Finite Group Databasebeta

← C3×#27648 #10646255

Subgroups — C3×#27648 #10646255

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#484C212yesmaximal
2C3×C4×C22⋊Q8C211yesmaximal, characteristic
3C2×C3×#384C211yesmaximal, characteristic
4C3×#2083C212yesmaximal, complemented
5C3×#2072C212yesmaximal, complemented
6C3×#1947C212yesmaximal, complemented
7C3×#1824C212yesmaximal
8C3×#1803C212yesmaximal, complemented
9C3×#1797C211yesmaximal, characteristic
10#27648C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×C22⋊Q8C2211yescharacteristic
12C2×C3×C4×Q8C2211yescharacteristic
13C22×C3×C4⋊C4—21no—
14C22×C3×C4⋊C4—21no—
15C22×C3×C4⋊C4C2212yes—
16C22×C3×C42C2211yescharacteristic
17C3×#484—28no—
18C2×C3×#107C2214yes—
19C2×C3×C4⋊C8C2212yes—
20C2×C3×#103C2214yes—
21C2×C3×C22⋊C8C2212yes—
22C3×#391C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×#387—21no—
25C3×#385—21no—
26C3×#384—21no—
27C3×#384—21no—
28C3×#384C2212yes—
29C3×#384C2212yes—
30C3×#380C2211yescharacteristic
31C3×C4×C4⋊C4—21no—
32C3×C4×C4⋊C4C2211yescharacteristic
33C3×C4×C22⋊C4—21no—
34C3×#340C2214yes—
35C3×#340C2214yes—
36C3×#336C2212yes—
37C2×#484C2×C312yes—
38C4×C22⋊Q8C2×C311yescharacteristic
39C2×#384C2×C311yescharacteristic
40#2083C2×C312yescomplemented
41#2072C2×C312yescomplemented
42#1947C2×C312yescomplemented
43#1824C2×C312yes—
44#1803C2×C312yescomplemented
45#1797C2×C311yescharacteristic
46C22×C3×Q8C2311yescharacteristic
47C23×C3×C4—22no—
48C23×C3×C4D411yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C22×C3×C8C2314yes—
52C3×C22⋊Q8—22no—
53C3×C22⋊Q8D414yes—
54C3×C4×Q8—24no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—44no—
66C2×C3×C4⋊C4—44no—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2314yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C42—24no—
74C2×C3×C42—21no—
75C2×C3×C42—21no—
76C2×C3×C42D411yescharacteristic
77C2×C3×C42D411yescharacteristic
78C2×C3×C42C2311yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C3×#107—28no—
81C3×C4⋊C8—24no—
82C3×#103—28no—
83C3×C22⋊C8—24no—
84C3×#95—21no—
85C3×#95—21no—
86C3×#95—21no—
87C3×#95—21no—
88C3×#95—21no—
89C2×C22⋊Q8C22×C311yescharacteristic
90C2×C4×Q8C22×C311yescharacteristic
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4C22×C312yes—
94C22×C42C22×C311yescharacteristic
95#484—28no—
96C2×#107C22×C314yes—
97C2×C4⋊C8C22×C312yes—
98C2×#103C22×C314yes—
99C2×C22⋊C8C22×C312yes—
100#391C22×C311yescharacteristic
101#389C22×C311yescharacteristic
102#387—21no—
103#385—21no—
104#384—21no—
105#384—21no—
106#384C22×C312yes—
107#384C22×C312yes—
108#380C22×C311yescharacteristic
109C4×C4⋊C4—21no—
110C4×C4⋊C4C22×C311yescharacteristic
111C4×C22⋊C4—21no—
112#340C22×C314yes—
113#340C22×C314yes—
114#336C22×C312yes—
115C24×C3C2×D411yescharacteristic
116C2×C3×Q8—21no—
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8C2×D412yes—
120C22×C3×C4—22no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—24no—
132C22×C3×C4—44no—
133C22×C3×C4—44no—
134C22×C3×C4—44no—
135C22×C3×C4Q1614yes—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4#4112yes—
143C22×C3×C4C2411yescharacteristic
144C2×C3×C8—24no—
145C2×C3×C8—24no—
146C2×C3×C8—24no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—44no—
154C3×C4⋊C4—44no—
155C3×C4⋊C4—44no—
156C3×C4⋊C4—42no—
157C3×C4⋊C4—42no—
158C3×C4⋊C4—44no—
159C3×C4⋊C4—44no—
160C3×C4⋊C4C2×D412yes—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—42no—
163C3×C42—22no—
164C3×C42—22no—
165C3×C42—22no—
166C3×C42—42no—
167C3×C42—42no—
168C3×C42#4114yes—
169C22×Q8C23×C311yescharacteristic
170C23×C4—22no—
171C23×C4C3×D411yescharacteristic
172C23×C4C3×D411yescharacteristic
173C23×C4C23×C311yescharacteristic
174C22×C8C23×C314yes—
175C22⋊Q8—22no—
176C22⋊Q8C3×D414yes—
177C4×Q8—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4C23×C314yes—
191C2×C4⋊C4C23×C311yescharacteristic
192C2×C4⋊C4C23×C311yescharacteristic
193C2×C4⋊C4C23×C311yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C42—24no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42C3×D411yescharacteristic
200C2×C42C3×D411yescharacteristic
201C2×C42C23×C311yescharacteristic
202C2×C42C23×C311yescharacteristic
203#107—28no—
204C4⋊C8—24no—
205#103—28no—
206C22⋊C8—24no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212C23×C3—21no—
213C23×C3—21no—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3#12112yes—
