Finite Group Databasebeta

← C3×#29627 #10648234

Subgroups — C3×#29627 #10648234

Order 768.

1326 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#451C212yesmaximal
3C3×#2855C211yesmaximal, characteristic
4C3×#2043C214yesmaximal, complemented
5C3×#2016C214yesmaximal, complemented
6C3×#1817C211yesmaximal, characteristic
7C3×#1787C212yesmaximal
8#29627C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×Q8C2212yes—
10C22×C3×Q16C2214yes—
11C2×C3×C4⋊Q8—28no—
12C2×C3×C4⋊Q8—22no—
13C2×C3×C4⋊Q8C2214yes—
14C2×C3×C22⋊Q8C2211yescharacteristic
15C22×C3×C4⋊C4—21no—
16C22×C3×C4⋊C4—21no—
17C22×C3×C42C2211yescharacteristic
18C3×#451—28no—
19C2×C3×#103C2214yes—
20C2×C3×#103C2218yes—
21C2×C3×C22⋊C8C2212yes—
22C3×#391C2212yes—
23C3×#389C2212yes—
24C3×#387—22no—
25C3×#385—21no—
26C3×#381—21no—
27C3×#380C2211yescharacteristic
28C3×#336C2214yes—
29C22×C4⋊Q8C2×C311yescharacteristic
30C2×#451C2×C312yes—
31#2855C2×C311yescharacteristic
32#2043C2×C314yescomplemented
33#2016C2×C314yescomplemented
34#1817C2×C311yescharacteristic
35#1787C2×C312yes—
36C22×C3×Q8—24no—
37C22×C3×Q8—22no—
38C22×C3×Q8—48no—
39C22×C3×Q8D418yes—
40C22×C3×Q8C2311yescharacteristic
41C22×C3×Q8C2314yes—
42C23×C3×C4—22no—
43C23×C3×C4D411yescharacteristic
44C23×C3×C4D411yescharacteristic
45C23×C3×C4C2311yescharacteristic
46C2×C3×Q16—28no—
47C2×C3×Q16—48no—
48C22×C3×C8C2314yes—
49C3×C4⋊Q8—216no—
50C3×C4⋊Q8—216no—
51C3×C22⋊Q8—24no—
52C2×C3×C4⋊C4—28no—
53C2×C3×C4⋊C4—28no—
54C2×C3×C4⋊C4—24no—
55C2×C3×C4⋊C4—24no—
56C2×C3×C4⋊C4—22no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C4⋊C4C2311yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C22⋊C4—21no—
62C2×C3×C42—24no—
63C2×C3×C42D412yes—
64C2×C3×C42C2312yes—
65C3×#103—28no—
66C3×#103—48no—
67C3×C22⋊C8—24no—
68C3×#95—21no—
69C3×#95—22no—
70C3×#95—22no—
71C3×#95—22no—
72C23×Q8C22×C312yes—
73C22×Q16C22×C314yes—
74C2×C4⋊Q8—28no—
75C2×C4⋊Q8—22no—
76C2×C4⋊Q8C22×C314yes—
77C2×C22⋊Q8C22×C311yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C42C22×C311yescharacteristic
81#451—28no—
82C2×#103C22×C314yes—
83C2×#103C22×C318yes—
84C2×C22⋊C8C22×C312yes—
85#391C22×C312yes—
86#389C22×C312yes—
87#387—22no—
88#385—21no—
89#381—21no—
90#380C22×C311yescharacteristic
91#336C22×C314yes—
92C24×C3C2×D411yescharacteristic
93C2×C3×Q8—28no—
94C2×C3×Q8—24no—
95C2×C3×Q8—22no—
96C2×C3×Q8—28no—
97C2×C3×Q8—28no—
98C2×C3×Q8—24no—
99C2×C3×Q8—416no—
100C2×C3×Q8—48no—
101C2×C3×Q8—416no—
102C2×C3×Q8C2×D418yes—
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—21no—
113C22×C3×C4—48no—
114C22×C3×C4Q1614yescomplemented
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D414yes—
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D412yes—
119C22×C3×C4C2411yescharacteristic
120C3×Q16—416no—
121C2×C3×C8—24no—
122C2×C3×C8—28no—
123C3×C4⋊C4—216no—
124C3×C4⋊C4—216no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4—42no—
128C3×C4⋊C4—44no—
129C3×C22⋊C4—42no—
130C3×C22⋊C4—42no—
131C3×C42—24no—
132C3×C42—24no—
133C22×Q8—24no—
134C22×Q8—22no—
135C22×Q8—48no—
136C22×Q8C3×D418yes—
137C22×Q8C23×C311yescharacteristic
138C22×Q8C23×C314yes—
139C23×C4—22no—
140C23×C4C3×D411yescharacteristic
141C23×C4C3×D411yescharacteristic
142C23×C4C23×C311yescharacteristic
143C2×Q16—28no—
144C2×Q16—48no—
145C22×C8C23×C314yes—
146C4⋊Q8—216no—
147C4⋊Q8—216no—
148C22⋊Q8—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4C23×C311yescharacteristic
