Finite Group Databasebeta

← C3×#29624 #10648231

Subgroups — C3×#29624 #10648231

Order 768.

1388 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×C3×#448C212yesmaximal
3C3×#2854C211yesmaximal, characteristic
4C3×#2043C214yesmaximal, complemented
5C3×#2013C214yesmaximal, complemented
6C3×#1817C211yesmaximal, characteristic, complemented
7C3×#1787C212yesmaximal, complemented
8#29624C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×Q8C2212yes—
10C22×C3×SD16C2214yes—
11C2×C3×C4⋊Q8—28no—
12C2×C3×C4⋊Q8—22no—
13C2×C3×C4⋊Q8C2214yes—
14C2×C3×#123—21no—
15C2×C3×#121C2211yescharacteristic
16C22×C3×C4⋊C4—21no—
17C22×C3×C4⋊C4—21no—
18C22×C3×C42C2211yescharacteristic
19C3×#448—28no—
20C2×C3×#103C2214yes—
21C2×C3×#103C2218yes—
22C2×C3×C22⋊C8C2212yes—
23C3×#390C2212yes—
24C3×#389C2212yes—
25C3×#388—22no—
26C3×#386—21no—
27C3×#380C2211yescharacteristic
28C3×#336C2214yes—
29C22×C4⋊Q8C2×C311yescharacteristic, complemented
30C2×#448C2×C312yes—
31#2854C2×C311yescharacteristic
32#2043C2×C314yescomplemented
33#2013C2×C314yescomplemented
34#1817C2×C311yescharacteristic, complemented
35#1787C2×C312yescomplemented
36C22×C3×Q8—24no—
37C22×C3×Q8—22no—
38C22×C3×Q8—48no—
39C22×C3×Q8D418yes—
40C22×C3×Q8C2314yes—
41C22×C3×D4—21no—
42C22×C3×D4C2311yescharacteristic
43C23×C3×C4—22no—
44C23×C3×C4D411yescharacteristic
45C23×C3×C4D411yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C2×C3×SD16—28no—
48C2×C3×SD16—48no—
49C22×C3×C8C2314yes—
50C3×C4⋊Q8—216no—
51C3×C4⋊Q8—216no—
52C3×#123—44no—
53C3×#121—24no—
54C2×C3×C4⋊C4—28no—
55C2×C3×C4⋊C4—28no—
56C2×C3×C4⋊C4—24no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C4⋊C4C2311yescharacteristic
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4—21no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C42—24no—
66C2×C3×C42D412yes—
67C2×C3×C42C2312yes—
68C3×#103—28no—
69C3×#103—48no—
70C3×C22⋊C8—24no—
71C3×#95—21no—
72C3×#95—22no—
73C23×Q8C22×C312yes—
74C22×SD16C22×C314yes—
75C2×C4⋊Q8—28no—
76C2×C4⋊Q8—22no—
77C2×C4⋊Q8C22×C314yes—
78C2×#123—21no—
79C2×#121C22×C311yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C42C22×C311yescharacteristic
83#448—28no—
84C2×#103C22×C314yes—
85C2×#103C22×C318yes—
86C2×C22⋊C8C22×C312yes—
87#390C22×C312yes—
88#389C22×C312yes—
89#388—22no—
90#386—21no—
91#380C22×C311yescharacteristic
92#336C22×C314yes—
93C24×C3—21no—
94C24×C3C2×D411yescharacteristic
95C2×C3×Q8—28no—
96C2×C3×Q8—24no—
97C2×C3×Q8—28no—
98C2×C3×Q8—28no—
99C2×C3×Q8—416no—
100C2×C3×Q8—48no—
101C2×C3×Q8—416no—
102C2×C3×Q8C2×D418yes—
103C2×C3×D4—22no—
104C2×C3×D4—24no—
105C2×C3×D4—42no—
106C2×C3×D4—42no—
107C2×C3×D4—42no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—24no—
112C22×C3×C4—24no—
113C22×C3×C4—28no—
114C22×C3×C4—24no—
115C22×C3×C4—22no—
116C22×C3×C4—21no—
117C22×C3×C4—48no—
118C22×C3×C4SD1614yescomplemented
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D414yes—
121C22×C3×C4C2×D412yes—
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2411yescharacteristic
124C3×SD16—416no—
125C2×C3×C8—24no—
126C2×C3×C8—28no—
127C3×C4⋊C4—216no—
128C3×C4⋊C4—216no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—24no—
131C3×C4⋊C4—44no—
132C3×C22⋊C4—42no—
133C3×C22⋊C4—42no—
134C3×C22⋊C4—44no—
135C3×C22⋊C4—44no—
136C3×C42—24no—
137C3×C42—24no—
138C22×Q8—24no—
139C22×Q8—22no—
140C22×Q8—48no—
141C22×Q8C3×D418yes—
142C22×Q8C23×C314yes—
143C22×D4—21no—
144C22×D4C23×C311yescharacteristic
145C23×C4—22no—
146C23×C4C3×D411yescharacteristic
147C23×C4C3×D411yescharacteristic
148C23×C4C23×C311yescharacteristic
149C2×SD16—28no—
150C2×SD16—48no—
151C22×C8C23×C314yes—
152C4⋊Q8—216no—
153C4⋊Q8—216no—
154#123—44no—
155#121—24no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4C23×C311yescharacteristic
162C2×C4⋊C4C23×C312yes—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C42—24no—
168C2×C42C3×D412yes—
