Finite Group Databasebeta

← C3×#10255 #10628862

Subgroups — C3×#10255 #10628862

Order 768.

582 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3245C211yesmaximal, characteristic
2C3×#3037C212yesmaximal, complemented
3C3×#1984C211yesmaximal, characteristic, complemented
4C3×#1492C212yesmaximal, complemented
5C3×#1292C211yesmaximal, characteristic, complemented
6#10255C311yesmaximal, characteristic, direct product factor, Sylow
7C2×C3×#137—22no—
8C3×#532C2212yes—
9C2×C3×C4×Q8C2211yescharacteristic
10C3×#474—22no—
11C3×#466—22no—
12C3×#464—24no—
13C3×#462—22no—
14C3×#460—22no—
15C3×#459—22no—
16C3×#423C2212yes—
17C2×C3×C4⋊C8C2211yescharacteristic
18C3×#417—22no—
19C2×C3×C4×C8C2211yescharacteristic
20C3×#336—22no—
21C3×D4⋊C8—24no—
22#3245C2×C311yescharacteristic
23#3037C2×C312yescomplemented
24#1984C2×C311yescharacteristic, complemented
25#1492C2×C312yescomplemented
26#1292C2×C311yescharacteristic, complemented
27C2×C3×#41—22no—
28C22×C3×Q8D411yescharacteristic
29C3×#137—44no—
30C3×#137—44no—
31C2×C3×Q16—24no—
32C2×C3×SD16—24no—
33C2×C3×D8—22no—
34C2×C3×M16—22no—
35C22×C3×C8—21no—
36C22×C3×C8D412yes—
37C3×C4⋊Q8D412yes—
38C3×#127D412yes—
39C3×#124—22no—
40C3×#121—22no—
41C3×#121—22no—
42C3×C4×Q8—22no—
43C3×C4×Q8—22no—
44C3×C4×D4—22no—
45C3×C4×D4—22no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C3×#107—22no—
51C3×C4⋊C8—22no—
52C3×C4⋊C8—22no—
53C3×C4⋊C8D414yes—
54C3×#103—24no—
55C3×#102—24no—
56C3×#102—24no—
57C3×C4×C8—22no—
58C2×#137—22no—
59#532C22×C312yes—
60C2×C4×Q8C22×C311yescharacteristic
61#474—22no—
62#466—22no—
63#464—24no—
64#462—22no—
65#460—22no—
66#459—22no—
67#423C22×C312yes—
68C2×C4⋊C8C22×C311yescharacteristic
69#417—22no—
70C2×C4×C8C22×C311yescharacteristic
71#336—22no—
72D4⋊C8—24no—
73C3×#41—42no—
74C3×#41—42no—
75C3×#41—42no—
76C3×#41—42no—
77C2×C3×Q8—21no—
78C2×C3×Q8—22no—
79C2×C3×Q8—22no—
80C2×C3×Q8—22no—
81C2×C3×Q8D812yescomplemented
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—42no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—42no—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4#4111yescharacteristic
92C3×Q16—48no—
93C3×SD16—48no—
94C3×D8—82no—
95C3×M16—42no—
96C3×M16—42no—
97C2×C3×C8—22no—
98C2×C3×C8—22no—
99C2×C3×C8—24no—
100C2×C3×C8—41no—
101C2×C3×C8—41no—
102C2×C3×C8—42no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—42no—
107C3×C22⋊C4—42no—
108C3×C22⋊C4—42no—
109C3×C42—41no—
110C3×C42—41no—
111C3×C42C2×D412yes—
112C3×C42C2×D412yes—
113C2×#41—22no—
114C22×Q8C3×D411yescharacteristic
115#137—44no—
116#137—44no—
117C2×Q16—24no—
118C2×SD16—24no—
119C2×D8—22no—
120C2×M16—22no—
121C22×C8—21no—
122C22×C8C3×D412yes—
123C4⋊Q8C3×D412yes—
124#127C3×D412yes—
125#124—22no—
126#121—22no—
127#121—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×D4—22no—
131C4×D4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4C3×D411yescharacteristic
134C2×C42—21no—
135C2×C42C23×C311yescharacteristic
136#107—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8C3×D414yes—
140#103—24no—
141#102—24no—
142#102—24no—
143C4×C8—22no—
144C23×C3—42no—
145C23×C3#12111yescharacteristic
146C3×Q8—24no—
147C3×Q8—42no—
148C3×Q8—42no—
149C3×Q8—42no—
150C3×D4—44no—
151C3×D4—42no—
152C3×D4—42no—
153C3×D4—42no—
154C3×D4—42no—
