Finite Group Databasebeta

← C3×#23953 #10642560

Subgroups — C3×#23953 #10642560

Order 768.

1166 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3493C211yesmaximal, characteristic
2C3×#3004C212yesmaximal, complemented
3C3×#2310C211yesmaximal, characteristic, complemented
4C3×#2028C212yesmaximal, complemented
5C3×#2007C212yesmaximal, complemented
6C3×#1897C212yesmaximal, complemented
7C3×#1882C214yesmaximal, complemented
8C3×#1822C211yesmaximal, characteristic, complemented
9#23953C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×Q8C2211yescharacteristic
11C2×C3×#137C2214yes—
12C3×#556—22no—
13C3×#555—22no—
14C3×#553—22no—
15C3×#552—22no—
16C3×#544C2211yescharacteristic
17C3×#538—22no—
18C3×#537C2211yescharacteristic
19C3×#529—22no—
20C3×#525C2212yes—
21C3×#520C2212yes—
22C2×C3×#117C2211yescharacteristic
23C2×C3×#117C2211yescharacteristic
24C3×#451—22no—
25C3×#451—22no—
26C3×#448—22no—
27C3×#448—22no—
28C2×C3×#104C2214yes—
29C3×#419C2214yes—
30C3×#407C2212yes—
31C3×#391—22no—
32C3×#391C2212yes—
33C3×#391C2212yes—
34C3×#381—22no—
35C3×#343C2212yes—
36C3×#343C2212yes—
37C3×#339C2214yes—
38#3493C2×C311yescharacteristic
39#3004C2×C312yescomplemented
40#2310C2×C311yescharacteristic, complemented
41#2028C2×C312yescomplemented
42#2007C2×C312yescomplemented
43#1897C2×C312yescomplemented
44#1882C2×C314yescomplemented
45#1822C2×C311yescharacteristic, complemented
46C2×C3×#41C2312yes—
47C22×C3×Q8—24no—
48C22×C3×Q8—21no—
49C22×C3×Q8—22no—
50C22×C3×Q8—22no—
51C22×C3×Q8—42no—
52C22×C3×Q8C2312yes—
53C23×C3×C4—21no—
54C23×C3×C4C2311yescharacteristic
55C3×#137—28no—
56C3×#137—44no—
57C2×C3×Q16—24no—
58C2×C3×SD16—24no—
59C2×C3×M16C2314yes—
60C3×C4⋊Q8—22no—
61C3×#126—24no—
62C3×#126—24no—
63C3×#125—22no—
64C3×#125—22no—
65C3×#124—22no—
66C3×#124—24no—
67C3×#123—24no—
68C3×#123—22no—
69C3×C22⋊Q8—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8D412yes—
72C3×C22⋊Q8D412yes—
73C3×#121—24no—
74C3×#121—22no—
75C3×#121—22no—
76C3×#121D412yes—
77C3×#121D412yes—
78C3×#120—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—24no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×#117—24no—
87C3×#117—22no—
88C3×#117—22no—
89C3×#117—22no—
90C3×#117—42no—
91C3×#117C2312yes—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4D412yes—
98C2×C3×C42D412yes—
99C2×C3×C42C2312yes—
100C3×#104—44no—
101C3×#103—24no—
102C3×#103—24no—
103C3×C22⋊C8—22no—
104C3×C22⋊C8—22no—
105C3×#95—24no—
106C3×#95—24no—
107C23×Q8C22×C311yescharacteristic
108C2×#137C22×C314yes—
109#556—22no—
110#555—22no—
111#553—22no—
112#552—22no—
113#544C22×C311yescharacteristic
114#538—22no—
115#537C22×C311yescharacteristic
116#529—22no—
117#525C22×C312yes—
118#520C22×C312yes—
119C2×#117C22×C311yescharacteristic
120C2×#117C22×C311yescharacteristic
121#451—22no—
122#451—22no—
123#448—22no—
124#448—22no—
125C2×#104C22×C314yes—
126#419C22×C314yes—
127#407C22×C312yes—
128#391—22no—
129#391C22×C312yes—
130#391C22×C312yes—
131#381—22no—
132#343C22×C312yes—
133#343C22×C312yes—
134#339C22×C314yes—
135C24×C3C2×D411yescharacteristic
136C3×#41—24no—
137C3×#41—24no—
138C2×C3×Q8—24no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—24no—
142C2×C3×Q8—24no—
143C2×C3×Q8—48no—
144C2×C3×Q8—48no—
145C2×C3×Q8—42no—
146C2×C3×Q8—41no—
147C2×C3×Q8—41no—
148C2×C3×Q8C2×D412yes—
149C2×C3×Q8#4114yes—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4C2×D412yes—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—24no—
158C22×C3×C4—22no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—42no—
163C22×C3×C4—42no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2411yescharacteristic
170C3×Q16—48no—
171C3×SD16—48no—
172C3×M16—24no—
173C3×M16—24no—
174C2×C3×C8—24no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4C2×D412yes—
185C3×C4⋊C4C2×D412yes—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—24no—
188C3×C22⋊C4—24no—
189C3×C22⋊C4—22no—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—22no—
193C3×C22⋊C4—22no—
194C3×C22⋊C4—42no—
195C3×C42—24no—
196C3×C42—22no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—24no—
200C2×#41C23×C312yes—
201C22×Q8—24no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—42no—
206C22×Q8C23×C312yes—
207C23×C4—21no—
208C23×C4C23×C311yescharacteristic
209#137—28no—
210#137—44no—
211C2×Q16—24no—
212C2×SD16—24no—
213C2×M16C23×C314yes—
214C4⋊Q8—22no—
215#126—24no—
216#126—24no—
217#125—22no—
218#125—22no—
219#124—22no—
220#124—24no—
221#123—24no—
222#123—22no—
223C22⋊Q8—22no—