169C2×C42C23×C312yes—
170#103—28no—
171#103—48no—
172C22⋊C8—24no—
173#95—21no—
174#95—22no—
175C23×C3—22no—
176C23×C3—22no—
177C23×C3—41no—
178C23×C3—42no—
179C23×C3—41no—
180C23×C3—42no—
181C23×C3—41no—
182C23×C3#12012yes—
183C23×C3C2×SD1612yes—
184C23×C3#13712yes—
185C23×C3C22×D411yescharacteristic
186C3×Q8—28no—
187C3×Q8—28no—
188C3×Q8—416no—
189C3×Q8—48no—
190C3×D4—44no—
191C3×D4—84no—
192C2×C3×C4—28no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—22no—
196C2×C3×C4—22no—
197C2×C3×C4—22no—
198C2×C3×C4—28no—
199C2×C3×C4—28no—
200C2×C3×C4—28no—
201C2×C3×C4—416no—
202C2×C3×C4—48no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—41no—
209C2×C3×C4#12012yes—
210C2×C3×C4C2×SD1614yes—
211C2×C3×C4#13714yes—
212C2×C3×C4C22×D412yes—
213C2×C3×C4H(5,𝔽2)14yes—
214C3×C8—44no—
215C24—21no—
216C24C2×C3×D411yescharacteristic
217C2×Q8—28no—
218C2×Q8—24no—
219C2×Q8—28no—
220C2×Q8—28no—
221C2×Q8—416no—
222C2×Q8—48no—
223C2×Q8—416no—
224C2×Q8C2×C3×D418yes—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—28no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—48no—
240C22×C4C3×SD1614yescomplemented
241C22×C4C2×C3×D411yescharacteristic
242C22×C4C2×C3×D414yes—
243C22×C4C2×C3×D412yes—
244C22×C4C2×C3×D412yes—
245C22×C4C24×C311yescharacteristic
246SD16—416no—
247C2×C8—24no—
248C2×C8—28no—
249C4⋊C4—216no—
250C4⋊C4—216no—
251C4⋊C4—22no—
252C4⋊C4—24no—
253C4⋊C4—44no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C42—24no—
259C42—24no—
260C22×C3—22no—
261C22×C3—24no—
262C22×C3—24no—
263C22×C3—22no—
264C22×C3—81no—
265C22×C3—82no—
266C22×C3—82no—
267C22×C3—81no—
268C22×C3—81no—
269C22×C3#44814yes—
270C22×C3C2×#12011yescharacteristic
271C22×C3#53512yes—
272C22×C3C22×SD1611yescharacteristic
273C22×C3C2×#13712yes—
274C22×C3C2×#13711yescharacteristic
275C3×C4—22no—
276C3×C4—24no—
277C3×C4—24no—
278C3×C4—48no—
279C3×C4—82no—
280C3×C4—81no—
281C3×C4#53414yes—
282C23—22no—
283C23—22no—
284C23—41no—
285C23—42no—
286C23—41no—
287C23—41no—
288C23—42no—
289C23C3×#12012yes—
290C23C2×C3×SD1612yes—
291C23C3×#13712yes—
292C23C22×C3×D411yescharacteristic
293Q8—28no—
294Q8—28no—
295Q8—416no—
296Q8—48no—
297D4—44no—
298D4—84no—
299C2×C4—28no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—28no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—28no—
307C2×C4—28no—
308C2×C4—416no—
309C2×C4—48no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4C3×#12012yes—
317C2×C4C2×C3×SD1614yes—
318C2×C4C3×#13714yes—
319C2×C4C22×C3×D412yes—
320C2×C4C3×H(5,𝔽2)14yes—
321C8—44no—
322C2×C3—24no—
323C2×C3—161no—
324C2×C3#261311yescharacteristic
325C2×C3C2×#44811yescharacteristic
326C2×C3#300411yescharacteristic
327C2×C3#320612yes—
328C2×C3#321812yes—
329C22—24no—
330C22—24no—
331C22—22no—
332C22—22no—
333C22—82no—
334C22—82no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22C3×#44814yes—
339C22C2×C3×#12011yescharacteristic
340C22C3×#53512yes—
341C22C22×C3×SD1611yescharacteristic
342C22C2×C3×#13712yes—
343C22C2×C3×#13711yescharacteristic
344C4—24no—
345C4—24no—
346C4—22no—
347C4—48no—
348C4—82no—
349C4—81no—
350C4C3×#53414yes—
351C3#2962411yescharacteristic, direct product factor, Sylow
352C2—24no—
353C2—161no—
354C2C3×#261311yescharacteristic
355C2C2×C3×#44811yescharacteristic
356C2C3×#300411yescharacteristic
357C2C3×#320612yes—
358C2C3×#321812yes—
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×#2043C2semidirect
5. C3×#2013C2semidirect
6. C3×#1817C2semidirect
7. C3×#1787C2semidirect
8. #29624C3direct (N × H)
29. C22×C4⋊Q8C2×C3semidirect
32. #2043C2×C3semidirect
33. #2013C2×C3semidirect
34. #1817C2×C3semidirect
35. #1787C2×C3semidirect
118. C22×C3×C4SD16semidirect
240. C22×C4C3×SD16semidirect
351. C3#29624direct (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×#4482
C3×#28542
C3×#20432
C3×#20132
C3×#18172
C3×#17872
#296243