217C23×C3C2×Q1612yes—
218C23×C3#13712yes—
219C23×C3C22×D411yescharacteristic
220C3×Q8—24no—
221C3×Q8—42no—
222C2×C3×C4—28no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—28no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—44no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—44no—
245C2×C3×C4—44no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—88no—
250C2×C3×C4#12112yes—
251C2×C3×C4#12312yes—
252C2×C3×C4#12312yes—
253C2×C3×C4C2×Q1612yes—
254C2×C3×C4C2×Q1612yes—
255C2×C3×C4AGL(1,ℤ/8)12yes—
256C2×C3×C4AGL(1,ℤ/8)12yes—
257C2×C3×C4C22×D411yescharacteristic
258C2×C3×C4C2×#4112yes—
259C2×C3×C4C2×#4111yescharacteristic
260C2×C3×C42−1+412yes—
261C3×C8—44no—
262C24C2×C3×D411yescharacteristic
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8C2×C3×D412yes—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—24no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4C3×Q1614yes—
283C22×C4C2×C3×D411yescharacteristic
284C22×C4C2×C3×D412yes—
285C22×C4C2×C3×D411yescharacteristic
286C22×C4C2×C3×D411yescharacteristic
287C22×C4C2×C3×D411yescharacteristic
288C22×C4C2×C3×D411yescharacteristic
289C22×C4C3×#4112yes—
290C22×C4C24×C311yescharacteristic
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C4⋊C4—24no—
295C4⋊C4—21no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4C2×C3×D412yes—
308C22⋊C4—24no—
309C22⋊C4—42no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—42no—
314C42—42no—
315C42C3×#4114yes—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3#46214yes—
323C22×C3C2×#12111yescharacteristic
324C22×C3C2×#12311yescharacteristic
325C22×C3#53611yescharacteristic
326C22×C3C22×Q1611yescharacteristic
327C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
328C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
329C22×C3C2×#13711yescharacteristic
330C3×C4—22no—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—42no—
335C3×C4—42no—
336C3×C4—81no—
337C3×C4—81no—
338C3×C4—84no—
339C3×C4#48414yescomplemented
340C3×C4#54712yes—
341C3×C4#55212yes—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23C3×#12112yes—
347C23C2×C3×Q1612yes—
348C23C3×#13712yes—
349C23C22×C3×D411yescharacteristic
350Q8—24no—
351Q8—42no—
352C2×C4—28no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—28no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—44no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—88no—
380C2×C4C3×#12112yes—
381C2×C4C3×#12312yes—
382C2×C4C3×#12312yes—
383C2×C4C2×C3×Q1612yes—
384C2×C4C2×C3×Q1612yes—
385C2×C4C3×AGL(1,ℤ/8)12yes—
386C2×C4C3×AGL(1,ℤ/8)12yes—
387C2×C4C22×C3×D411yescharacteristic
388C2×C4C2×C3×#4112yes—
389C2×C4C2×C3×#4111yescharacteristic
390C2×C4C3×2−1+412yes—
391C8—44no—
392C2×C3—24no—
393C2×C3#248611yescharacteristic
394C2×C3C2×#46211yescharacteristic
395C2×C3#303811yescharacteristic
396C2×C3C2×#48411yescharacteristic
397C2×C3#309211yescharacteristic
398C2×C3#323011yescharacteristic
399C2×C3#323911yescharacteristic
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22C3×#46214yes—
407C22C2×C3×#12111yescharacteristic
408C22C2×C3×#12311yescharacteristic
409C22C3×#53611yescharacteristic
410C22C22×C3×Q1611yescharacteristic
411C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
412C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
413C22C2×C3×#13711yescharacteristic
414C4—22no—
415C4—22no—
416C4—22no—
417C4—22no—
418C4—42no—
419C4—42no—
420C4—81no—
421C4—84no—
422C4—81no—
423C4C3×#48414yescomplemented
424C4C3×#54712yes—
425C4C3×#55212yes—
426C3#2764811yescharacteristic, direct product factor, Sylow
427C2—24no—
428C2C3×#248611yescharacteristic
429C2C2×C3×#46211yescharacteristic
430C2C3×#303811yescharacteristic
431C2C2×C3×#48411yescharacteristic
432C2C3×#309211yescharacteristic
433C2C3×#323011yescharacteristic
434C2C3×#323911yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#2083C2semidirect
5. C3×#2072C2semidirect
6. C3×#1947C2semidirect
8. C3×#1803C2semidirect
10. #27648C3direct (N × H)
40. #2083C2×C3semidirect
41. #2072C2×C3semidirect
42. #1947C2×C3semidirect
44. #1803C2×C3semidirect
339. C3×C4#484semidirect
423. C4C3×#484semidirect
426. C3#27648direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4842
C3×C4×C22⋊Q82
C2×C3×#3842
C3×#20832
C3×#20722
C3×#19472
C3×#18242
C3×#18032
C3×#17972
#276483