Finite Group Databasebeta

← C3×#23944 #10642551

Subgroups — C3×#23944 #10642551

Order 768.

1206 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic, complemented
2C2×C3×#452C212yesmaximal
3C3×#2311C211yesmaximal, characteristic
4C3×#2026C211yesmaximal, characteristic, complemented
5C3×#2025C211yesmaximal, characteristic, complemented
6C3×#2016C211yesmaximal, characteristic, complemented
7C3×#2013C211yesmaximal, characteristic, complemented
8C3×#1880C212yesmaximal, complemented
9C3×#1878C212yesmaximal, complemented
10C3×#1872C212yesmaximal, complemented
11C3×#1818C211yesmaximal, characteristic, complemented
12#23944C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×Q16C2212yes—
15C22×C3×SD16C2212yes—
16C3×#541—28no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#126—22no—
19C2×C3×#125—22no—
20C2×C3×#125C2211yescharacteristic
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C4×Q8C2212yes—
23C2×C3×#117C2211yescharacteristic
24C22×C3×C4⋊C4C2211yescharacteristic
25C3×#452—28no—
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#102C2212yes—
30C2×C3×C22⋊C8C2212yes—
31C3×#391C2211yescharacteristic
32C3×#391C2211yescharacteristic
33C3×#390C2211yescharacteristic
34C3×#390C2211yescharacteristic
35C3×#386—22no—
36C3×#380—22no—
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#336C2211yescharacteristic
40C3×#336C2211yescharacteristic
41C3×#336C2212yes—
42C2×#541C2×C311yescharacteristic, complemented
43C2×#452C2×C312yes—
44#2311C2×C311yescharacteristic
45#2026C2×C311yescharacteristic, complemented
46#2025C2×C311yescharacteristic, complemented
47#2016C2×C311yescharacteristic, complemented
48#2013C2×C311yescharacteristic, complemented
49#1880C2×C312yescomplemented
50#1878C2×C312yescomplemented
51#1872C2×C312yescomplemented
52#1818C2×C311yescharacteristic, complemented
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—24no—
56C2×C3×#41—42no—
57C22×C3×Q8C2312yes—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4—21no—
63C23×C3×C4C2311yescharacteristic
64C2×C3×Q16—24no—
65C2×C3×Q16—24no—
66C2×C3×Q16—42no—
67C2×C3×SD16—24no—
68C2×C3×SD16—24no—
69C2×C3×SD16—42no—
70C22×C3×C8C2312yes—
71C22×C3×C8C2312yes—
72C3×C4⋊Q8—24no—
73C3×#126—216no—
74C3×#125—216no—
75C3×#125—24no—
76C3×C22⋊Q8—24no—
77C3×C22⋊Q8D418yes—
78C3×C4×Q8—24no—
79C3×C4×Q8—24no—
80C3×#117—24no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—42no—
87C2×C3×C4⋊C4D412yes—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—22no—
93C2×C3×C22⋊C4—22no—
94C2×C3×C22⋊C4D412yes—
95C2×C3×C42—22no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C3×#103—24no—
99C3×#103—42no—
100C3×#103—42no—
101C3×#102—24no—
102C3×C22⋊C8—24no—
103C3×#95—22no—
104C3×#95—22no—
105C22×#41C22×C311yescharacteristic
106C22×Q16C22×C312yes—
107C22×SD16C22×C312yes—
108#541—28no—
109C2×C4⋊Q8C22×C311yescharacteristic
110C2×#126—22no—
111C2×#125—22no—
112C2×#125C22×C311yescharacteristic
113C2×C22⋊Q8C22×C312yes—
114C2×C4×Q8C22×C312yes—
115C2×#117C22×C311yescharacteristic
116C22×C4⋊C4C22×C311yescharacteristic
117#452—28no—
118C2×#103C22×C312yes—
119C2×#103C22×C312yes—
120C2×#103C22×C312yes—
121C2×#102C22×C312yes—
122C2×C22⋊C8C22×C312yes—
123#391C22×C311yescharacteristic
124#391C22×C311yescharacteristic
125#390C22×C311yescharacteristic
126#390C22×C311yescharacteristic
127#386—22no—
128#380—22no—
129#340C22×C312yes—
130#340C22×C312yes—
131#336C22×C312yes—
132#336C22×C311yescharacteristic
133#336C22×C311yescharacteristic
134C24×C3—21no—
135C24×C3C2×D411yescharacteristic
136C3×#41—48no—
137C3×#41—48no—
138C2×C3×Q8—22no—
139C2×C3×Q8—21no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×Q8—22no—
144C2×C3×Q8—22no—
145C2×C3×Q8—22no—
146C2×C3×Q8C2×D414yes—
147C2×C3×Q8#4112yes—
148C2×C3×D4—24no—
149C2×C3×D4—22no—
150C2×C3×D4—21no—
151C2×C3×D4—22no—
152C2×C3×D4—42no—
153C2×C3×D4—42no—
154C2×C3×D4#4112yes—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—24no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—42no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4—82no—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D414yes—
172C22×C3×C4C2411yescharacteristic
173C3×Q16—48no—
174C3×SD16—48no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C3×C4⋊C4—28no—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—28no—
184C3×C4⋊C4—28no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—22no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—42no—
190C3×C4⋊C4C2×D414yes—
191C3×C22⋊C4—28no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—82no—
194C3×C22⋊C4—82no—
195C3×C42—28no—
196C3×C42—22no—
197C3×C42—22no—
198C2×#41—22no—
199C2×#41—22no—
200C2×#41—24no—
201C2×#41—42no—
202C22×Q8C23×C312yes—
203C22×Q8C23×C311yescharacteristic
