Finite Group Databasebeta

← C3×#29560 #10648167

Subgroups — C3×#29560 #10648167

Order 768.

796 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
1C3×#2862C211yesmaximal, characteristic, complemented
2C3×#2293C211yesmaximal, characteristic
3C3×#2057C211yesmaximal, characteristic, complemented
4C3×#2043C211yesmaximal, characteristic, complemented
5C3×#2023C211yesmaximal, characteristic
6C3×#2013C211yesmaximal, characteristic
7C3×#1973C212yesmaximal
8C3×#1935C212yesmaximal, complemented
9C3×#1928C212yesmaximal, complemented
10C3×#1877C212yesmaximal
11C3×#1745C211yesmaximal, characteristic, complemented
12#29560C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C2×C3×#117—21no—
18C2×C3×#107C2212yes—
19C2×C3×#103C2212yes—
20C2×C3×#103C2212yes—
21C2×C3×#103C2212yes—
22C2×C3×#102C2212yes—
23C2×C3×#97C2212yes—
24C3×#391—21no—
25C3×#391—21no—
26C3×#391C2212yes—
27C3×#391C2212yes—
28C3×#390C2211yescharacteristic
29C3×#390C2211yescharacteristic
30C3×#389C2211yescharacteristic
31C3×#389C2211yescharacteristic
32C3×#388—21no—
33C3×#388—21no—
34C3×#382—21no—
35C3×#382—21no—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C22⋊C4—21no—
38C3×C43C2211yescharacteristic
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#2862C2×C311yescharacteristic, complemented
45#2293C2×C311yescharacteristic
46#2057C2×C311yescharacteristic, complemented
47#2043C2×C311yescharacteristic, complemented
48#2023C2×C311yescharacteristic
49#2013C2×C311yescharacteristic
50#1973C2×C312yes—
51#1935C2×C312yescomplemented
52#1928C2×C312yescomplemented
53#1877C2×C312yes—
54#1745C2×C311yescharacteristic, complemented
55C22×C3×Q8—21no—
56C22×C3×Q8C2312yes—
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C2×C3×SD16—24no—
60C2×C3×SD16—24no—
61C2×C3×SD16—42no—
62C22×C3×C8C2312yes—
63C22×C3×C8C2312yes—
64C3×C4⋊Q8—42no—
65C3×C4⋊Q8—42no—
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C3×#117—44no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C42—21no—
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D411yescharacteristic
86C2×C3×C42D411yescharacteristic
87C2×C3×C42D411yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C3×#107—42no—
92C3×#103—24no—
93C3×#103—24no—
94C3×#103—42no—
95C3×#102—42no—
96C3×#97—24no—
97C3×#95—21no—
98C3×#95—21no—
99C3×#95—21no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—22no—
103C3×#95—22no—
104C22×SD16C22×C312yes—
105C2×C4⋊Q8C22×C311yescharacteristic
106C2×C4⋊Q8C22×C311yescharacteristic
107C2×C4×D4C22×C311yescharacteristic
108C2×#117—21no—
109C2×#107C22×C312yes—
110C2×#103C22×C312yes—
111C2×#103C22×C312yes—
112C2×#103C22×C312yes—
113C2×#102C22×C312yes—
114C2×#97C22×C312yes—
115#391—21no—
116#391—21no—
117#391C22×C312yes—
118#391C22×C312yes—
119#390C22×C311yescharacteristic
120#390C22×C311yescharacteristic
121#389C22×C311yescharacteristic
122#389C22×C311yescharacteristic
123#388—21no—
124#388—21no—
125#382—21no—
126#382—21no—
127C4×C4⋊C4C22×C311yescharacteristic
128C4×C22⋊C4—21no—
129C43C22×C311yescharacteristic
130#340C22×C312yes—
131#340C22×C312yes—
132#336C22×C311yescharacteristic
133#336C22×C312yes—
134#336C22×C311yescharacteristic
135C24×C3—21no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—42no—
143C2×C3×Q8—42no—
144C2×C3×Q8—42no—
145C2×C3×D4—22no—
146C2×C3×D4—21no—
147C2×C3×D4—22no—
148C2×C3×D4#4112yes—
149C22×C3×C4—24no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—42no—
156C22×C3×C4—42no—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C3×SD16—48no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8#4114yes—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
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—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4#4112yes—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—42no—
184C3×C22⋊C4—42no—
185C3×C22⋊C4—42no—
186C3×C22⋊C4—42no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—22no—
191C3×C42—21no—
192C3×C42—21no—
193C3×C42—22no—
194C3×C42—41no—
195C3×C42—42no—
196C22×Q8—21no—
197C22×Q8C23×C312yes—
198C22×D4C23×C311yescharacteristic
199C23×C4—21no—
200C2×SD16—24no—
201C2×SD16—24no—
202C2×SD16—42no—
203C22×C8C23×C312yes—
204C22×C8C23×C312yes—
205C4⋊Q8—42no—
206C4⋊Q8—42no—
207C4×D4—22no—
208C4×D4—22no—
209#117—44no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C23×C312yes—
216C2×C4⋊C4C23×C311yescharacteristic
217C2×C4⋊C4C23×C311yescharacteristic
218C2×C4⋊C4C23×C311yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C42—21no—
225C2×C42C3×D411yescharacteristic
226C2×C42C3×D411yescharacteristic
