Finite Group Databasebeta

← C3×#29661 #10648268

Subgroups — C3×#29661 #10648268

Order 768.

1272 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#533C211yesmaximal, characteristic, complemented
2C2×C3×#460C212yesmaximal, complemented
3C3×#2856C211yesmaximal, characteristic
4C3×#2023C212yesmaximal, complemented
5C3×#2013C212yesmaximal, complemented
6C3×#2000C214yesmaximal, complemented
7C3×#1803C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic, complemented
9#29661C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×SD16C2214yes—
11C3×#533—28no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#127C2211yescharacteristic
15C2×C3×#125—21no—
16C2×C3×C22⋊Q8—21no—
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×#121—21no—
19C2×C3×#121—21no—
20C2×C3×#121C2212yes—
21C2×C3×C4×Q8—21no—
22C2×C3×C4×Q8C2212yes—
23C2×C3×#117—21no—
24C22×C3×C42C2211yescharacteristic
25C3×#460—28no—
26C2×C3×C4⋊C8C2212yes—
27C2×C3×#103C2214yes—
28C2×C3×#102C2214yes—
29C2×C3×C22⋊C8C2212yes—
30C3×#390—21no—
31C3×#390—21no—
32C3×#390C2212yes—
33C3×#384C2211yescharacteristic
34C3×#340C2214yes—
35C3×#336C2212yes—
36C2×#533C2×C311yescharacteristic, complemented
37C2×#460C2×C312yescomplemented
38#2856C2×C311yescharacteristic
39#2023C2×C312yescomplemented
40#2013C2×C312yescomplemented
41#2000C2×C314yescomplemented
42#1803C2×C312yescomplemented
43#1797C2×C311yescharacteristic, complemented
44C22×C3×Q8—21no—
45C22×C3×Q8C2312yes—
46C22×C3×D4—22no—
47C22×C3×D4—21no—
48C22×C3×D4—21no—
49C22×C3×D4C2312yes—
50C23×C3×C4D411yescharacteristic
51C23×C3×C4D411yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C2×C3×SD16—28no—
54C2×C3×SD16—44no—
55C2×C3×SD16—44no—
56C22×C3×C8C2314yes—
57C3×C4⋊Q8—24no—
58C3×C4⋊Q8—24no—
59C3×#127—24no—
60C3×#125—28no—
61C3×C22⋊Q8—28no—
62C3×C22⋊Q8—24no—
63C3×C22⋊Q8—24no—
64C3×#121—44no—
65C3×#121—44no—
66C3×#121—44no—
67C3×C4×Q8—28no—
68C3×C4×Q8—24no—
69C3×C4×Q8D418yescomplemented
70C3×#117—28no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—21no—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C42—22no—
85C2×C3×C42—24no—
86C2×C3×C42D411yescharacteristic
87C2×C3×C42D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C3×C4⋊C8—24no—
91C3×#103—44no—
92C3×#102—28no—
93C3×C22⋊C8—42no—
94C22×SD16C22×C314yes—
95#533—28no—
96C2×C4⋊Q8C22×C311yescharacteristic
97C2×C4⋊Q8C22×C311yescharacteristic
98C2×#127C22×C311yescharacteristic
99C2×#125—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8C22×C312yes—
102C2×#121—21no—
103C2×#121—21no—
104C2×#121C22×C312yes—
105C2×C4×Q8—21no—
106C2×C4×Q8C22×C312yes—
107C2×#117—21no—
108C22×C42C22×C311yescharacteristic
109#460—28no—
110C2×C4⋊C8C22×C312yes—
111C2×#103C22×C314yes—
112C2×#102C22×C314yes—
113C2×C22⋊C8C22×C312yes—
114#390—21no—
115#390—21no—
116#390C22×C312yes—
117#384C22×C311yescharacteristic
118#340C22×C314yes—
119#336C22×C312yes—
120C24×C3—22no—
121C24×C3C2×D411yescharacteristic
122C2×C3×Q8—24no—
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—22no—
127C2×C3×Q8—22no—
128C2×C3×Q8—42no—
129C2×C3×Q8—42no—
130C2×C3×Q8C2×D414yes—
131C2×C3×D4—24no—
