Finite Group Databasebeta

← C3×#29578 #10648185

Subgroups — C3×#29578 #10648185

Order 768.

732 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2859C211yesmaximal, characteristic, complemented
2C3×#2293C211yesmaximal, characteristic
3C3×#2099C211yesmaximal, characteristic, complemented
4C3×#2093C211yesmaximal, characteristic
5C3×#2084C211yesmaximal, characteristic, complemented
6C3×#2071C211yesmaximal, characteristic
7C3×#1944C212yesmaximal, complemented
8C3×#1936C212yesmaximal
9C3×#1921C212yesmaximal, complemented
10C3×#1864C212yesmaximal
11C3×#1745C211yesmaximal, characteristic, complemented
12#29578C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#125—21no—
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×#117—21no—
16C2×C3×#107C2212yes—
17C2×C3×#106C2212yes—
18C2×C3×#102C2212yes—
19C2×C3×#102C2212yes—
20C2×C3×#97C2212yes—
21C3×#390C2211yescharacteristic
22C3×#390C2211yescharacteristic
23C3×#389—21no—
24C3×#389—21no—
25C3×#389C2211yescharacteristic
26C3×#389C2212yes—
27C3×#389C2211yescharacteristic
28C3×#389C2212yes—
29C3×#388—21no—
30C3×#388—21no—
31C3×#384C2211yescharacteristic
32C3×#384C2211yescharacteristic
33C3×#382—21no—
34C3×C4×C4⋊C4C2211yescharacteristic
35C3×C4×C22⋊C4—21no—
36C3×C43C2211yescharacteristic
37C3×#340C2212yes—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#2859C2×C311yescharacteristic, complemented
45#2293C2×C311yescharacteristic
46#2099C2×C311yescharacteristic, complemented
47#2093C2×C311yescharacteristic
48#2084C2×C311yescharacteristic, complemented
49#2071C2×C311yescharacteristic
50#1944C2×C312yescomplemented
51#1936C2×C312yes—
52#1921C2×C312yescomplemented
53#1864C2×C312yes—
54#1745C2×C311yescharacteristic, complemented
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C22×C3×C8C2312yes—
58C22×C3×C8C2312yes—
59C3×#125—44no—
60C3×C4×D4—22no—
61C3×C4×D4—22no—
62C3×#117—44no—
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—21no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4C2311yescharacteristic
72C2×C3×C4⋊C4C2311yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C42—21no—
82C2×C3×C42D411yescharacteristic
83C2×C3×C42D411yescharacteristic
84C2×C3×C42D411yescharacteristic
85C2×C3×C42D411yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C3×#107—24no—
90C3×#106—24no—
91C3×#102—24no—
92C3×#102—24no—
93C3×#97—24no—
94C3×#95—21no—
95C3×#95—21no—
96C3×#95—21no—
97C3×#95—22no—
98C3×#95—22no—
99C2×#125—21no—
100C2×C4×D4C22×C311yescharacteristic
101C2×#117—21no—
102C2×#107C22×C312yes—
103C2×#106C22×C312yes—
104C2×#102C22×C312yes—
105C2×#102C22×C312yes—
106C2×#97C22×C312yes—
107#390C22×C311yescharacteristic
108#390C22×C311yescharacteristic
109#389—21no—
110#389—21no—
111#389C22×C311yescharacteristic
112#389C22×C312yes—
113#389C22×C312yes—
114#389C22×C311yescharacteristic
115#388—21no—
116#388—21no—
117#384C22×C311yescharacteristic
118#384C22×C311yescharacteristic
119#382—21no—
120C4×C4⋊C4C22×C311yescharacteristic
121C4×C22⋊C4—21no—
122C43C22×C311yescharacteristic
123#340C22×C312yes—
124#340C22×C312yes—
125#340C22×C312yes—
126#340C22×C312yes—
127#336C22×C311yescharacteristic
128#336C22×C311yescharacteristic
129#336C22×C312yes—
130C24×C3—21no—
131C2×C3×D4—22no—
132C2×C3×D4—21no—
133C2×C3×D4—22no—
134C2×C3×D4#4112yes—
135C22×C3×C4—24no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2411yescharacteristic
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C2×C3×C8#4114yes—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—42no—
164C3×C4⋊C4—42no—
165C3×C4⋊C4—44no—
166C3×C4⋊C4—44no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—42no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4#4112yes—
172C3×C22⋊C4—24no—
173C3×C22⋊C4—42no—
174C3×C22⋊C4—42no—
175C3×C22⋊C4—42no—
176C3×C22⋊C4—42no—
177C3×C42—22no—
178C3×C42—22no—
179C3×C42—22no—
180C3×C42—22no—
181C3×C42—21no—
182C3×C42—21no—
183C3×C42—22no—
184C3×C42—41no—
185C3×C42—42no—
186C22×D4C23×C311yescharacteristic
187C23×C4—21no—
188C22×C8C23×C312yes—
189C22×C8C23×C312yes—
190#125—44no—
191C4×D4—22no—
192C4×D4—22no—
193#117—44no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C23×C311yescharacteristic
203C2×C4⋊C4C23×C311yescharacteristic
204C2×C4⋊C4C23×C312yes—
205C2×C4⋊C4C23×C312yes—
206C2×C4⋊C4C23×C311yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C42—21no—
