Finite Group Databasebeta

← C3×#24694 #10643301

Subgroups — C3×#24694 #10643301

Order 768.

836 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#477C211yesmaximal, characteristic, complemented
2C2×C3×#440C211yesmaximal, characteristic
3C3×#2542C211yesmaximal, characteristic
4C3×#2356C211yesmaximal, characteristic, complemented
5C3×#2072C212yesmaximal, complemented
6C3×#2055C212yesmaximal
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1947C212yesmaximal, complemented
9C3×#1940C211yesmaximal, characteristic, complemented
10C3×#1925C212yesmaximal
11C3×#1804C211yesmaximal, characteristic
12#24694C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8—22no—
16C2×C3×C4×Q8C2211yescharacteristic
17C2×C3×C4×D4C2211yescharacteristic
18C3×#477—24no—
19C3×#440—24no—
20C2×C3×#107C2211yescharacteristic
21C2×C3×#107C2212yes—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2212yes—
25C2×C3×#103C2211yescharacteristic
26C2×C3×#102C2212yes—
27C2×C3×#102C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C3×#399—21no—
30C3×#399—21no—
31C3×#391C2212yes—
32C3×#390C2212yes—
33C3×#387—22no—
34C3×#384C2212yes—
35C3×#383—21no—
36C3×#382C2211yescharacteristic
37C3×C4×C4⋊C4C2212yes—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×#376—21no—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#477C2×C311yescharacteristic, complemented
45C2×#440C2×C311yescharacteristic
46#2542C2×C311yescharacteristic
47#2356C2×C311yescharacteristic, complemented
48#2072C2×C312yescomplemented
49#2055C2×C312yes—
50#1993C2×C311yescharacteristic, complemented
51#1947C2×C312yescomplemented
52#1940C2×C311yescharacteristic, complemented
53#1925C2×C312yes—
54#1804C2×C311yescharacteristic
55C22×C3×Q8—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C2×C3×SD16—22no—
60C2×C3×SD16—22no—
61C2×C3×SD16—22no—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2311yescharacteristic
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×C22⋊Q8—48no—
68C3×C4×Q8—22no—
69C3×C4×Q8D414yes—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4D412yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C42—21no—
85C2×C3×C42—22no—
86C2×C3×C42C2312yes—
87C2×C3×C42C2311yescharacteristic
88C3×#107—22no—
89C3×#107—24no—
90C3×C4⋊C8—24no—
91C3×C4⋊C8—22no—
92C3×#103—22no—
93C3×#103—24no—
94C3×#102—24no—
95C3×#102—22no—
96C3×#97—22no—
97C3×#95—21no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—21no—
101C3×#95—22no—
102C3×#95D411yescharacteristic
103C3×#95D411yescharacteristic
104C22×SD16C22×C311yescharacteristic
105C2×C4⋊Q8C22×C311yescharacteristic
106C2×C22⋊Q8—22no—
107C2×C4×Q8C22×C311yescharacteristic
108C2×C4×D4C22×C311yescharacteristic
109#477—24no—
110#440—24no—
111C2×#107C22×C312yes—
112C2×#107C22×C311yescharacteristic
113C2×C4⋊C8C22×C311yescharacteristic
114C2×C4⋊C8C22×C312yes—
115C2×#103C22×C312yes—
116C2×#103C22×C311yescharacteristic
117C2×#102C22×C312yes—
118C2×#102C22×C311yescharacteristic
119C2×#97C22×C311yescharacteristic
120#399—21no—
121#399—21no—
122#391C22×C312yes—
123#390C22×C312yes—
124#387—22no—
125#384C22×C312yes—
126#383—21no—
127#382C22×C311yescharacteristic
128C4×C4⋊C4C22×C312yes—
129C4×C4⋊C4C22×C311yescharacteristic
130#376—21no—
131#340C22×C312yes—
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C312yes—
135C24×C3—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—21no—
141C2×C3×Q8—44no—
142C2×C3×Q8—42no—
143C2×C3×Q8C2×D412yes—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4#4112yes—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—44no—
156C22×C3×C4—42no—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2411yescharacteristic
162C3×SD16—28no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8#4112yes—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—44no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—44no—
184C3×C4⋊C4C2×D412yes—
185C3×C22⋊C4—44no—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—44no—
188C3×C42—24no—
189C3×C42—22no—
190C3×C42—22no—
191C3×C42—44no—
192C3×C42C2×D412yes—
193C3×C42#4112yes—
194C22×Q8—21no—
195C22×Q8C23×C311yescharacteristic
196C22×D4C23×C311yescharacteristic
197C23×C4—21no—
198C2×SD16—22no—
199C2×SD16—22no—
200C2×SD16—22no—
201C22×C8C23×C311yescharacteristic
202C22×C8C23×C312yes—
203C22×C8C23×C311yescharacteristic
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C22⋊Q8—48no—
