Finite Group Databasebeta

← C3×#29638 #10648245

Subgroups — C3×#29638 #10648245

Order 768.

1292 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#480C212yesmaximal
3C3×#2854C211yesmaximal, characteristic
4C3×#2057C214yesmaximal, complemented
5C3×#2023C214yesmaximal, complemented
6C3×#1823C212yesmaximal, complemented
7C3×#1817C211yesmaximal, characteristic, complemented
8#29638C311yesmaximal, 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×#480—28no—
19C2×C3×#107C2214yes—
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×#480C2×C312yes—
31#2854C2×C311yescharacteristic
32#2057C2×C314yescomplemented
33#2023C2×C314yescomplemented
34#1823C2×C312yescomplemented
35#1817C2×C311yescharacteristic, complemented
36C22×C3×Q8—28no—
37C22×C3×Q8—24no—
38C22×C3×Q8—44no—
39C22×C3×Q8—44no—
40C22×C3×D4—21no—
41C22×C3×D4C2311yescharacteristic
42C23×C3×C4—22no—
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C22×C3×C8C2314yes—
47C3×C4⋊Q8—216no—
48C3×C4⋊Q8—216no—
49C3×#123—44no—
50C3×#121—24no—
51C2×C3×C4⋊C4—28no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4D418yes—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C4⋊C4C2311yescharacteristic
59C2×C3×C4⋊C4C2314yes—
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—21no—
63C2×C3×C22⋊C4—22no—
64C2×C3×C42—24no—
65C2×C3×C42D412yes—
66C2×C3×C42C2312yes—
67C3×#107—28no—
68C3×#102—28no—
69C3×C22⋊C8—24no—
70C3×#95—21no—
71C3×#95—22no—
72C23×Q8—21no—
73C2×C4⋊Q8—28no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8C22×C314yes—
76C2×#123—21no—
77C2×#121C22×C311yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4C22×C312yes—
80C22×C42C22×C311yescharacteristic
81#480—28no—
82C2×#107C22×C314yes—
83C2×#102C22×C314yes—
84C2×C22⋊C8C22×C312yes—
85#390C22×C312yes—
86#389C22×C312yes—
87#388—22no—
88#386—21no—
89#380C22×C311yescharacteristic
90#340C22×C318yes—
91#336C22×C314yes—
92C24×C3—21no—
93C24×C3C2×D411yescharacteristic
94C2×C3×Q8—216no—
95C2×C3×Q8—48no—
96C2×C3×Q8—432no—
97C2×C3×Q8—48no—
98C2×C3×D4—22no—
99C2×C3×D4—24no—
100C2×C3×D4—42no—
101C2×C3×D4—42no—
102C2×C3×D4—42no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—24no—
107C22×C3×C4—24no—
108C22×C3×C4—28no—
109C22×C3×C4—24no—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—44no—
113C22×C3×C4—44no—
114C22×C3×C4D814yes—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D414yes—
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D412yes—
119C22×C3×C4C2411yescharacteristic
120C2×C3×C8—24no—
121C2×C3×C8—28no—
122C3×C4⋊C4—216no—
123C3×C4⋊C4—28no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4—44no—
128C3×C4⋊C4C2×D418yes—
129C3×C22⋊C4—42no—
130C3×C22⋊C4—42no—
131C3×C22⋊C4—44no—
132C3×C22⋊C4—44no—
133C3×C42—24no—
134C3×C42—24no—
135C22×Q8—28no—
136C22×Q8—24no—
137C22×Q8—44no—
138C22×Q8—44no—
139C22×D4—21no—
140C22×D4C23×C311yescharacteristic
141C23×C4—22no—
142C23×C4C3×D411yescharacteristic
143C23×C4C3×D411yescharacteristic
144C23×C4C23×C311yescharacteristic
145C22×C8C23×C314yes—
146C4⋊Q8—216no—
147C4⋊Q8—216no—
148#123—44no—
149#121—24no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C3×D418yes—
156C2×C4⋊C4C23×C311yescharacteristic
157C2×C4⋊C4C23×C312yes—
158C2×C4⋊C4C23×C314yes—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C42—24no—
164C2×C42C3×D412yes—
165C2×C42C23×C312yes—
166#107—28no—
167#102—28no—
168C22⋊C8—24no—
169#95—21no—
170#95—22no—
171C23×C3—22no—
172C23×C3—22no—
173C23×C3—41no—
174C23×C3—42no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—42no—
