Finite Group Databasebeta

← C3×#32906 #10651513

Subgroups — C3×#32906 #10651513

Order 768.

1436 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Q8C211yesmaximal, characteristic
2C2×C3×#462C212yesmaximal, complemented
3C2×C3×#451C212yesmaximal
4C3×#2857C211yesmaximal, characteristic
5C3×#1878C214yesmaximal, complemented
6C3×#1872C212yesmaximal
7C3×#1803C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic
9#32906C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×Q8C2211yescharacteristic
11C22×C3×Q16C2214yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C22⋊Q8C2212yes—
14C2×C3×C4×Q8—28no—
15C2×C3×C4×Q8—21no—
16C2×C3×C4×Q8—21no—
17C2×C3×C4×Q8C2212yes—
18C2×C3×C4×Q8C2212yes—
19C22×C3×C4⋊C4—21no—
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C42—21no—
22C22×C3×C42C2211yescharacteristic
23C3×#462—28no—
24C3×#451—28no—
25C2×C3×C4⋊C8C2212yes—
26C2×C3×#103C2214yes—
27C2×C3×#103C2214yes—
28C2×C3×C22⋊C8C2212yes—
29C3×#391C2212yes—
30C3×#389—21no—
31C3×#389—21no—
32C3×#384C2211yescharacteristic
33C3×#380—21no—
34C3×#380—21no—
35C3×#340C2214yes—
36C3×#336C2212yes—
37C22×C4×Q8C2×C311yescharacteristic
38C2×#462C2×C312yescomplemented
39C2×#451C2×C312yes—
40#2857C2×C311yescharacteristic
41#1878C2×C314yescomplemented
42#1872C2×C312yes—
43#1803C2×C312yescomplemented
44#1797C2×C311yescharacteristic
45C22×C3×Q8—24no—
46C22×C3×Q8—24no—
47C22×C3×Q8—21no—
48C22×C3×Q8—22no—
49C22×C3×Q8D414yes—
50C22×C3×Q8C2312yes—
51C22×C3×Q8C2312yes—
52C23×C3×C4—21no—
53C23×C3×C4—21no—
54C23×C3×C4D411yescharacteristic
55C23×C3×C4D411yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C2×C3×Q16—28no—
58C2×C3×Q16—28no—
59C2×C3×Q16—28no—
60C22×C3×C8C2314yes—
61C3×C4⋊Q8—24no—
62C3×C22⋊Q8—24no—
63C3×C22⋊Q8—24no—
64C3×C4×Q8—28no—
65C3×C4×Q8—28no—
66C3×C4×Q8—24no—
67C3×C4×Q8—24no—
68C3×C4×Q8—24no—
69C3×C4×Q8D418yes—
70C2×C3×C4⋊C4—28no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—28no—
84C2×C3×C42—22no—
85C2×C3×C42—22no—
86C2×C3×C42—24no—
87C2×C3×C42D411yescharacteristic
88C2×C3×C42D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C3×C4⋊C8—24no—
92C3×#103—28no—
93C3×#103—28no—
94C3×C22⋊C8—24no—
95C3×#95—22no—
96C3×#95—22no—
97C23×Q8C22×C311yescharacteristic
98C22×Q16C22×C314yes—
99C2×C4⋊Q8C22×C311yescharacteristic
100C2×C22⋊Q8C22×C312yes—
101C2×C4×Q8—28no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×C4×Q8C22×C312yes—
105C2×C4×Q8C22×C312yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C22×C311yescharacteristic
108C22×C42—21no—
109C22×C42C22×C311yescharacteristic
110#462—28no—
111#451—28no—
112C2×C4⋊C8C22×C312yes—
113C2×#103C22×C314yes—
114C2×#103C22×C314yes—
115C2×C22⋊C8C22×C312yes—
116#391C22×C312yes—
117#389—21no—
118#389—21no—
119#384C22×C311yescharacteristic
120#380—21no—
121#380—21no—
122#340C22×C314yes—
123#336C22×C312yes—
124C24×C3C2×D411yescharacteristic
125C2×C3×Q8—24no—
126C2×C3×Q8—216no—
127C2×C3×Q8—24no—
128C2×C3×Q8—216no—
129C2×C3×Q8—24no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—24no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—24no—
137C2×C3×Q8C2×D414yes—
138C2×C3×Q8C2×D414yes—
139C2×C3×Q8#4114yes—
140C22×C3×C4—24no—
141C22×C3×C4—24no—
142C22×C3×C4—24no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—24no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—41no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4Q1614yescomplemented
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4#4112yes—
165C22×C3×C4C2411yescharacteristic
166C3×Q16—416no—
167C2×C3×C8—24no—
168C2×C3×C8—24no—
169C2×C3×C8—24no—
170C3×C4⋊C4—28no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—28no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—44no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4C2×D414yes—
182C3×C22⋊C4—44no—
183C3×C42—28no—
184C3×C42—28no—
185C3×C42—22no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42C2×D414yes—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—21no—
192C22×Q8—22no—
193C22×Q8C3×D414yes—
194C22×Q8C23×C312yes—
195C22×Q8C23×C312yes—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4C3×D411yescharacteristic
199C23×C4C3×D411yescharacteristic
200C23×C4C23×C311yescharacteristic
201C2×Q16—28no—
202C2×Q16—28no—
203C2×Q16—28no—
204C22×C8C23×C314yes—
205C4⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—24no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—24no—
213C4×Q8C3×D418yes—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C23×C312yes—
225C2×C4⋊C4C23×C312yes—
226C2×C22⋊C4—22no—
227C2×C42—28no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42C3×D411yescharacteristic
232C2×C42C3×D411yescharacteristic
233C2×C42C23×C311yescharacteristic
234C2×C42C23×C311yescharacteristic
