Finite Group Databasebeta

← C3×#32892 #10651499

Subgroups — C3×#32892 #10651499

Order 768.

3580 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×2−1+4C211yesmaximal, characteristic
2C2×C3×#456C214yesmaximal, complemented
3C3×#3007C212yesmaximal, complemented
4C3×#2434C211yesmaximal, characteristic, complemented
5C3×#1793C214yesmaximal, complemented
6C3×#1789C212yesmaximal, complemented
7C3×#1784C211yesmaximal, characteristic, complemented
8#32892C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×2−1+4—216no—
10C2×C3×2−1+4—24no—
11C2×C3×2−1+4—22no—
12C2×C3×2−1+4C2214yes—
13C22×C3×#41—22no—
14C22×C3×#41—22no—
15C22×C3×#41C2212yes—
16C23×C3×Q8—22no—
17C23×C3×Q8C2211yescharacteristic
18C2×C3×#137C2214yescomplemented
19C2×C3×C4⋊Q8C2212yescomplemented
20C2×C3×C22⋊Q8C2211yescharacteristic
21C3×#456—216no—
22C3×#452—24no—
23C3×#451—22no—
24C3×#450—22no—
25C2×C3×#104C2218yescomplemented
26C3×#417C2214yescomplemented
27C2×C3×#101C2214yescomplemented
28C3×#407C2212yes—
29C3×#395—22no—
30C3×#391C2212yescomplemented
31C3×#385—22no—
32C3×C4×C22⋊C4C2211yescharacteristic
33C22×2−1+4C2×C311yescharacteristic
34C2×#456C2×C314yescomplemented
35#3007C2×C312yescomplemented
36#2434C2×C311yescharacteristic, complemented
37#1793C2×C314yescomplemented
38#1789C2×C312yescomplemented
39#1784C2×C311yescharacteristic, complemented
40C3×2−1+4—232no—
41C3×2−1+4—232no—
42C3×2−1+4—232no—
43C3×2−1+4—216no—
44C3×2−1+4—28no—
45C3×2−1+4D4116yes—
46C2×C3×#41—232no—
47C2×C3×#41—28no—
48C2×C3×#41—28no—
49C2×C3×#41—232no—
50C2×C3×#41—216no—
51C2×C3×#41—28no—
52C2×C3×#41—28no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41D414yes—
56C2×C3×#41C2314yes—
57C22×C3×Q8—22no—
58C22×C3×Q8—232no—
59C22×C3×Q8—28no—
60C22×C3×Q8—28no—
61C22×C3×Q8—28no—
62C22×C3×Q8—22no—
63C22×C3×Q8—22no—
64C22×C3×Q8D412yes—
65C22×C3×Q8D412yes—
66C22×C3×Q8C2311yescharacteristic
67C22×C3×Q8C2312yes—
68C22×C3×D4—22no—
69C22×C3×D4—22no—
70C22×C3×D4D412yes—
71C23×C3×C4—24no—
72C23×C3×C4—22no—
73C23×C3×C4D412yes—
74C23×C3×C4C2311yescharacteristic
75C3×#137—216no—
76C2×C3×Q16—24no—
77C2×C3×SD16—24no—
78C2×C3×M16C2314yes—
79C3×C4⋊Q8—28no—
80C3×C22⋊Q8—22no—
81C3×C22⋊Q8—22no—
82C2×C3×C4⋊C4—24no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2312yes—
88C3×#104—216no—
89C3×#103—24no—
90C3×#102—24no—
91C3×#101—28no—
92C3×C22⋊C8—22no—
93C3×C22⋊C8—22no—
94C3×#95—21no—
95C3×#95—24no—
96C2×2−1+4—216no—
97C2×2−1+4—24no—
98C2×2−1+4—22no—
99C2×2−1+4C22×C314yes—
100C22×#41—22no—
101C22×#41—22no—
102C22×#41C22×C312yes—
103C23×Q8—22no—
104C23×Q8C22×C311yescharacteristic
105C2×#137C22×C314yescomplemented
106C2×C4⋊Q8C22×C312yescomplemented
107C2×C22⋊Q8C22×C311yescharacteristic
108#456—216no—
109#452—24no—
110#451—22no—
111#450—22no—
112C2×#104C22×C318yescomplemented
113#417C22×C314yescomplemented
114C2×#101C22×C314yescomplemented
115#407C22×C312yes—
116#395—22no—
117#391C22×C312yescomplemented
118#385—22no—
119C4×C22⋊C4C22×C311yescharacteristic
120C24×C3—22no—
121C24×C3C2×D411yescharacteristic
122C3×#41—264no—
123C3×#41—264no—
124C3×#41—232no—
125C3×#41—216no—
126C3×#41—232no—
127C3×#41—216no—
128C3×#41—28no—
129C3×#41—232no—
130C3×#41—28no—
131C3×#41—232no—
132C3×#41—28no—
133C3×#41C2×D4116yes—
134C2×C3×Q8—216no—
