Finite Group Databasebeta

← C3×#24199 #10642806

Subgroups — C3×#24199 #10642806

Order 768.

1084 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×C3×#461C211yesmaximal, characteristic
3C2×C3×#440C211yesmaximal, characteristic
4C3×#2580C211yesmaximal, characteristic
5C3×#2043C212yesmaximal, complemented
6C3×#2013C212yesmaximal
7C3×#1985C211yesmaximal, characteristic, complemented
8C3×#1960C211yesmaximal, characteristic, complemented
9C3×#1880C212yesmaximal
10C3×#1863C212yesmaximal, complemented
11C3×#1805C211yesmaximal, characteristic, complemented
12#24199C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C3×#557—24no—
16C3×#557—24no—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#125—22no—
20C2×C3×#125—21no—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×D4C2211yescharacteristic
26C3×#461—24no—
27C3×#440—24no—
28C2×C3×#107C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2212yes—
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2212yes—
33C2×C3×#103C2212yes—
34C2×C3×#103C2212yes—
35C2×C3×#102C2211yescharacteristic
36C2×C3×#97C2211yescharacteristic
37C3×#399—21no—
38C3×#393—21no—
39C3×#390C2212yes—
40C3×#389C2212yes—
41C3×#385—22no—
42C3×#382C2211yescharacteristic
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×#557C2×C311yescharacteristic, complemented
47C2×#461C2×C311yescharacteristic
48C2×#440C2×C311yescharacteristic
49#2580C2×C311yescharacteristic
50#2043C2×C312yescomplemented
51#2013C2×C312yes—
52#1985C2×C311yescharacteristic, complemented
53#1960C2×C311yescharacteristic, complemented
54#1880C2×C312yes—
55#1863C2×C312yescomplemented
56#1805C2×C311yescharacteristic, complemented
57C22×C3×Q8D411yescharacteristic
58C22×C3×Q8C2312yes—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C2×C3×SD16—22no—
63C2×C3×SD16—22no—
64C2×C3×SD16—24no—
65C2×C3×SD16—24no—
66C2×C3×SD16—22no—
67C2×C3×SD16—42no—
68C22×C3×C8C2312yes—
69C22×C3×C8C2311yescharacteristic
70C22×C3×C8C2311yescharacteristic
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8—24no—
73C3×C4⋊Q8—22no—
74C3×C4⋊Q8D414yes—
75C3×#125—216no—
76C3×#125—28no—
77C3×C4×Q8—28no—
78C3×C4×Q8—24no—
79C3×C4×Q8—24no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8—22no—
83C3×C4×Q8D414yes—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C2×C3×C4⋊C4—22no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4D412yes—
94C2×C3×C4⋊C4D411yescharacteristic
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C4⋊C4C2311yescharacteristic
98C2×C3×C22⋊C4—22no—
99C2×C3×C22⋊C4—21no—
100C2×C3×C22⋊C4—22no—
101C2×C3×C42—22no—
102C2×C3×C42—21no—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105C3×#107—22no—
106C3×C4⋊C8—22no—
107C3×#103—22no—
108C3×#103—24no—
109C3×#103—24no—
110C3×#103—42no—
111C3×#103—42no—
112C3×#102—22no—
113C3×#97—22no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95—22no—
117C22×SD16C22×C311yescharacteristic
118C22×SD16C22×C312yes—
119#557—24no—
120#557—24no—
121C2×C4⋊Q8C22×C312yes—
122C2×C4⋊Q8C22×C311yescharacteristic
123C2×#125—22no—
124C2×#125—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C22×C312yes—
127C2×C4×Q8C22×C311yescharacteristic
128C2×C4×Q8C22×C311yescharacteristic
129C2×C4×D4C22×C311yescharacteristic
130#461—24no—
131#440—24no—
132C2×#107C22×C311yescharacteristic
133C2×C4⋊C8C22×C311yescharacteristic
134C2×#103C22×C312yes—
135C2×#103C22×C312yes—
136C2×#103C22×C311yescharacteristic
