Finite Group Databasebeta

← C3×#23949 #10642556

Subgroups — C3×#23949 #10642556

Order 768.

1258 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3492C211yesmaximal, characteristic
2C3×#3004C212yesmaximal, complemented
3C2×C3×#391C211yesmaximal, characteristic, complemented
4C3×#2028C212yesmaximal, complemented
5C3×#2007C212yesmaximal, complemented
6C3×#1896C212yesmaximal, complemented
7C3×#1882C214yesmaximal, complemented
8C3×#1821C211yesmaximal, characteristic, complemented
9#23949C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×Q8C2211yescharacteristic
11C2×C3×#137C2214yes—
12C3×#554—22no—
13C3×#553—22no—
14C3×#552—22no—
15C3×#547—22no—
16C3×#544C2211yescharacteristic
17C3×#539—22no—
18C3×#537C2211yescharacteristic
19C3×#525—22no—
20C3×#525C2212yes—
21C3×#520C2212yes—
22C22×C3×C4⋊C4C2211yescharacteristic
23C22×C3×C42C2211yescharacteristic
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—
34C2×C3×#95—22no—
35C3×#343C2212yes—
36C3×#343C2212yes—
37C3×#339C2214yes—
38#3492C2×C311yescharacteristic
39#3004C2×C312yescomplemented
40C2×#391C2×C311yescharacteristic, complemented
41#2028C2×C312yescomplemented
42#2007C2×C312yescomplemented
43#1896C2×C312yescomplemented
44#1882C2×C314yescomplemented
45#1821C2×C311yescharacteristic, complemented
46C2×C3×#41—22no—
47C2×C3×#41C2312yes—
48C22×C3×Q8—24no—
49C22×C3×Q8—21no—
50C22×C3×Q8—22no—
51C22×C3×Q8—22no—
52C22×C3×Q8—42no—
53C22×C3×Q8C2312yes—
54C23×C3×C4—21no—
55C23×C3×C4—21no—
56C23×C3×C4D412yes—
57C23×C3×C4C2311yescharacteristic
58C3×#137—28no—
59C3×#137—44no—
60C2×C3×Q16—24no—
61C2×C3×SD16—24no—
62C2×C3×M16C2314yes—
63C3×C4⋊Q8—22no—
64C3×#126—24no—
65C3×#125—24no—
66C3×#125—22no—
67C3×#123—24no—
68C3×#123—24no—
69C3×#123—22no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—24no—
72C3×C22⋊Q8—22no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8D412yes—
75C3×C22⋊Q8D412yes—
76C3×#121—22no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#121D412yes—
80C3×#121D412yes—
81C3×#120—22no—
82C3×C4×Q8—22no—
83C3×C4×D4—24no—
84C3×C4×D4—22no—
85C3×C4×D4—24no—
86C3×C4×D4—22no—
87C3×C4×D4—22no—
88C3×#117—22no—
89C3×#117—22no—
90C3×#117C2312yes—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4D412yes—
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42—21no—
98C2×C3×C42—42no—
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#554—22no—
110#553—22no—
111#552—22no—
112#547—22no—
113#544C22×C311yescharacteristic
114#539—22no—
115#537C22×C311yescharacteristic
116#525—22no—
117#525C22×C312yes—
118#520C22×C312yes—
119C22×C4⋊C4C22×C311yescharacteristic
120C22×C42C22×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—
131C2×#95—22no—
132#343C22×C312yes—
133#343C22×C312yes—
134#339C22×C314yes—
135C24×C3C2×D411yescharacteristic
136C3×#41—24no—
137C3×#41—24no—
138C3×#41—84no—
139C2×C3×Q8—24no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—24no—
143C2×C3×Q8—24no—
144C2×C3×Q8—48no—
145C2×C3×Q8—48no—
146C2×C3×Q8—42no—
147C2×C3×Q8—42no—
148C2×C3×Q8C2×D412yes—
149C2×C3×Q8#4114yescomplemented
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—24no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4—42no—
168C22×C3×C4—41no—
169C22×C3×C4—41no—
170C22×C3×C4—44no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D412yes—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2411yescharacteristic
177C3×Q16—48no—
178C3×SD16—48no—
179C3×M16—24no—
180C3×M16—24no—
181C2×C3×C8—24no—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4C2×D412yes—
192C3×C4⋊C4C2×D412yes—
193C3×C22⋊C4—24no—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—22no—
196C3×C22⋊C4—22no—
197C3×C22⋊C4—22no—
198C3×C22⋊C4—22no—
199C3×C22⋊C4—22no—
200C3×C42—24no—
201C3×C42—22no—
202C3×C42—24no—
203C3×C42—42no—
204C2×#41—22no—
205C2×#41C23×C312yes—
206C22×Q8—24no—
207C22×Q8—21no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—42no—
211C22×Q8C23×C312yes—
212C23×C4—21no—
213C23×C4—21no—
214C23×C4C3×D412yes—
215C23×C4C23×C311yescharacteristic
216#137—28no—
217#137—44no—
218C2×Q16—24no—
219C2×SD16—24no—
220C2×M16C23×C314yes—
221C4⋊Q8—22no—
222#126—24no—
223#125—24no—
224#125—22no—
225#123—24no—
226#123—24no—
227#123—22no—
228C22⋊Q8—22no—
229C22⋊Q8—24no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8C3×D412yes—
233C22⋊Q8C3×D412yes—
234#121—22no—
235#121—22no—
236#121—22no—
237#121C3×D412yes—
238#121C3×D412yes—
239#120—22no—
240C4×Q8—22no—
241C4×D4—24no—
242C4×D4—22no—