213C2×C42C3×D411yescharacteristic
214C2×C42C3×D411yescharacteristic
215C2×C42C3×D411yescharacteristic
216C2×C42C3×D411yescharacteristic
217C2×C42C23×C311yescharacteristic
218C2×C42C23×C311yescharacteristic
219C2×C42C23×C311yescharacteristic
220#107—24no—
221#106—24no—
222#102—24no—
223#102—24no—
224#97—24no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
230C23×C3—22no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3—41no—
235C23×C3—41no—
236C23×C3—41no—
237C23×C3C22×D411yescharacteristic
238C3×D4—24no—
239C3×D4—42no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—22no—
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—21no—
256C2×C3×C4—48no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4AGL(1,ℤ/8)12yes—
272C2×C3×C4AGL(1,ℤ/8)12yes—
273C2×C3×C4#13712yes—
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4C2×#4112yes—
277C2×C3×C42−1+412yes—
278C3×C8—24no—
279C3×C8—42no—
280C24—21no—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—22no—
284C2×D4C3×#4112yes—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4C2×C3×D411yescharacteristic
294C22×C4C2×C3×D411yescharacteristic
295C22×C4C2×C3×D411yescharacteristic
296C22×C4C2×C3×D411yescharacteristic
297C22×C4C2×C3×D411yescharacteristic
298C22×C4C2×C3×D411yescharacteristic
299C22×C4C24×C311yescharacteristic
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8C3×#4114yes—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4C3×#4112yes—
322C22⋊C4—24no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—21no—
332C42—21no—
333C42—22no—
334C42—41no—
335C42—42no—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3#52511yescharacteristic
344C22×C3#53211yescharacteristic
345C22×C3#53611yescharacteristic
346C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
347C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
348C22×C3C2×#13711yescharacteristic
349C22×C3#57511yescharacteristic
350C3×C4—22no—
351C3×C4—22no—
352C3×C4—22no—
353C3×C4—42no—
354C3×C4—41no—
355C3×C4—41no—
356C3×C4—41no—
357C3×C4—82no—
358C3×C4—81no—
359C3×C4—82no—
360C3×C4—81no—
361C3×C4#55212yes—
362C3×C4#55212yes—
363C23—22no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23C22×C3×D411yescharacteristic
371D4—24no—
372D4—42no—
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—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—48no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4C3×AGL(1,ℤ/8)12yes—
405C2×C4C3×AGL(1,ℤ/8)12yes—
406C2×C4C3×#13712yes—
407C2×C4C2×C3×#4111yescharacteristic
408C2×C4C2×C3×#4111yescharacteristic
409C2×C4C2×C3×#4112yes—
410C2×C4C3×2−1+412yes—
411C8—24no—
412C8—42no—
413C2×C3—82no—
414C2×C3#261111yescharacteristic
415C2×C3#311011yescharacteristic
416C2×C3#311511yescharacteristic
417C2×C3#317211yescharacteristic
418C2×C3#318411yescharacteristic
419C2×C3#323411yescharacteristic
420C2×C3#323811yescharacteristic
421C22—42no—
422C22—42no—
423C22—42no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22C3×#52511yescharacteristic
429C22C3×#53211yescharacteristic
430C22C3×#53611yescharacteristic
431C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
432C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
433C22C2×C3×#13711yescharacteristic
434C22C3×#57511yescharacteristic
435C4—22no—
436C4—22no—
437C4—22no—
438C4—41no—
439C4—41no—
440C4—42no—
441C4—41no—
442C4—81no—
443C4—82no—
444C4—81no—
445C4—82no—
446C4C3×#55212yes—
447C4C3×#55212yes—
448C3#2957811yescharacteristic, direct product factor, Sylow
449C2—82no—
450C2C3×#261111yescharacteristic
451C2C3×#311011yescharacteristic
452C2C3×#311511yescharacteristic
453C2C3×#317211yescharacteristic
454C2C3×#318411yescharacteristic
455C2C3×#323411yescharacteristic
456C2C3×#323811yescharacteristic
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×#2859C2semidirect
3. C3×#2099C2semidirect
5. C3×#2084C2semidirect
7. C3×#1944C2semidirect
9. C3×#1921C2semidirect
11. C3×#1745C2semidirect
12. #29578C3direct (N × H)
44. #2859C2×C3semidirect
46. #2099C2×C3semidirect
48. #2084C2×C3semidirect
50. #1944C2×C3semidirect
52. #1921C2×C3semidirect
54. #1745C2×C3semidirect
448. C3#29578direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28592
C3×#22932
C3×#20992
C3×#20932
C3×#20842
C3×#20712
C3×#19442
C3×#19362
C3×#19212
C3×#18642
C3×#17452
#295783