Finite Group Databasebeta

← C3×#29640 #10648247

Subgroups — C3×#29640 #10648247

Order 768.

1076 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
1C22×C3×#125C211yesmaximal, characteristic, complemented
2C2×C3×#481C211yesmaximal, characteristic
3C2×C3×#480C211yesmaximal, characteristic
4C3×#2854C211yesmaximal, characteristic
5C3×#2059C214yesmaximal, complemented
6C3×#2025C214yesmaximal, complemented
7C3×#1824C211yesmaximal, characteristic, complemented
8C3×#1823C211yesmaximal, characteristic, complemented
9C3×#1817C211yesmaximal, characteristic, complemented
10#29640C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#125—24no—
12C2×C3×#125—24no—
13C2×C3×#125—22no—
14C2×C3×#125C2214yes—
15C2×C3×#123—21no—
16C2×C3×#121C2211yescharacteristic
17C22×C3×C4⋊C4—21no—
18C22×C3×C4⋊C4—21no—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C4⋊C4C2211yescharacteristic
21C22×C3×C42C2211yescharacteristic
22C3×#481—24no—
23C3×#480—24no—
24C2×C3×#107C2212yes—
25C2×C3×#106C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#390C2212yes—
31C3×#389C2212yes—
32C3×#388—22no—
33C3×#386—21no—
34C3×#380C2211yescharacteristic
35C3×#340C2214yes—
36C3×#340C2214yes—
37C3×#336C2214yes—
38C22×#125C2×C311yescharacteristic, complemented
39C2×#481C2×C311yescharacteristic
40C2×#480C2×C311yescharacteristic
41#2854C2×C311yescharacteristic
42#2059C2×C314yescomplemented
43#2025C2×C314yescomplemented
44#1824C2×C311yescharacteristic, complemented
45#1823C2×C311yescharacteristic, complemented
46#1817C2×C311yescharacteristic, complemented
47C22×C3×D4—21no—
48C22×C3×D4C2311yescharacteristic
49C23×C3×C4—21no—
50C23×C3×C4—21no—
51C23×C3×C4D411yescharacteristic
52C23×C3×C4D411yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C22×C3×C8C2312yes—
55C22×C3×C8C2312yes—
56C3×#125—216no—
57C3×#125—28no—
58C3×#125—28no—
59C3×#123—44no—
60C3×#121—22no—
61C3×#121—22no—
62C2×C3×C4⋊C4—28no—
63C2×C3×C4⋊C4—28no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—22no—
71C2×C3×C4⋊C4D414yes—
72C2×C3×C4⋊C4D414yes—
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2311yescharacteristic
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C22⋊C4—21no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C42—22no—
82C2×C3×C42—22no—
83C2×C3×C42D412yes—
84C2×C3×C42C2312yes—
85C3×#107—24no—
86C3×#106—24no—
87C3×#102—24no—
88C3×#102—24no—
89C3×C22⋊C8—22no—
90C3×C22⋊C8—22no—
91C3×#95—21no—
92C3×#95—22no—
93C2×#125—24no—
94C2×#125—24no—
95C2×#125—22no—
96C2×#125C22×C314yes—
97C2×#123—21no—
98C2×#121C22×C311yescharacteristic
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4C22×C311yescharacteristic
102C22×C4⋊C4C22×C311yescharacteristic
103C22×C42C22×C311yescharacteristic
104#481—24no—
105#480—24no—
106C2×#107C22×C312yes—
107C2×#106C22×C312yes—
108C2×#102C22×C312yes—
109C2×#102C22×C312yes—
110C2×C22⋊C8C22×C311yescharacteristic
111C2×C22⋊C8C22×C311yescharacteristic
112#390C22×C312yes—
113#389C22×C312yes—
114#388—22no—
115#386—21no—
116#380C22×C311yescharacteristic
117#340C22×C314yes—
118#340C22×C314yes—
119#336C22×C314yes—
120C24×C3—21no—
121C24×C3C2×D411yescharacteristic
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—24no—
125C2×C3×D4—42no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C22×C3×C4—24no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4—42no—
140C22×C3×C4—44no—
141C22×C3×C4—42no—
142C22×C3×C4—42no—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D412yes—
147C22×C3×C4C2×D412yes—
148C22×C3×C4C2411yescharacteristic
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8—24no—
152C2×C3×C8—24no—
153C3×C4⋊C4—216no—
154C3×C4⋊C4—216no—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—44no—
163C3×C4⋊C4C2×D414yes—
164C3×C4⋊C4C2×D414yes—
165C3×C22⋊C4—44no—
166C3×C22⋊C4—44no—
167C3×C22⋊C4—42no—
168C3×C22⋊C4—42no—
169C3×C42—24no—
170C3×C42—22no—
171C3×C42—22no—
172C22×D4—21no—
173C22×D4C23×C311yescharacteristic
174C23×C4—21no—
175C23×C4—21no—
176C23×C4C3×D411yescharacteristic
177C23×C4C3×D411yescharacteristic
178C23×C4C23×C311yescharacteristic
179C22×C8C23×C312yes—
180C22×C8C23×C312yes—
181#125—216no—
182#125—28no—
183#125—28no—
184#123—44no—
185#121—22no—
186#121—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C3×D414yes—