132C2×C3×D4—24no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—42no—
136C2×C3×D4—42no—
137C2×C3×D4—44no—
138C2×C3×D4—44no—
139C2×C3×D4—44no—
140C2×C3×D4—42no—
141C2×C3×D4—42no—
142C2×C3×D4—42no—
143C2×C3×D4—42no—
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×C4—21no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—41no—
156C22×C3×C4SD1614yescomplemented
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C3×SD16—416no—
164C2×C3×C8—24no—
165C2×C3×C8—24no—
166C2×C3×C8—24no—
167C3×C4⋊C4—24no—
168C3×C4⋊C4—24no—
169C3×C4⋊C4—28no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—28no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4C2×D414yes—
177C3×C22⋊C4—28no—
178C3×C22⋊C4—44no—
179C3×C22⋊C4—44no—
180C3×C22⋊C4—44no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42—28no—
184C3×C42—22no—
185C3×C42C2×D414yes—
186C22×Q8—21no—
187C22×Q8C23×C312yes—
188C22×D4—22no—
189C22×D4—21no—
190C22×D4—21no—
191C22×D4C23×C312yes—
192C23×C4C3×D411yescharacteristic
193C23×C4C3×D411yescharacteristic
194C23×C4C23×C311yescharacteristic
195C2×SD16—28no—
196C2×SD16—44no—
197C2×SD16—44no—
198C22×C8C23×C314yes—
199C4⋊Q8—24no—
200C4⋊Q8—24no—
201#127—24no—
202#125—28no—
203C22⋊Q8—28no—
204C22⋊Q8—24no—
205C22⋊Q8—24no—
206#121—44no—
207#121—44no—
208#121—44no—
209C4×Q8—28no—
210C4×Q8—24no—
211C4×Q8C3×D418yescomplemented
212#117—28no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C23×C312yes—
221C2×C4⋊C4C23×C312yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42C3×D411yescharacteristic
229C2×C42C3×D411yescharacteristic
230C2×C42C23×C311yescharacteristic
231C2×C42C23×C311yescharacteristic
232C4⋊C8—24no—
233#103—44no—
234#102—28no—
235C22⋊C8—42no—
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—42no—
249C23×C3C2×SD1612yes—
250C23×C3C22×D411yescharacteristic
251C3×Q8—24no—
252C3×Q8—24no—
253C3×Q8—48no—
254C3×Q8—42no—
255C3×Q8—42no—
256C3×D4—44no—
257C3×D4—44no—
258C3×D4—84no—
259C3×D4—88no—
260C3×D4—84no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—28no—
268C2×C3×C4—21no—
269C2×C3×C4—24no—
270C2×C3×C4—24no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—44no—
274C2×C3×C4—42no—
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×C4C2×SD1612yes—
286C2×C3×C4C2×SD1612yes—
287C2×C3×C4AGL(1,ℤ/8)12yes—
288C2×C3×C4#13712yes—
289C2×C3×C4C22×D412yes—
290C2×C3×C4H(5,𝔽2)11yescharacteristic
291C2×C3×C4H(5,𝔽2)12yes—
292C2×C3×C42−1+411yescharacteristic
293C3×C8—44no—
294C24—22no—
295C24C2×C3×D411yescharacteristic
296C2×Q8—24no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8C2×C3×D414yes—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4C3×SD1614yescomplemented
331C22×C4C2×C3×D411yescharacteristic
332C22×C4C2×C3×D411yescharacteristic
333C22×C4C2×C3×D411yescharacteristic
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C24×C311yescharacteristic
337SD16—416no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—28no—
344C4⋊C4—24no—
345C4⋊C4—28no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4C2×C3×D414yes—
351C22⋊C4—28no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C42—22no—
