Finite Group Databasebeta

← C3×#29617 #10648224

Subgroups — C3×#29617 #10648224

Order 768.

1358 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#537C211yesmaximal, characteristic, complemented
2C3×#3006C212yesmaximal, complemented
3C3×#2504C211yesmaximal, characteristic, complemented
4C3×#2040C214yesmaximal, complemented
5C3×#2006C214yesmaximal, complemented
6C3×#1821C211yesmaximal, characteristic, complemented
7C3×#1789C212yesmaximal, complemented
8#29617C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×#41C2212yes—
10C2×C3×AGL(1,ℤ/8)C2214yescomplemented
11C3×#537—24no—
12C3×#537—22no—
13C3×#537C2214yes—
14C2×C3×C22⋊Q8—22no—
15C2×C3×#121—22no—
16C2×C3×#121—22no—
17C2×C3×#121C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C3×#449—24no—
20C3×#448—22no—
21C3×#447—22no—
22C2×C3×#104C2218yescomplemented
23C3×#417C2214yescomplemented
24C3×#407C2212yes—
25C3×#397—22no—
26C3×#390C2212yescomplemented
27C3×#386—22no—
28C3×#382C2212yescomplemented
29C3×C4×C22⋊C4C2211yescharacteristic
30C3×#343C2214yescomplemented
31C2×#537C2×C311yescharacteristic, complemented
32#3006C2×C312yescomplemented
33#2504C2×C311yescharacteristic, complemented
34#2040C2×C314yescomplemented
35#2006C2×C314yescomplemented
36#1821C2×C311yescharacteristic, complemented
37#1789C2×C312yescomplemented
38C2×C3×#41—22no—
39C2×C3×#41—22no—
40C2×C3×#41—48no—
41C2×C3×#41D414yes—
42C2×C3×#41C2314yes—
43C22×C3×Q8D412yes—
44C22×C3×D4—22no—
45C22×C3×D4—22no—
46C22×C3×D4—21no—
47C22×C3×D4D412yes—
48C22×C3×D4C2311yescharacteristic
49C23×C3×C4—22no—
50C23×C3×C4D412yes—
51C23×C3×C4C2311yescharacteristic
52C3×AGL(1,ℤ/8)—48no—
53C2×C3×SD16—24no—
54C2×C3×D8—24no—
55C2×C3×M16C2314yes—
56C3×C22⋊Q8—28no—
57C3×C22⋊Q8—28no—
58C3×#121—28no—
59C3×#121—28no—
60C3×#121—28no—
61C3×#121—28no—
62C3×#121—22no—
63C3×#121—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4D412yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C42C2312yes—
75C3×#104—48no—
76C3×#103—24no—
77C3×#102—24no—
78C3×C22⋊C8—22no—
79C3×C22⋊C8—22no—
80C3×#95—21no—
81C3×#95—24no—
82C22×#41C22×C312yes—
83C2×AGL(1,ℤ/8)C22×C314yescomplemented
84#537—24no—
85#537—22no—
86#537C22×C314yes—
87C2×C22⋊Q8—22no—
88C2×#121—22no—
89C2×#121—22no—
90C2×#121C22×C311yescharacteristic
91C22×C4⋊C4C22×C311yescharacteristic
92#449—24no—
93#448—22no—
94#447—22no—
95C2×#104C22×C318yescomplemented
96#417C22×C314yescomplemented
97#407C22×C312yes—
98#397—22no—
99#390C22×C312yescomplemented
100#386—22no—
101#382C22×C312yescomplemented
102C4×C22⋊C4C22×C311yescharacteristic
103#343C22×C314yescomplemented
104C24×C3—22no—
105C24×C3—21no—
106C24×C3C2×D411yescharacteristic
107C3×#41—28no—
108C3×#41—28no—
109C3×#41—44no—
110C3×#41—44no—
111C3×#41—48no—
112C3×#41—48no—
113C2×C3×Q8—22no—
114C2×C3×Q8—44no—
115C2×C3×Q8C2×D414yes—
116C2×C3×D4—24no—
117C2×C3×D4—24no—
118C2×C3×D4—28no—
119C2×C3×D4—22no—
120C2×C3×D4—22no—
121C2×C3×D4—24no—
122C2×C3×D4—44no—
123C2×C3×D4—48no—
124C2×C3×D4—44no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
127C2×C3×D4—84no—
128C2×C3×D4C2×D414yes—
129C22×C3×C4—24no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—24no—
134C22×C3×C4—24no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—48no—
138C22×C3×C4—82no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D414yes—
143C22×C3×C4C2411yescharacteristic
144C3×SD16—48no—
145C3×D8—48no—
146C3×M16—28no—
147C2×C3×C8—24no—
148C3×C4⋊C4—28no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—22no—
152C3×C22⋊C4—216no—
153C3×C22⋊C4—42no—
154C3×C22⋊C4—42no—
155C3×C22⋊C4—84no—
156C3×C42—24no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—48no—
160C2×#41C3×D414yes—
161C2×#41C23×C314yes—
162C22×Q8C3×D412yes—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4C3×D412yes—
167C22×D4C23×C311yescharacteristic
168C23×C4—22no—
169C23×C4C3×D412yes—
170C23×C4C23×C311yescharacteristic
171AGL(1,ℤ/8)—48no—
172C2×SD16—24no—
173C2×D8—24no—
174C2×M16C23×C314yes—
175C22⋊Q8—28no—
176C22⋊Q8—28no—
177#121—28no—
178#121—28no—
179#121—28no—
180#121—28no—
181#121—22no—
182#121—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C3×D412yes—
187C2×C4⋊C4C23×C311yescharacteristic
188C2×C4⋊C4C23×C312yes—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—24no—
193C2×C42C23×C312yes—
194#104—48no—
195#103—24no—
196#102—24no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199#95—21no—