155C3×D4—82no—
156C3×D4—82no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—22no—
160C2×C3×C4—41no—
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—42no—
164C2×C3×C4—41no—
165C2×C3×C4—42no—
166C2×C3×C4—82no—
167C2×C3×C4—82no—
168C2×C3×C4#12012yes—
169C2×C3×C4#12011yescharacteristic
170C2×C3×C4#12112yes—
171C2×C3×C4#12711yescharacteristic
172C2×C3×C4C2×D811yescharacteristic
173C2×C3×C4#13511yescharacteristic
174C2×C3×C4AGL(1,ℤ/8)12yes—
175C3×C8—42no—
176C3×C8—44no—
177C3×C8—42no—
178#41—42no—
179#41—42no—
180#41—42no—
181#41—42no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8C3×D812yescomplemented
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—42no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—42no—
194C22×C4C2×C3×D411yescharacteristic
195C22×C4C2×C3×D411yescharacteristic
196C22×C4C3×#4111yescharacteristic
197Q16—48no—
198SD16—48no—
199D8—82no—
200M16—42no—
201M16—42no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—24no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—42no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—42no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C42—41no—
215C42—41no—
216C42C2×C3×D412yes—
217C42C2×C3×D412yes—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—82no—
221C22×C3—82no—
222C22×C3—82no—
223C22×C3#39211yescharacteristic
224C22×C3#46611yescharacteristic
225C22×C3#46911yescharacteristic
226C3×C4—21no—
227C3×C4—22no—
228C3×C4—42no—
229C3×C4—82no—
230C3×C4—81no—
231C3×C4#44712yes—
232C3×C4#44912yes—
233C3×C4#45511yescharacteristic
234C3×C4#45611yescharacteristic
235C23—42no—
236C23C3×#12111yescharacteristic
237Q8—24no—
238Q8—42no—
239Q8—42no—
240Q8—42no—
241D4—44no—
242D4—42no—
243D4—42no—
244D4—42no—
245D4—42no—
246D4—82no—
247D4—82no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4C3×#12012yes—
260C2×C4C3×#12011yescharacteristic
261C2×C4C3×#12112yes—
262C2×C4C3×#12711yescharacteristic
263C2×C4C2×C3×D811yescharacteristic
264C2×C4C3×#13511yescharacteristic
265C2×C4C3×AGL(1,ℤ/8)12yes—
266C8—42no—
267C8—44no—
268C8—42no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—162no—
272C2×C3#199911yescharacteristic
273C2×C3#200711yescharacteristic
274C2×C3#201711yescharacteristic
275C22—21no—
276C22—21no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22C3×#39211yescharacteristic
281C22C3×#46611yescharacteristic
282C22C3×#46911yescharacteristic
283C4—21no—
284C4—22no—
285C4—42no—
286C4—82no—
287C4—81no—
288C4C3×#44712yes—
289C4C3×#44912yes—
290C4C3×#45511yescharacteristic
291C4C3×#45611yescharacteristic
292C3#1025511yescharacteristic, direct product factor, Sylow
293C2—21no—
294C2—21no—
295C2—162no—
296C2C3×#199911yescharacteristic
297C2C3×#200711yescharacteristic
298C2C3×#201711yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC3×#3245C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#3037C2semidirect
3. C3×#1984C2semidirect
4. C3×#1492C2semidirect
5. C3×#1292C2semidirect
6. #10255C3direct (N × H)
23. #3037C2×C3semidirect
24. #1984C2×C3semidirect
25. #1492C2×C3semidirect
26. #1292C2×C3semidirect
81. C2×C3×Q8D8semidirect
186. C2×Q8C3×D8semidirect
292. C3#10255direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32452
C3×#30372
C3×#19842
C3×#14922
C3×#12922
#102553