135C2×C3×Q8—216no—
136C2×C3×Q8—28no—
137C2×C3×Q8—264no—
138C2×C3×Q8—232no—
139C2×C3×Q8—216no—
140C2×C3×Q8—28no—
141C2×C3×Q8—232no—
142C2×C3×Q8—22no—
143C2×C3×Q8—24no—
144C2×C3×Q8—28no—
145C2×C3×Q8—24no—
146C2×C3×Q8—24no—
147C2×C3×Q8—22no—
148C2×C3×Q8—24no—
149C2×C3×Q8C2×D414yes—
150C2×C3×Q8C2×D418yes—
151C2×C3×D4—216no—
152C2×C3×D4—216no—
153C2×C3×D4—28no—
154C2×C3×D4—24no—
155C2×C3×D4—24no—
156C2×C3×D4—28no—
157C2×C3×D4—22no—
158C2×C3×D4C2×D414yes—
159C22×C3×C4—232no—
160C22×C3×C4—216no—
161C22×C3×C4—216no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—24no—
166C22×C3×C4—28no—
167C22×C3×C4—24no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4C2×D414yes—
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2411yescharacteristic
177C3×Q16—48no—
178C3×SD16—48no—
179C3×M16—28no—
180C2×C3×C8—24no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—48no—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—41no—
186C3×C22⋊C4—41no—
187C3×C42—24no—
1882−1+4—232no—
1892−1+4—232no—
1902−1+4—232no—
1912−1+4—216no—
1922−1+4—28no—
1932−1+4C3×D4116yes—
194C2×#41—232no—
195C2×#41—28no—
196C2×#41—28no—
197C2×#41—232no—
198C2×#41—216no—
199C2×#41—28no—
200C2×#41—28no—
201C2×#41—22no—
202C2×#41—22no—
203C2×#41C3×D414yes—
204C2×#41C23×C314yes—
205C22×Q8—22no—
206C22×Q8—232no—
207C22×Q8—28no—
208C22×Q8—28no—
209C22×Q8—28no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8C3×D412yes—
213C22×Q8C3×D412yes—
214C22×Q8C23×C311yescharacteristic
215C22×Q8C23×C312yes—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4C3×D412yes—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4C3×D412yes—
222C23×C4C23×C311yescharacteristic
223#137—216no—
224C2×Q16—24no—
225C2×SD16—24no—
226C2×M16C23×C314yes—
227C4⋊Q8—28no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C42C23×C312yes—
236#104—216no—
237#103—24no—
238#102—24no—
239#101—28no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242#95—21no—
243#95—24no—
244C23×C3—28no—
245C23×C3—22no—
246C23×C3—24no—
247C23×C3—22no—
248C23×C3—41no—
249C23×C3—44no—
250C23×C3—44no—
251C23×C3#12011yescharacteristic
252C23×C3#12011yescharacteristic
253C23×C3C22×D411yescharacteristic
254C3×Q8—232no—
255C3×Q8—28no—
256C3×Q8—216no—
257C3×Q8—24no—
258C3×Q8—216no—
259C3×Q8—44no—
260C3×Q8#12018yes—
261C3×D4—216no—
262C3×D4—232no—
263C3×D4—28no—
264C3×D4—216no—
265C3×D4—24no—
266C3×D4#12018yes—
267C2×C3×C4—28no—
268C2×C3×C4—28no—
269C2×C3×C4—264no—
270C2×C3×C4—216no—
271C2×C3×C4—28no—
272C2×C3×C4—24no—
273C2×C3×C4—22no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—216no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4#12018yes—
287C2×C3×C4#12012yes—
288C2×C3×C4C22×D414yes—
289C2×C3×C4C22×D412yes—
290C3×C8—44no—
291C24—22no—
292C24C2×C3×D411yescharacteristic
293#41—264no—
294#41—264no—
295#41—232no—
296#41—216no—
297#41—232no—
298#41—216no—
299#41—28no—
300#41—232no—
301#41—28no—
302#41—232no—
303#41—28no—
304#41C2×C3×D4116yes—
305C2×Q8—216no—
306C2×Q8—216no—
307C2×Q8—28no—
308C2×Q8—264no—
309C2×Q8—232no—
310C2×Q8—216no—
311C2×Q8—28no—
312C2×Q8—232no—
313C2×Q8—22no—
314C2×Q8—24no—
315C2×Q8—28no—
316C2×Q8—24no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8C2×C3×D414yes—
321C2×Q8C2×C3×D418yes—
322C2×D4—216no—
