Finite Group Databasebeta

← C3×#32931 #10651538

Subgroups — C3×#32931 #10651538

Order 768.

1630 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 366 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
1C24×C3×C8C211yesmaximal, characteristic
2C2×C3×#467C213yesmaximal, complemented
3C3×#2848C212yesmaximal
4C3×#1892C216yesmaximal, complemented
5C3×#1808C213yesmaximal, complemented
6#32931C311yesmaximal, characteristic, direct product factor, Sylow
7C24×C3×C4C2211yescharacteristic
8C22×C3×Q16C2211yescharacteristic, complemented
9C23×C3×C8—212no—
10C23×C3×C8C2213yes—
11C23×C3×C8C2213yes—
12C2×C3×C22⋊Q8C2216yes—
13C3×#467—212no—
14C2×C3×#107C2213yescomplemented
15C2×C3×#103C2216yescomplemented
16C3×#385—26no—
17C3×#380C2216yes—
18C3×#379—22no—
19C3×#340C2216yescomplemented
20C24×C8C2×C311yescharacteristic
21C2×#467C2×C313yescomplemented
22#2848C2×C312yes—
23#1892C2×C316yescomplemented
24#1808C2×C313yescomplemented
25C25×C3D411yescharacteristic
26C22×C3×Q8C2312yes—
27C23×C3×C4—26no—
28C23×C3×C4—26no—
29C23×C3×C4D413yes—
30C23×C3×C4C2313yes—
31C2×C3×Q16—26no—
32C22×C3×C8—232no—
33C22×C3×C8—26no—
34C22×C3×C8—212no—
35C22×C3×C8—212no—
36C22×C3×C8D4112yes—
37C22×C3×C8C2313yes—
38C22×C3×C8C2311yescharacteristic
39C3×C22⋊Q8—212no—
40C3×C22⋊Q8—212no—
41C2×C3×C4⋊C4—26no—
42C2×C3×C4⋊C4C2316yes—
43C2×C3×C22⋊C4—26no—
44C2×C3×C22⋊C4—26no—
45C3×#107—26no—
46C3×#103—212no—
47C3×#95—26no—
48C24×C4C22×C311yescharacteristic
49C22×Q16C22×C311yescharacteristic, complemented
50C23×C8—212no—
51C23×C8C22×C313yes—
52C23×C8C22×C313yes—
53C2×C22⋊Q8C22×C316yes—
54#467—212no—
55C2×#107C22×C313yescomplemented
56C2×#103C22×C316yescomplemented
57#385—26no—
58#380C22×C316yes—
59#379—22no—
60#340C22×C316yescomplemented
61C24×C3—26no—
62C24×C3—26no—
63C24×C3Q1614yescomplemented
64C24×C3C2×D413yes—
65C2×C3×Q8—26no—
66C2×C3×Q8—26no—
67C22×C3×C4—28no—
68C22×C3×C4—224no—
69C22×C3×C4—26no—
70C22×C3×C4—212no—
71C22×C3×C4—26no—
72C22×C3×C4—26no—
73C22×C3×C4—26no—
74C22×C3×C4—22no—
75C22×C3×C4C2×D413yes—
76C22×C3×C4C2×D416yes—
77C22×C3×C4#4116yes—
78C22×C3×C4C2411yescharacteristic
79C3×Q16—44no—
80C2×C3×C8—224no—
81C2×C3×C8—26no—
82C2×C3×C8—224no—
83C2×C3×C8C2×D416yes—
84C2×C3×C8C2×D416yes—
85C3×C4⋊C4—26no—
86C3×C4⋊C4—26no—
87C3×C4⋊C4—412no—
88C3×C22⋊C4—412no—
89C3×C22⋊C4—412no—
90C25C3×D411yescharacteristic
91C22×Q8C23×C312yes—
92C23×C4—26no—
93C23×C4—26no—
94C23×C4C3×D413yes—
95C23×C4C23×C313yes—
96C2×Q16—26no—
97C22×C8—232no—
98C22×C8—26no—
99C22×C8—212no—
100C22×C8—212no—
101C22×C8C3×D4112yes—
102C22×C8C23×C313yes—
103C22×C8C23×C311yescharacteristic
104C22⋊Q8—212no—
105C22⋊Q8—212no—
106C2×C4⋊C4—26no—
107C2×C4⋊C4C23×C316yes—
108C2×C22⋊C4—26no—
109C2×C22⋊C4—26no—
110#107—26no—
111#103—212no—
112#95—26no—
113C23×C3—28no—
114C23×C3—224no—
115C23×C3—26no—
116C23×C3—26no—
