Finite Group Databasebeta

← C3×#33048 #10651655

Subgroups — C3×#33048 #10651655

Order 768.

1054 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#486C211yesmaximal, characteristic, complemented
2C2×C3×C4×M16C211yesmaximal, characteristic, complemented
3C3×#2857C211yesmaximal, characteristic, complemented
4C3×#2856C211yesmaximal, characteristic
5C3×#1990C211yesmaximal, characteristic, complemented
6C3×#1957C211yesmaximal, characteristic, complemented
7C3×#1936C214yesmaximal, complemented
8C3×#1895C214yesmaximal, complemented
9C3×#1834C211yesmaximal, characteristic, complemented
10#33048C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×M16C2212yes—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×#127C2211yescharacteristic
14C2×C3×C22⋊Q8—21no—
15C2×C3×#121—21no—
16C2×C3×#121—21no—
17C2×C3×#121C2212yes—
18C22×C3×C42C2211yescharacteristic
19C3×#486—24no—
20C2×C3×#102C2214yes—
21C2×C3×#102C2214yes—
22C3×C4×M16—24no—
23C2×C3×#97C2212yes—
24C2×C3×C4×C8C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C3×#391—21no—
27C3×#390—21no—
28C3×#390—21no—
29C3×#390C2212yes—
30C3×#389—21no—
31C3×#389C2212yes—
32C3×#384C2211yescharacteristic
33C3×#384C2211yescharacteristic
34C3×#380—21no—
35C3×#380C2212yes—
36C3×#340C2214yes—
37C3×#340C2214yes—
38C2×#486C2×C311yescharacteristic, complemented
39C2×C4×M16C2×C311yescharacteristic, complemented
40#2857C2×C311yescharacteristic, complemented
41#2856C2×C311yescharacteristic
42#1990C2×C311yescharacteristic, complemented
43#1957C2×C311yescharacteristic, complemented
44#1936C2×C314yescomplemented
45#1895C2×C314yescomplemented
46#1834C2×C311yescharacteristic, complemented
47C22×C3×Q8—21no—
48C22×C3×D4—22no—
49C22×C3×D4—21no—
50C22×C3×D4—21no—
51C22×C3×D4C2312yes—
52C23×C3×C4D411yescharacteristic
53C23×C3×C4D411yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C2×C3×M16—24no—
56C2×C3×M16—24no—
57C2×C3×M16—24no—
58C22×C3×C8C2312yes—
59C22×C3×C8C2312yes—
60C3×C4⋊Q8—22no—
61C3×C4⋊Q8—22no—
62C3×#127—22no—
63C3×#127—22no—
64C3×C22⋊Q8—44no—
65C3×#121—28no—
66C3×#121—44no—
67C3×#121—44no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C42—22no—
82C2×C3×C42—22no—
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C3×#102—28no—
88C3×#102—28no—
89C3×#97—24no—
90C3×C4×C8—22no—
91C3×C4×C8D414yescomplemented
92C3×#95—22no—
93C3×#95—22no—
94C22×M16C22×C312yes—
95C2×C4⋊Q8C22×C311yescharacteristic
96C2×#127C22×C311yescharacteristic
97C2×C22⋊Q8—21no—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121C22×C312yes—
101C22×C42C22×C311yescharacteristic
102#486—24no—
103C2×#102C22×C314yes—
104C2×#102C22×C314yes—
105C4×M16—24no—
106C2×#97C22×C312yes—
107C2×C4×C8C22×C311yescharacteristic
108C2×C4×C8C22×C311yescharacteristic
109#391—21no—
110#390—21no—
111#390—21no—
112#390C22×C312yes—
113#389—21no—
114#389C22×C312yes—
115#384C22×C311yescharacteristic
116#384C22×C311yescharacteristic
117#380—21no—
118#380C22×C312yes—
119#340C22×C314yes—
120#340C22×C314yes—
121C24×C3—22no—
122C24×C3C2×D411yescharacteristic
123C2×C3×Q8—42no—
124C2×C3×Q8—42no—
125C2×C3×Q8—42no—
126C2×C3×D4—22no—
127C2×C3×D4—24no—
128C2×C3×D4—24no—
129C2×C3×D4—22no—
130C2×C3×D4—44no—
131C2×C3×D4—44no—
132C2×C3×D4—44no—
133C2×C3×D4—42no—
134C2×C3×D4—42no—
135C2×C3×D4—42no—
136C2×C3×D4—42no—
137C2×C3×D4—42no—
138C2×C3×D4—42no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4C2×D411yescharacteristic
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D411yescharacteristic
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4#4114yes—
157C22×C3×C4C2411yescharacteristic
158C3×M16—216no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8—22no—
162C2×C3×C8—22no—
163C2×C3×C8C2×D414yes—
164C2×C3×C8#4114yes—
165C3×C4⋊C4—22no—
166C3×C4⋊C4—24no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—44no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C22⋊C4—44no—
176C3×C22⋊C4—44no—
177C3×C22⋊C4—44no—
178C3×C22⋊C4—44no—
179C3×C42—22no—
180C3×C42—22no—
181C3×C42—21no—
182C3×C42—21no—
183C3×C42C2×D412yes—
184C3×C42#4112yes—
185C22×Q8—21no—
186C22×D4—22no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4C23×C312yes—
190C23×C4C3×D411yescharacteristic
191C23×C4C3×D411yescharacteristic
192C23×C4C23×C311yescharacteristic
193C2×M16—24no—
194C2×M16—24no—
195C2×M16—24no—
196C22×C8C23×C312yes—
197C22×C8C23×C312yes—
198C4⋊Q8—22no—
199C4⋊Q8—22no—
200#127—22no—
201#127—22no—
202C22⋊Q8—44no—
203#121—28no—
204#121—44no—
205#121—44no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C23×C312yes—
213C2×C4⋊C4C23×C312yes—
214C2×C4⋊C4C23×C312yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C3×D411yescharacteristic
222C2×C42C3×D411yescharacteristic