207C4×Q8—22no—
208C4×Q8C3×D414yes—
209C4×D4—22no—
210C4×D4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C3×D412yes—
216C2×C4⋊C4C23×C311yescharacteristic
217C2×C4⋊C4C23×C312yes—
218C2×C4⋊C4C23×C312yes—
219C2×C4⋊C4C23×C311yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C42—21no—
224C2×C42—22no—
225C2×C42C23×C312yes—
226C2×C42C23×C311yescharacteristic
227#107—22no—
228#107—24no—
229C4⋊C8—24no—
230C4⋊C8—22no—
231#103—22no—
232#103—24no—
233#102—24no—
234#102—22no—
235#97—22no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95—22no—
241#95C3×D411yescharacteristic
242#95C3×D411yescharacteristic
243C23×C3—22no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3C22×D411yescharacteristic
249C3×Q8—22no—
250C3×Q8—22no—
251C3×Q8—41no—
252C3×Q8—41no—
253C3×Q8—84no—
254C3×D4—22no—
255C3×D4—22no—
256C3×D4—41no—
257C3×D4—41no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—84no—
279C2×C3×C4#12112yes—
280C2×C3×C4#12112yes—
281C2×C3×C4AGL(1,ℤ/8)14yes—
282C2×C3×C4C22×D411yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C42−1+411yescharacteristic
287C2×C3×C42−1+411yescharacteristic
288C3×C8—22no—
289C3×C8—42no—
290C3×C8—41no—
291C24—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—44no—
298C2×Q8—42no—
299C2×Q8C2×C3×D412yes—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4C3×#4112yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4C2×C3×D412yes—
314C22×C4C2×C3×D411yescharacteristic
315C22×C4C2×C3×D411yescharacteristic
316C22×C4C2×C3×D412yes—
317C22×C4C24×C311yescharacteristic
318SD16—28no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8C3×#4112yes—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4C2×C3×D412yes—
341C22⋊C4—44no—
342C22⋊C4—42no—
343C22⋊C4—44no—
344C42—24no—
345C42—22no—
346C42—22no—
347C42—44no—
348C42C2×C3×D412yes—
349C42C3×#4112yes—
350C22×C3—44no—
351C22×C3—42no—
352C22×C3—82no—
353C22×C3—81no—
354C22×C3—81no—
355C22×C3C2×#12111yescharacteristic
356C22×C3#54711yescharacteristic
357C22×C3#54811yescharacteristic
358C22×C3C2×AGL(1,ℤ/8)12yes—
359C22×C3#57812yes—
360C3×C4—24no—
361C3×C4—22no—
362C3×C4—22no—
363C3×C4—42no—
364C3×C4—42no—
365C3×C4—42no—
366C3×C4—82no—
367C3×C4—81no—
368C3×C4—82no—
369C3×C4#52511yescharacteristic
370C3×C4#53611yescharacteristic
371C3×C4#55211yescharacteristic
372C3×C4#55611yescharacteristic
373C23—22no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23C22×C3×D411yescharacteristic
379Q8—22no—
380Q8—22no—
381Q8—41no—
382Q8—41no—
383Q8—84no—
384D4—22no—
385D4—22no—
386D4—41no—
387D4—41no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—84no—
409C2×C4C3×#12112yes—
410C2×C4C3×#12112yes—
411C2×C4C3×AGL(1,ℤ/8)14yes—
412C2×C4C22×C3×D411yescharacteristic
413C2×C4C2×C3×#4111yescharacteristic
414C2×C4C2×C3×#4111yescharacteristic
415C2×C4C2×C3×#4111yescharacteristic
416C2×C4C3×2−1+411yescharacteristic
417C2×C4C3×2−1+411yescharacteristic
418C8—22no—
419C8—42no—
420C8—41no—
421C2×C3—82no—
422C2×C3#241511yescharacteristic
423C2×C3#303811yescharacteristic
424C2×C3#304411yescharacteristic
425C2×C3#334312yes—
426C2×C3#336712yes—
427C22—42no—
428C22—44no—
429C22—81no—
430C22—81no—
431C22—82no—
432C22C2×C3×#12111yescharacteristic
433C22C3×#54711yescharacteristic
434C22C3×#54811yescharacteristic
435C22C2×C3×AGL(1,ℤ/8)12yes—
436C22C3×#57812yes—
437C4—22no—
438C4—22no—
439C4—24no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—81no—
444C4—82no—
445C4—82no—
446C4C3×#52511yescharacteristic
447C4C3×#53611yescharacteristic
448C4C3×#55211yescharacteristic
449C4C3×#55611yescharacteristic
450C3#2469411yescharacteristic, direct product factor, Sylow
451C2—82no—
452C2C3×#241511yescharacteristic
453C2C3×#303811yescharacteristic
454C2C3×#304411yescharacteristic
455C2C3×#334312yes—
456C2C3×#336712yes—
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
1. C2×C3×#477C2semidirect
4. C3×#2356C2semidirect
5. C3×#2072C2semidirect
7. C3×#1993C2semidirect
8. C3×#1947C2semidirect
9. C3×#1940C2semidirect
12. #24694C3direct (N × H)
44. C2×#477C2×C3semidirect
47. #2356C2×C3semidirect
48. #2072C2×C3semidirect
50. #1993C2×C3semidirect
51. #1947C2×C3semidirect
52. #1940C2×C3semidirect
450. C3#24694direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4772
C2×C3×#4402
C3×#25422
C3×#23562
C3×#20722
C3×#20552
C3×#19932
C3×#19472
C3×#19402
C3×#19252
C3×#18042
#246943