117C23×C3—224no—
118C23×C3#12116yes—
119C23×C3C2×Q1616yes—
120C23×C3#13516yes—
121C23×C3C22×D411yescharacteristic
122C3×Q8—42no—
123C3×Q8—46no—
124C2×C3×C4—224no—
125C2×C3×C4—26no—
126C2×C3×C4—26no—
127C2×C3×C4—26no—
128C2×C3×C4—212no—
129C2×C3×C4—46no—
130C2×C3×C4—46no—
131C2×C3×C4—412no—
132C2×C3×C4#12116yes—
133C2×C3×C4C22×D413yes—
134C2×C3×C4C2×#4113yes—
135C3×C8—26no—
136C3×C8#12014yes—
137C24—26no—
138C24—26no—
139C24C3×Q1614yescomplemented
140C24C2×C3×D413yes—
141C2×Q8—26no—
142C2×Q8—26no—
143C22×C4—28no—
144C22×C4—224no—
145C22×C4—26no—
146C22×C4—212no—
147C22×C4—26no—
148C22×C4—26no—
149C22×C4—26no—
150C22×C4—22no—
151C22×C4C2×C3×D413yes—
152C22×C4C2×C3×D416yes—
153C22×C4C3×#4116yes—
154C22×C4C24×C311yescharacteristic
155Q16—44no—
156C2×C8—224no—
157C2×C8—26no—
158C2×C8—224no—
159C2×C8C2×C3×D416yes—
160C2×C8C2×C3×D416yes—
161C4⋊C4—26no—
162C4⋊C4—26no—
163C4⋊C4—412no—
164C22⋊C4—412no—
165C22⋊C4—412no—
166C22×C3—232no—
167C22×C3—26no—
168C22×C3—212no—
169C22×C3—26no—
170C22×C3—212no—
171C22×C3#467112yescomplemented
172C22×C3C2×#12113yes—
173C22×C3C22×Q1611yescharacteristic
174C22×C3C2×#13513yes—
175C3×C4—26no—
176C3×C4—82no—
177C3×C4—86no—
178C3×C4C2×#12011yescharacteristic
179C3×C4#52513yes—
180C23—28no—
181C23—224no—
182C23—26no—
183C23—26no—
184C23—224no—
185C23C3×#12116yes—
186C23C2×C3×Q1616yes—
187C23C3×#13516yes—
188C23C22×C3×D411yescharacteristic
189Q8—42no—
190Q8—46no—
191C2×C4—224no—
192C2×C4—26no—
193C2×C4—26no—
194C2×C4—26no—
195C2×C4—212no—
196C2×C4—412no—
197C2×C4—46no—
198C2×C4—46no—
199C2×C4C3×#12116yes—
200C2×C4C22×C3×D413yes—
201C2×C4C2×C3×#4113yes—
202C8—26no—
203C8C3×#12014yes—
204C2×C3—212no—
205C2×C3#284711yescharacteristic
206C2×C3C2×#46713yes—
207C2×C3#305013yes—
208C22—232no—
209C22—212no—
210C22—26no—
211C22—212no—
212C22—26no—
213C22C3×#467112yescomplemented
214C22C2×C3×#12113yes—
215C22C22×C3×Q1611yescharacteristic
216C22C2×C3×#13513yes—
217C4—26no—
218C4—86no—
219C4—82no—
220C4C2×C3×#12011yescharacteristic
221C4C3×#52513yes—
222C3#3293111yescharacteristic, direct product factor, Sylow
223C2—212no—
224C2C3×#284711yescharacteristic
225C2C2×C3×#46713yes—
226C2C3×#305013yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#467C2semidirect
4. C3×#1892C2semidirect
5. C3×#1808C2semidirect
6. #32931C3direct (N × H)
8. C22×C3×Q16C22semidirect
14. C2×C3×#107C22semidirect
15. C2×C3×#103C22semidirect
19. C3×#340C22semidirect
21. C2×#467C2×C3semidirect
23. #1892C2×C3semidirect
24. #1808C2×C3semidirect
49. C22×Q16C22×C3semidirect
55. C2×#107C22×C3semidirect
56. C2×#103C22×C3semidirect
60. #340C22×C3semidirect
63. C24×C3Q16semidirect
139. C24C3×Q16semidirect
171. C22×C3#467semidirect
213. C22C3×#467semidirect
222. C3#32931direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×C82
C2×C3×#4672
C3×#28482
C3×#18922
C3×#18082
#329313