Finite Group Databasebeta

← C3×#29639 #10648246

Subgroups — C3×#29639 #10648246

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#481C212yesmaximal
3C3×#2854C211yesmaximal, characteristic
4C3×#2058C214yesmaximal, complemented
5C3×#2023C214yesmaximal, complemented
6C3×#1824C212yesmaximal, complemented
7C3×#1817C211yesmaximal, characteristic, complemented
8#29639C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×Q8—21no—
10C2×C3×C4⋊Q8—28no—
11C2×C3×C4⋊Q8—22no—
12C2×C3×C4⋊Q8C2214yes—
13C2×C3×#123—21no—
14C2×C3×#121C2211yescharacteristic
15C22×C3×C4⋊C4—21no—
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C42C2211yescharacteristic
18C3×#481—28no—
19C2×C3×#106C2214yes—
20C2×C3×#102C2214yes—
21C2×C3×C22⋊C8C2212yes—
22C3×#390C2212yes—
23C3×#389C2212yes—
24C3×#388—22no—
25C3×#386—21no—
26C3×#380C2211yescharacteristic
27C3×#340C2218yes—
28C3×#336C2214yes—
29C22×C4⋊Q8C2×C311yescharacteristic, complemented
30C2×#481C2×C312yes—
31#2854C2×C311yescharacteristic
32#2058C2×C314yescomplemented
33#2023C2×C314yescomplemented
34#1824C2×C312yescomplemented
35#1817C2×C311yescharacteristic, complemented
36C22×C3×Q8—28no—
37C22×C3×Q8—24no—
38C22×C3×Q8—48no—
39C22×C3×D4—21no—
40C22×C3×D4C2311yescharacteristic
41C23×C3×C4—22no—
42C23×C3×C4D411yescharacteristic
43C23×C3×C4D411yescharacteristic
44C23×C3×C4C2311yescharacteristic
45C22×C3×C8C2314yes—
46C3×C4⋊Q8—216no—
47C3×C4⋊Q8—216no—
48C3×#123—44no—
49C3×#121—24no—
50C2×C3×C4⋊C4—28no—
51C2×C3×C4⋊C4—24no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—24no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4D418yes—
56C2×C3×C4⋊C4C2312yes—
57C2×C3×C4⋊C4C2311yescharacteristic
58C2×C3×C4⋊C4C2314yes—
59C2×C3×C22⋊C4—21no—
60C2×C3×C22⋊C4—22no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C42—24no—
64C2×C3×C42D412yes—
65C2×C3×C42C2312yes—
66C3×#106—28no—
67C3×#102—28no—
68C3×C22⋊C8—24no—
69C3×#95—21no—
70C3×#95—22no—
71C23×Q8—21no—
72C2×C4⋊Q8—28no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8C22×C314yes—
75C2×#123—21no—
76C2×#121C22×C311yescharacteristic
77C22×C4⋊C4—21no—
78C22×C4⋊C4C22×C312yes—
79C22×C42C22×C311yescharacteristic
80#481—28no—
81C2×#106C22×C314yes—
82C2×#102C22×C314yes—
83C2×C22⋊C8C22×C312yes—
84#390C22×C312yes—
85#389C22×C312yes—
86#388—22no—
87#386—21no—
88#380C22×C311yescharacteristic
89#340C22×C318yes—
90#336C22×C314yes—
91C24×C3—21no—
92C24×C3C2×D411yescharacteristic
93C2×C3×Q8—216no—
94C2×C3×Q8—432no—
95C2×C3×Q8—48no—
96C2×C3×Q8—48no—
97C2×C3×D4—22no—
98C2×C3×D4—24no—
99C2×C3×D4—42no—
100C2×C3×D4—42no—
101C2×C3×D4—42no—
102C22×C3×C4—22no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4—24no—
106C22×C3×C4—24no—
107C22×C3×C4—28no—
108C22×C3×C4—24no—
109C22×C3×C4—22no—
110C22×C3×C4—21no—
111C22×C3×C4—48no—
112C22×C3×C4SD1614yes—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4C2×D414yes—
115C22×C3×C4C2×D412yes—
116C22×C3×C4C2×D412yes—
117C22×C3×C4C2411yescharacteristic
118C2×C3×C8—24no—
119C2×C3×C8—28no—
120C3×C4⋊C4—216no—
121C3×C4⋊C4—28no—
122C3×C4⋊C4—24no—
123C3×C4⋊C4—22no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—44no—
126C3×C4⋊C4C2×D418yes—
127C3×C22⋊C4—42no—
128C3×C22⋊C4—42no—
129C3×C22⋊C4—44no—
130C3×C22⋊C4—44no—
131C3×C42—24no—
132C3×C42—24no—
133C22×Q8—28no—
134C22×Q8—24no—
135C22×Q8—48no—
136C22×D4—21no—
137C22×D4C23×C311yescharacteristic
138C23×C4—22no—
139C23×C4C3×D411yescharacteristic
140C23×C4C3×D411yescharacteristic
141C23×C4C23×C311yescharacteristic
142C22×C8C23×C314yes—
143C4⋊Q8—216no—
144C4⋊Q8—216no—
145#123—44no—
146#121—24no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C3×D418yes—
153C2×C4⋊C4C23×C311yescharacteristic
154C2×C4⋊C4C23×C312yes—
