Finite Group Databasebeta

← C3×#29654 #10648261

Subgroups — C3×#29654 #10648261

Order 768.

948 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3458C211yesmaximal, characteristic, complemented
2C3×#3094C212yesmaximal
3C3×#2500C211yesmaximal, characteristic
4C3×#2061C212yesmaximal, complemented
5C3×#2060C212yesmaximal, complemented
6C3×#2027C212yesmaximal, complemented
7C3×#2027C212yesmaximal, complemented
8C3×#1826C212yesmaximal, complemented
9C3×#1822C211yesmaximal, characteristic, complemented
10#29654C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#544—22no—
12C3×#544—22no—
13C3×#544C2212yes—
14C3×#544C2212yes—
15C3×#543—22no—
16C3×#533—24no—
17C3×#533—22no—
18C2×C3×#121C2211yescharacteristic
19C2×C3×#117C2212yes—
20C2×C3×#117C2211yescharacteristic
21C3×#482—24no—
22C3×#481—22no—
23C3×#480—22no—
24C3×#428C2214yes—
25C3×#418C2214yes—
26C3×#407C2212yes—
27C3×#394—22no—
28C3×#390C2211yescharacteristic
29C3×#390C2211yescharacteristic
30C3×#388—22no—
31C3×#384C2211yescharacteristic
32C3×#382C2211yescharacteristic
33C3×C4×C22⋊C4C2211yescharacteristic
34C3×#343C2212yes—
35C3×#343C2212yes—
36C3×#339C2214yes—
37C3×#339C2214yes—
38#3458C2×C311yescharacteristic, complemented
39#3094C2×C312yes—
40#2500C2×C311yescharacteristic
41#2061C2×C312yescomplemented
42#2060C2×C312yescomplemented
43#2027C2×C312yescomplemented
44#2027C2×C312yescomplemented
45#1826C2×C312yescomplemented
46#1822C2×C311yescharacteristic, complemented
47C22×C3×D4—21no—
48C22×C3×D4C2311yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C2×C3×M16C2314yes—
51C3×C4⋊Q8—22no—
52C3×C4⋊Q8—22no—
53C3×C4⋊Q8—24no—
54C3×C4⋊Q8—24no—
55C3×#125—24no—
56C3×#125—24no—
57C3×#125—24no—
58C3×C22⋊Q8—28no—
59C3×C22⋊Q8—28no—
60C3×C22⋊Q8—24no—
61C3×C22⋊Q8—24no—
62C3×#121—22no—
63C3×#121—22no—
64C3×C4×Q8—24no—
65C3×C4×Q8—24no—
66C3×#117—22no—
67C3×#117—22no—
68C3×#117—22no—
69C3×#117—22no—
70C3×#117D414yes—
71C3×#117C2314yes—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4D412yes—
74C2×C3×C4⋊C4D412yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4D412yes—
84C2×C3×C42—22no—
85C2×C3×C42D412yes—
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C3×#107—24no—
89C3×#106—24no—
90C3×#102—24no—
91C3×#102—24no—
92C3×C22⋊C8—22no—
93C3×C22⋊C8—22no—
94C3×#95—21no—
95C3×#95—22no—
96#544—22no—
97#544—22no—
98#544C22×C312yes—
99#544C22×C312yes—
100#543—22no—
101#533—24no—
102#533—22no—
103C2×#121C22×C311yescharacteristic
104C2×#117C22×C312yes—
105C2×#117C22×C311yescharacteristic
106#482—24no—
107#481—22no—
108#480—22no—
109#428C22×C314yes—
110#418C22×C314yes—
111#407C22×C312yes—
112#394—22no—
113#390C22×C311yescharacteristic
114#390C22×C311yescharacteristic
115#388—22no—
116#384C22×C311yescharacteristic
117#382C22×C311yescharacteristic
118C4×C22⋊C4C22×C311yescharacteristic
119#343C22×C312yes—
120#343C22×C312yes—
121#339C22×C314yes—
122#339C22×C314yes—
123C24×C3—21no—
124C24×C3C2×D411yescharacteristic
125C2×C3×Q8—24no—
126C2×C3×Q8—24no—
127C2×C3×D4—22no—
128C2×C3×D4—22no—
129C2×C3×D4—22no—
130C2×C3×D4—42no—
131C2×C3×D4—82no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2411yescharacteristic
145C3×M16—24no—
146C3×M16—24no—
147C2×C3×C8—24no—
148C3×C4⋊C4—28no—
149C3×C4⋊C4—28no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—82no—
158C3×C4⋊C4C2×D414yes—
159C3×C4⋊C4C2×D414yes—
160C3×C22⋊C4—24no—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—24no—
163C3×C22⋊C4—44no—
164C3×C22⋊C4—42no—
165C3×C22⋊C4—41no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—82no—
168C3×C42—24no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42—24no—
172C3×C42—22no—
173C3×C42—22no—
174C22×D4—21no—
175C22×D4C23×C311yescharacteristic
176C23×C4C23×C311yescharacteristic
177C2×M16C23×C314yes—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—24no—
181C4⋊Q8—24no—
182#125—24no—
183#125—24no—
184#125—24no—
185C22⋊Q8—28no—
186C22⋊Q8—28no—
187C22⋊Q8—24no—
188C22⋊Q8—24no—
189#121—22no—
190#121—22no—
191C4×Q8—24no—
192C4×Q8—24no—
193#117—22no—
194#117—22no—
195#117—22no—
196#117—22no—
197#117C3×D414yes—
