Finite Group Databasebeta

← C3×#10258 #10628865

Subgroups — C3×#10258 #10628865

Order 768.

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

normal subgroups only

This group has 526 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×#3249C211yesmaximal, characteristic, complemented
2C3×#3077C212yesmaximal
3C3×#1954C211yesmaximal, characteristic
4C3×#1493C212yesmaximal, complemented
5C3×#1292C211yesmaximal, characteristic, complemented
6#10258C311yesmaximal, characteristic, direct product factor, Sylow
7C3×#533C2212yes—
8C2×C3×C4×D4C2211yescharacteristic
9C3×#478—24no—
10C3×#477—22no—
11C3×#476—22no—
12C3×#474—22no—
13C3×#467—22no—
14C3×#462—22no—
15C3×#428—22no—
16C3×#423C2212yes—
17C2×C3×C4⋊C8C2211yescharacteristic
18C3×#418—22no—
19C2×C3×C22⋊C8—22no—
20C2×C3×C4×C8C2211yescharacteristic
21C3×Q8⋊C8—24no—
22#3249C2×C311yescharacteristic, complemented
23#3077C2×C312yes—
24#1954C2×C311yescharacteristic
25#1493C2×C312yescomplemented
26#1292C2×C311yescharacteristic, complemented
27C22×C3×D4D411yescharacteristic
28C23×C3×C4—21no—
29C2×C3×Q16—22no—
30C2×C3×M16—22no—
31C22×C3×C8—21no—
32C22×C3×C8D412yes—
33C3×C4⋊Q8D412yes—
34C3×C4⋊Q8D412yes—
35C3×#125—22no—
36C3×C22⋊Q8—22no—
37C3×C22⋊Q8—22no—
38C3×C4×Q8—22no—
39C3×C4×Q8—22no—
40C3×C4×D4—22no—
41C3×C4×D4—22no—
42C3×#117—22no—
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C42C2311yescharacteristic
46C3×#107—24no—
47C3×#107—22no—
48C3×#106—24no—
49C3×C4⋊C8—22no—
50C3×C4⋊C8—22no—
51C3×C4⋊C8D414yes—
52C3×#103—24no—
53C3×#102—24no—
54C3×#102—24no—
55C3×C22⋊C8—44no—
56C3×C4×C8—22no—
57#533C22×C312yes—
58C2×C4×D4C22×C311yescharacteristic
59#478—24no—
60#477—22no—
61#476—22no—
62#474—22no—
63#467—22no—
64#462—22no—
65#428—22no—
66#423C22×C312yes—
67C2×C4⋊C8C22×C311yescharacteristic
68#418—22no—
69C2×C22⋊C8—22no—
70C2×C4×C8C22×C311yescharacteristic
71Q8⋊C8—24no—
72C24×C3—21no—
73C2×C3×Q8—22no—
74C2×C3×Q8—22no—
75C2×C3×D4—22no—
76C2×C3×D4—21no—
77C2×C3×D4—22no—
78C2×C3×D4Q1612yescomplemented
79C22×C3×C4—21no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—41no—
83C22×C3×C4—41no—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4#4111yescharacteristic
87C3×Q16—82no—
88C3×M16—42no—
89C3×M16—42no—
90C2×C3×C8—22no—
91C2×C3×C8—24no—
92C2×C3×C8—22no—
93C2×C3×C8—41no—
94C2×C3×C8—41no—
95C2×C3×C8—42no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—22no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—42no—
102C3×C4⋊C4—42no—
103C3×C22⋊C4—42no—
104C3×C22⋊C4—42no—
105C3×C42—42no—
106C3×C42C2×D412yes—
107C3×C42C2×D412yes—
108C22×D4C3×D411yescharacteristic
109C23×C4—21no—
110C2×Q16—22no—
111C2×M16—22no—
112C22×C8—21no—
113C22×C8C3×D412yes—
114C4⋊Q8C3×D412yes—
115C4⋊Q8C3×D412yes—
116#125—22no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×D4—22no—
122C4×D4—22no—
123#117—22no—
124C2×C4⋊C4C3×D411yescharacteristic
125C2×C22⋊C4—21no—
126C2×C42C23×C311yescharacteristic
127#107—24no—
128#107—22no—
129#106—24no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8C3×D414yes—
133#103—24no—
134#102—24no—
135#102—24no—
136C22⋊C8—44no—
137C4×C8—22no—
138C23×C3—22no—
139C23×C3—42no—
140C23×C3—42no—
141C23×C3—41no—
