Finite Group Databasebeta

← C3×#24332 #10642939

Subgroups — C3×#24332 #10642939

Order 768.

1002 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic
2C2×C3×#475C211yesmaximal, characteristic
3C2×C3×C4×Q16C211yesmaximal, characteristic
4C3×#2582C211yesmaximal, characteristic
5C3×#2059C212yesmaximal
6C3×#2026C212yesmaximal
7C3×#1990C211yesmaximal, characteristic
8C3×#1984C211yesmaximal, characteristic
9C3×#1878C212yesmaximal
10C3×#1865C212yesmaximal
11C3×#1803C211yesmaximal, characteristic
12#24332C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2211yescharacteristic
14C3×#557—24no—
15C3×#557—24no—
16C2×C3×C4⋊Q8—21no—
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125—22no—
19C2×C3×#125C2212yes—
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C3×#475—24no—
26C3×C4×Q16—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#106C2212yes—
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×C4×C8C2211yescharacteristic
35C3×#398—22no—
36C3×#391—21no—
37C3×#391C2212yes—
38C3×#389—21no—
39C3×#389C2212yes—
40C3×#384C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#557C2×C311yescharacteristic
47C2×#475C2×C311yescharacteristic
48C2×C4×Q16C2×C311yescharacteristic
49#2582C2×C311yescharacteristic
50#2059C2×C312yes—
51#2026C2×C312yes—
52#1990C2×C311yescharacteristic
53#1984C2×C311yescharacteristic
54#1878C2×C312yes—
55#1865C2×C312yes—
56#1803C2×C311yescharacteristic
57C22×C3×Q8—21no—
58C22×C3×Q8D411yescharacteristic
59C22×C3×Q8C2311yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C2×C3×Q16—22no—
62C2×C3×Q16—22no—
63C2×C3×Q16Q814yescomplemented
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2312yes—
66C22×C3×C8C2311yescharacteristic
67C3×C4⋊Q8—28no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8D414yes—
70C3×#125—216no—
71C3×#125—24no—
72C3×#125—24no—
73C3×C4×Q8—28no—
74C3×C4×Q8—24no—
75C3×C4×Q8—24no—
76C3×C4×Q8—22no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8D414yes—
82C2×C3×C4⋊C4—22no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D412yes—
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C42—22no—
98C2×C3×C42—21no—
99C2×C3×C42—21no—
100C2×C3×C42C2312yes—
101C2×C3×C42C2311yescharacteristic
102C3×#107—22no—
103C3×#106—24no—
104C3×C4⋊C8—22no—
105C3×#103—22no—
106C3×#103—24no—
107C3×#103—22no—
108C3×#103—24no—
109C3×C4×C8—22no—
110C3×#95—22no—
111C3×#95—22no—
112C3×#95—22no—
113C22×Q16C22×C311yescharacteristic
114#557—24no—
115#557—24no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8C22×C311yescharacteristic
118C2×#125—22no—
119C2×#125C22×C312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8C22×C311yescharacteristic
122C2×C4×Q8C22×C312yes—
123C2×C4×Q8C22×C311yescharacteristic
124C2×C4×Q8C22×C311yescharacteristic
125#475—24no—
126C4×Q16—24no—
127C2×#107C22×C311yescharacteristic
128C2×#106C22×C312yes—
129C2×C4⋊C8C22×C311yescharacteristic
130C2×#103C22×C312yes—
131C2×#103C22×C312yes—
132C2×#103C22×C311yescharacteristic
133C2×#103C22×C311yescharacteristic
134C2×C4×C8C22×C311yescharacteristic
135#398—22no—
136#391—21no—
137#391C22×C312yes—
138#389—21no—
139#389C22×C312yes—
140#384C22×C311yescharacteristic
141#340C22×C312yes—
142#340C22×C312yes—
143#340C22×C312yes—
144#336C22×C312yes—
145#336C22×C312yes—
146C2×C3×Q8—22no—
147C2×C3×Q8—24no—
148C2×C3×Q8—22no—
149C2×C3×Q8—22no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—42no—
159C2×C3×Q8—42no—
160C2×C3×Q8C2×D412yes—
161C2×C3×Q8C2×Q812yes—
162C2×C3×Q8C2×Q812yes—
163C2×C3×Q8#4112yes—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C3×Q16—28no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—22no—
184C2×C3×C8C2×Q812yes—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—28no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4—42no—
200C3×C4⋊C4—81no—
201C3×C4⋊C4—82no—
202C3×C4⋊C4C2×D414yes—
203C3×C4⋊C4C2×D412yes—
204C3×C42—28no—
205C3×C42—24no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—42no—
209C3×C42C2×D412yes—
210C3×C42C2×D412yes—
211C22×Q8—21no—
212C22×Q8C3×D411yescharacteristic
213C22×Q8C23×C311yescharacteristic
214C22×Q8C23×C311yescharacteristic
215C2×Q16—22no—
216C2×Q16—22no—
217C2×Q16C3×Q814yescomplemented
218C22×C8C23×C311yescharacteristic
219C22×C8C23×C312yes—
220C22×C8C23×C311yescharacteristic
221C4⋊Q8—28no—
222C4⋊Q8—22no—
223C4⋊Q8C3×D414yes—
224#125—216no—
225#125—24no—
226#125—24no—
227C4×Q8—28no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8C3×D414yes—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4C3×D411yescharacteristic
