Finite Group Databasebeta

← C3×#29713 #10648320

Subgroups — C3×#29713 #10648320

Order 768.

1152 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#448C212yesmaximal, complemented
2C3×#2856C211yesmaximal, characteristic
3C2×C3×#391C211yesmaximal, characteristic, complemented
4C3×#2072C212yesmaximal, complemented
5C3×#2055C214yesmaximal, complemented
6C3×#2014C212yesmaximal, complemented
7C3×#1874C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic, complemented
9#29713C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×Q8C2211yescharacteristic
11C22×C3×SD16C2214yes—
12C2×C3×#127C2211yescharacteristic
13C2×C3×#121—21no—
14C2×C3×#121—21no—
15C2×C3×#121C2212yes—
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C3×#448—28no—
19C2×C3×C4⋊C8C2212yes—
20C2×C3×#103C2214yes—
21C2×C3×#102C2214yes—
22C2×C3×C22⋊C8C2212yes—
23C3×#391—21no—
24C3×#391—21no—
25C3×#391C2212yes—
26C3×#391C2212yes—
27C3×#390—21no—
28C3×#390—21no—
29C3×#390C2212yes—
30C3×#384C2211yescharacteristic
31C2×C3×#95—21no—
32C2×C3×#95—21no—
33C3×#340C2214yes—
34C3×#336C2212yes—
35C2×#448C2×C312yescomplemented
36#2856C2×C311yescharacteristic
37C2×#391C2×C311yescharacteristic, complemented
38#2072C2×C312yescomplemented
39#2055C2×C314yescomplemented
40#2014C2×C312yescomplemented
41#1874C2×C312yescomplemented
42#1797C2×C311yescharacteristic, complemented
43C22×C3×Q8—21no—
44C22×C3×Q8—22no—
45C22×C3×Q8—42no—
46C22×C3×Q8—42no—
47C22×C3×Q8D414yescomplemented
48C22×C3×Q8C2312yes—
49C22×C3×D4—22no—
50C22×C3×D4—21no—
51C22×C3×D4—21no—
52C22×C3×D4C2312yes—
53C23×C3×C4—21no—
54C23×C3×C4—21no—
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C2×C3×SD16—28no—
59C2×C3×SD16—44no—
60C2×C3×SD16—44no—
61C22×C3×C8C2314yes—
62C3×#127—42no—
63C3×#121—24no—
64C3×#121—24no—
65C3×#121—44no—
66C3×#121—44no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—42no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—22no—
76C2×C3×C42—42no—
77C2×C3×C42D411yescharacteristic
78C2×C3×C42D411yescharacteristic
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C3×C4⋊C8—42no—
82C3×#103—28no—
83C3×#102—44no—
84C3×C22⋊C8—24no—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95—22no—
88C3×#95—22no—
89C23×Q8C22×C311yescharacteristic
90C22×SD16C22×C314yes—
91C2×#127C22×C311yescharacteristic
92C2×#121—21no—
93C2×#121—21no—
94C2×#121C22×C312yes—
95C22×C4⋊C4C22×C311yescharacteristic
96C22×C42C22×C311yescharacteristic
97#448—28no—
98C2×C4⋊C8C22×C312yes—
99C2×#103C22×C314yes—
100C2×#102C22×C314yes—
101C2×C22⋊C8C22×C312yes—
102#391—21no—
103#391—21no—
104#391C22×C312yes—
105#391C22×C312yes—
106#390—21no—
107#390—21no—
108#390C22×C312yes—
109#384C22×C311yescharacteristic
110C2×#95—21no—
111C2×#95—21no—
112#340C22×C314yes—
113#336C22×C312yes—
114C24×C3—22no—
115C24×C3C2×D411yescharacteristic
116C2×C3×Q8—24no—
117C2×C3×Q8—22no—
118C2×C3×Q8—24no—
119C2×C3×Q8—24no—
120C2×C3×Q8—48no—
121C2×C3×Q8—42no—
122C2×C3×Q8—42no—
123C2×C3×Q8—48no—
124C2×C3×Q8C2×D414yes—
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4—24no—
128C2×C3×D4—24no—
129C2×C3×D4—42no—
130C2×C3×D4—42no—
131C2×C3×D4—42no—
132C2×C3×D4—42no—
133C2×C3×D4—44no—
134C2×C3×D4—44no—
135C2×C3×D4—44no—
136C2×C3×D4—42no—
137C2×C3×D4—42no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—24no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—41no—
147C22×C3×C4—41no—
148C22×C3×C4—41no—
149C22×C3×C4—42no—
150C22×C3×C4—42no—
151C22×C3×C4—42no—
152C22×C3×C4—42no—
153C22×C3×C4—42no—
154C22×C3×C4SD1614yescomplemented
155C22×C3×C4C2×D411yescharacteristic
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4#4112yes—
162C22×C3×C4C2411yescharacteristic
163C3×SD16—416no—
164C2×C3×C8—24no—
165C2×C3×C8—24no—
166C2×C3×C8—24no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—42no—
173C3×C22⋊C4—44no—
174C3×C22⋊C4—44no—
175C3×C22⋊C4—44no—
176C3×C42—22no—
177C3×C42—22no—
178C3×C42—41no—
179C3×C42—41no—
180C22×Q8—21no—
181C22×Q8—22no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8C3×D414yescomplemented
185C22×Q8C23×C312yes—
186C22×D4—22no—
187C22×D4—21no—
188C22×D4—21no—
189C22×D4C23×C312yes—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4C3×D411yescharacteristic
193C23×C4C3×D411yescharacteristic
194C23×C4C23×C311yescharacteristic
195C2×SD16—28no—
196C2×SD16—44no—
197C2×SD16—44no—
198C22×C8C23×C314yes—
199#127—42no—
200#121—24no—
201#121—24no—
202#121—44no—
203#121—44no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C23×C312yes—
209C2×C4⋊C4C23×C312yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C42—42no—
214C2×C42C3×D411yescharacteristic
215C2×C42C3×D411yescharacteristic
216C2×C42C23×C311yescharacteristic
217C2×C42C23×C311yescharacteristic
218C4⋊C8—42no—
