Finite Group Databasebeta

← C3×#29616 #10648223

Subgroups — C3×#29616 #10648223

Order 768.

1414 conjugacy classes of subgroups.

The subgroup lattice has 418 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×#2501C211yesmaximal, characteristic, complemented
4C3×#2039C214yesmaximal, complemented
5C3×#2006C214yesmaximal, complemented
6C3×#1821C211yesmaximal, characteristic, complemented
7C3×#1789C212yesmaximal, complemented
8#29616C311yesmaximal, 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×#123—22no—
15C2×C3×C22⋊Q8—22no—
16C2×C3×#121—22no—
17C2×C3×#121—22no—
18C2×C3×#121C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C3×#449—24no—
21C3×#448—22no—
22C3×#447—22no—
23C2×C3×#104C2218yescomplemented
24C3×#417C2214yescomplemented
25C3×#407C2212yes—
26C3×#399—22no—
27C3×#390C2212yescomplemented
28C3×#384C2212yescomplemented
29C3×C4×C22⋊C4C2211yescharacteristic
30C3×#343C2214yescomplemented
31C2×#537C2×C311yescharacteristic, complemented
32#3006C2×C312yescomplemented
33#2501C2×C311yescharacteristic, complemented
34#2039C2×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×#123—48no—
57C3×C22⋊Q8—28no—
58C3×C22⋊Q8—28no—
59C3×#121—28no—
60C3×#121—28no—
61C3×#121—28no—
62C3×#121—28no—
63C3×#121—22no—
64C3×#121—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—24no—
69C2×C3×C4⋊C4D412yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—21no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C42C2312yes—
77C3×#104—48no—
78C3×#103—24no—
79C3×#102—24no—
80C3×C22⋊C8—22no—
81C3×C22⋊C8—22no—
82C3×#95—21no—
83C22×#41C22×C312yes—
84C2×AGL(1,ℤ/8)C22×C314yescomplemented
85#537—24no—
86#537—22no—
87#537C22×C314yes—
88C2×#123—22no—
89C2×C22⋊Q8—22no—
90C2×#121—22no—
91C2×#121—22no—
92C2×#121C22×C311yescharacteristic
93C22×C4⋊C4C22×C311yescharacteristic
94#449—24no—
95#448—22no—
96#447—22no—
97C2×#104C22×C318yescomplemented
98#417C22×C314yescomplemented
99#407C22×C312yes—
100#399—22no—
101#390C22×C312yescomplemented
102#384C22×C312yescomplemented
103C4×C22⋊C4C22×C311yescharacteristic
104#343C22×C314yescomplemented
105C24×C3—22no—
106C24×C3—21no—
107C24×C3C2×D411yescharacteristic
108C3×#41—28no—
109C3×#41—28no—
110C3×#41—44no—
111C3×#41—44no—
112C3×#41—48no—
113C3×#41—48no—
114C2×C3×Q8—22no—
115C2×C3×Q8—44no—
116C2×C3×Q8C2×D414yes—
117C2×C3×D4—24no—
118C2×C3×D4—24no—
119C2×C3×D4—28no—
120C2×C3×D4—22no—
121C2×C3×D4—22no—
122C2×C3×D4—24no—
123C2×C3×D4—44no—
124C2×C3×D4—48no—
125C2×C3×D4—44no—
126C2×C3×D4—41no—
127C2×C3×D4—41no—
128C2×C3×D4—84no—
129C2×C3×D4C2×D414yes—
130C22×C3×C4—24no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—24no—
136C22×C3×C4—24no—
137C22×C3×C4—21no—
138C22×C3×C4—48no—
139C22×C3×C4—44no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D414yes—
144C22×C3×C4C2411yescharacteristic
145C3×SD16—48no—
146C3×D8—48no—
147C3×M16—28no—
148C2×C3×C8—24no—
149C3×C4⋊C4—28no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—48no—
154C3×C22⋊C4—216no—
155C3×C22⋊C4—24no—
156C3×C22⋊C4—48no—
157C3×C22⋊C4—42no—
158C3×C42—24no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—48no—
162C2×#41C3×D414yes—
163C2×#41C23×C314yes—
164C22×Q8C3×D412yes—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—21no—
168C22×D4C3×D412yes—
169C22×D4C23×C311yescharacteristic
170C23×C4—22no—
171C23×C4C3×D412yes—
172C23×C4C23×C311yescharacteristic
173AGL(1,ℤ/8)—48no—
174C2×SD16—24no—
175C2×D8—24no—
176C2×M16C23×C314yes—
177#123—48no—
178C22⋊Q8—28no—
179C22⋊Q8—28no—
180#121—28no—
181#121—28no—
182#121—28no—
183#121—28no—
184#121—22no—
185#121—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4C3×D412yes—
191C2×C4⋊C4C23×C311yescharacteristic
192C2×C4⋊C4C23×C312yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—24no—
197C2×C42C23×C312yes—
198#104—48no—
199#103—24no—
200#102—24no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203#95—21no—
204C23×C3—22no—
205C23×C3—22no—
206C23×C3—24no—
207C23×C3—41no—
208C23×C3—44no—
209C23×C3—44no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3—44no—
213C23×C3—41no—
214C23×C3—42no—