227C2×C42C3×D411yescharacteristic
228C2×C42C3×D411yescharacteristic
229C2×C42C23×C311yescharacteristic
230C2×C42C23×C311yescharacteristic
231C2×C42C23×C311yescharacteristic
232#107—42no—
233#103—24no—
234#103—24no—
235#103—42no—
236#102—42no—
237#97—24no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245C23×C3—22no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3C22×D411yescharacteristic
253C3×Q8—42no—
254C3×Q8—42no—
255C3×Q8—42no—
256C3×Q8—42no—
257C3×Q8—84no—
258C3×D4—24no—
259C3×D4—42no—
260C2×C3×C4—21no—
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—22no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—48no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—42no—
291C2×C3×C4AGL(1,ℤ/8)12yes—
292C2×C3×C4#13712yes—
293C2×C3×C4#13712yes—
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4C2×#4112yes—
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4H(5,𝔽2)12yes—
298C3×C8—24no—
299C3×C8—42no—
300C24—21no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—42no—
308C2×Q8—42no—
309C2×Q8—42no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—22no—
313C2×D4C3×#4112yes—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4C2×C3×D411yescharacteristic
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C2×C3×D411yescharacteristic
325C22×C4C2×C3×D411yescharacteristic
326C22×C4C2×C3×D411yescharacteristic
327C22×C4C2×C3×D411yescharacteristic
328C22×C4C24×C311yescharacteristic
329SD16—48no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8C3×#4114yes—
336C4⋊C4—21no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4C3×#4112yes—
347C22⋊C4—24no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—21no—
357C42—21no—
358C42—22no—
359C42—41no—
360C42—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—81no—
365C22×C3—81no—
366C22×C3—81no—
367C22×C3—81no—
368C22×C3#52511yescharacteristic
369C22×C3#53211yescharacteristic
370C22×C3#53511yescharacteristic
371C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
372C22×C3C2×#13711yescharacteristic
373C22×C3C2×#13711yescharacteristic
374C22×C3#57511yescharacteristic
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—22no—
378C3×C4—42no—
379C3×C4—41no—
380C3×C4—41no—
381C3×C4—41no—
382C3×C4—82no—
383C3×C4—81no—
384C3×C4—81no—
385C3×C4—82no—
386C3×C4#55112yes—
387C3×C4#55512yes—
388C23—22no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23C22×C3×D411yescharacteristic
396Q8—42no—
397Q8—42no—
398Q8—42no—
399Q8—42no—
400Q8—84no—
401D4—24no—
402D4—42no—
403C2×C4—21no—
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—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—48no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4C3×AGL(1,ℤ/8)12yes—
435C2×C4C3×#13712yes—
436C2×C4C3×#13712yes—
437C2×C4C2×C3×#4111yescharacteristic
438C2×C4C2×C3×#4112yes—
439C2×C4C2×C3×#4111yescharacteristic
440C2×C4C3×H(5,𝔽2)12yes—
441C8—24no—
442C8—42no—
443C2×C3—82no—
444C2×C3#261011yescharacteristic
445C2×C3#311011yescharacteristic
446C2×C3#311611yescharacteristic
447C2×C3#317211yescharacteristic
448C2×C3#318211yescharacteristic
449C2×C3#321311yescharacteristic
450C2×C3#321811yescharacteristic
451C22—42no—
452C22—42no—
453C22—42no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22C3×#52511yescharacteristic
459C22C3×#53211yescharacteristic
460C22C3×#53511yescharacteristic
461C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
462C22C2×C3×#13711yescharacteristic
463C22C2×C3×#13711yescharacteristic
464C22C3×#57511yescharacteristic
465C4—22no—
466C4—22no—
467C4—22no—
468C4—41no—
469C4—41no—
470C4—42no—
471C4—41no—
472C4—81no—
473C4—82no—
474C4—81no—
475C4—82no—
476C4C3×#55112yes—
477C4C3×#55512yes—
478C3#2956011yescharacteristic, direct product factor, Sylow
479C2—82no—
480C2C3×#261011yescharacteristic
481C2C3×#311011yescharacteristic
482C2C3×#311611yescharacteristic
483C2C3×#317211yescharacteristic
484C2C3×#318211yescharacteristic
485C2C3×#321311yescharacteristic
486C2C3×#321811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C43 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2862C2semidirect
3. C3×#2057C2semidirect
4. C3×#2043C2semidirect
8. C3×#1935C2semidirect
9. C3×#1928C2semidirect
11. C3×#1745C2semidirect
12. #29560C3direct (N × H)
44. #2862C2×C3semidirect
46. #2057C2×C3semidirect
47. #2043C2×C3semidirect
51. #1935C2×C3semidirect
52. #1928C2×C3semidirect
54. #1745C2×C3semidirect
478. C3#29560direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28622
C3×#22932
C3×#20572
C3×#20432
C3×#20232
C3×#20132
C3×#19732
C3×#19352
C3×#19282
C3×#18772
C3×#17452
#295603