223C2×C42C23×C311yescharacteristic
224C2×C42C23×C311yescharacteristic
225#102—28no—
226#102—28no—
227#97—24no—
228C4×C8—22no—
229C4×C8C3×D414yescomplemented
230#95—22no—
231#95—22no—
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—21no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—42no—
239C23×C3—42no—
240C23×C3—42no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3#12412yes—
246C23×C3C22×D411yescharacteristic
247C3×Q8—84no—
248C3×D4—44no—
249C3×D4—44no—
250C3×D4—88no—
251C3×D4—84no—
252C3×D4—84no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—44no—
264C2×C3×C4—44no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4#12112yes—
275C2×C3×C4#12112yes—
276C2×C3×C4#12412yes—
277C2×C3×C4AGL(1,ℤ/8)12yes—
278C2×C3×C4AGL(1,ℤ/8)12yes—
279C2×C3×C4AGL(1,ℤ/8)12yes—
280C2×C3×C4#13712yes—
281C2×C3×C4C22×D411yescharacteristic
282C2×C3×C4C2×#4112yes—
283C2×C3×C4C2×#4112yes—
284C2×C3×C4C2×#4111yescharacteristic
285C3×C8—24no—
286C3×C8—24no—
287C24—22no—
288C24C2×C3×D411yescharacteristic
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×D4—22no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—22no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C2×C3×D411yescharacteristic
319C22×C4C2×C3×D411yescharacteristic
320C22×C4C2×C3×D411yescharacteristic
321C22×C4C2×C3×D411yescharacteristic
322C22×C4C3×#4114yes—
323C22×C4C24×C311yescharacteristic
324M16—216no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8C2×C3×D414yes—
330C2×C8C3×#4114yes—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C42—22no—
346C42—22no—
347C42—21no—
348C42—21no—
349C42C2×C3×D412yes—
350C42C3×#4112yes—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—41no—
355C22×C3—41no—
356C22×C3—41no—
357C22×C3—41no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3—82no—
364C22×C3—82no—
365C22×C3C2×#12111yescharacteristic
366C22×C3C2×#12411yescharacteristic
367C22×C3#53211yescharacteristic
368C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
369C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
370C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C3×C4—22no—
373C3×C4—41no—
374C3×C4—41no—
375C3×C4—82no—
376C3×C4—82no—
377C3×C4—82no—
378C3×C4#46814yes—
379C3×C4#46914yes—
380C3×C4#52512yes—
381C3×C4#52912yes—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23C3×#12412yes—
396C23C22×C3×D411yescharacteristic
397Q8—84no—
398D4—44no—
399D4—44no—
400D4—88no—
401D4—84no—
402D4—84no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4C3×#12112yes—
425C2×C4C3×#12112yes—
426C2×C4C3×#12412yes—
427C2×C4C3×AGL(1,ℤ/8)12yes—
428C2×C4C3×AGL(1,ℤ/8)12yes—
429C2×C4C3×AGL(1,ℤ/8)12yes—
430C2×C4C3×#13712yes—
431C2×C4C22×C3×D411yescharacteristic
432C2×C4C2×C3×#4112yes—
433C2×C4C2×C3×#4112yes—
434C2×C4C2×C3×#4111yescharacteristic
435C8—24no—
436C8—24no—
437C2×C3—42no—
438C2×C3—162no—
439C2×C3#285011yescharacteristic
440C2×C3C2×#46811yescharacteristic
441C2×C3C2×#46911yescharacteristic
442C2×C3#313811yescharacteristic
443C2×C3#314111yescharacteristic
444C2×C3#318211yescharacteristic
445C2×C3#318411yescharacteristic
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22C2×C3×#12111yescharacteristic
461C22C2×C3×#12411yescharacteristic
462C22C3×#53211yescharacteristic
463C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
464C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
465C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
466C22C2×C3×#13711yescharacteristic
467C4—22no—
468C4—41no—
469C4—41no—
470C4—82no—
471C4—82no—
472C4—82no—
473C4C3×#46814yes—
474C4C3×#46914yes—
475C4C3×#52512yes—
476C4C3×#52912yes—
477C3#3304811yescharacteristic, direct product factor, Sylow
478C2—42no—
479C2—162no—
480C2C3×#285011yescharacteristic
481C2C2×C3×#46811yescharacteristic
482C2C2×C3×#46911yescharacteristic
483C2C3×#313811yescharacteristic
484C2C3×#314111yescharacteristic
485C2C3×#318211yescharacteristic
486C2C3×#318411yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#2856C2
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
1. C2×C3×#486C2semidirect
2. C2×C3×C4×M16C2semidirect
3. C3×#2857C2semidirect
5. C3×#1990C2semidirect
6. C3×#1957C2semidirect
7. C3×#1936C2semidirect
8. C3×#1895C2semidirect
9. C3×#1834C2semidirect
10. #33048C3direct (N × H)
38. C2×#486C2×C3semidirect
39. C2×C4×M16C2×C3semidirect
40. #2857C2×C3semidirect
42. #1990C2×C3semidirect
43. #1957C2×C3semidirect
44. #1936C2×C3semidirect
45. #1895C2×C3semidirect
46. #1834C2×C3semidirect
91. C3×C4×C8D4semidirect
229. C4×C8C3×D4semidirect
477. C3#33048direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4862
C2×C3×C4×M162
C3×#28572
C3×#28562
C3×#19902
C3×#19572
C3×#19362
C3×#18952
C3×#18342
#330483