Finite Group Databasebeta

← C3×#29518 #10648125

Subgroups — C3×#29518 #10648125

Order 768.

722 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2860C211yesmaximal, characteristic
2C3×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3C3×#2099C211yesmaximal, characteristic, complemented
4C3×#2093C211yesmaximal, characteristic
5C3×#2058C211yesmaximal, characteristic, complemented
6C3×#2023C211yesmaximal, characteristic
7C3×#1944C212yesmaximal, complemented
8C3×#1936C212yesmaximal
9C3×#1926C212yesmaximal
10C3×#1921C212yesmaximal, complemented
11C3×#1745C211yesmaximal, characteristic, complemented
12#29518C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#126—21no—
15C2×C3×C4×Q8—21no—
16C2×C3×#107C2212yes—
17C2×C3×#106C2212yes—
18C2×C3×#102C2212yes—
19C2×C3×#102C2212yes—
20C2×C3×#97C2212yes—
21C3×#391—21no—
22C3×#390C2211yescharacteristic
23C3×#390C2211yescharacteristic
24C3×#390C2211yescharacteristic
25C3×#389—21no—
26C3×#389C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#389C2212yes—
29C3×#389C2211yescharacteristic
30C3×#388—21no—
31C3×#388—21no—
32C3×#388—21no—
33C3×#384C2211yescharacteristic
34C3×C4×C4⋊C4—21no—
35C3×C4×C4⋊C4C2212yes—
36C3×C43C2211yescharacteristic
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#2860C2×C311yescharacteristic
45C4×C4⋊Q8C2×C311yescharacteristic, complemented
46#2099C2×C311yescharacteristic, complemented
47#2093C2×C311yescharacteristic
48#2058C2×C311yescharacteristic, complemented
49#2023C2×C311yescharacteristic
50#1944C2×C312yescomplemented
51#1936C2×C312yes—
52#1926C2×C312yes—
53#1921C2×C312yescomplemented
54#1745C2×C311yescharacteristic, complemented
55C22×C3×Q8—21no—
56C22×C3×D4C2311yescharacteristic
57C22×C3×C8C2312yes—
58C22×C3×C8C2312yes—
59C3×C4⋊Q8—24no—
60C3×#126—44no—
61C3×C4×Q8—44no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C22⋊C4—21no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C42—22no—
81C2×C3×C42D411yescharacteristic
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C3×#107—24no—
89C3×#106—24no—
90C3×#102—24no—
91C3×#102—24no—
92C3×#97—24no—
93C3×#95—22no—
94C3×#95—22no—
95C3×#95—21no—
96C3×#95—21no—
97C3×#95—21no—
98C2×C4⋊Q8C22×C311yescharacteristic
99C2×#126—21no—
100C2×C4×Q8—21no—
101C2×#107C22×C312yes—
102C2×#106C22×C312yes—
103C2×#102C22×C312yes—
104C2×#102C22×C312yes—
105C2×#97C22×C312yes—
106#391—21no—
107#390C22×C311yescharacteristic
108#390C22×C311yescharacteristic
109#390C22×C311yescharacteristic
110#389—21no—
111#389C22×C312yes—
112#389C22×C311yescharacteristic
113#389C22×C311yescharacteristic
114#389C22×C311yescharacteristic
115#388—21no—
116#388—21no—
117#388—21no—
118#384C22×C311yescharacteristic
119C4×C4⋊C4—21no—
120C4×C4⋊C4C22×C312yes—
121C43C22×C311yescharacteristic
122#340C22×C312yes—
123#340C22×C312yes—
124#340C22×C312yes—
125#340C22×C312yes—
126#336C22×C311yescharacteristic
127#336C22×C311yescharacteristic
128#336C22×C312yes—
129C24×C3—21no—
130C2×C3×Q8—24no—
131C2×C3×Q8—42no—
132C2×C3×Q8—42no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8#4114yes—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—44no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—42no—
166C3×C4⋊C4—42no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4#4114yes—
170C3×C22⋊C4—42no—
171C3×C22⋊C4—42no—
172C3×C22⋊C4—42no—
173C3×C22⋊C4—42no—
174C3×C22⋊C4—42no—
175C3×C22⋊C4—42no—
176C3×C42—22no—
177C3×C42—22no—
178C3×C42—22no—
179C3×C42—22no—
180C3×C42—22no—
181C3×C42—21no—
182C3×C42—21no—
183C3×C42—41no—
184C3×C42—44no—
185C22×Q8—21no—
186C22×D4C23×C311yescharacteristic
187C22×C8C23×C312yes—
188C22×C8C23×C312yes—
189C4⋊Q8—24no—
190#126—44no—
191C4×Q8—44no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C23×C311yescharacteristic
200C2×C4⋊C4C23×C312yes—
201C2×C4⋊C4C23×C311yescharacteristic
202C2×C4⋊C4C23×C311yescharacteristic
203C2×C4⋊C4C23×C312yes—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C42—22no—
211C2×C42C3×D411yescharacteristic
212C2×C42C3×D411yescharacteristic
213C2×C42C3×D411yescharacteristic