204C22×D4—21no—
205C22×D4C23×C311yescharacteristic
206C23×C4—21no—
207C23×C4—21no—
208C23×C4C23×C311yescharacteristic
209C2×Q16—24no—
210C2×Q16—24no—
211C2×Q16—42no—
212C2×SD16—24no—
213C2×SD16—24no—
214C2×SD16—42no—
215C22×C8C23×C312yes—
216C22×C8C23×C312yes—
217C4⋊Q8—24no—
218#126—216no—
219#125—216no—
220#125—24no—
221C22⋊Q8—24no—
222C22⋊Q8C3×D418yes—
223C4×Q8—24no—
224C4×Q8—24no—
225#117—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C3×D412yes—
233C2×C4⋊C4C23×C311yescharacteristic
234C2×C4⋊C4C23×C311yescharacteristic
235C2×C4⋊C4C23×C312yes—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4C3×D412yes—
240C2×C42—22no—
241C2×C42C23×C311yescharacteristic
242C2×C42C23×C311yescharacteristic
243#103—24no—
244#103—42no—
245#103—42no—
246#102—24no—
247C22⋊C8—24no—
248#95—22no—
249#95—22no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—22no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3#12012yes—
260C23×C3C22×D411yescharacteristic
261C3×Q8—24no—
262C3×Q8—24no—
263C3×Q8—24no—
264C3×Q8—42no—
265C3×Q8—42no—
266C3×Q8—42no—
267C3×D4—24no—
268C3×D4—48no—
269C3×D4—42no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—48no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—84no—
295C2×C3×C4#12012yes—
296C2×C3×C4C22×D412yes—
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C4H(5,𝔽2)12yes—
300C3×C8—42no—
301C3×C8—42no—
302C24—21no—
303C24C2×C3×D411yescharacteristic
304#41—48no—
305#41—48no—
306C2×Q8—22no—
307C2×Q8—21no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8C2×C3×D414yes—
315C2×Q8C3×#4112yes—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—22no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4C3×#4112yes—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—82no—
336C22×C4C2×C3×D412yes—
337C22×C4C2×C3×D412yes—
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×D414yes—
340C22×C4C24×C311yescharacteristic
341Q16—48no—
342SD16—48no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C4⋊C4—28no—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—28no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—42no—
358C4⋊C4C2×C3×D414yes—
359C22⋊C4—28no—
360C22⋊C4—24no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C42—28no—
364C42—22no—
365C42—22no—
366C22×C3—22no—
367C22×C3—24no—
368C22×C3—42no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—44no—
372C22×C3—42no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3—82no—
376C22×C3C2×#12011yescharacteristic
377C22×C3#52511yescharacteristic
378C22×C3#53511yescharacteristic
379C22×C3#57712yes—
380C22×C3#57812yes—
381C3×C4—22no—
382C3×C4—42no—
383C3×C4—44no—
384C3×C4—42no—
385C3×C4—42no—
386C3×C4—44no—
387C3×C4—81no—
388C3×C4—81no—
389C3×C4#54612yes—
390C3×C4#54612yes—
391C23—22no—
392C23—22no—
393C23—21no—
394C23—21no—
395C23—22no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23C3×#12012yes—
401C23C22×C3×D411yescharacteristic
402Q8—24no—
403Q8—24no—
404Q8—24no—
405Q8—42no—
406Q8—42no—
407Q8—42no—
408D4—24no—
409D4—48no—
410D4—42no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—84no—
436C2×C4C3×#12012yes—
437C2×C4C22×C3×D412yes—
438C2×C4C2×C3×#4111yescharacteristic
439C2×C4C2×C3×#4111yescharacteristic
440C2×C4C3×H(5,𝔽2)12yes—
441C8—42no—
442C8—42no—
443C2×C3—42no—
444C2×C3—82no—
445C2×C3#240411yescharacteristic
446C2×C3#301312yes—
447C2×C3#312012yes—
448C2×C3#322211yescharacteristic
449C2×C3#322311yescharacteristic
450C22—22no—
451C22—24no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—44no—
457C22—81no—
458C22—81no—
459C22—82no—
460C22C2×C3×#12011yescharacteristic
461C22C3×#52511yescharacteristic
462C22C3×#53511yescharacteristic
463C22C3×#57712yes—
464C22C3×#57812yes—
465C4—22no—
466C4—42no—
467C4—44no—
468C4—42no—
469C4—42no—
470C4—44no—
471C4—81no—
472C4—81no—
473C4C3×#54612yes—
474C4C3×#54612yes—
475C3#2394411yescharacteristic, direct product factor, Sylow
476C2—42no—
477C2—82no—
478C2C3×#240411yescharacteristic
479C2C3×#301312yes—
480C2C3×#312012yes—
481C2C3×#322211yescharacteristic
482C2C3×#322311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#541C2semidirect
4. C3×#2026C2semidirect
5. C3×#2025C2semidirect
6. C3×#2016C2semidirect
7. C3×#2013C2semidirect
8. C3×#1880C2semidirect
9. C3×#1878C2semidirect
10. C3×#1872C2semidirect
11. C3×#1818C2semidirect
12. #23944C3direct (N × H)
42. C2×#541C2×C3semidirect
45. #2026C2×C3semidirect
46. #2025C2×C3semidirect
47. #2016C2×C3semidirect
48. #2013C2×C3semidirect
49. #1880C2×C3semidirect
50. #1878C2×C3semidirect
51. #1872C2×C3semidirect
52. #1818C2×C3semidirect
475. C3#23944direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×C3×#4522
C3×#23112
C3×#20262
C3×#20252
C3×#20162
C3×#20132
C3×#18802
C3×#18782
C3×#18722
C3×#18182
#239443