Finite Group Databasebeta

← C3×#24333 #10642940

Subgroups — C3×#24333 #10642940

Order 768.

1034 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q82C211yesmaximal, characteristic
2C2×C3×#477C211yesmaximal, characteristic
3C2×C3×C4×Q16C211yesmaximal, characteristic
4C3×#2582C211yesmaximal, characteristic
5C3×#2057C212yesmaximal
6C3×#2024C212yesmaximal
7C3×#1990C211yesmaximal, characteristic
8C3×#1984C211yesmaximal, characteristic
9C3×#1878C212yesmaximal
10C3×#1863C212yesmaximal
11C3×#1803C211yesmaximal, characteristic
12#24333C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2211yescharacteristic
14C3×Q82—24no—
15C3×Q82—24no—
16C2×C3×C4⋊Q8—21no—
17C2×C3×C4⋊Q8—22no—
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C3×#477—24no—
26C3×C4×Q16—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#107C2212yes—
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×Q82C2×C311yescharacteristic
47C2×#477C2×C311yescharacteristic
48C2×C4×Q16C2×C311yescharacteristic
49#2582C2×C311yescharacteristic
50#2057C2×C312yes—
51#2024C2×C312yes—
52#1990C2×C311yescharacteristic
53#1984C2×C311yescharacteristic
54#1878C2×C312yes—
55#1863C2×C312yes—
56#1803C2×C311yescharacteristic
57C22×C3×Q8—22no—
58C22×C3×Q8—21no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C2×C3×Q16—22no—
63C2×C3×Q16—22no—
64C2×C3×Q16Q814yescomplemented
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2312yes—
67C22×C3×C8C2311yescharacteristic
68C3×C4⋊Q8—216no—
69C3×C4⋊Q8—28no—
70C3×C4⋊Q8—24no—
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8D414yes—
74C3×C4×Q8—28no—
75C3×C4×Q8—24no—
76C3×C4×Q8—24no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8D414yes—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
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×#107—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
114Q82—24no—
115Q82—24no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—22no—
118C2×C4⋊Q8C22×C312yes—
119C2×C4⋊Q8C22×C311yescharacteristic
120C2×C4×Q8—21no—
121C2×C4×Q8C22×C311yescharacteristic
122C2×C4×Q8C22×C312yes—
123C2×C4×Q8C22×C311yescharacteristic
124C2×C4×Q8C22×C311yescharacteristic
125#477—24no—
126C4×Q16—24no—
127C2×#107C22×C312yes—
128C2×#107C22×C311yescharacteristic
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—28no—
147C2×C3×Q8—22no—
148C2×C3×Q8—24no—
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—81no—
159C2×C3×Q8—81no—
160C2×C3×Q8—82no—
161C2×C3×Q8—82no—
162C2×C3×Q8Q1614yescomplemented
163C2×C3×Q8C2×D412yes—
164C2×C3×Q8C2×Q812yes—
165C2×C3×Q8C2×Q812yes—
166C2×C3×Q8#4112yes—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2×D411yescharacteristic
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D411yescharacteristic
177C22×C3×C4C2411yescharacteristic
178C3×Q16—28no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—21no—
182C2×C3×C8—22no—
183C2×C3×C8—22no—
184C2×C3×C8—21no—
185C2×C3×C8—21no—
186C2×C3×C8—22no—
187C2×C3×C8C2×Q812yes—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—22no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—42no—
201C3×C4⋊C4—42no—
202C3×C4⋊C4—81no—
203C3×C4⋊C4—82no—
204C3×C4⋊C4C2×D414yes—
205C3×C4⋊C4C2×D412yes—
206C3×C42—28no—
207C3×C42—24no—
208C3×C42—22no—
209C3×C42—22no—
210C3×C42—42no—
211C3×C42C2×D412yes—
212C3×C42C2×D412yes—
213C22×Q8—22no—
214C22×Q8—21no—
215C22×Q8C3×D411yescharacteristic
216C22×Q8C23×C311yescharacteristic
217C22×Q8C23×C311yescharacteristic
218C2×Q16—22no—
219C2×Q16—22no—
220C2×Q16C3×Q814yescomplemented
221C22×C8C23×C311yescharacteristic
222C22×C8C23×C312yes—
223C22×C8C23×C311yescharacteristic
224C4⋊Q8—216no—
225C4⋊Q8—28no—
226C4⋊Q8—24no—
227C4⋊Q8—24no—
228C4⋊Q8—22no—
229C4⋊Q8C3×D414yes—
230C4×Q8—28no—
231C4×Q8—24no—
232C4×Q8—24no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8C3×D414yes—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4C3×D411yescharacteristic
248C2×C4⋊C4C3×D412yes—
249C2×C4⋊C4C23×C311yescharacteristic
250C2×C4⋊C4C23×C312yes—
251C2×C4⋊C4C23×C312yes—
252C2×C4⋊C4C23×C311yescharacteristic
253C2×C42—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42C23×C312yes—
257C2×C42C23×C311yescharacteristic
258#107—22no—
259#107—24no—
260C4⋊C8—22no—
261#103—22no—
262#103—24no—
263#103—22no—
264#103—24no—