214C2×C42C3×D411yescharacteristic
215C2×C42C23×C311yescharacteristic
216C2×C42C23×C311yescharacteristic
217C2×C42C23×C311yescharacteristic
218#107—24no—
219#106—24no—
220#102—24no—
221#102—24no—
222#97—24no—
223#95—22no—
224#95—22no—
225#95—21no—
226#95—21no—
227#95—21no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3C22×D411yescharacteristic
236C3×Q8—48no—
237C3×D4—42no—
238C3×D4—42no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—24no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
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—42no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4AGL(1,ℤ/8)12yes—
270C2×C3×C4AGL(1,ℤ/8)12yes—
271C2×C3×C4#13712yes—
272C2×C3×C4C2×#4112yes—
273C2×C3×C4C2×#4112yes—
274C2×C3×C42−1+411yescharacteristic
275C2×C3×C42−1+411yescharacteristic
276C3×C8—24no—
277C3×C8—42no—
278C24—21no—
279C2×Q8—24no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4C2×C3×D411yescharacteristic
292C22×C4C2×C3×D411yescharacteristic
293C22×C4C2×C3×D411yescharacteristic
294C22×C4C2×C3×D411yescharacteristic
295C22×C4C2×C3×D411yescharacteristic
296C22×C4C2×C3×D411yescharacteristic
297C22×C4C24×C311yescharacteristic
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8C3×#4114yes—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4C3×#4114yes—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—21no—
331C42—21no—
332C42—41no—
333C42—44no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3#52511yescharacteristic
342C22×C3#53211yescharacteristic
343C22×C3#53611yescharacteristic
344C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
345C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
346C22×C3C2×#13711yescharacteristic
347C22×C3#57511yescharacteristic
348C3×C4—22no—
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—44no—
352C3×C4—41no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4—81no—
356C3×C4—82no—
357C3×C4—81no—
358C3×C4—81no—
359C3×C4#55212yes—
360C3×C4#55612yes—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23C22×C3×D411yescharacteristic
369Q8—48no—
370D4—42no—
371D4—42no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4C3×AGL(1,ℤ/8)12yes—
403C2×C4C3×AGL(1,ℤ/8)12yes—
404C2×C4C3×#13712yes—
405C2×C4C2×C3×#4112yes—
406C2×C4C2×C3×#4112yes—
407C2×C4C3×2−1+411yescharacteristic
408C2×C4C3×2−1+411yescharacteristic
409C8—24no—
410C8—42no—
411C2×C3—161no—
412C2×C3#260411yescharacteristic
413C2×C3#311211yescharacteristic
414C2×C3#311411yescharacteristic
415C2×C3#317211yescharacteristic
416C2×C3#318411yescharacteristic
417C2×C3#323511yescharacteristic
418C2×C3#323911yescharacteristic
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22C3×#52511yescharacteristic
427C22C3×#53211yescharacteristic
428C22C3×#53611yescharacteristic
429C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
430C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
431C22C2×C3×#13711yescharacteristic
432C22C3×#57511yescharacteristic
433C4—22no—
434C4—22no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—44no—
440C4—82no—
441C4—81no—
442C4—81no—
443C4—81no—
444C4C3×#55212yes—
445C4C3×#55612yes—
446C3#2951811yescharacteristic, direct product factor, Sylow
447C2—161no—
448C2C3×#260411yescharacteristic
449C2C3×#311211yescharacteristic
450C2C3×#311411yescharacteristic
451C2C3×#317211yescharacteristic
452C2C3×#318411yescharacteristic
453C2C3×#323511yescharacteristic
454C2C3×#323911yescharacteristic
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
2. C3×C4×C4⋊Q8C2semidirect
3. C3×#2099C2semidirect
5. C3×#2058C2semidirect
7. C3×#1944C2semidirect
10. C3×#1921C2semidirect
11. C3×#1745C2semidirect
12. #29518C3direct (N × H)
45. C4×C4⋊Q8C2×C3semidirect
46. #2099C2×C3semidirect
48. #2058C2×C3semidirect
50. #1944C2×C3semidirect
53. #1921C2×C3semidirect
54. #1745C2×C3semidirect
446. C3#29518direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28602
C3×C4×C4⋊Q82
C3×#20992
C3×#20932
C3×#20582
C3×#20232
C3×#19442
C3×#19362
C3×#19262
C3×#19212
C3×#17452
#295183