235C4⋊C8—24no—
236#103—28no—
237#103—28no—
238C22⋊C8—24no—
239#95—22no—
240#95—22no—
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3#12112yes—
246C23×C3C2×Q1612yes—
247C23×C3#13712yes—
248C23×C3C22×D411yescharacteristic
249C3×Q8—216no—
250C3×Q8—24no—
251C3×Q8—24no—
252C3×Q8—24no—
253C3×Q8—44no—
254C3×Q8—44no—
255C3×Q8#12118yes—
256C2×C3×C4—216no—
257C2×C3×C4—21no—
258C2×C3×C4—24no—
259C2×C3×C4—21no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—28no—
269C2×C3×C4—21no—
270C2×C3×C4—24no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—44no—
274C2×C3×C4—48no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4#12012yes—
284C2×C3×C4#12012yes—
285C2×C3×C4#12112yes—
286C2×C3×C4C2×Q1612yes—
287C2×C3×C4C2×Q1612yes—
288C2×C3×C4#13712yes—
289C2×C3×C4C22×D411yescharacteristic
290C2×C3×C4C22×D412yes—
291C2×C3×C4C2×#4112yes—
292C2×C3×C4C2×#4111yescharacteristic
293C3×C8—44no—
294C24C2×C3×D411yescharacteristic
295C2×Q8—24no—
296C2×Q8—216no—
297C2×Q8—24no—
298C2×Q8—216no—
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8C2×C3×D414yes—
308C2×Q8C2×C3×D414yes—
309C2×Q8C3×#4114yes—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4C3×Q1614yescomplemented
328C22×C4C2×C3×D411yescharacteristic
329C22×C4C2×C3×D411yescharacteristic
330C22×C4C2×C3×D411yescharacteristic
331C22×C4C2×C3×D412yes—
332C22×C4C2×C3×D411yescharacteristic
333C22×C4C2×C3×D411yescharacteristic
334C22×C4C3×#4112yes—
335C22×C4C24×C311yescharacteristic
336Q16—416no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C4⋊C4—28no—
341C4⋊C4—24no—
342C4⋊C4—28no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4C2×C3×D414yes—
352C22⋊C4—44no—
353C42—28no—
354C42—28no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42C2×C3×D414yes—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—22no—
364C22×C3—22no—
365C22×C3#46214yescomplemented
366C22×C3C2×#12011yescharacteristic
367C22×C3C2×#12111yescharacteristic
368C22×C3#52511yescharacteristic
369C22×C3C22×Q1611yescharacteristic
370C22×C3C2×#13711yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C22×C3#57511yescharacteristic
373C3×C4—28no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—41no—
378C3×C4—44no—
379C3×C4—82no—
380C3×C4—82no—
381C3×C4#45114yescomplemented
382C3×C4C2×#12112yes—
383C3×C4#52512yes—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23C3×#12112yes—
389C23C2×C3×Q1612yes—
390C23C3×#13712yes—
391C23C22×C3×D411yescharacteristic
392Q8—216no—
393Q8—24no—
394Q8—24no—
395Q8—24no—
396Q8—44no—
397Q8—44no—
398Q8C3×#12118yes—
399C2×C4—216no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—24no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—28no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—44no—
417C2×C4—48no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4C3×#12012yes—
427C2×C4C3×#12012yes—
428C2×C4C3×#12112yes—
429C2×C4C2×C3×Q1612yes—
430C2×C4C2×C3×Q1612yes—
431C2×C4C3×#13712yes—
432C2×C4C22×C3×D411yescharacteristic
433C2×C4C22×C3×D412yes—
434C2×C4C2×C3×#4112yes—
435C2×C4C2×C3×#4111yescharacteristic
436C8—44no—
437C2×C3—24no—
438C2×C3#284711yescharacteristic
439C2×C3C2×#45111yescharacteristic
440C2×C3#300711yescharacteristic
441C2×C3C2×#46211yescharacteristic
442C2×C3#303711yescharacteristic
443C2×C3#310411yescharacteristic
444C2×C3#311311yescharacteristic
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22C3×#46214yescomplemented
452C22C2×C3×#12011yescharacteristic
453C22C2×C3×#12111yescharacteristic
454C22C3×#52511yescharacteristic
455C22C22×C3×Q1611yescharacteristic
456C22C2×C3×#13711yescharacteristic
457C22C2×C3×#13711yescharacteristic
458C22C3×#57511yescharacteristic
459C4—22no—
460C4—22no—
461C4—28no—
462C4—22no—
463C4—41no—
464C4—44no—
465C4—82no—
466C4—82no—
467C4C3×#45114yescomplemented
468C4C2×C3×#12112yes—
469C4C3×#52512yes—
470C3#3290611yescharacteristic, direct product factor, Sylow
471C2—24no—
472C2C3×#284711yescharacteristic
473C2C2×C3×#45111yescharacteristic
474C2C3×#300711yescharacteristic
475C2C2×C3×#46211yescharacteristic
476C2C3×#303711yescharacteristic
477C2C3×#310411yescharacteristic
478C2C3×#311311yescharacteristic
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
2. C2×C3×#462C2semidirect
5. C3×#1878C2semidirect
7. C3×#1803C2semidirect
9. #32906C3direct (N × H)
38. C2×#462C2×C3semidirect
41. #1878C2×C3semidirect
43. #1803C2×C3semidirect
157. C22×C3×C4Q16semidirect
327. C22×C4C3×Q16semidirect
365. C22×C3#462semidirect
381. C3×C4#451semidirect
451. C22C3×#462semidirect
467. C4C3×#451semidirect
470. C3#32906direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×Q82
C2×C3×#4622
C2×C3×#4512
C3×#28572
C3×#18782
C3×#18722
C3×#18032
C3×#17972
#329063