Finite Group Databasebeta

← C3×#24646 #10643253

Subgroups — C3×#24646 #10643253

Order 768.

864 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#461C211yesmaximal, characteristic
2C2×C3×#440C211yesmaximal, characteristic
3C3×#2546C211yesmaximal, characteristic, complemented
4C3×#2357C211yesmaximal, characteristic
5C3×#2055C212yesmaximal
6C3×#2013C212yesmaximal
7C3×#1981C212yesmaximal, complemented
8C3×#1960C211yesmaximal, characteristic, complemented
9C3×#1940C211yesmaximal, characteristic, complemented
10C3×#1928C212yesmaximal, complemented
11C3×#1805C211yesmaximal, characteristic, complemented
12#24646C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C3×#461—24no—
20C3×#440—24no—
21C2×C3×#107C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C3×#398—21no—
32C3×#398—21no—
33C3×#391C2212yes—
34C3×#390C2212yes—
35C3×#389—22no—
36C3×#388—22no—
37C3×#382C2211yescharacteristic
38C3×#381—21no—
39C3×C4×C4⋊C4C2212yes—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×C4×C22⋊C4—21no—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#461C2×C311yescharacteristic
45C2×#440C2×C311yescharacteristic
46#2546C2×C311yescharacteristic, complemented
47#2357C2×C311yescharacteristic
48#2055C2×C312yes—
49#2013C2×C312yes—
50#1981C2×C312yescomplemented
51#1960C2×C311yescharacteristic, complemented
52#1940C2×C311yescharacteristic, complemented
53#1928C2×C312yescomplemented
54#1805C2×C311yescharacteristic, complemented
55C22×C3×Q8C2312yes—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C2×C3×SD16—22no—
60C2×C3×SD16—22no—
61C2×C3×SD16—24no—
62C2×C3×SD16—22no—
63C2×C3×SD16—42no—
64C2×C3×SD16—42no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×C4⋊Q8—24no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×C4⋊Q8—42no—
72C3×C4×Q8—22no—
73C3×C4×Q8—22no—
74C3×C4×D4—22no—
75C3×C4×D4D414yes—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4D412yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C42—21no—
89C2×C3×C42—22no—
90C2×C3×C42C2312yes—
91C2×C3×C42C2311yescharacteristic
92C3×#107—22no—
93C3×C4⋊C8—22no—
94C3×C4⋊C8—42no—
95C3×#103—24no—
96C3×#103—22no—
97C3×#103—24no—
98C3×#103—42no—
99C3×#103—42no—
100C3×#102—22no—
101C3×#97—22no—
102C3×#95—21no—
103C3×#95—22no—
104C3×#95—21no—
105C3×#95—22no—
106C3×#95—22no—
107C3×#95D411yescharacteristic
108C3×#95D411yescharacteristic
109C22×SD16C22×C311yescharacteristic
110C22×SD16C22×C312yes—
111C2×C4⋊Q8C22×C311yescharacteristic
112C2×C4⋊Q8C22×C312yes—
113C2×C4×Q8C22×C311yescharacteristic
114C2×C4×D4C22×C311yescharacteristic
115#461—24no—
116#440—24no—
117C2×#107C22×C311yescharacteristic
118C2×C4⋊C8C22×C311yescharacteristic
119C2×C4⋊C8C22×C312yes—
120C2×#103C22×C311yescharacteristic
121C2×#103C22×C312yes—
122C2×#103C22×C312yes—
123C2×#103C22×C312yes—
124C2×#103C22×C312yes—
125C2×#102C22×C311yescharacteristic
126C2×#97C22×C311yescharacteristic
127#398—21no—
128#398—21no—
129#391C22×C312yes—
130#390C22×C312yes—
131#389—22no—
132#388—22no—
133#382C22×C311yescharacteristic
134#381—21no—
135C4×C4⋊C4C22×C312yes—
136C4×C4⋊C4C22×C311yescharacteristic
137C4×C22⋊C4—21no—
138#340C22×C312yes—
139#336C22×C312yes—
140C24×C3—21no—
141C2×C3×Q8—21no—
142C2×C3×Q8—21no—
143C2×C3×Q8—22no—
144C2×C3×Q8—21no—
145C2×C3×Q8—21no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8—22no—
149C2×C3×Q8—21no—
150C2×C3×Q8—22no—
151C2×C3×Q8—22no—
152C2×C3×Q8#4112yes—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4C2×D412yes—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—24no—
162C22×C3×C4—21no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—82no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2411yescharacteristic
171C3×SD16—28no—
172C3×SD16—48no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—22no—
177C2×C3×C8—21no—
178C2×C3×C8—22no—
179C2×C3×C8—21no—
180C2×C3×C8—22no—
181C2×C3×C8#4112yes—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—22no—
188C3×C4⋊C4—44no—
189C3×C4⋊C4—44no—
190C3×C4⋊C4—42no—
191C3×C4⋊C4—42no—
192C3×C4⋊C4—42no—
193C3×C4⋊C4C2×D412yes—
194C3×C22⋊C4—24no—
195C3×C22⋊C4—44no—
196C3×C22⋊C4—82no—
197C3×C42—22no—
198C3×C42—22no—
199C3×C42—42no—
200C3×C42—82no—
201C3×C42C2×D412yes—
202C3×C42#4112yes—
203C22×Q8C23×C312yes—
204C22×Q8C23×C311yescharacteristic
205C22×D4C23×C311yescharacteristic
206C23×C4—21no—
207C2×SD16—22no—
208C2×SD16—22no—
209C2×SD16—24no—
210C2×SD16—22no—
211C2×SD16—42no—
212C2×SD16—42no—
213C22×C8C23×C312yes—
214C22×C8C23×C311yescharacteristic
215C22×C8C23×C311yescharacteristic