197C2×C4⋊C4C3×D414yes—
198C2×C4⋊C4C23×C311yescharacteristic
199C2×C4⋊C4C23×C312yes—
200C2×C4⋊C4C23×C312yes—
201C2×C4⋊C4C23×C312yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42C3×D412yes—
209C2×C42C23×C312yes—
210#107—24no—
211#106—24no—
212#102—24no—
213#102—24no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216#95—21no—
217#95—22no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—41no—
221C23×C3—42no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3#12012yes—
226C23×C3#13512yes—
227C23×C3AGL(1,ℤ/8)12yes—
228C23×C3C22×D411yescharacteristic
229C3×D4—42no—
230C3×D4—42no—
231C3×D4—84no—
232C2×C3×C4—22no—
233C2×C3×C4—24no—
234C2×C3×C4—24no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—24no—
238C2×C3×C4—24no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—44no—
244C2×C3×C4—48no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—44no—
248C2×C3×C4—44no—
249C2×C3×C4—48no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4#12012yes—
257C2×C3×C4C22×D411yescharacteristic
258C2×C3×C4C22×D411yescharacteristic
259C2×C3×C42−1+414yes—
260C3×C8—42no—
261C3×C8—42no—
262C24—21no—
263C24C2×C3×D411yescharacteristic
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—24no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4C2×C3×D412yes—
286C22×C4C2×C3×D412yes—
287C22×C4C2×C3×D411yescharacteristic
288C22×C4C2×C3×D412yes—
289C22×C4C2×C3×D412yes—
290C22×C4C24×C311yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—24no—
294C2×C8—24no—
295C4⋊C4—216no—
296C4⋊C4—216no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—24no—
304C4⋊C4—44no—
305C4⋊C4C2×C3×D414yes—
306C4⋊C4C2×C3×D414yes—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C42—24no—
312C42—22no—
313C42—22no—
314C22×C3—22no—
315C22×C3—24no—
316C22×C3—24no—
317C22×C3—22no—
318C22×C3—81no—
319C22×C3—82no—
320C22×C3—82no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3#44914yes—
324C22×C3C2×#12011yescharacteristic
325C22×C3#53612yes—
326C22×C3C2×#13511yescharacteristic
327C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
328C22×C3#57512yes—
329C3×C4—22no—
330C3×C4—42no—
331C3×C4—44no—
332C3×C4—42no—
333C3×C4—44no—
334C3×C4—82no—
335C3×C4—81no—
336C3×C4#53612yes—
337C3×C4#53612yes—
338C23—22no—
339C23—22no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—41no—
344C23—42no—
345C23C3×#12012yes—
346C23C3×#13512yes—
347C23C3×AGL(1,ℤ/8)12yes—
348C23C22×C3×D411yescharacteristic
349D4—42no—
350D4—42no—
351D4—84no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—44no—
364C2×C4—48no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—48no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4C3×#12012yes—
377C2×C4C22×C3×D411yescharacteristic
378C2×C4C22×C3×D411yescharacteristic
379C2×C4C3×2−1+414yes—
380C8—42no—
381C8—42no—
382C2×C3—24no—
383C2×C3—161no—
384C2×C3#261411yescharacteristic
385C2×C3C2×#44911yescharacteristic
386C2×C3#300511yescharacteristic
387C2×C3#323212yes—
388C2×C3#323412yes—
389C22—24no—
390C22—24no—
391C22—22no—
392C22—22no—
393C22—82no—
394C22—82no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22C3×#44914yes—
399C22C2×C3×#12011yescharacteristic
400C22C3×#53612yes—
401C22C2×C3×#13511yescharacteristic
402C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
403C22C3×#57512yes—
404C4—22no—
405C4—42no—
406C4—44no—
407C4—42no—
408C4—44no—
409C4—82no—
410C4—81no—
411C4C3×#53612yes—
412C4C3×#53612yes—
413C3#2964011yescharacteristic, direct product factor, Sylow
414C2—24no—
415C2—161no—
416C2C3×#261411yescharacteristic
417C2C2×C3×#44911yescharacteristic
418C2C3×#300511yescharacteristic
419C2C3×#323212yes—
420C2C3×#323412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×#125C2semidirect
5. C3×#2059C2semidirect
6. C3×#2025C2semidirect
7. C3×#1824C2semidirect
8. C3×#1823C2semidirect
9. C3×#1817C2semidirect
10. #29640C3direct (N × H)
38. C22×#125C2×C3semidirect
42. #2059C2×C3semidirect
43. #2025C2×C3semidirect
44. #1824C2×C3semidirect
45. #1823C2×C3semidirect
46. #1817C2×C3semidirect
413. C3#29640direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C2×C3×#4812
C2×C3×#4802
C3×#28542
C3×#20592
C3×#20252
C3×#18242
C3×#18232
C3×#18172
#296403