137C2×#103C22×C312yes—
138C2×#103C22×C312yes—
139C2×#102C22×C311yescharacteristic
140C2×#97C22×C311yescharacteristic
141#399—21no—
142#393—21no—
143#390C22×C312yes—
144#389C22×C312yes—
145#385—22no—
146#382C22×C311yescharacteristic
147#340C22×C312yes—
148#336C22×C312yes—
149#336C22×C312yes—
150C24×C3—21no—
151C2×C3×Q8—22no—
152C2×C3×Q8—24no—
153C2×C3×Q8—22no—
154C2×C3×Q8—21no—
155C2×C3×Q8—22no—
156C2×C3×Q8—21no—
157C2×C3×Q8—22no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—22no—
161C2×C3×Q8—21no—
162C2×C3×Q8—22no—
163C2×C3×Q8C2×D414yes—
164C2×C3×Q8C2×D412yes—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4#4112yes—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—21no—
176C22×C3×C4—42no—
177C22×C3×C4—82no—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D411yescharacteristic
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2411yescharacteristic
183C3×SD16—28no—
184C3×SD16—48no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—21no—
191C2×C3×C8—22no—
192C2×C3×C8—21no—
193C2×C3×C8#4112yes—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—28no—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—24no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—21no—
203C3×C4⋊C4—21no—
204C3×C4⋊C4—21no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—82no—
208C3×C4⋊C4C2×D412yes—
209C3×C22⋊C4—42no—
210C3×C22⋊C4—44no—
211C3×C22⋊C4—82no—
212C3×C42—28no—
213C3×C42—24no—
214C3×C42—22no—
215C3×C42—22no—
216C3×C42C2×D412yes—
217C3×C42C2×D412yes—
218C22×Q8C3×D411yescharacteristic
219C22×Q8C23×C312yes—
220C22×Q8C23×C311yescharacteristic
221C22×D4C23×C311yescharacteristic
222C23×C4—21no—
223C2×SD16—22no—
224C2×SD16—22no—
225C2×SD16—24no—
226C2×SD16—24no—
227C2×SD16—22no—
228C2×SD16—42no—
229C22×C8C23×C312yes—
230C22×C8C23×C311yescharacteristic
231C22×C8C23×C311yescharacteristic
232C4⋊Q8—24no—
233C4⋊Q8—24no—
234C4⋊Q8—22no—
235C4⋊Q8C3×D414yes—
236#125—216no—
237#125—28no—
238C4×Q8—28no—
239C4×Q8—24no—
240C4×Q8—24no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8C3×D414yes—
245C4×D4—22no—
246C4×D4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4C3×D412yes—
255C2×C4⋊C4C3×D411yescharacteristic
256C2×C4⋊C4C23×C311yescharacteristic
257C2×C4⋊C4C23×C312yes—
258C2×C4⋊C4C23×C311yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42C23×C312yes—
265C2×C42C23×C311yescharacteristic
266#107—22no—
267C4⋊C8—22no—
268#103—22no—
269#103—24no—
270#103—24no—
271#103—42no—
272#103—42no—
273#102—22no—
274#97—22no—
275#95—21no—
276#95—21no—
277#95—22no—
278C23×C3—22no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—42no—
282C23×C3—42no—
283C23×C3C22×D411yescharacteristic
284C3×Q8—28no—
285C3×Q8—22no—
286C3×Q8—22no—
287C3×Q8—24no—
288C3×Q8—24no—
289C3×Q8—42no—
290C3×Q8—42no—
291C3×Q8—41no—
292C3×Q8—41no—
293C3×D4—22no—
294C3×D4—22no—
295C3×D4—41no—
296C3×D4—41no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—41no—
313C2×C3×C4—42no—
314C2×C3×C4—42no—
315C2×C3×C4—42no—
316C2×C3×C4—42no—
317C2×C3×C4—84no—
318C2×C3×C4#12012yes—
319C2×C3×C4#12012yes—
320C2×C3×C4#13714yes—
321C2×C3×C4C22×D411yescharacteristic
322C2×C3×C4C22×D411yescharacteristic
323C2×C3×C4C2×#4111yescharacteristic
324C2×C3×C4C2×#4111yescharacteristic
325C2×C3×C4H(5,𝔽2)11yescharacteristic
326C2×C3×C42−1+411yescharacteristic
327C3×C8—22no—
328C3×C8—41no—
329C3×C8—42no—
330C24—21no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—22no—
334C2×Q8—21no—