178C23×C3#12012yes—
179C23×C3C2×D812yes—
180C23×C3AGL(1,ℤ/8)12yes—
181C23×C3C22×D411yescharacteristic
182C3×Q8—432no—
183C3×D4—44no—
184C3×D4—84no—
185C2×C3×C4—28no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—28no—
192C2×C3×C4—28no—
193C2×C3×C4—28no—
194C2×C3×C4—44no—
195C2×C3×C4—44no—
196C2×C3×C4—416no—
197C2×C3×C4—41no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4#12012yes—
204C2×C3×C4C2×D814yes—
205C2×C3×C4#13714yes—
206C2×C3×C4C22×D412yes—
207C2×C3×C42−1+414yes—
208C3×C8—44no—
209C24—21no—
210C24C2×C3×D411yescharacteristic
211C2×Q8—216no—
212C2×Q8—48no—
213C2×Q8—432no—
214C2×Q8—48no—
215C2×D4—22no—
216C2×D4—24no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—28no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4C3×D814yes—
232C22×C4C2×C3×D411yescharacteristic
233C22×C4C2×C3×D414yes—
234C22×C4C2×C3×D412yes—
235C22×C4C2×C3×D412yes—
236C22×C4C24×C311yescharacteristic
237C2×C8—24no—
238C2×C8—28no—
239C4⋊C4—216no—
240C4⋊C4—28no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—24no—
244C4⋊C4—44no—
245C4⋊C4C2×C3×D418yes—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C22⋊C4—44no—
250C42—24no—
251C42—24no—
252C22×C3—22no—
253C22×C3—24no—
254C22×C3—24no—
255C22×C3—22no—
256C22×C3—81no—
257C22×C3—82no—
258C22×C3—82no—
259C22×C3—81no—
260C22×C3—81no—
261C22×C3#44714yes—
262C22×C3C2×#12011yescharacteristic
263C22×C3#53612yes—
264C22×C3C22×D811yescharacteristic
265C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C3C2×#13712yes—
267C3×C4—22no—
268C3×C4—24no—
269C3×C4—24no—
270C3×C4—48no—
271C3×C4—82no—
272C3×C4—81no—
273C3×C4#53614yes—
274C23—22no—
275C23—22no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—41no—
280C23—42no—
281C23C3×#12012yes—
282C23C2×C3×D812yes—
283C23C3×AGL(1,ℤ/8)12yes—
284C23C22×C3×D411yescharacteristic
285Q8—432no—
286D4—44no—
287D4—84no—
288C2×C4—28no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—28no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—28no—
296C2×C4—28no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—416no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4C3×#12012yes—
307C2×C4C2×C3×D814yes—
308C2×C4C3×#13714yes—
309C2×C4C22×C3×D412yes—
310C2×C4C3×2−1+414yes—
311C8—44no—
312C2×C3—24no—
313C2×C3—161no—
314C2×C3#261411yescharacteristic
315C2×C3C2×#44711yescharacteristic
316C2×C3#300411yescharacteristic
317C2×C3#322612yes—
318C2×C3#323812yes—
319C22—24no—
320C22—24no—
321C22—22no—
322C22—22no—
323C22—82no—
324C22—82no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22C3×#44714yes—
329C22C2×C3×#12011yescharacteristic
330C22C3×#53612yes—
331C22C22×C3×D811yescharacteristic
332C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
333C22C2×C3×#13712yes—
334C4—24no—
335C4—24no—
336C4—22no—
337C4—48no—
338C4—82no—
339C4—81no—
340C4C3×#53614yes—
341C3#2963811yescharacteristic, direct product factor, Sylow
342C2—24no—
343C2—161no—
344C2C3×#261411yescharacteristic
345C2C2×C3×#44711yescharacteristic
346C2C3×#300411yescharacteristic
347C2C3×#322612yes—
348C2C3×#323812yes—
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×#2057C2semidirect
5. C3×#2023C2semidirect
6. C3×#1823C2semidirect
7. C3×#1817C2semidirect
8. #29638C3direct (N × H)
29. C22×C4⋊Q8C2×C3semidirect
32. #2057C2×C3semidirect
33. #2023C2×C3semidirect
34. #1823C2×C3semidirect
35. #1817C2×C3semidirect
341. C3#29638direct (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×#4802
C3×#28542
C3×#20572
C3×#20232
C3×#18232
C3×#18172
#296383