323C2×D4—216no—
324C2×D4—28no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—28no—
328C2×D4—22no—
329C2×D4C2×C3×D414yes—
330C22×C4—232no—
331C22×C4—216no—
332C22×C4—216no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4C2×C3×D414yes—
344C22×C4C2×C3×D411yescharacteristic
345C22×C4C2×C3×D412yes—
346C22×C4C2×C3×D412yes—
347C22×C4C24×C311yescharacteristic
348Q16—48no—
349SD16—48no—
350M16—28no—
351C2×C8—24no—
352C4⋊C4—22no—
353C4⋊C4—42no—
354C4⋊C4—48no—
355C22⋊C4—24no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C42—24no—
359C22×C3—21no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—28no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—44no—
368C22×C3—416no—
369C22×C3—44no—
370C22×C3—44no—
371C22×C3#45612yescomplemented
372C22×C3C2×#12012yes—
373C22×C3C2×#12011yescharacteristic
374C3×C4—28no—
375C3×C4—28no—
376C3×C4—22no—
377C3×C4—44no—
378C3×C4—81no—
379C3×C4—81no—
380C3×C4C2×#12014yes—
381C23—28no—
382C23—22no—
383C23—22no—
384C23—24no—
385C23—41no—
386C23—44no—
387C23—44no—
388C23C3×#12011yescharacteristic
389C23C3×#12011yescharacteristic
390C23C22×C3×D411yescharacteristic
391Q8—232no—
392Q8—28no—
393Q8—216no—
394Q8—24no—
395Q8—216no—
396Q8—44no—
397Q8C3×#12018yes—
398D4—216no—
399D4—232no—
400D4—28no—
401D4—216no—
402D4—24no—
403D4C3×#12018yes—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—264no—
407C2×C4—216no—
408C2×C4—28no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—216no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4C3×#12012yes—
424C2×C4C3×#12018yes—
425C2×C4C22×C3×D414yes—
426C2×C4C22×C3×D412yes—
427C8—44no—
428C2×C3—22no—
429C2×C3—22no—
430C2×C3—41no—
431C2×C3—48no—
432C2×C3#284611yescharacteristic
433C2×C3C2×#45611yescharacteristic
434C2×C3#302011yescharacteristic
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—28no—
443C22—44no—
444C22—416no—
445C22—44no—
446C22—44no—
447C22C3×#45612yescomplemented
448C22C2×C3×#12012yes—
449C22C2×C3×#12011yescharacteristic
450C4—28no—
451C4—28no—
452C4—22no—
453C4—44no—
454C4—81no—
455C4—81no—
456C4C2×C3×#12014yes—
457C3#3289211yescharacteristic, direct product factor, Sylow
458C2—22no—
459C2—22no—
460C2—41no—
461C2—48no—
462C2C3×#284611yescharacteristic
463C2C2×C3×#45611yescharacteristic
464C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×2−1+4C2
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×#456C2semidirect
3. C3×#3007C2semidirect
4. C3×#2434C2semidirect
5. C3×#1793C2semidirect
6. C3×#1789C2semidirect
7. C3×#1784C2semidirect
8. #32892C3direct (N × H)
18. C2×C3×#137C22semidirect
19. C2×C3×C4⋊Q8C22semidirect
25. C2×C3×#104C22semidirect
26. C3×#417C22semidirect
27. C2×C3×#101C22semidirect
30. C3×#391C22semidirect
34. C2×#456C2×C3semidirect
35. #3007C2×C3semidirect
36. #2434C2×C3semidirect
37. #1793C2×C3semidirect
38. #1789C2×C3semidirect
39. #1784C2×C3semidirect
105. C2×#137C22×C3semidirect
106. C2×C4⋊Q8C22×C3semidirect
112. C2×#104C22×C3semidirect
113. #417C22×C3semidirect
114. C2×#101C22×C3semidirect
117. #391C22×C3semidirect
371. C22×C3#456semidirect
447. C22C3×#456semidirect
457. C3#32892direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×2−1+42
C2×C3×#4562
C3×#30072
C3×#24342
C3×#17932
C3×#17892
C3×#17842
#328923