335C2×Q8—22no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—22no—
341C2×Q8—21no—
342C2×Q8—22no—
343C2×Q8C2×C3×D414yes—
344C2×Q8C2×C3×D412yes—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4C3×#4112yes—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—42no—
357C22×C4—82no—
358C22×C4C2×C3×D412yes—
359C22×C4C2×C3×D411yescharacteristic
360C22×C4C2×C3×D411yescharacteristic
361C22×C4C2×C3×D412yes—
362C22×C4C24×C311yescharacteristic
363SD16—28no—
364SD16—48no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—21no—
371C2×C8—22no—
372C2×C8—21no—
373C2×C8C3×#4112yes—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—82no—
388C4⋊C4C2×C3×D412yes—
389C22⋊C4—42no—
390C22⋊C4—44no—
391C22⋊C4—82no—
392C42—28no—
393C42—24no—
394C42—22no—
395C42—22no—
396C42C2×C3×D412yes—
397C42C2×C3×D412yes—
398C22×C3—44no—
399C22×C3—42no—
400C22×C3—82no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3C2×#12011yescharacteristic
404C22×C3#54811yescharacteristic
405C22×C3#55011yescharacteristic
406C22×C3C2×#13712yes—
407C22×C3#57712yes—
408C3×C4—22no—
409C3×C4—24no—
410C3×C4—22no—
411C3×C4—42no—
412C3×C4—44no—
413C3×C4—42no—
414C3×C4—42no—
415C3×C4—82no—
416C3×C4—81no—
417C3×C4#52511yescharacteristic
418C3×C4#53711yescharacteristic
419C3×C4#54611yescharacteristic
420C3×C4#54711yescharacteristic
421C23—22no—
422C23—42no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23C22×C3×D411yescharacteristic
427Q8—28no—
428Q8—22no—
429Q8—22no—
430Q8—24no—
431Q8—24no—
432Q8—42no—
433Q8—42no—
434Q8—41no—
435Q8—41no—
436D4—22no—
437D4—22no—
438D4—41no—
439D4—41no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—84no—
461C2×C4C3×#12012yes—
462C2×C4C3×#12012yes—
463C2×C4C3×#13714yes—
464C2×C4C22×C3×D411yescharacteristic
465C2×C4C22×C3×D411yescharacteristic
466C2×C4C2×C3×#4111yescharacteristic
467C2×C4C2×C3×#4111yescharacteristic
468C2×C4C3×H(5,𝔽2)11yescharacteristic
469C2×C4C3×2−1+411yescharacteristic
470C8—22no—
471C8—41no—
472C8—42no—
473C2×C3—82no—
474C2×C3#240711yescharacteristic
475C2×C3#300411yescharacteristic
476C2×C3#301011yescharacteristic
477C2×C3#337212yes—
478C2×C3#340012yes—
479C22—42no—
480C22—44no—
481C22—82no—
482C22—81no—
483C22—81no—
484C22C2×C3×#12011yescharacteristic
485C22C3×#54811yescharacteristic
486C22C3×#55011yescharacteristic
487C22C2×C3×#13712yes—
488C22C3×#57712yes—
489C4—22no—
490C4—24no—
491C4—22no—
492C4—42no—
493C4—44no—
494C4—42no—
495C4—42no—
496C4—82no—
497C4—81no—
498C4C3×#52511yescharacteristic
499C4C3×#53711yescharacteristic
500C4C3×#54611yescharacteristic
501C4C3×#54711yescharacteristic
502C3#2419911yescharacteristic, direct product factor, Sylow
503C2—82no—
504C2C3×#240711yescharacteristic
505C2C3×#300411yescharacteristic
506C2C3×#301011yescharacteristic
507C2C3×#337212yes—
508C2C3×#340012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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
1. C2×C3×#557C2semidirect
5. C3×#2043C2semidirect
7. C3×#1985C2semidirect
8. C3×#1960C2semidirect
10. C3×#1863C2semidirect
11. C3×#1805C2semidirect
12. #24199C3direct (N × H)
46. C2×#557C2×C3semidirect
50. #2043C2×C3semidirect
52. #1985C2×C3semidirect
53. #1960C2×C3semidirect
55. #1863C2×C3semidirect
56. #1805C2×C3semidirect
502. C3#24199direct (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×#4612
C2×C3×#4402
C3×#25802
C3×#20432
C3×#20132
C3×#19852
C3×#19602
C3×#18802
C3×#18632
C3×#18052
#241993