142C23×C3—41no—
143C23×C3#12111yescharacteristic
144C3×Q8—44no—
145C3×Q8—42no—
146C3×Q8—42no—
147C3×D4—24no—
148C3×D4—42no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—41no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—41no—
156C2×C3×C4—82no—
157C2×C3×C4—81no—
158C2×C3×C4—81no—
159C2×C3×C4#12012yes—
160C2×C3×C4#12011yescharacteristic
161C2×C3×C4#12112yes—
162C2×C3×C4#12711yescharacteristic
163C2×C3×C4C2×Q1611yescharacteristic
164C2×C3×C4#13511yescharacteristic
165C2×C3×C4#13712yes—
166C3×C8—42no—
167C3×C8—44no—
168C3×C8—42no—
169C24—21no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×D4—22no—
173C2×D4—21no—
174C2×D4—22no—
175C2×D4C3×Q1612yescomplemented
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4C2×C3×D411yescharacteristic
182C22×C4C2×C3×D411yescharacteristic
183C22×C4C3×#4111yescharacteristic
184Q16—82no—
185M16—42no—
186M16—42no—
187C2×C8—22no—
188C2×C8—24no—
189C2×C8—22no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—42no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C42—42no—
203C42C2×C3×D412yes—
204C42C2×C3×D412yes—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—44no—
208C22×C3—42no—
209C22×C3—82no—
210C22×C3—81no—
211C22×C3—81no—
212C22×C3#39211yescharacteristic
213C22×C3#46711yescharacteristic
214C22×C3#47011yescharacteristic
215C3×C4—21no—
216C3×C4—22no—
217C3×C4—81no—
218C3×C4—82no—
219C3×C4—82no—
220C3×C4#45112yes—
221C3×C4#45212yes—
222C3×C4#45311yescharacteristic
223C3×C4#45411yescharacteristic
224C23—22no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23C3×#12111yescharacteristic
230Q8—44no—
231Q8—42no—
232Q8—42no—
233D4—24no—
234D4—42no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4C3×#12012yes—
246C2×C4C3×#12011yescharacteristic
247C2×C4C3×#12112yes—
248C2×C4C3×#12711yescharacteristic
249C2×C4C2×C3×Q1611yescharacteristic
250C2×C4C3×#13511yescharacteristic
251C2×C4C3×#13712yes—
252C8—42no—
253C8—44no—
254C8—42no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—82no—
258C2×C3#200111yescharacteristic
259C2×C3#200611yescharacteristic
260C2×C3#201711yescharacteristic
261C22—21no—
262C22—21no—
263C22—44no—
264C22—42no—
265C22—82no—
266C22—81no—
267C22—81no—
268C22C3×#39211yescharacteristic
269C22C3×#46711yescharacteristic
270C22C3×#47011yescharacteristic
271C4—21no—
272C4—22no—
273C4—81no—
274C4—82no—
275C4—82no—
276C4C3×#45112yes—
277C4C3×#45212yes—
278C4C3×#45311yescharacteristic
279C4C3×#45411yescharacteristic
280C3#1025811yescharacteristic, direct product factor, Sylow
281C2—21no—
282C2—21no—
283C2—82no—
284C2C3×#200111yescharacteristic
285C2C3×#200611yescharacteristic
286C2C3×#201711yescharacteristic
subgroupG/H
center, socleC22×C3#392
commutator, Frattini, cosocleC2×C42C23×C3
prime radicalC22×C3×D4D4
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
1. C3×#3249C2semidirect
4. C3×#1493C2semidirect
5. C3×#1292C2semidirect
6. #10258C3direct (N × H)
22. #3249C2×C3semidirect
25. #1493C2×C3semidirect
26. #1292C2×C3semidirect
78. C2×C3×D4Q16semidirect
175. C2×D4C3×Q16semidirect
280. C3#10258direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32492
C3×#30772
C3×#19542
C3×#14932
C3×#12922
#102583