155C2×C4⋊C4C23×C314yes—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C42—24no—
161C2×C42C3×D412yes—
162C2×C42C23×C312yes—
163#106—28no—
164#102—28no—
165C22⋊C8—24no—
166#95—21no—
167#95—22no—
168C23×C3—22no—
169C23×C3—22no—
170C23×C3—41no—
171C23×C3—42no—
172C23×C3—41no—
173C23×C3—42no—
174C23×C3—41no—
175C23×C3#12012yes—
176C23×C3C2×SD1612yes—
177C23×C3#13712yes—
178C23×C3C22×D411yescharacteristic
179C3×Q8—432no—
180C3×D4—44no—
181C3×D4—84no—
182C2×C3×C4—28no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—28no—
189C2×C3×C4—28no—
190C2×C3×C4—28no—
191C2×C3×C4—416no—
192C2×C3×C4—48no—
193C2×C3×C4—41no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—41no—
199C2×C3×C4#12012yes—
200C2×C3×C4C2×SD1614yes—
201C2×C3×C4AGL(1,ℤ/8)14yes—
202C2×C3×C4C22×D412yes—
203C2×C3×C42−1+414yes—
204C3×C8—44no—
205C24—21no—
206C24C2×C3×D411yescharacteristic
207C2×Q8—216no—
208C2×Q8—432no—
209C2×Q8—48no—
210C2×Q8—48no—
211C2×D4—22no—
212C2×D4—24no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—28no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—48no—
226C22×C4C3×SD1614yes—
227C22×C4C2×C3×D411yescharacteristic
228C22×C4C2×C3×D414yes—
229C22×C4C2×C3×D412yes—
230C22×C4C2×C3×D412yes—
231C22×C4C24×C311yescharacteristic
232C2×C8—24no—
233C2×C8—28no—
234C4⋊C4—216no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—44no—
240C4⋊C4C2×C3×D418yes—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C42—24no—
246C42—24no—
247C22×C3—22no—
248C22×C3—24no—
249C22×C3—24no—
250C22×C3—22no—
251C22×C3—81no—
252C22×C3—82no—
253C22×C3—82no—
254C22×C3—81no—
255C22×C3—81no—
256C22×C3#44814yes—
257C22×C3C2×#12011yescharacteristic
258C22×C3#53612yes—
259C22×C3C22×SD1611yescharacteristic
260C22×C3C2×AGL(1,ℤ/8)12yes—
261C22×C3C2×#13711yescharacteristic
262C3×C4—22no—
263C3×C4—24no—
264C3×C4—24no—
265C3×C4—48no—
266C3×C4—82no—
267C3×C4—81no—
268C3×C4#53614yes—
269C23—22no—
270C23—22no—
271C23—41no—
272C23—42no—
273C23—41no—
274C23—41no—
275C23—42no—
276C23C3×#12012yes—
277C23C2×C3×SD1612yes—
278C23C3×#13712yes—
279C23C22×C3×D411yescharacteristic
280Q8—432no—
281D4—44no—
282D4—84no—
283C2×C4—28no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—28no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—28no—
291C2×C4—28no—
292C2×C4—416no—
293C2×C4—48no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4C3×#12012yes—
301C2×C4C2×C3×SD1614yes—
302C2×C4C3×AGL(1,ℤ/8)14yes—
303C2×C4C22×C3×D412yes—
304C2×C4C3×2−1+414yes—
305C8—44no—
306C2×C3—24no—
307C2×C3—161no—
308C2×C3#261411yescharacteristic
309C2×C3C2×#44811yescharacteristic
310C2×C3#300311yescharacteristic
311C2×C3#322712yes—
312C2×C3#323912yes—
313C22—24no—
314C22—24no—
315C22—22no—
316C22—22no—
317C22—82no—
318C22—82no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22C3×#44814yes—
323C22C2×C3×#12011yescharacteristic
324C22C3×#53612yes—
325C22C22×C3×SD1611yescharacteristic
326C22C2×C3×AGL(1,ℤ/8)12yes—
327C22C2×C3×#13711yescharacteristic
328C4—24no—
329C4—24no—
330C4—22no—
331C4—48no—
332C4—82no—
333C4—81no—
334C4C3×#53614yes—
335C3#2963911yescharacteristic, direct product factor, Sylow
336C2—24no—
337C2—161no—
338C2C3×#261411yescharacteristic
339C2C2×C3×#44811yescharacteristic
340C2C3×#300311yescharacteristic
341C2C3×#322712yes—
342C2C3×#323912yes—
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×C4⋊Q8C2semidirect
4. C3×#2058C2semidirect
5. C3×#2023C2semidirect
6. C3×#1824C2semidirect
7. C3×#1817C2semidirect
8. #29639C3direct (N × H)
29. C22×C4⋊Q8C2×C3semidirect
32. #2058C2×C3semidirect
33. #2023C2×C3semidirect
34. #1824C2×C3semidirect
35. #1817C2×C3semidirect
335. C3#29639direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×C3×#4812
C3×#28542
C3×#20582
C3×#20232
C3×#18242
C3×#18172
#296393