342C2×D4—232no—
343C2×D4—28no—
344C2×D4—24no—
345C2×D4—28no—
346C2×D4—232no—
347C2×D4—24no—
348C2×D4—22no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4C2×C3×D418yes—
352C2×D4C2×C3×D414yes—
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4C2×C3×D411yescharacteristic
364C22×C4C2×C3×D412yes—
365C22×C4C2×C3×D412yes—
366C22×C4C24×C311yescharacteristic
367Q16—48no—
368SD16—48no—
369M16—28no—
370C2×C8—24no—
371C4⋊C4—22no—
372C4⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—48no—
377C22⋊C4—84no—
378C42—24no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—28no—
385C22×C3—28no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—44no—
389C22×C3—48no—
390C22×C3—416no—
391C22×C3—416no—
392C22×C3—44no—
393C22×C3—44no—
394C22×C3—44no—
395C22×C3—44no—
396C22×C3—84no—
397C22×C3—88no—
398C22×C3—832no—
399C22×C3#45512yescomplemented
400C22×C3C2×#12012yes—
401C22×C3C2×#12011yescharacteristic
402C3×C4—28no—
403C3×C4—22no—
404C3×C4—82no—
405C3×C4—81no—
406C3×C4—81no—
407C3×C4C2×#12014yes—
408C23—28no—
409C23—28no—
410C23—28no—
411C23—232no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—41no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—82no—
425C23—82no—
426C23—816no—
427C23—832no—
428C23—88no—
429C23—88no—
430C23C3×#12011yescharacteristic
431C23C3×#12011yescharacteristic
432C23C3×#12018yes—
433C23C22×C3×D411yescharacteristic
434Q8—28no—
435Q8—24no—
436Q8—44no—
437Q8C3×#12018yes—
438D4—264no—
439D4—216no—
440D4—216no—
441D4—28no—
442D4—216no—
443D4—216no—
444D4—24no—
445D4C3×#12018yes—
446C2×C4—28no—
447C2×C4—232no—
448C2×C4—216no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—216no—
453C2×C4—24no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4C3×#12012yes—
462C2×C4C22×C3×D414yes—
463C2×C4C22×C3×D412yes—
464C8—44no—
465C2×C3—22no—
466C2×C3—22no—
467C2×C3—41no—
468C2×C3—48no—
469C2×C3—48no—
470C2×C3#284611yescharacteristic
471C2×C3C2×#45511yescharacteristic
472C2×C3#301911yescharacteristic
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—28no—
481C22—28no—
482C22—44no—
483C22—48no—
484C22—416no—
485C22—416no—
486C22—44no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—84no—
491C22—88no—
492C22—832no—
493C22C3×#45512yescomplemented
494C22C2×C3×#12012yes—
495C22C2×C3×#12011yescharacteristic
496C4—28no—
497C4—22no—
498C4—82no—
499C4—81no—
500C4—81no—
501C4C2×C3×#12014yes—
502C3#3289111yescharacteristic, direct product factor, Sylow
503C2—22no—
504C2—22no—
505C2—41no—
506C2—48no—
507C2—48no—
508C2C3×#284611yescharacteristic
509C2C2×C3×#45511yescharacteristic
510C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×H(5,𝔽2)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#455C2semidirect
3. C3×#3007C2semidirect
4. C3×#2435C2semidirect
5. C3×#1792C2semidirect
6. C3×#1789C2semidirect
7. C3×#1783C2semidirect
8. #32891C3direct (N × H)
18. C2×C3×#137C22semidirect
19. C2×C3×#124C22semidirect
20. C2×C3×C22⋊Q8C22semidirect
25. C2×C3×#104C22semidirect
26. C3×#417C22semidirect
27. C2×C3×#100C22semidirect
28. C3×#407C22semidirect
30. C3×#390C22semidirect
32. C3×C4×C22⋊C4C22semidirect
34. C2×#455C2×C3semidirect
35. #3007C2×C3semidirect
36. #2435C2×C3semidirect
37. #1792C2×C3semidirect
38. #1789C2×C3semidirect
39. #1783C2×C3semidirect
57. C22×C3×Q8C23semidirect
78. C2×C3×M16C23semidirect
83. C2×C3×C4⋊C4C23semidirect
88. C2×C3×C42C23semidirect
105. C2×#137C22×C3semidirect
106. C2×#124C22×C3semidirect
107. C2×C22⋊Q8C22×C3semidirect
112. C2×#104C22×C3semidirect
113. #417C22×C3semidirect
114. C2×#100C22×C3semidirect
115. #407C22×C3semidirect
117. #390C22×C3semidirect
119. C4×C22⋊C4C22×C3semidirect
210. C22×Q8C23×C3semidirect
231. C2×M16C23×C3semidirect
236. C2×C4⋊C4C23×C3semidirect
241. C2×C42C23×C3semidirect
399. C22×C3#455semidirect
493. C22C3×#455semidirect
502. C3#32891direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×H(5,𝔽2)2
C2×C3×#4552
C3×#30072
C3×#24352
C3×#17922
C3×#17892
C3×#17832
#328913