215C23×C3#12011yescharacteristic
216C23×C3#12011yescharacteristic
217C23×C3C22×D411yescharacteristic
218C3×Q8—24no—
219C3×Q8—24no—
220C3×Q8—44no—
221C3×D4—24no—
222C3×D4—24no—
223C3×D4—44no—
224C3×D4—48no—
225C3×D4—48no—
226C3×D4—44no—
227C3×D4—88no—
228C3×D4—82no—
229C3×D4—82no—
230C2×C3×C4—22no—
231C2×C3×C4—24no—
232C2×C3×C4—24no—
233C2×C3×C4—24no—
234C2×C3×C4—24no—
235C2×C3×C4—24no—
236C2×C3×C4—48no—
237C2×C3×C4—42no—
238C2×C3×C4—44no—
239C2×C3×C4—42no—
240C2×C3×C4—44no—
241C2×C3×C4—48no—
242C2×C3×C4—44no—
243C2×C3×C4—42no—
244C2×C3×C4—41no—
245C2×C3×C4—82no—
246C2×C3×C4#12012yes—
247C2×C3×C4C22×D412yes—
248C2×C3×C4H(5,𝔽2)14yes—
249C3×C8—44no—
250C24—22no—
251C24—21no—
252C24C2×C3×D411yescharacteristic
253#41—28no—
254#41—28no—
255#41—44no—
256#41—44no—
257#41—48no—
258#41—48no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×Q8C2×C3×D414yes—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—28no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—24no—
268C2×D4—44no—
269C2×D4—48no—
270C2×D4—44no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—84no—
274C2×D4C2×C3×D414yes—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—48no—
284C22×C4—44no—
285C22×C4C2×C3×D411yescharacteristic
286C22×C4C2×C3×D412yes—
287C22×C4C2×C3×D412yes—
288C22×C4C2×C3×D414yes—
289C22×C4C24×C311yescharacteristic
290SD16—48no—
291D8—48no—
292M16—28no—
293C2×C8—24no—
294C4⋊C4—28no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—48no—
299C22⋊C4—216no—
300C22⋊C4—24no—
301C22⋊C4—48no—
302C22⋊C4—42no—
303C42—24no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—44no—
312C22×C3—44no—
313C22×C3—44no—
314C22×C3—44no—
315C22×C3—88no—
316C22×C3—84no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3#45312yes—
321C22×C3C2×#12011yescharacteristic
322C22×C3#53512yes—
323C3×C4—22no—
324C3×C4—44no—
325C3×C4—44no—
326C3×C4—44no—
327C3×C4—81no—
328C3×C4#53414yes—
329C23—22no—
330C23—22no—
331C23—24no—
332C23—41no—
333C23—44no—
334C23—44no—
335C23—42no—
336C23—42no—
337C23—44no—
338C23—41no—
339C23—42no—
340C23C3×#12011yescharacteristic
341C23C3×#12011yescharacteristic
342C23C22×C3×D411yescharacteristic
343Q8—24no—
344Q8—24no—
345Q8—44no—
346D4—24no—
347D4—24no—
348D4—44no—
349D4—48no—
350D4—48no—
351D4—44no—
352D4—88no—
353D4—82no—
354D4—82no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—48no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—48no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—82no—
371C2×C4C3×#12012yes—
372C2×C4C22×C3×D412yes—
373C2×C4C3×H(5,𝔽2)14yes—
374C8—44no—
375C2×C3—22no—
376C2×C3—22no—
377C2×C3—41no—
378C2×C3—84no—
379C2×C3—161no—
380C2×C3#261311yescharacteristic
381C2×C3C2×#45311yescharacteristic
382C2×C3#302011yescharacteristic
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—88no—
395C22—84no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22C3×#45312yes—
400C22C2×C3×#12011yescharacteristic
401C22C3×#53512yes—
402C4—22no—
403C4—44no—
404C4—44no—
405C4—44no—
406C4—81no—
407C4C3×#53414yes—
408C3#2961611yescharacteristic, direct product factor, Sylow
409C2—22no—
410C2—22no—
411C2—41no—
412C2—84no—
413C2—161no—
414C2C3×#261311yescharacteristic
415C2C2×C3×#45311yescharacteristic
416C2C3×#302011yescharacteristic
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×#2501C2semidirect
4. C3×#2039C2semidirect
5. C3×#2006C2semidirect
6. C3×#1821C2semidirect
7. C3×#1789C2semidirect
8. #29616C3direct (N × H)
10. C2×C3×AGL(1,ℤ/8)C22semidirect
23. C2×C3×#104C22semidirect
24. C3×#417C22semidirect
27. C3×#390C22semidirect
28. C3×#384C22semidirect
30. C3×#343C22semidirect
31. C2×#537C2×C3semidirect
32. #3006C2×C3semidirect
33. #2501C2×C3semidirect
34. #2039C2×C3semidirect
35. #2006C2×C3semidirect
36. #1821C2×C3semidirect
37. #1789C2×C3semidirect
84. C2×AGL(1,ℤ/8)C22×C3semidirect
97. C2×#104C22×C3semidirect
98. #417C22×C3semidirect
101. #390C22×C3semidirect
102. #384C22×C3semidirect
104. #343C22×C3semidirect
408. C3#29616direct (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×#25012
C3×#20392
C3×#20062
C3×#18212
C3×#17892
#296163