243C4×D4—24no—
244C4×D4—22no—
245C4×D4—22no—
246#117—22no—
247#117—22no—
248#117C23×C312yes—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4C3×D412yes—
253C2×C4⋊C4C23×C311yescharacteristic
254C2×C4⋊C4C23×C311yescharacteristic
255C2×C42—21no—
256C2×C42—42no—
257C2×C42C23×C312yes—
258#104—44no—
259#103—24no—
260#103—24no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263#95—24no—
264#95—24no—
265C23×C3—21no—
266C23×C3—21no—
267C23×C3—22no—
268C23×C3—22no—
269C23×C3—22no—
270C23×C3#12011yescharacteristic
271C23×C3#12011yescharacteristic
272C23×C3C22×D411yescharacteristic
273C3×Q8—28no—
274C3×Q8—24no—
275C3×Q8—48no—
276C3×Q8—44no—
277C3×Q8—82no—
278C3×D4—24no—
279C3×D4—42no—
280C3×D4—42no—
281C3×D4—44no—
282C3×D4—44no—
283C3×D4—82no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—24no—
287C2×C3×C4—24no—
288C2×C3×C4—24no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—24no—
295C2×C3×C4—48no—
296C2×C3×C4—42no—
297C2×C3×C4—44no—
298C2×C3×C4—44no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—44no—
303C2×C3×C4—44no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4—82no—
307C2×C3×C4—84no—
308C2×C3×C4#12012yes—
309C2×C3×C4C22×D412yes—
310C2×C3×C4C2×#4112yes—
311C2×C3×C4H(5,𝔽2)12yes—
312C3×C8—44no—
313C24C2×C3×D411yescharacteristic
314#41—24no—
315#41—24no—
316#41—84no—
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—24no—
321C2×Q8—24no—
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8C2×C3×D412yes—
327C2×Q8C3×#4114yescomplemented
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4C2×C3×D412yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—44no—
349C22×C4C2×C3×D411yescharacteristic
350C22×C4C2×C3×D412yes—
351C22×C4C2×C3×D412yes—
352C22×C4C2×C3×D412yes—
353C22×C4C2×C3×D412yes—
354C22×C4C24×C311yescharacteristic
355Q16—48no—
356SD16—48no—
357M16—24no—
358M16—24no—
359C2×C8—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4C2×C3×D412yes—
370C4⋊C4C2×C3×D412yes—
371C22⋊C4—24no—
372C22⋊C4—24no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C42—24no—
379C42—22no—
380C42—24no—
381C42—42no—
382C22×C3—24no—
383C22×C3—24no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—42no—
391C22×C3—42no—
392C22×C3—42no—
393C22×C3#45512yes—
394C22×C3#45612yes—
395C22×C3C2×#12011yescharacteristic
396C22×C3#52511yescharacteristic
397C22×C3#53511yescharacteristic
398C3×C4—22no—
399C3×C4—44no—
400C3×C4—44no—
401C3×C4—42no—
402C3×C4—42no—
403C3×C4—42no—
404C3×C4—42no—
405C3×C4—42no—
406C3×C4#54614yes—
407C23—22no—
408C23—22no—
409C23—21no—
410C23—21no—
411C23—22no—
412C23C3×#12011yescharacteristic
413C23C3×#12011yescharacteristic
414C23C22×C3×D411yescharacteristic
415Q8—28no—
416Q8—24no—
417Q8—48no—
418Q8—44no—
419Q8—82no—
420D4—24no—
421D4—44no—
422D4—44no—
423D4—42no—
424D4—42no—
425D4—82no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—48no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—82no—
449C2×C4—84no—
450C2×C4C3×#12012yes—
451C2×C4C22×C3×D412yes—
452C2×C4C2×C3×#4112yes—
453C2×C4C3×H(5,𝔽2)12yes—
454C8—44no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—22no—
458C2×C3—22no—
459C2×C3—82no—
460C2×C3#240411yescharacteristic
461C2×C3C2×#45511yescharacteristic
462C2×C3C2×#45611yescharacteristic
463C22—24no—
464C22—24no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22C3×#45512yes—
475C22C3×#45612yes—
476C22C2×C3×#12011yescharacteristic
477C22C3×#52511yescharacteristic
478C22C3×#53511yescharacteristic
479C4—22no—
480C4—44no—
481C4—44no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4C3×#54614yes—
488C3#2394911yescharacteristic, direct product factor, Sylow
489C2—21no—
490C2—22no—
491C2—22no—
492C2—21no—
493C2—82no—
494C2C3×#240411yescharacteristic
495C2C2×C3×#45511yescharacteristic
496C2C2×C3×#45611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3492C2
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. C2×C3×#391C2semidirect
4. C3×#2028C2semidirect
5. C3×#2007C2semidirect
6. C3×#1896C2semidirect
7. C3×#1882C2semidirect
8. C3×#1821C2semidirect
9. #23949C3direct (N × H)
39. #3004C2×C3semidirect
40. C2×#391C2×C3semidirect
41. #2028C2×C3semidirect
42. #2007C2×C3semidirect
43. #1896C2×C3semidirect
44. #1882C2×C3semidirect
45. #1821C2×C3semidirect
149. C2×C3×Q8#41semidirect
327. C2×Q8C3×#41semidirect
488. C3#23949direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34922
C3×#30042
C2×C3×#3912
C3×#20282
C3×#20072
C3×#18962
C3×#18822
C3×#18212
#239493