Finite Group Databasebeta

← C3×#32984 #10651591

Subgroups — C3×#32984 #10651591

Order 768.

1430 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#117C211yesmaximal, characteristic
2C2×C3×#484C211yesmaximal, characteristic, complemented
3C2×C3×#483C211yesmaximal, characteristic, complemented
4C3×#2848C211yesmaximal, characteristic
5C3×#1873C214yesmaximal, complemented
6C3×#1854C214yesmaximal, complemented
7C3×#1824C211yesmaximal, characteristic, complemented
8C3×#1823C211yesmaximal, characteristic, complemented
9C3×#1779C211yesmaximal, characteristic
10#32984C311yesmaximal, characteristic, direct product factor, Sylow
11C24×C3×C4C2211yescharacteristic
12C2×C3×C22⋊Q8C2212yes—
13C2×C3×C22⋊Q8C2211yescharacteristic
14C2×C3×#117—22no—
15C2×C3×#117—24no—
16C2×C3×#117—24no—
17C2×C3×#117C2214yes—
18C22×C3×C4⋊C4C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4—21no—
21C22×C3×C42—21no—
22C3×#484—24no—
23C3×#483—24no—
24C2×C3×#107C2212yescomplemented
25C2×C3×#106C2212yescomplemented
26C2×C3×#103C2212yescomplemented
27C2×C3×#103C2212yescomplemented
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2214yes—
31C3×#385—21no—
32C3×#385—22no—
33C3×#380C2211yescharacteristic
34C3×#380C2212yes—
35C3×#379—21no—
36C3×#340C2214yescomplemented
37C3×#340C2214yescomplemented
38C22×#117C2×C311yescharacteristic
39C2×#484C2×C311yescharacteristic, complemented
40C2×#483C2×C311yescharacteristic, complemented
41#2848C2×C311yescharacteristic
42#1873C2×C314yescomplemented
43#1854C2×C314yescomplemented
44#1824C2×C311yescharacteristic, complemented
45#1823C2×C311yescharacteristic, complemented
46#1779C2×C311yescharacteristic
47C25×C3D411yescharacteristic
48C22×C3×Q8C2311yescharacteristic
49C23×C3×C4—22no—
50C23×C3×C4—22no—
51C23×C3×C4—28no—
52C23×C3×C4—21no—
53C23×C3×C4—21no—
54C23×C3×C4D411yescharacteristic
55C23×C3×C4D412yes—
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2312yes—
58C22×C3×C8C2312yes—
59C22×C3×C8C2312yes—
60C3×C22⋊Q8—22no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—28no—
63C3×#117—216no—
64C3×#117—28no—
65C3×#117—28no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4D414yes—
73C2×C3×C4⋊C4D414yes—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C22⋊C4—28no—
79C2×C3×C22⋊C4—24no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42—28no—
85C2×C3×C42—24no—
86C3×#107—24no—
87C3×#106—24no—
88C3×#103—24no—
89C3×#103—24no—
90C3×C22⋊C8—22no—
91C3×C22⋊C8—22no—
92C3×C22⋊C8—28no—
93C3×#95—21no—
94C3×#95—22no—
95C24×C4C22×C311yescharacteristic
96C2×C22⋊Q8C22×C311yescharacteristic
97C2×C22⋊Q8C22×C312yes—
98C2×#117—22no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117C22×C314yes—
102C22×C4⋊C4C22×C311yescharacteristic
103C22×C4⋊C4C22×C311yescharacteristic
104C22×C22⋊C4—21no—
105C22×C42—21no—
106#484—24no—
107#483—24no—
108C2×#107C22×C312yescomplemented
109C2×#106C22×C312yescomplemented
110C2×#103C22×C312yescomplemented
111C2×#103C22×C312yescomplemented
112C2×C22⋊C8C22×C311yescharacteristic
113C2×C22⋊C8C22×C311yescharacteristic
114C2×C22⋊C8C22×C314yes—
115#385—22no—
116#385—21no—
117#380C22×C311yescharacteristic
118#380C22×C312yes—
119#379—21no—
120#340C22×C314yescomplemented
121#340C22×C314yescomplemented
122C24×C3—22no—
123C24×C3—24no—
124C24×C3—22no—
125C24×C3—22no—
126C24×C3—42no—
127C24×C3C2×D412yes—
128C24×C3C2×D411yescharacteristic
129C2×C3×Q8—21no—
130C2×C3×Q8—21no—
131C2×C3×Q8—24no—
132C22×C3×C4—216no—
133C22×C3×C4—216no—
134C22×C3×C4—22no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—24no—
138C22×C3×C4—24no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—24no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—24no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—44no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4#4118yes—
158C22×C3×C4C2411yescharacteristic
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—24no—
162C2×C3×C8—24no—
163C3×C4⋊C4—24no—
164C3×C4⋊C4—24no—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—44no—
172C3×C4⋊C4C2×D414yes—
173C3×C4⋊C4C2×D414yes—
174C3×C22⋊C4—216no—
175C3×C22⋊C4—44no—
176C3×C22⋊C4—42no—
177C3×C22⋊C4—44no—
178C3×C22⋊C4—42no—
179C3×C42—216no—
180C25C3×D411yescharacteristic
181C22×Q8C23×C311yescharacteristic
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—28no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4C3×D412yes—
188C23×C4C3×D411yescharacteristic
189C23×C4C23×C311yescharacteristic
190C23×C4C23×C312yes—
191C22×C8C23×C312yes—
192C22×C8C23×C312yes—
193C22⋊Q8—22no—
194C22⋊Q8—22no—
195C22⋊Q8—28no—
196#117—216no—
197#117—28no—
198#117—28no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4C3×D414yes—
206C2×C4⋊C4C3×D414yes—
207C2×C4⋊C4C23×C312yes—
