Finite Group Databasebeta

← C3×#29619 #10648226

Subgroups — C3×#29619 #10648226

Order 768.

1332 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#537C211yesmaximal, characteristic
2C3×#3007C212yesmaximal, complemented
3C3×#2502C211yesmaximal, characteristic, complemented
4C3×#2039C214yesmaximal, complemented
5C3×#2007C214yesmaximal, complemented
6C3×#1821C211yesmaximal, characteristic, complemented
7C3×#1789C212yesmaximal, complemented
8#29619C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×#41C2212yes—
10C2×C3×#137C2214yescomplemented
11C3×#537—24no—
12C3×#537—22no—
13C3×#537C2214yes—
14C2×C3×C22⋊Q8—22no—
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×#121—22no—
17C2×C3×#121—22no—
18C22×C3×C4⋊C4C2211yescharacteristic
19C3×#452—24no—
20C3×#451—22no—
21C3×#450—22no—
22C2×C3×#104C2218yescomplemented
23C3×#417C2214yescomplemented
24C3×#407C2212yes—
25C3×#398—22no—
26C3×#391C2212yescomplemented
27C3×#385—22no—
28C3×#384C2212yescomplemented
29C3×C4×C22⋊C4C2211yescharacteristic
30C3×#343C2214yescomplemented
31C2×#537C2×C311yescharacteristic
32#3007C2×C312yescomplemented
33#2502C2×C311yescharacteristic, complemented
34#2039C2×C314yescomplemented
35#2007C2×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×Q8C2311yescharacteristic
45C22×C3×D4—22no—
46C22×C3×D4—22no—
47C22×C3×D4D412yes—
48C23×C3×C4—22no—
49C23×C3×C4D412yes—
50C23×C3×C4C2311yescharacteristic
51C3×#137—48no—
52C2×C3×Q16—24no—
53C2×C3×SD16—24no—
54C2×C3×M16C2314yes—
55C3×C22⋊Q8—28no—
56C3×C22⋊Q8—28no—
57C3×C22⋊Q8—22no—
58C3×C22⋊Q8—22no—
59C3×#121—28no—
60C3×#121—28no—
61C3×#121—28no—
62C3×#121—28no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4D412yes—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C22⋊C4—21no—
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×#137C22×C314yescomplemented
84#537—24no—
85#537—22no—
86#537C22×C314yes—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8C22×C311yescharacteristic
89C2×#121—22no—
90C2×#121—22no—
91C22×C4⋊C4C22×C311yescharacteristic
92#452—24no—
93#451—22no—
94#450—22no—
95C2×#104C22×C318yescomplemented
96#417C22×C314yescomplemented
97#407C22×C312yes—
98#398—22no—
99#391C22×C312yescomplemented
100#385—22no—
101#384C22×C312yescomplemented
102C4×C22⋊C4C22×C311yescharacteristic
103#343C22×C314yescomplemented
104C24×C3—22no—
105C24×C3C2×D411yescharacteristic
106C3×#41—28no—
107C3×#41—28no—
108C3×#41—44no—
109C3×#41—44no—
110C3×#41—48no—
111C3×#41—48no—
112C2×C3×Q8—22no—
113C2×C3×Q8—22no—
114C2×C3×Q8—24no—
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—44no—
122C2×C3×D4—48no—
123C2×C3×D4—84no—
124C2×C3×D4C2×D414yes—
125C22×C3×C4—24no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—24no—
131C22×C3×C4—24no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—48no—
135C22×C3×C4—82no—
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2×D414yes—
140C22×C3×C4C2411yescharacteristic
141C3×Q16—48no—
142C3×SD16—48no—
143C3×M16—28no—
144C2×C3×C8—24no—
145C3×C4⋊C4—28no—
146C3×C4⋊C4—24no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—42no—
150C3×C4⋊C4—84no—
151C3×C22⋊C4—24no—
152C3×C22⋊C4—216no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C42—24no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—48no—
159C2×#41C3×D414yes—
160C2×#41C23×C314yes—
161C22×Q8C3×D412yes—
162C22×Q8C23×C311yescharacteristic
163C22×D4—22no—
164C22×D4—22no—
165C22×D4C3×D412yes—
166C23×C4—22no—
167C23×C4C3×D412yes—
168C23×C4C23×C311yescharacteristic
169#137—48no—
170C2×Q16—24no—
171C2×SD16—24no—
172C2×M16C23×C314yes—
173C22⋊Q8—28no—
174C22⋊Q8—28no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177#121—28no—
178#121—28no—
179#121—28no—
180#121—28no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—24no—
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×C42C23×C312yes—
193#104—48no—
194#103—24no—
195#102—24no—
196C22⋊C8—22no—
197C22⋊C8—22no—
198#95—21no—