156C2×C4⋊C4C23×C312yes—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C42—24no—
160C2×C42C3×D412yes—
161C2×C42C23×C312yes—
162#103—28no—
163#103—48no—
164C22⋊C8—24no—
165#95—21no—
166#95—22no—
167#95—22no—
168#95—22no—
169C23×C3—22no—
170C23×C3—22no—
171C23×C3#12012yes—
172C23×C3C2×Q1612yes—
173C23×C3#13712yes—
174C23×C3C22×D411yescharacteristic
175C3×Q8—28no—
176C3×Q8—28no—
177C3×Q8—416no—
178C3×Q8—48no—
179C3×Q8—44no—
180C2×C3×C4—28no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—28no—
187C2×C3×C4—28no—
188C2×C3×C4—28no—
189C2×C3×C4—416no—
190C2×C3×C4—48no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4#12012yes—
199C2×C3×C4C2×Q1614yes—
200C2×C3×C4#13714yes—
201C2×C3×C4C22×D412yes—
202C2×C3×C4H(5,𝔽2)14yes—
203C3×C8—44no—
204C24C2×C3×D411yescharacteristic
205C2×Q8—28no—
206C2×Q8—24no—
207C2×Q8—22no—
208C2×Q8—28no—
209C2×Q8—28no—
210C2×Q8—24no—
211C2×Q8—416no—
212C2×Q8—48no—
213C2×Q8—416no—
214C2×Q8C2×C3×D418yes—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—28no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—48no—
226C22×C4C3×Q1614yescomplemented
227C22×C4C2×C3×D411yescharacteristic
228C22×C4C2×C3×D414yes—
229C22×C4C2×C3×D412yes—
230C22×C4C2×C3×D412yes—
231C22×C4C24×C311yescharacteristic
232Q16—416no—
233C2×C8—24no—
234C2×C8—28no—
235C4⋊C4—216no—
236C4⋊C4—216no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C42—24no—
244C42—24no—
245C22×C3—22no—
246C22×C3—24no—
247C22×C3—24no—
248C22×C3—22no—
249C22×C3#45114yes—
250C22×C3C2×#12011yescharacteristic
251C22×C3#53512yes—
252C22×C3C22×Q1611yescharacteristic
253C22×C3C2×#13712yes—
254C22×C3C2×#13711yescharacteristic
255C3×C4—22no—
256C3×C4—24no—
257C3×C4—24no—
258C3×C4—48no—
259C3×C4—81no—
260C3×C4—82no—
261C3×C4—81no—
262C3×C4#53414yes—
263C23—22no—
264C23—22no—
265C23C3×#12012yes—
266C23C2×C3×Q1612yes—
267C23C3×#13712yes—
268C23C22×C3×D411yescharacteristic
269Q8—28no—
270Q8—28no—
271Q8—416no—
272Q8—48no—
273Q8—44no—
274C2×C4—28no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—28no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—416no—
284C2×C4—48no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4C3×#12012yes—
293C2×C4C2×C3×Q1614yes—
294C2×C4C3×#13714yes—
295C2×C4C22×C3×D412yes—
296C2×C4C3×H(5,𝔽2)14yes—
297C8—44no—
298C2×C3—24no—
299C2×C3#261311yescharacteristic
300C2×C3C2×#45111yescharacteristic
301C2×C3#300411yescharacteristic
302C2×C3#320912yes—
303C2×C3#321812yes—
304C22—24no—
305C22—24no—
306C22—22no—
307C22—22no—
308C22C3×#45114yes—
309C22C2×C3×#12011yescharacteristic
310C22C3×#53512yes—
311C22C22×C3×Q1611yescharacteristic
312C22C2×C3×#13712yes—
313C22C2×C3×#13711yescharacteristic
314C4—24no—
315C4—24no—
316C4—22no—
317C4—48no—
318C4—82no—
319C4—81no—
320C4—81no—
321C4C3×#53414yes—
322C3#2962711yescharacteristic, direct product factor, Sylow
323C2—24no—
324C2C3×#261311yescharacteristic
325C2C2×C3×#45111yescharacteristic
326C2C3×#300411yescharacteristic
327C2C3×#320912yes—
328C2C3×#321812yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
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
4. C3×#2043C2semidirect
5. C3×#2016C2semidirect
8. #29627C3direct (N × H)
32. #2043C2×C3semidirect
33. #2016C2×C3semidirect
114. C22×C3×C4Q16semidirect
226. C22×C4C3×Q16semidirect
322. C3#29627direct (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×#4512
C3×#28552
C3×#20432
C3×#20162
C3×#18172
C3×#17872
#296273