Finite Group Databasebeta

← C3×#29256 #10647863

Subgroups — C3×#29256 #10647863

Order 768.

884 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#127C211yesmaximal, characteristic
2C3×C4×#125C211yesmaximal, characteristic, complemented
3C3×#2092C212yesmaximal, complemented
4C3×#2025C212yesmaximal, complemented
5C3×#1982C211yesmaximal, characteristic
6C3×#1982C211yesmaximal, characteristic
7C3×#1948C211yesmaximal, characteristic, complemented
8C3×#1947C211yesmaximal, characteristic, complemented
9C3×#1864C212yesmaximal, complemented
10C3×#1862C212yesmaximal, complemented
11C3×#1831C211yesmaximal, characteristic, complemented
12#29256C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic
15C2×C3×C4×D4—21no—
16C2×C3×C4×D4—21no—
17C2×C3×C4×D4C2212yes—
18C2×C3×#107C2212yes—
19C2×C3×#106C2212yes—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#102C2212yes—
23C2×C3×#102C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×#102C2212yes—
26C3×#390—21no—
27C3×#390—21no—
28C3×#390C2212yes—
29C3×#389C2211yescharacteristic
30C3×#389C2211yescharacteristic
31C3×#387—21no—
32C3×#387—21no—
33C3×#384C2211yescharacteristic
34C3×#382C2211yescharacteristic
35C3×C4×C4⋊C4—21no—
36C3×C4×C4⋊C4—21no—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×C43C2211yescharacteristic
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#336C2212yes—
43C3×#336C2212yes—
44C4×#127C2×C311yescharacteristic
45C4×#125C2×C311yescharacteristic, complemented
46#2092C2×C312yescomplemented
47#2025C2×C312yescomplemented
48#1982C2×C311yescharacteristic
49#1982C2×C311yescharacteristic
50#1948C2×C311yescharacteristic, complemented
51#1947C2×C311yescharacteristic, complemented
52#1864C2×C312yescomplemented
53#1862C2×C312yescomplemented
54#1831C2×C311yescharacteristic, complemented
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4C2312yes—
58C23×C3×C4—22no—
59C22×C3×C8C2312yes—
60C22×C3×C8C2312yes—
61C3×#127—22no—
62C3×#127Q814yes—
63C3×#125—22no—
64C3×#125—22no—
65C3×C4×D4—42no—
66C3×C4×D4—42no—
67C3×C4×D4—44no—
68C3×C4×D4—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⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—21no—
84C2×C3×C42—21no—
85C2×C3×C42D411yescharacteristic
86C2×C3×C42D411yescharacteristic
87C2×C3×C42D411yescharacteristic
88C2×C3×C42D411yescharacteristic
89C2×C3×C42C2311yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×#107—42no—
93C3×#106—42no—
94C3×C4⋊C8—22no—
95C3×C4⋊C8—22no—
96C3×#102—24no—
97C3×#102—24no—
98C3×#102—42no—
99C3×#102—42no—
100C3×#95—21no—
101C3×#95—21no—
102C3×#95—21no—
103C3×#95—21no—
104C3×#95—21no—
105C3×#95—21no—
106C2×#127C22×C311yescharacteristic
107C2×#125C22×C311yescharacteristic
108C2×C4×D4—21no—
109C2×C4×D4—21no—
110C2×C4×D4C22×C312yes—
111C2×#107C22×C312yes—
112C2×#106C22×C312yes—
113C2×C4⋊C8C22×C311yescharacteristic
114C2×C4⋊C8C22×C311yescharacteristic
115C2×#102C22×C312yes—
116C2×#102C22×C312yes—
117C2×#102C22×C312yes—
118C2×#102C22×C312yes—
119#390—21no—
120#390—21no—
121#390C22×C312yes—
122#389C22×C311yescharacteristic
123#389C22×C311yescharacteristic
124#387—21no—
125#387—21no—
126#384C22×C311yescharacteristic
127#382C22×C311yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C22×C311yescharacteristic
131C4×C4⋊C4C22×C311yescharacteristic
132C43C22×C311yescharacteristic
133#340C22×C312yes—
134#340C22×C312yes—
135#336C22×C312yes—
136#336C22×C312yes—
137C24×C3—22no—
138C2×C3×D4—24no—
139C2×C3×D4—24no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—42no—
146C2×C3×D4—42no—
147C2×C3×D4—42no—
148C2×C3×D4—42no—
149C2×C3×D4C2×Q814yes—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—44no—
156C22×C3×C4—44no—
157C22×C3×C4—44no—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4#4112yes—
186C3×C4⋊C4#4112yes—
187C3×C22⋊C4—44no—
188C3×C22⋊C4—44no—
189C3×C22⋊C4—44no—
190C3×C42—22no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—21no—
194C3×C42—21no—
195C3×C42—21no—
196C3×C42—21no—
197C3×C42—41no—
198C3×C42—42no—
199C3×C42—42no—
200C3×C42C2×Q812yes—
201C3×C42#4112yes—
202C22×D4—21no—
203C22×D4—21no—
204C22×D4C23×C312yes—
205C23×C4—22no—
206C22×C8C23×C312yes—
207C22×C8C23×C312yes—
208#127—22no—
209#127C3×Q814yes—
210#125—22no—
211#125—22no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—44no—
215C4×D4—44no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C23×C312yes—
223C2×C4⋊C4C23×C311yescharacteristic
224C2×C4⋊C4C23×C311yescharacteristic
225C2×C4⋊C4C23×C311yescharacteristic
226C2×C4⋊C4C23×C311yescharacteristic
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42C3×D411yescharacteristic
233C2×C42C3×D411yescharacteristic
234C2×C42C3×D411yescharacteristic
235C2×C42C3×D411yescharacteristic
236C2×C42C23×C311yescharacteristic