216C4⋊Q8—24no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—42no—
220C4×Q8—22no—
221C4×Q8—22no—
222C4×D4—22no—
223C4×D4C3×D414yes—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C3×D412yes—
230C2×C4⋊C4C23×C311yescharacteristic
231C2×C4⋊C4C23×C312yes—
232C2×C4⋊C4C23×C311yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42C23×C312yes—
239C2×C42C23×C311yescharacteristic
240#107—22no—
241C4⋊C8—22no—
242C4⋊C8—42no—
243#103—24no—
244#103—22no—
245#103—24no—
246#103—42no—
247#103—42no—
248#102—22no—
249#97—22no—
250#95—21no—
251#95—22no—
252#95—21no—
253#95—22no—
254#95—22no—
255#95C3×D411yescharacteristic
256#95C3×D411yescharacteristic
257C23×C3—22no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3C22×D411yescharacteristic
263C3×Q8—22no—
264C3×Q8—22no—
265C3×Q8—41no—
266C3×Q8—41no—
267C3×Q8—42no—
268C3×Q8—42no—
269C3×Q8—42no—
270C3×Q8—42no—
271C3×D4—22no—
272C3×D4—22no—
273C3×D4—41no—
274C3×D4—41no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—84no—
296C2×C3×C4#12112yes—
297C2×C3×C4#12112yes—
298C2×C3×C4#13714yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4H(5,𝔽2)11yescharacteristic
304C2×C3×C4H(5,𝔽2)11yescharacteristic
305C3×C8—22no—
306C3×C8—41no—
307C3×C8—42no—
308C24—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—22no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8C3×#4112yes—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—21no—
326C2×D4C2×C3×D412yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—82no—
334C22×C4C2×C3×D412yes—
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C2×C3×D411yescharacteristic
337C22×C4C2×C3×D412yes—
338C22×C4C24×C311yescharacteristic
339SD16—28no—
340SD16—48no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8C3×#4112yes—
350C4⋊C4—24no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4C2×C3×D412yes—
362C22⋊C4—24no—
363C22⋊C4—44no—
364C22⋊C4—82no—
365C42—22no—
366C42—22no—
367C42—42no—
368C42—82no—
369C42C2×C3×D412yes—
370C42C3×#4112yes—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—82no—
375C22×C3—82no—
376C22×C3C2×#12111yescharacteristic
377C22×C3#54611yescharacteristic
378C22×C3#55011yescharacteristic
379C22×C3C2×#13712yes—
380C22×C3#57712yes—
381C3×C4—22no—
382C3×C4—22no—
383C3×C4—24no—
384C3×C4—42no—
385C3×C4—42no—
386C3×C4—42no—
387C3×C4—82no—
388C3×C4—82no—
389C3×C4—81no—
390C3×C4#52511yescharacteristic
391C3×C4#53511yescharacteristic
392C3×C4#55311yescharacteristic
393C3×C4#55511yescharacteristic
394C23—22no—
395C23—42no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23C22×C3×D411yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—41no—
403Q8—41no—
404Q8—42no—
405Q8—42no—
406Q8—42no—
407Q8—42no—
408D4—22no—
409D4—22no—
410D4—41no—
411D4—41no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—84no—
433C2×C4C3×#12112yes—
434C2×C4C3×#12112yes—
435C2×C4C3×#13714yes—
436C2×C4C22×C3×D411yescharacteristic
437C2×C4C2×C3×#4111yescharacteristic
438C2×C4C2×C3×#4111yescharacteristic
439C2×C4C2×C3×#4111yescharacteristic
440C2×C4C3×H(5,𝔽2)11yescharacteristic
441C2×C4C3×H(5,𝔽2)11yescharacteristic
442C8—22no—
443C8—42no—
444C8—41no—
445C2×C3—82no—
446C2×C3#241411yescharacteristic
447C2×C3#303911yescharacteristic
448C2×C3#304211yescharacteristic
449C2×C3#332012yes—
450C2×C3#340012yes—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—82no—
455C22—82no—
456C22C2×C3×#12111yescharacteristic
457C22C3×#54611yescharacteristic
458C22C3×#55011yescharacteristic
459C22C2×C3×#13712yes—
460C22C3×#57712yes—
461C4—22no—
462C4—22no—
463C4—24no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—82no—
468C4—82no—
469C4—81no—
470C4C3×#52511yescharacteristic
471C4C3×#53511yescharacteristic
472C4C3×#55311yescharacteristic
473C4C3×#55511yescharacteristic
474C3#2464611yescharacteristic, direct product factor, Sylow
475C2—82no—
476C2C3×#241411yescharacteristic
477C2C3×#303911yescharacteristic
478C2C3×#304211yescharacteristic
479C2C3×#332012yes—
480C2C3×#340012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2546C2semidirect
7. C3×#1981C2semidirect
8. C3×#1960C2semidirect
9. C3×#1940C2semidirect
10. C3×#1928C2semidirect
11. C3×#1805C2semidirect
12. #24646C3direct (N × H)
46. #2546C2×C3semidirect
50. #1981C2×C3semidirect
51. #1960C2×C3semidirect
52. #1940C2×C3semidirect
53. #1928C2×C3semidirect
54. #1805C2×C3semidirect
474. C3#24646direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4612
C2×C3×#4402
C3×#25462
C3×#23572
C3×#20552
C3×#20132
C3×#19812
C3×#19602
C3×#19402
C3×#19282
C3×#18052
#246463