200#95—24no—
201C23×C3—22no—
202C23×C3—22no—
203C23×C3—24no—
204C23×C3—41no—
205C23×C3—44no—
206C23×C3—44no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3—44no—
210C23×C3—41no—
211C23×C3—42no—
212C23×C3#12011yescharacteristic
213C23×C3#12011yescharacteristic
214C23×C3C22×D411yescharacteristic
215C3×Q8—24no—
216C3×Q8—24no—
217C3×Q8—44no—
218C3×D4—24no—
219C3×D4—24no—
220C3×D4—44no—
221C3×D4—48no—
222C3×D4—48no—
223C3×D4—44no—
224C3×D4—88no—
225C3×D4—82no—
226C3×D4—82no—
227C2×C3×C4—22no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—48no—
233C2×C3×C4—44no—
234C2×C3×C4—44no—
235C2×C3×C4—48no—
236C2×C3×C4—44no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—41no—
241C2×C3×C4—42no—
242C2×C3×C4—82no—
243C2×C3×C4#12012yes—
244C2×C3×C4C22×D412yes—
245C2×C3×C4H(5,𝔽2)14yes—
246C3×C8—44no—
247C24—22no—
248C24—21no—
249C24C2×C3×D411yescharacteristic
250#41—28no—
251#41—28no—
252#41—44no—
253#41—44no—
254#41—48no—
255#41—48no—
256C2×Q8—22no—
257C2×Q8—44no—
258C2×Q8C2×C3×D414yes—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—28no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—44no—
266C2×D4—48no—
267C2×D4—44no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—84no—
271C2×D4C2×C3×D414yes—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—48no—
281C22×C4—82no—
282C22×C4C2×C3×D411yescharacteristic
283C22×C4C2×C3×D412yes—
284C22×C4C2×C3×D412yes—
285C22×C4C2×C3×D414yes—
286C22×C4C24×C311yescharacteristic
287SD16—48no—
288D8—48no—
289M16—28no—
290C2×C8—24no—
291C4⋊C4—28no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C22⋊C4—216no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—84no—
299C42—24no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—44no—
308C22×C3—44no—
309C22×C3—44no—
310C22×C3—44no—
311C22×C3—88no—
312C22×C3—84no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3#45412yes—
317C22×C3C2×#12011yescharacteristic
318C22×C3#53512yes—
319C3×C4—22no—
320C3×C4—44no—
321C3×C4—44no—
322C3×C4—82no—
323C3×C4—81no—
324C3×C4#53414yes—
325C23—22no—
326C23—22no—
327C23—24no—
328C23—41no—
329C23—44no—
330C23—44no—
331C23—42no—
332C23—42no—
333C23—44no—
334C23—41no—
335C23—42no—
336C23C3×#12011yescharacteristic
337C23C3×#12011yescharacteristic
338C23C22×C3×D411yescharacteristic
339Q8—24no—
340Q8—24no—
341Q8—44no—
342D4—24no—
343D4—24no—
344D4—44no—
345D4—48no—
346D4—48no—
347D4—44no—
348D4—88no—
349D4—82no—
350D4—82no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—48no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—48no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4C3×#12012yes—
368C2×C4C22×C3×D412yes—
369C2×C4C3×H(5,𝔽2)14yes—
370C8—44no—
371C2×C3—22no—
372C2×C3—22no—
373C2×C3—41no—
374C2×C3—84no—
375C2×C3—161no—
376C2×C3#261311yescharacteristic
377C2×C3C2×#45411yescharacteristic
378C2×C3#301911yescharacteristic
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—44no—
387C22—44no—
388C22—44no—
389C22—44no—
390C22—88no—
391C22—84no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22C3×#45412yes—
396C22C2×C3×#12011yescharacteristic
397C22C3×#53512yes—
398C4—22no—
399C4—44no—
400C4—44no—
401C4—82no—
402C4—81no—
403C4C3×#53414yes—
404C3#2961711yescharacteristic, direct product factor, Sylow
405C2—22no—
406C2—22no—
407C2—41no—
408C2—84no—
409C2—161no—
410C2C3×#261311yescharacteristic
411C2C2×C3×#45411yescharacteristic
412C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
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. C2×C3×#537C2semidirect
2. C3×#3006C2semidirect
3. C3×#2504C2semidirect
4. C3×#2040C2semidirect
5. C3×#2006C2semidirect
6. C3×#1821C2semidirect
7. C3×#1789C2semidirect
8. #29617C3direct (N × H)
10. C2×C3×AGL(1,ℤ/8)C22semidirect
22. C2×C3×#104C22semidirect
23. C3×#417C22semidirect
26. C3×#390C22semidirect
28. C3×#382C22semidirect
30. C3×#343C22semidirect
31. C2×#537C2×C3semidirect
32. #3006C2×C3semidirect
33. #2504C2×C3semidirect
34. #2040C2×C3semidirect
35. #2006C2×C3semidirect
36. #1821C2×C3semidirect
37. #1789C2×C3semidirect
83. C2×AGL(1,ℤ/8)C22×C3semidirect
95. C2×#104C22×C3semidirect
96. #417C22×C3semidirect
99. #390C22×C3semidirect
101. #382C22×C3semidirect
103. #343C22×C3semidirect
404. C3#29617direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5372
C3×#30062
C3×#25042
C3×#20402
C3×#20062
C3×#18212
C3×#17892
#296173