237C2×C42C23×C311yescharacteristic
238C2×C42C23×C311yescharacteristic
239#107—42no—
240#106—42no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243#102—24no—
244#102—24no—
245#102—42no—
246#102—42no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253C23×C3—24no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3C22×D411yescharacteristic
261C3×D4—24no—
262C3×D4—24no—
263C3×D4—48no—
264C3×D4—42no—
265C3×D4—42no—
266C3×D4—48no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—42no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—88no—
296C2×C3×C4C22⋊Q812yes—
297C2×C3×C4C22⋊Q812yes—
298C2×C3×C4#13512yes—
299C2×C3×C4AGL(1,ℤ/8)12yes—
300C2×C3×C4AGL(1,ℤ/8)12yes—
301C2×C3×C4C22×Q811yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C4C2×#4111yescharacteristic
305C2×C3×C4H(5,𝔽2)11yescharacteristic
306C2×C3×C4H(5,𝔽2)11yescharacteristic
307C3×C8—42no—
308C3×C8—42no—
309C24—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4C2×C3×Q814yes—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4C2×C3×D411yescharacteristic
331C22×C4C2×C3×D411yescharacteristic
332C22×C4C2×C3×D411yescharacteristic
333C22×C4C2×C3×D411yescharacteristic
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C24×C311yescharacteristic
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4C3×#4112yes—
358C4⋊C4C3×#4112yes—
359C22⋊C4—44no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—41no—
370C42—42no—
371C42—42no—
372C42C2×C3×Q812yes—
373C42C3×#4112yes—
374C22×C3—44no—
375C22×C3—44no—
376C22×C3—44no—
377C22×C3—82no—
378C22×C3—82no—
379C22×C3—82no—
380C22×C3—82no—
381C22×C3C2×C22⋊Q811yescharacteristic
382C22×C3#52511yescharacteristic
383C22×C3#53511yescharacteristic
384C22×C3C2×#13511yescharacteristic
385C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
386C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
387C22×C3#57511yescharacteristic
388C3×C4—22no—
389C3×C4—22no—
390C3×C4—41no—
391C3×C4—42no—
392C3×C4—41no—
393C3×C4—42no—
394C3×C4—41no—
395C3×C4—81no—
396C3×C4—81no—
397C3×C4—82no—
398C3×C4#47814yes—
399C3×C4#55411yescharacteristic
400C3×C4#55411yescharacteristic
401C3×C4#55911yescharacteristic
402C3×C4#55911yescharacteristic
403C23—24no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23C22×C3×D411yescharacteristic
411D4—24no—
412D4—24no—
413D4—48no—
414D4—42no—
415D4—42no—
416D4—48no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—88no—
446C2×C4C3×C22⋊Q812yes—
447C2×C4C3×C22⋊Q812yes—
448C2×C4C3×#13512yes—
449C2×C4C3×AGL(1,ℤ/8)12yes—
450C2×C4C3×AGL(1,ℤ/8)12yes—
451C2×C4C22×C3×Q811yescharacteristic
452C2×C4C2×C3×#4111yescharacteristic
453C2×C4C2×C3×#4111yescharacteristic
454C2×C4C2×C3×#4111yescharacteristic
455C2×C4C3×H(5,𝔽2)11yescharacteristic
456C2×C4C3×H(5,𝔽2)11yescharacteristic
457C8—42no—
458C8—42no—
459C2×C3—84no—
460C2×C3#258011yescharacteristic
461C2×C3C2×#47811yescharacteristic
462C2×C3#307711yescharacteristic
463C2×C3#310611yescharacteristic
464C2×C3#311511yescharacteristic
465C2×C3#321211yescharacteristic
466C2×C3#321711yescharacteristic
467C22—44no—
468C22—44no—
469C22—44no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22C2×C3×C22⋊Q811yescharacteristic
475C22C3×#52511yescharacteristic
476C22C3×#53511yescharacteristic
477C22C2×C3×#13511yescharacteristic
478C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
479C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
480C22C3×#57511yescharacteristic
481C4—22no—
482C4—22no—
483C4—41no—
484C4—41no—
485C4—42no—
486C4—41no—
487C4—42no—
488C4—81no—
489C4—82no—
490C4—81no—
491C4C3×#47814yes—
492C4C3×#55411yescharacteristic
493C4C3×#55411yescharacteristic
494C4C3×#55911yescharacteristic
495C4C3×#55911yescharacteristic
496C3#2925611yescharacteristic, direct product factor, Sylow
497C2—84no—
498C2C3×#258011yescharacteristic
499C2C2×C3×#47811yescharacteristic
500C2C3×#307711yescharacteristic
501C2C3×#310611yescharacteristic
502C2C3×#311511yescharacteristic
503C2C3×#321211yescharacteristic
504C2C3×#321711yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#127C22
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
2. C3×C4×#125C2semidirect
3. C3×#2092C2semidirect
4. C3×#2025C2semidirect
7. C3×#1948C2semidirect
8. C3×#1947C2semidirect
9. C3×#1864C2semidirect
10. C3×#1862C2semidirect
11. C3×#1831C2semidirect
12. #29256C3direct (N × H)
45. C4×#125C2×C3semidirect
46. #2092C2×C3semidirect
47. #2025C2×C3semidirect
50. #1948C2×C3semidirect
51. #1947C2×C3semidirect
52. #1864C2×C3semidirect
53. #1862C2×C3semidirect
54. #1831C2×C3semidirect
496. C3#29256direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#1272
C3×C4×#1252
C3×#20922
C3×#20252
C3×#19822
C3×#19482
C3×#19472
C3×#18642
C3×#18622
C3×#18312
#292563