356C42—22no—
357C42—28no—
358C42—22no—
359C42C2×C3×D414yes—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—22no—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3—82no—
370C22×C3—82no—
371C22×C3—82no—
372C22×C3—82no—
373C22×C3—82no—
374C22×C3C2×#12011yescharacteristic
375C22×C3#53511yescharacteristic
376C22×C3#53711yescharacteristic
377C22×C3C22×SD1611yescharacteristic
378C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
379C22×C3C2×#13711yescharacteristic
380C22×C3#57511yescharacteristic
381C3×C4—22no—
382C3×C4—22no—
383C3×C4—22no—
384C3×C4—41no—
385C3×C4—44no—
386C3×C4—44no—
387C3×C4—82no—
388C3×C4#44814yes—
389C3×C4#53512yes—
390C3×C4#53712yes—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23C2×C3×SD1612yes—
405C23C22×C3×D411yescharacteristic
406Q8—24no—
407Q8—24no—
408Q8—48no—
409Q8—42no—
410Q8—42no—
411D4—44no—
412D4—44no—
413D4—84no—
414D4—88no—
415D4—84no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—28no—
423C2×C4—21no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4C3×#12012yes—
439C2×C4C3×#12012yes—
440C2×C4C2×C3×SD1612yes—
441C2×C4C2×C3×SD1612yes—
442C2×C4C3×AGL(1,ℤ/8)12yes—
443C2×C4C3×#13712yes—
444C2×C4C22×C3×D412yes—
445C2×C4C3×H(5,𝔽2)11yescharacteristic
446C2×C4C3×H(5,𝔽2)12yes—
447C2×C4C3×2−1+411yescharacteristic
448C8—44no—
449C2×C3—42no—
450C2×C3—162no—
451C2×C3#261511yescharacteristic
452C2×C3C2×#44811yescharacteristic
453C2×C3#300611yescharacteristic
454C2×C3#320811yescharacteristic
455C2×C3#321611yescharacteristic
456C2×C3#324611yescharacteristic
457C2×C3#325411yescharacteristic
458C22—22no—
459C22—22no—
460C22—22no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22C2×C3×#12011yescharacteristic
473C22C3×#53511yescharacteristic
474C22C3×#53711yescharacteristic
475C22C22×C3×SD1611yescharacteristic
476C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
477C22C2×C3×#13711yescharacteristic
478C22C3×#57511yescharacteristic
479C4—22no—
480C4—22no—
481C4—22no—
482C4—44no—
483C4—41no—
484C4—44no—
485C4—82no—
486C4C3×#44814yes—
487C4C3×#53512yes—
488C4C3×#53712yes—
489C3#2966111yescharacteristic, direct product factor, Sylow
490C2—42no—
491C2—162no—
492C2C3×#261511yescharacteristic
493C2C2×C3×#44811yescharacteristic
494C2C3×#300611yescharacteristic
495C2C3×#320811yescharacteristic
496C2C3×#321611yescharacteristic
497C2C3×#324611yescharacteristic
498C2C3×#325411yescharacteristic
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×#533C2semidirect
2. C2×C3×#460C2semidirect
4. C3×#2023C2semidirect
5. C3×#2013C2semidirect
6. C3×#2000C2semidirect
7. C3×#1803C2semidirect
8. C3×#1797C2semidirect
9. #29661C3direct (N × H)
36. C2×#533C2×C3semidirect
37. C2×#460C2×C3semidirect
39. #2023C2×C3semidirect
40. #2013C2×C3semidirect
41. #2000C2×C3semidirect
42. #1803C2×C3semidirect
43. #1797C2×C3semidirect
69. C3×C4×Q8D4semidirect
156. C22×C3×C4SD16semidirect
211. C4×Q8C3×D4semidirect
330. C22×C4C3×SD16semidirect
489. C3#29661direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5332
C2×C3×#4602
C3×#28562
C3×#20232
C3×#20132
C3×#20002
C3×#18032
C3×#17972
#296613