265C4×C8—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269C23×C3C22×D411yescharacteristic
270C3×Q8—22no—
271C3×Q8—22no—
272C3×Q8—22no—
273C3×Q8—28no—
274C3×Q8—22no—
275C3×Q8—22no—
276C3×Q8—41no—
277C3×Q8—41no—
278C3×Q8—84no—
279C3×Q8—88no—
280C3×Q8C22⋊Q814yes—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—22no—
289C2×C3×C4—24no—
290C2×C3×C4—22no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—41no—
295C2×C3×C4—41no—
296C2×C3×C4—42no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—41no—
300C2×C3×C4—41no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4#12012yes—
307C2×C3×C4#12012yes—
308C2×C3×C4C2×Q1614yes—
309C2×C3×C4C2×Q1612yes—
310C2×C3×C4#13712yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×Q811yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C42−1+411yescharacteristic
316C2×C3×C42−1+411yescharacteristic
317C3×C8—22no—
318C3×C8—41no—
319C3×C8—42no—
320C2×Q8—28no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—22no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—81no—
333C2×Q8—81no—
334C2×Q8—82no—
335C2×Q8—82no—
336C2×Q8C3×Q1614yescomplemented
337C2×Q8C2×C3×D412yes—
338C2×Q8C2×C3×Q812yes—
339C2×Q8C2×C3×Q812yes—
340C2×Q8C3×#4112yes—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4C2×C3×D412yes—
348C22×C4C2×C3×D411yescharacteristic
349C22×C4C2×C3×D412yes—
350C22×C4C2×C3×D411yescharacteristic
351C22×C4C24×C311yescharacteristic
352Q16—28no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—22no—
361C2×C8C2×C3×Q812yes—
362C4⋊C4—28no—
363C4⋊C4—28no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—81no—
377C4⋊C4—82no—
378C4⋊C4C2×C3×D414yes—
379C4⋊C4C2×C3×D412yes—
380C42—28no—
381C42—24no—
382C42—22no—
383C42—22no—
384C42—42no—
385C42C2×C3×D412yes—
386C42C2×C3×D412yes—
387C22×C3C2×#12011yescharacteristic
388C22×C3#54811yescharacteristic
389C22×C3D4×Q811yescharacteristic
390C22×C3C22×Q1611yescharacteristic
391C22×C3C2×#13711yescharacteristic
392C22×C3#57611yescharacteristic
393C22×C3#57711yescharacteristic
394C3×C4—24no—
395C3×C4—24no—
396C3×C4—22no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—44no—
400C3×C4—42no—
401C3×C4—82no—
402C3×C4—81no—
403C3×C4#45114yes—
404C3×C4C2×C22⋊Q811yescharacteristic
405C3×C4#53611yescharacteristic
406C3×C4#54811yescharacteristic
407C3×C4D4×Q811yescharacteristic
408C23C22×C3×D411yescharacteristic
409Q8—28no—
410Q8—22no—
411Q8—22no—
412Q8—22no—
413Q8—22no—
414Q8—22no—
415Q8—41no—
416Q8—41no—
417Q8—84no—
418Q8—88no—
419Q8C3×C22⋊Q814yes—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4C3×#12012yes—
446C2×C4C3×#12012yes—
447C2×C4C2×C3×Q1614yes—
448C2×C4C2×C3×Q1612yes—
449C2×C4C3×#13712yes—
450C2×C4C22×C3×D411yescharacteristic
451C2×C4C22×C3×D411yescharacteristic
452C2×C4C22×C3×Q811yescharacteristic
453C2×C4C2×C3×#4111yescharacteristic
454C2×C4C3×2−1+411yescharacteristic
455C2×C4C3×2−1+411yescharacteristic
456C8—22no—
457C8—41no—
458C8—42no—
459C2×C3#240911yescharacteristic
460C2×C3C2×#45111yescharacteristic
461C2×C3#301211yescharacteristic
462C2×C3#336311yescharacteristic
463C2×C3#337511yescharacteristic
464C2×C3Q8×Q1611yescharacteristic
465C2×C3#338711yescharacteristic
466C22C2×C3×#12011yescharacteristic
467C22C3×#54811yescharacteristic
468C22C3×D4×Q811yescharacteristic
469C22C22×C3×Q1611yescharacteristic
470C22C2×C3×#13711yescharacteristic
471C22C3×#57611yescharacteristic
472C22C3×#57711yescharacteristic
473C4—24no—
474C4—24no—
475C4—22no—
476C4—42no—
477C4—42no—
478C4—44no—
479C4—42no—
480C4—81no—
481C4—82no—
482C4C3×#45114yes—
483C4C2×C3×C22⋊Q811yescharacteristic
484C4C3×#53611yescharacteristic
485C4C3×#54811yescharacteristic
486C4C3×D4×Q811yescharacteristic
487C3#2433311yescharacteristic, direct product factor, Sylow
488C2C3×#240911yescharacteristic
489C2C2×C3×#45111yescharacteristic
490C2C3×#301211yescharacteristic
491C2C3×#336311yescharacteristic
492C2C3×#337511yescharacteristic
493C2C3×Q8×Q1611yescharacteristic
494C2C3×#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. #24333C3direct (N × H)
64. C2×C3×Q16Q8semidirect
162. C2×C3×Q8Q16semidirect
220. C2×Q16C3×Q8semidirect
336. C2×Q8C3×Q16semidirect
487. C3#24333direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q822
C2×C3×#4772
C2×C3×C4×Q162
C3×#25822
C3×#20572
C3×#20242
C3×#19902
C3×#19842
C3×#18782
C3×#18632
C3×#18032
#243333