198#117C23×C314yes—
199C2×C4⋊C4—22no—
200C2×C4⋊C4C3×D412yes—
201C2×C4⋊C4C3×D412yes—
202C2×C4⋊C4C23×C311yescharacteristic
203C2×C4⋊C4C23×C311yescharacteristic
204C2×C4⋊C4C23×C311yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4C3×D412yes—
211C2×C42—22no—
212C2×C42C3×D412yes—
213C2×C42C23×C311yescharacteristic
214C2×C42C23×C311yescharacteristic
215#107—24no—
216#106—24no—
217#102—24no—
218#102—24no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221#95—21no—
222#95—22no—
223C23×C3—21no—
224C23×C3—21no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—42no—
228C23×C3—42no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3#12012yes—
233C23×C3C22×D411yescharacteristic
234C3×Q8—84no—
235C3×Q8—84no—
236C3×D4—44no—
237C3×D4—84no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—24no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—82no—
257C2×C3×C4—82no—
258C2×C3×C4#12011yescharacteristic
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4C22×D412yes—
261C2×C3×C42−1+412yes—
262C2×C3×C42−1+412yes—
263C3×C8—44no—
264C24—21no—
265C24C2×C3×D411yescharacteristic
266C2×Q8—24no—
267C2×Q8—24no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—42no—
272C2×D4—82no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4C2×C3×D411yescharacteristic
281C22×C4C2×C3×D412yes—
282C22×C4C2×C3×D412yes—
283C22×C4C2×C3×D412yes—
284C22×C4C2×C3×D412yes—
285C22×C4C24×C311yescharacteristic
286M16—24no—
287M16—24no—
288C2×C8—24no—
289C4⋊C4—28no—
290C4⋊C4—28no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—82no—
299C4⋊C4C2×C3×D414yes—
300C4⋊C4C2×C3×D414yes—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—44no—
305C22⋊C4—42no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—82no—
309C42—24no—
310C42—22no—
311C42—22no—
312C42—24no—
313C42—22no—
314C42—22no—
315C22×C3—21no—
316C22×C3—21no—
317C22×C3—22no—
318C22×C3—22no—
319C22×C3—22no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3—82no—
323C22×C3—82no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3C2×#12011yescharacteristic
328C22×C3#53611yescharacteristic
329C22×C3#53611yescharacteristic
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—42no—
333C3×C4—42no—
334C3×C4—44no—
335C3×C4—42no—
336C3×C4—44no—
337C3×C4—81no—
338C3×C4—81no—
339C3×C4#53614yes—
340C23—21no—
341C23—21no—
342C23—41no—
343C23—41no—
344C23—42no—
345C23—42no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23C3×#12012yes—
350C23C22×C3×D411yescharacteristic
351Q8—84no—
352Q8—84no—
353D4—44no—
354D4—84no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4C3×#12011yescharacteristic
376C2×C4C3×#12011yescharacteristic
377C2×C4C22×C3×D412yes—
378C2×C4C3×2−1+412yes—
379C2×C4C3×2−1+412yes—
380C8—44no—
381C2×C3—21no—
382C2×C3—21no—
383C2×C3—42no—
384C2×C3—161no—
385C2×C3#261411yescharacteristic
386C2×C3#301811yescharacteristic
387C2×C3#302111yescharacteristic
388C22—21no—
389C22—21no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—81no—
394C22—81no—
395C22—82no—
396C22—82no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22C2×C3×#12011yescharacteristic
401C22C3×#53611yescharacteristic
402C22C3×#53611yescharacteristic
403C4—21no—
404C4—21no—
405C4—42no—
406C4—42no—
407C4—44no—
408C4—42no—
409C4—44no—
410C4—81no—
411C4—81no—
412C4C3×#53614yes—
413C3#2965411yescharacteristic, direct product factor, Sylow
414C2—21no—
415C2—21no—
416C2—42no—
417C2—161no—
418C2C3×#261411yescharacteristic
419C2C3×#301811yescharacteristic
420C2C3×#302111yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3458C2semidirect
4. C3×#2061C2semidirect
5. C3×#2060C2semidirect
6. C3×#2027C2semidirect
7. C3×#2027C2semidirect
8. C3×#1826C2semidirect
9. C3×#1822C2semidirect
10. #29654C3direct (N × H)
38. #3458C2×C3semidirect
41. #2061C2×C3semidirect
42. #2060C2×C3semidirect
43. #2027C2×C3semidirect
44. #2027C2×C3semidirect
45. #1826C2×C3semidirect
46. #1822C2×C3semidirect
413. C3#29654direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34582
C3×#30942
C3×#25002
C3×#20612
C3×#20602
C3×#20272
C3×#18262
C3×#18222
#296543