219#103—28no—
220#102—44no—
221C22⋊C8—24no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226C23×C3—21no—
227C23×C3—21no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—42no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—42no—
237C23×C3#12112yes—
238C23×C3C2×SD1612yes—
239C23×C3AGL(1,ℤ/8)12yes—
240C23×C3C22×D411yescharacteristic
241C3×Q8—28no—
242C3×Q8—48no—
243C3×Q8—44no—
244C3×D4—44no—
245C3×D4—44no—
246C3×D4—84no—
247C3×D4—84no—
248C3×D4—88no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—24no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—44no—
264C2×C3×C4—44no—
265C2×C3×C4—48no—
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—84no—
275C2×C3×C4#12112yes—
276C2×C3×C4C2×SD1612yes—
277C2×C3×C4C2×SD1612yes—
278C2×C3×C4#13712yes—
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4H(5,𝔽2)12yes—
282C2×C3×C4H(5,𝔽2)12yes—
283C3×C8—44no—
284C24—22no—
285C24C2×C3×D411yescharacteristic
286C2×Q8—24no—
287C2×Q8—22no—
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—48no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—48no—
294C2×Q8C2×C3×D414yes—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—44no—
306C2×D4—42no—
307C2×D4—42no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4C3×SD1614yescomplemented
325C22×C4C2×C3×D411yescharacteristic
326C22×C4C2×C3×D411yescharacteristic
327C22×C4C2×C3×D412yes—
328C22×C4C2×C3×D411yescharacteristic
329C22×C4C2×C3×D411yescharacteristic
330C22×C4C2×C3×D411yescharacteristic
331C22×C4C3×#4112yes—
332C22×C4C24×C311yescharacteristic
333SD16—416no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C42—22no—
347C42—22no—
348C42—41no—
349C42—41no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—82no—
357C22×C3—82no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—82no—
361C22×C3—82no—
362C22×C3—82no—
363C22×C3#46014yes—
364C22×C3C2×#12111yescharacteristic
365C22×C3#53411yescharacteristic
366C22×C3#53511yescharacteristic
367C22×C3C22×SD1611yescharacteristic
368C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
369C22×C3C2×#13711yescharacteristic
370C22×C3#57511yescharacteristic
371C3×C4—22no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—44no—
376C3×C4—41no—
377C3×C4—82no—
378C3×C4—82no—
379C3×C4#53412yes—
380C3×C4#53812yes—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23C3×#12112yes—
393C23C2×C3×SD1612yes—
394C23C3×AGL(1,ℤ/8)12yes—
395C23C22×C3×D411yescharacteristic
396Q8—28no—
397Q8—48no—
398Q8—44no—
399D4—44no—
400D4—44no—
401D4—84no—
402D4—84no—
403D4—88no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—24no—
417C2×C4—21no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—48no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—84no—
430C2×C4C3×#12112yes—
431C2×C4C2×C3×SD1612yes—
432C2×C4C2×C3×SD1612yes—
433C2×C4C3×#13712yes—
434C2×C4C22×C3×D411yescharacteristic
435C2×C4C2×C3×#4111yescharacteristic
436C2×C4C3×H(5,𝔽2)12yes—
437C2×C4C3×H(5,𝔽2)12yes—
438C8—44no—
439C2×C3—24no—
440C2×C3—162no—
441C2×C3#261711yescharacteristic
442C2×C3C2×#46011yescharacteristic
443C2×C3#303711yescharacteristic
444C2×C3#318711yescharacteristic
445C2×C3#319311yescharacteristic
446C2×C3#320611yescharacteristic
447C2×C3#321311yescharacteristic
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22C3×#46014yes—
462C22C2×C3×#12111yescharacteristic
463C22C3×#53411yescharacteristic
464C22C3×#53511yescharacteristic
465C22C22×C3×SD1611yescharacteristic
466C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
467C22C2×C3×#13711yescharacteristic
468C22C3×#57511yescharacteristic
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—44no—
474C4—41no—
475C4—82no—
476C4—82no—
477C4C3×#53412yes—
478C4C3×#53812yes—
479C3#2971311yescharacteristic, direct product factor, Sylow
480C2—24no—
481C2—162no—
482C2C3×#261711yescharacteristic
483C2C2×C3×#46011yescharacteristic
484C2C3×#303711yescharacteristic
485C2C3×#318711yescharacteristic
486C2C3×#319311yescharacteristic
487C2C3×#320611yescharacteristic
488C2C3×#321311yescharacteristic
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×#448C2semidirect
3. C2×C3×#391C2semidirect
4. C3×#2072C2semidirect
5. C3×#2055C2semidirect
6. C3×#2014C2semidirect
7. C3×#1874C2semidirect
8. C3×#1797C2semidirect
9. #29713C3direct (N × H)
35. C2×#448C2×C3semidirect
37. C2×#391C2×C3semidirect
38. #2072C2×C3semidirect
39. #2055C2×C3semidirect
40. #2014C2×C3semidirect
41. #1874C2×C3semidirect
42. #1797C2×C3semidirect
47. C22×C3×Q8D4semidirect
154. C22×C3×C4SD16semidirect
184. C22×Q8C3×D4semidirect
324. C22×C4C3×SD16semidirect
479. C3#29713direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4482
C3×#28562
C2×C3×#3912
C3×#20722
C3×#20552
C3×#20142
C3×#18742
C3×#17972
#297133