208C2×C4⋊C4C23×C311yescharacteristic
209C2×C4⋊C4C23×C312yes—
210C2×C4⋊C4C23×C312yes—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C42—28no—
218C2×C42—24no—
219#107—24no—
220#106—24no—
221#103—24no—
222#103—24no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—28no—
226#95—21no—
227#95—22no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—28no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3—22no—
235C23×C3—22no—
236C23×C3—22no—
237C23×C3—48no—
238C23×C3—48no—
239C23×C3—44no—
240C23×C3—44no—
241C23×C3#12012yes—
242C23×C3#12314yes—
243C23×C3#13512yes—
244C23×C3#13712yes—
245C23×C3C22×D411yescharacteristic
246C3×Q8—42no—
247C3×Q8—42no—
248C2×C3×C4—216no—
249C2×C3×C4—28no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—28no—
256C2×C3×C4—28no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—48no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—44no—
263C2×C3×C4—48no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4#12012yes—
272C2×C3×C4#12314yes—
273C2×C3×C4C22×D411yescharacteristic
274C2×C3×C4C22×D411yescharacteristic
275C2×C3×C4C2×#4114yes—
276C3×C8—42no—
277C3×C8—42no—
278C24—22no—
279C24—24no—
280C24—22no—
281C24—22no—
282C24—42no—
283C24C2×C3×D412yes—
284C24C2×C3×D411yescharacteristic
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—24no—
288C22×C4—216no—
289C22×C4—216no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4C2×C3×D411yescharacteristic
310C22×C4C2×C3×D412yes—
311C22×C4C2×C3×D412yes—
312C22×C4C2×C3×D412yes—
313C22×C4C3×#4118yes—
314C22×C4C24×C311yescharacteristic
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—24no—
318C2×C8—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—24no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4C2×C3×D414yes—
329C4⋊C4C2×C3×D414yes—
330C22⋊C4—216no—
331C22⋊C4—44no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—42no—
335C42—216no—
336C22×C3—24no—
337C22×C3—24no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—24no—
341C22×C3—24no—
342C22×C3—24no—
343C22×C3—416no—
344C22×C3—42no—
345C22×C3—42no—
346C22×C3—42no—
347C22×C3—42no—
348C22×C3#45214yes—
349C22×C3C2×#12011yescharacteristic
350C22×C3C2×#12312yes—
351C22×C3C2×#13511yescharacteristic
352C22×C3C2×#13711yescharacteristic
353C22×C3#57512yes—
354C3×C4—22no—
355C3×C4—24no—
356C3×C4—44no—
357C3×C4—44no—
358C3×C4—82no—
359C3×C4—81no—
360C3×C4—81no—
361C3×C4#52512yes—
362C3×C4#52512yes—
363C23—22no—
364C23—22no—
365C23—28no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—48no—
373C23—48no—
374C23—44no—
375C23—44no—
376C23C3×#12012yes—
377C23C3×#12314yes—
378C23C3×#13512yes—
379C23C3×#13712yes—
380C23C22×C3×D411yescharacteristic
381Q8—42no—
382Q8—42no—
383C2×C4—216no—
384C2×C4—28no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—48no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4C3×#12012yes—
407C2×C4C3×#12314yes—
408C2×C4C22×C3×D411yescharacteristic
409C2×C4C22×C3×D411yescharacteristic
410C2×C4C2×C3×#4114yes—
411C8—42no—
412C8—42no—
413C2×C3—24no—
414C2×C3—44no—
415C2×C3#284911yescharacteristic
416C2×C3C2×#45211yescharacteristic
417C2×C3#300511yescharacteristic
418C2×C3#309112yes—
419C2×C3#309512yes—
420C22—22no—
421C22—24no—
422C22—24no—
423C22—22no—
424C22—24no—
425C22—24no—
426C22—24no—
427C22—416no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22C3×#45214yes—
433C22C2×C3×#12011yescharacteristic
434C22C2×C3×#12312yes—
435C22C2×C3×#13511yescharacteristic
436C22C2×C3×#13711yescharacteristic
437C22C3×#57512yes—
438C4—22no—
439C4—24no—
440C4—44no—
441C4—44no—
442C4—81no—
443C4—82no—
444C4—81no—
445C4C3×#52512yes—
446C4C3×#52512yes—
447C3#3298411yescharacteristic, direct product factor, Sylow
448C2—24no—
449C2—44no—
450C2C3×#284911yescharacteristic
451C2C2×C3×#45211yescharacteristic
452C2C3×#300511yescharacteristic
453C2C3×#309112yes—
454C2C3×#309512yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#484C2semidirect
3. C2×C3×#483C2semidirect
5. C3×#1873C2semidirect
6. C3×#1854C2semidirect
7. C3×#1824C2semidirect
8. C3×#1823C2semidirect
10. #32984C3direct (N × H)
24. C2×C3×#107C22semidirect
25. C2×C3×#106C22semidirect
26. C2×C3×#103C22semidirect
27. C2×C3×#103C22semidirect
36. C3×#340C22semidirect
37. C3×#340C22semidirect
39. C2×#484C2×C3semidirect
40. C2×#483C2×C3semidirect
42. #1873C2×C3semidirect
43. #1854C2×C3semidirect
44. #1824C2×C3semidirect
45. #1823C2×C3semidirect
108. C2×#107C22×C3semidirect
109. C2×#106C22×C3semidirect
110. C2×#103C22×C3semidirect
111. C2×#103C22×C3semidirect
120. #340C22×C3semidirect
121. #340C22×C3semidirect
447. C3#32984direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1172
C2×C3×#4842
C2×C3×#4832
C3×#28482
C3×#18732
C3×#18542
C3×#18242
C3×#18232
C3×#17792
#329843