199#95—24no—
200C23×C3—22no—
201C23×C3—22no—
202C23×C3—24no—
203C23×C3—41no—
204C23×C3—44no—
205C23×C3—44no—
206C23×C3—44no—
207C23×C3#12011yescharacteristic
208C23×C3#12011yescharacteristic
209C23×C3C22×D411yescharacteristic
210C3×Q8—24no—
211C3×Q8—24no—
212C3×Q8—44no—
213C3×Q8—44no—
214C3×D4—24no—
215C3×D4—24no—
216C3×D4—44no—
217C3×D4—48no—
218C3×D4—48no—
219C3×D4—88no—
220C2×C3×C4—22no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4—48no—
227C2×C3×C4—42no—
228C2×C3×C4—44no—
229C2×C3×C4—44no—
230C2×C3×C4—42no—
231C2×C3×C4—48no—
232C2×C3×C4—44no—
233C2×C3×C4—42no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—82no—
238C2×C3×C4#12012yes—
239C2×C3×C4C22×D412yes—
240C2×C3×C4H(5,𝔽2)14yes—
241C3×C8—44no—
242C24—22no—
243C24C2×C3×D411yescharacteristic
244#41—28no—
245#41—28no—
246#41—44no—
247#41—44no—
248#41—48no—
249#41—48no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—24no—
253C2×Q8—44no—
254C2×Q8C2×C3×D414yes—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—28no—
258C2×D4—22no—
259C2×D4—44no—
260C2×D4—48no—
261C2×D4—84no—
262C2×D4C2×C3×D414yes—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—48no—
273C22×C4—82no—
274C22×C4C2×C3×D411yescharacteristic
275C22×C4C2×C3×D412yes—
276C22×C4C2×C3×D412yes—
277C22×C4C2×C3×D414yes—
278C22×C4C24×C311yescharacteristic
279Q16—48no—
280SD16—48no—
281M16—28no—
282C2×C8—24no—
283C4⋊C4—28no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—42no—
288C4⋊C4—84no—
289C22⋊C4—216no—
290C22⋊C4—24no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C42—24no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—44no—
302C22×C3—44no—
303C22×C3—44no—
304C22×C3—44no—
305C22×C3—88no—
306C22×C3#45612yes—
307C22×C3C2×#12011yescharacteristic
308C22×C3#53512yes—
309C3×C4—22no—
310C3×C4—44no—
311C3×C4—44no—
312C3×C4—44no—
313C3×C4—81no—
314C3×C4—81no—
315C3×C4#53414yes—
316C23—22no—
317C23—22no—
318C23—24no—
319C23—41no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23C3×#12011yescharacteristic
324C23C3×#12011yescharacteristic
325C23C22×C3×D411yescharacteristic
326Q8—24no—
327Q8—24no—
328Q8—44no—
329Q8—44no—
330D4—24no—
331D4—24no—
332D4—44no—
333D4—48no—
334D4—48no—
335D4—88no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—48no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4C3×#12012yes—
355C2×C4C22×C3×D412yes—
356C2×C4C3×H(5,𝔽2)14yes—
357C8—44no—
358C2×C3—22no—
359C2×C3—22no—
360C2×C3—41no—
361C2×C3—84no—
362C2×C3#261311yescharacteristic
363C2×C3C2×#45611yescharacteristic
364C2×C3#301911yescharacteristic
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—44no—
373C22—44no—
374C22—44no—
375C22—44no—
376C22—88no—
377C22C3×#45612yes—
378C22C2×C3×#12011yescharacteristic
379C22C3×#53512yes—
380C4—22no—
381C4—44no—
382C4—44no—
383C4—44no—
384C4—81no—
385C4—81no—
386C4C3×#53414yes—
387C3#2961911yescharacteristic, direct product factor, Sylow
388C2—22no—
389C2—22no—
390C2—41no—
391C2—84no—
392C2C3×#261311yescharacteristic
393C2C2×C3×#45611yescharacteristic
394C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#537C2
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
2. C3×#3007C2semidirect
3. C3×#2502C2semidirect
4. C3×#2039C2semidirect
5. C3×#2007C2semidirect
6. C3×#1821C2semidirect
7. C3×#1789C2semidirect
8. #29619C3direct (N × H)
10. C2×C3×#137C22semidirect
22. C2×C3×#104C22semidirect
23. C3×#417C22semidirect
26. C3×#391C22semidirect
28. C3×#384C22semidirect
30. C3×#343C22semidirect
32. #3007C2×C3semidirect
33. #2502C2×C3semidirect
34. #2039C2×C3semidirect
35. #2007C2×C3semidirect
36. #1821C2×C3semidirect
37. #1789C2×C3semidirect
83. C2×#137C22×C3semidirect
95. C2×#104C22×C3semidirect
96. #417C22×C3semidirect
99. #391C22×C3semidirect
101. #384C22×C3semidirect
103. #343C22×C3semidirect
387. C3#29619direct (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×#30072
C3×#25022
C3×#20392
C3×#20072
C3×#18212
C3×#17892
#296193