246C2×C4⋊C4C3×D412yes—
247C2×C4⋊C4C23×C311yescharacteristic
248C2×C4⋊C4C23×C312yes—
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C312yes—
251C2×C42—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C23×C312yes—
255C2×C42C23×C311yescharacteristic
256#107—22no—
257#106—24no—
258C4⋊C8—22no—
259#103—22no—
260#103—24no—
261#103—22no—
262#103—24no—
263C4×C8—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267C23×C3C22×D411yescharacteristic
268C3×Q8—22no—
269C3×Q8—22no—
270C3×Q8—22no—
271C3×Q8—22no—
272C3×Q8—22no—
273C3×Q8—41no—
274C3×Q8—41no—
275C3×Q8—44no—
276C3×Q8—48no—
277C3×Q8C22⋊Q814yes—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—24no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—22no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4#12012yes—
305C2×C3×C4#12012yes—
306C2×C3×C4#13512yes—
307C2×C3×C4#13712yes—
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C22×D411yescharacteristic
310C2×C3×C4C22×Q811yescharacteristic
311C2×C3×C4C2×#4111yescharacteristic
312C2×C3×C42−1+411yescharacteristic
313C2×C3×C42−1+411yescharacteristic
314C3×C8—22no—
315C3×C8—41no—
316C3×C8—42no—
317C2×Q8—22no—
318C2×Q8—24no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8C2×C3×D412yes—
332C2×Q8C2×C3×Q812yes—
333C2×Q8C2×C3×Q812yes—
334C2×Q8C3×#4112yes—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4C2×C3×D412yes—
342C22×C4C2×C3×D412yes—
343C22×C4C2×C3×D411yescharacteristic
344C22×C4C2×C3×D411yescharacteristic
345C22×C4C24×C311yescharacteristic
346Q16—28no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8C2×C3×Q812yes—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—28no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—22no—
362C4⋊C4—24no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—81no—
372C4⋊C4—82no—
373C4⋊C4C2×C3×D414yes—
374C4⋊C4C2×C3×D412yes—
375C42—28no—
376C42—24no—
377C42—22no—
378C42—22no—
379C42—42no—
380C42C2×C3×D412yes—
381C42C2×C3×D412yes—
382C22×C3C2×#12011yescharacteristic
383C22×C3#54811yescharacteristic
384C22×C3D4×Q811yescharacteristic
385C22×C3C2×#13511yescharacteristic
386C22×C3C2×#13711yescharacteristic
387C22×C3#57711yescharacteristic
388C22×C3#57811yescharacteristic
389C3×C4—24no—
390C3×C4—22no—
391C3×C4—42no—
392C3×C4—42no—
393C3×C4—44no—
394C3×C4—42no—
395C3×C4—42no—
396C3×C4—82no—
397C3×C4—81no—
398C3×C4#45214yes—
399C3×C4C2×C22⋊Q811yescharacteristic
400C3×C4#53611yescharacteristic
401C3×C4#54811yescharacteristic
402C3×C4D4×Q811yescharacteristic
403C23C22×C3×D411yescharacteristic
404Q8—22no—
405Q8—22no—
406Q8—22no—
407Q8—22no—
408Q8—22no—
409Q8—44no—
410Q8—48no—
411Q8—41no—
412Q8—41no—
413Q8C3×C22⋊Q814yes—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4C3×#12012yes—
441C2×C4C3×#12012yes—
442C2×C4C3×#13512yes—
443C2×C4C3×#13712yes—
444C2×C4C22×C3×D411yescharacteristic
445C2×C4C22×C3×D411yescharacteristic
446C2×C4C22×C3×Q811yescharacteristic
447C2×C4C2×C3×#4111yescharacteristic
448C2×C4C3×2−1+411yescharacteristic
449C2×C4C3×2−1+411yescharacteristic
450C8—22no—
451C8—41no—
452C8—42no—
453C2×C3#240911yescharacteristic
454C2×C3C2×#45211yescharacteristic
455C2×C3#301311yescharacteristic
456C2×C3#336511yescharacteristic
457C2×C3#337411yescharacteristic
458C2×C3#338311yescharacteristic
459C2×C3#338711yescharacteristic
460C22C2×C3×#12011yescharacteristic
461C22C3×#54811yescharacteristic
462C22C3×D4×Q811yescharacteristic
463C22C2×C3×#13511yescharacteristic
464C22C2×C3×#13711yescharacteristic
465C22C3×#57711yescharacteristic
466C22C3×#57811yescharacteristic
467C4—24no—
468C4—22no—
469C4—42no—
470C4—42no—
471C4—44no—
472C4—42no—
473C4—42no—
474C4—81no—
475C4—82no—
476C4C3×#45214yes—
477C4C2×C3×C22⋊Q811yescharacteristic
478C4C3×#53611yescharacteristic
479C4C3×#54811yescharacteristic
480C4C3×D4×Q811yescharacteristic
481C3#2433211yescharacteristic, direct product factor, Sylow
482C2C3×#240911yescharacteristic
483C2C2×C3×#45211yescharacteristic
484C2C3×#301311yescharacteristic
485C2C3×#336511yescharacteristic
486C2C3×#337411yescharacteristic
487C2C3×#338311yescharacteristic
488C2C3×#338711yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
12. #24332C3direct (N × H)
63. C2×C3×Q16Q8semidirect
217. C2×Q16C3×Q8semidirect
481. C3#24332direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×C3×#4752
C2×C3×C4×Q162
C3×#25822
C3×#20592
C3×#20262
C3×#19902
C3×#19842
C3×#18782
C3×#18652
C3×#18032
#243323