224C22⋊Q8—22no—
225C22⋊Q8C3×D412yes—
226C22⋊Q8C3×D412yes—
227#121—24no—
228#121—22no—
229#121—22no—
230#121C3×D412yes—
231#121C3×D412yes—
232#120—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—24no—
238C4×D4—22no—
239C4×D4—22no—
240#117—24no—
241#117—22no—
242#117—22no—
243#117—22no—
244#117—42no—
245#117C23×C312yes—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C23×C311yescharacteristic
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4C3×D412yes—
252C2×C42C3×D412yes—
253C2×C42C23×C312yes—
254#104—44no—
255#103—24no—
256#103—24no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259#95—24no—
260#95—24no—
261C23×C3—21no—
262C23×C3—21no—
263C23×C3—22no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3#12012yes—
267C23×C3C22×D411yescharacteristic
268C3×Q8—28no—
269C3×Q8—24no—
270C3×Q8—44no—
271C3×Q8—44no—
272C3×Q8—44no—
273C3×Q8—82no—
274C3×D4—24no—
275C3×D4—42no—
276C3×D4—42no—
277C3×D4—82no—
278C3×D4—82no—
279C3×D4—82no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—24no—
283C2×C3×C4—24no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—48no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—44no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4—42no—
298C2×C3×C4—42no—
299C2×C3×C4—84no—
300C2×C3×C4#12011yescharacteristic
301C2×C3×C4#12011yescharacteristic
302C2×C3×C4C22×D412yes—
303C2×C3×C4C2×#4112yes—
304C2×C3×C4H(5,𝔽2)12yes—
305C3×C8—44no—
306C24C2×C3×D411yescharacteristic
307#41—24no—
308#41—24no—
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—42no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8C2×C3×D412yes—
320C2×Q8C3×#4114yes—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4C2×C3×D412yes—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C2×C3×D412yes—
337C22×C4C2×C3×D412yes—
338C22×C4C2×C3×D412yes—
339C22×C4C2×C3×D412yes—
340C22×C4C24×C311yescharacteristic
341Q16—48no—
342SD16—48no—
343M16—24no—
344M16—24no—
345C2×C8—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4C2×C3×D412yes—
356C4⋊C4C2×C3×D412yes—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—22no—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—42no—
366C42—24no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—24no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—22no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—42no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3C2×#12011yescharacteristic
382C22×C3#52511yescharacteristic
383C22×C3#53511yescharacteristic
384C3×C4—21no—
385C3×C4—21no—
386C3×C4—44no—
387C3×C4—42no—
388C3×C4—42no—
389C3×C4—42no—
390C3×C4—42no—
391C3×C4—42no—
392C3×C4—82no—
393C3×C4#54614yes—
394C23—21no—
395C23—21no—
396C23—22no—
397C23—41no—
398C23—41no—
399C23C3×#12012yes—
400C23C22×C3×D411yescharacteristic
401Q8—28no—
402Q8—24no—
403Q8—44no—
404Q8—44no—
405Q8—44no—
406Q8—82no—
407D4—24no—
408D4—42no—
409D4—42no—
410D4—82no—
411D4—82no—
412D4—82no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—48no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—84no—
433C2×C4C3×#12011yescharacteristic
434C2×C4C3×#12011yescharacteristic
435C2×C4C22×C3×D412yes—
436C2×C4C2×C3×#4112yes—
437C2×C4C3×H(5,𝔽2)12yes—
438C8—44no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—42no—
442C2×C3—82no—
443C2×C3#240411yescharacteristic
444C2×C3#302211yescharacteristic
445C2×C3#302211yescharacteristic
446C22—21no—
447C22—21no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22C2×C3×#12011yescharacteristic
457C22C3×#52511yescharacteristic
458C22C3×#53511yescharacteristic
459C4—21no—
460C4—21no—
461C4—44no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—82no—
468C4C3×#54614yes—
469C3#2395311yescharacteristic, direct product factor, Sylow
470C2—21no—
471C2—21no—
472C2—42no—
473C2—82no—
474C2C3×#240411yescharacteristic
475C2C3×#302211yescharacteristic
476C2C3×#302211yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3493C2
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. C3×#3004C2semidirect
3. C3×#2310C2semidirect
4. C3×#2028C2semidirect
5. C3×#2007C2semidirect
6. C3×#1897C2semidirect
7. C3×#1882C2semidirect
8. C3×#1822C2semidirect
9. #23953C3direct (N × H)
39. #3004C2×C3semidirect
40. #2310C2×C3semidirect
41. #2028C2×C3semidirect
42. #2007C2×C3semidirect
43. #1897C2×C3semidirect
44. #1882C2×C3semidirect
45. #1822C2×C3semidirect
469. C3#23953direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34932
C3×#30042
C3×#23102
C3×#20282
C3×#20072
C3×#18972
C3×#18822
C3×#18222
#239533