Finite Group Databasebeta

← C3×#7206 #10625813

Subgroups — C3×#7206 #10625813

Order 768.

284 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 102 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×#1984C211yesmaximal, characteristic
2C3×C4×#101C211yesmaximal, characteristic
3C3×#1461C211yesmaximal, characteristic
4#7206C311yesmaximal, characteristic, direct product factor, Sylow
5C2×C3×C4×Q8C2211yescharacteristic
6C3×#431—21no—
7C2×C3×C4⋊C8C411yescharacteristic
8C2×C3×#101C412yes—
9C3×C4×M16—21no—
10C2×C3×C4×C8C411yescharacteristic
11C3×#343—21no—
12C3×#336—21no—
13C3×#336—21no—
14C3×#324—21no—
15C3×#324C412yes—
16#1984C2×C311yescharacteristic
17C4×#101C2×C311yescharacteristic
18#1461C2×C311yescharacteristic
19C22×C3×Q8C2×C411yescharacteristic
20C2×C3×M16—22no—
21C22×C3×C8—21no—
22C22×C3×C8—21no—
23C3×C4×Q8—41no—
24C3×C4×Q8—41no—
25C2×C3×C4⋊C4C2×C411yescharacteristic
26C2×C3×C4⋊C4D411yescharacteristic
27C2×C3×C4⋊C4Q811yescharacteristic
28C2×C3×C42C2×C411yescharacteristic
29C2×C3×C42D411yescharacteristic
30C2×C3×C42D411yescharacteristic
31C3×C4⋊C8—41no—
32C3×C4⋊C8—41no—
33C3×#101—22no—
34C3×#101C814yescomplemented
35C3×C22⋊C8—22no—
36C3×#97—41no—
37C3×C4×C8—41no—
38C3×C4×C8—41no—
39C3×C4×C8—41no—
40C2×C4×Q8C22×C311yescharacteristic
41#431—21no—
42C2×C4⋊C8C3×C411yescharacteristic
43C2×#101C3×C412yes—
44C4×M16—21no—
45C2×C4×C8C3×C411yescharacteristic
46#343—21no—
47#336—21no—
48#336—21no—
49#324—21no—
50#324C3×C412yes—
51C2×C3×Q8—41no—
52C2×C3×Q8—41no—
53C2×C3×Q8C2×C812yes—
54C2×C3×Q8M1612yes—
55C22×C3×C4C4211yescharacteristic
56C22×C3×C4C22⋊C411yescharacteristic
57C22×C3×C4C22⋊C411yescharacteristic
58C22×C3×C4C22⋊C411yescharacteristic
59C22×C3×C4C4⋊C411yescharacteristic
60C22×C3×C4C4⋊C411yescharacteristic
61C22×C3×C4C4⋊C411yescharacteristic
62C3×M16—24no—
63C3×M16—42no—
64C2×C3×C8—22no—
65C2×C3×C8—42no—
66C2×C3×C8—41no—
67C2×C3×C8—41no—
68C2×C3×C8—42no—
69C2×C3×C8—42no—
70C2×C3×C8—41no—
71C3×C4⋊C4—41no—
72C3×C4⋊C4—41no—
73C3×C4⋊C4—41no—
74C3×C42—21no—
75C3×C42—21no—
76C3×C42—21no—
77C3×C42—21no—
78C3×C42—21no—
79C3×C42—21no—
80C22×Q8C2×C3×C411yescharacteristic
81C2×M16—22no—
82C22×C8—21no—
83C22×C8—21no—
84C4×Q8—41no—
85C4×Q8—41no—
86C2×C4⋊C4C2×C3×C411yescharacteristic
87C2×C4⋊C4C3×D411yescharacteristic
88C2×C4⋊C4C3×Q811yescharacteristic
89C2×C42C2×C3×C411yescharacteristic
90C2×C42C3×D411yescharacteristic
91C2×C42C3×D411yescharacteristic
92C4⋊C8—41no—
93C4⋊C8—41no—
94#101—22no—
95#101C3×C814yescomplemented
96C22⋊C8—22no—
97#97—41no—
98C4×C8—41no—
99C4×C8—41no—
100C4×C8—41no—
101C23×C3#9511yescharacteristic
102C3×Q8—42no—
103C3×Q8—42no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—21no—
114C2×C3×C4—21no—
115C2×C3×C4—41no—
116C2×C3×C4—41no—
117C2×C3×C4—41no—
118C2×C3×C4C4×C811yescharacteristic
119C2×C3×C4#9711yescharacteristic
120C2×C3×C4C22⋊C812yes—
121C2×C3×C4#9911yescharacteristic
122C2×C3×C4#9911yescharacteristic
123C2×C3×C4C4⋊C812yes—
124C3×C8—42no—
125C3×C8—44no—
126C3×C8—82no—
127C3×C8—81no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8C2×C3×C812yes—
131C2×Q8C3×M1612yes—
132C22×C4C3×C4211yescharacteristic
133C22×C4C3×C22⋊C411yescharacteristic
134C22×C4C3×C22⋊C411yescharacteristic
135C22×C4C3×C22⋊C411yescharacteristic
136C22×C4C3×C4⋊C411yescharacteristic
137C22×C4C3×C4⋊C411yescharacteristic
138C22×C4C3×C4⋊C411yescharacteristic
139M16—24no—
140M16—42no—
141C2×C8—22no—
142C2×C8—42no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C42—21no—
152C42—21no—
153C42—21no—
154C42—21no—
155C42—21no—
156C42—21no—
157C22×C3—21no—
158C22×C3—21no—
159C22×C3#33611yescharacteristic
160C22×C3#34211yescharacteristic
161C22×C3#34411yescharacteristic
162C3×C4—22no—
163C3×C4—21no—
164C3×C4—21no—
165C3×C4—22no—
166C3×C4—21no—
167C3×C4—41no—
168C3×C4—41no—
169C3×C4—41no—
170C3×C4#35511yescharacteristic
171C3×C4#35611yescharacteristic
172C23C3×#9511yescharacteristic
173Q8—42no—
174Q8—42no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4C3×C4×C811yescharacteristic
190C2×C4C3×#9711yescharacteristic
191C2×C4C3×C22⋊C812yes—
192C2×C4C3×#9911yescharacteristic
193C2×C4C3×#9911yescharacteristic
194C2×C4C3×C4⋊C812yes—
195C8—42no—
196C8—44no—
197C8—82no—
198C8—81no—
199C2×C3—21no—
200C2×C3—21no—
201C2×C3#128211yescharacteristic
202C2×C3#139411yescharacteristic
203C2×C3#139511yescharacteristic
204C22—21no—
205C22—21no—
206C22C3×#33611yescharacteristic
207C22C3×#34211yescharacteristic
208C22C3×#34411yescharacteristic
209C4—22no—
210C4—21no—
211C4—21no—
212C4—22no—
213C4—21no—
214C4—41no—
215C4—41no—
216C4—41no—
217C4C3×#35511yescharacteristic
218C4C3×#35611yescharacteristic
219C3#720611yescharacteristic, direct product factor, Sylow
220C2—21no—
221C2—21no—
222C2C3×#128211yescharacteristic
223C2C3×#139411yescharacteristic
224C2C3×#139511yescharacteristic
subgroupG/H
centerC2×C3×C4#99
commutatorC2×C4C3×C4×C8
Frattini, cosocleC2×C4×Q8C22×C3
socleC22×C3#342
prime radicalC23×C3#95
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
4. #7206C3direct (N × H)
34. C3×#101C8semidirect
95. #101C3×C8semidirect
219. C3#7206direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#19842
C3×C4×#1012
C3×#14612
#72063