Finite Group Databasebeta

← #10689656

Subgroups — #10689656

Order 768.

476 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 147 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
1#83230C211yesmaximal, characteristic
2C3×#1956C211yesmaximal, characteristic
3C2×#4546C211yesmaximal, characteristic
4#69567C212yesmaximal
5#67655C212yesmaximal
6#8929—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×#103—22no—
9C2×C3×C4×C8C2211yescharacteristic
10C3×#384C2211yescharacteristic
11#5229—22no—
12#4984—22no—
13Dic3×C4⋊C4—22no—
14C2×#807—22no—
15C4×C4⋊Dic3C2212yes—
16C2×C4⋊(C3⋊C8)C2212yes—
17#4546C414yescomplemented
18#4529—22no—
19#2369—31no—
20#1956D311yescharacteristic
21#1947—32no—
22C2×#333—31no—
23#1279—32no—
24C22×C3×Q8D412yes—
25C22×C3×C8—21no—
26C3×C4⋊Q8—22no—
27C3×C4⋊Q8C2×C412yes—
28C3×C4⋊Q8Q812yes—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—21no—
32C2×C3×C42C2311yescharacteristic
33C2×C4⋊Dic3—22no—
34C2×C4⋊Dic3D412yes—
35C2×C4⋊Dic3D412yes—
36C2×C4×Dic3—22no—
37C2×C4×Dic3—22no—
38C22×C3⋊C8—22no—
39C3×#103—44no—
40C3×C4×C8—22no—
41C3⋊#95—24no—
42C3⋊#95—22no—
43#807—44no—
44C4⋊(C3⋊C8)—22no—
45C4⋊(C3⋊C8)C2×C414yes—
46C2×C4⋊Q8C2×D311yescharacteristic
47C2×#107—62no—
48C2×C4⋊C8—32no—
49C2×#103—22no—
50C2×C4×C8C2×D311yescharacteristic
51#391—62no—
52#389—62no—
53#384C2×D311yescharacteristic
54C4×C4⋊C4—32no—
55C4×C4⋊C4—62no—
56#340—62no—
57#333—34no—
58C2×C3×Q8—22no—
59C2×C3×Q8—42no—
60C2×C3×Q8—42no—
61C2×C3×Q8C4⋊C414yes—
62C22×C3×C4—22no—
63C22×C3×C4—41no—
64C22×C3×C4C2×D412yes—
65C22×C3×C4C2×D411yescharacteristic
66C22×Dic3—22no—
67C22×Dic3—22no—
68C2×C3×C8—22no—
69C2×C3×C8—41no—
70C2×C3×C8—41no—
71C3×C4⋊C4—24no—
72C3×C4⋊C4—24no—
73C3×C4⋊C4—42no—
74C3×C4⋊C4—42no—
75C3×C4⋊C4—42no—
76C3×C42C22×C411yescharacteristic
77C3×C42C2×Q811yescharacteristic
78C3×C42#4112yes—
79C4⋊Dic3—22no—
80C4⋊Dic3—22no—
81C4⋊Dic3—22no—
82C4⋊Dic3—82no—
83C4⋊Dic3Q1614yes—
84C4×Dic3—42no—
85C4×Dic3—42no—
86C4×Dic3—82no—
87C2×C3⋊C8—24no—
88C2×C3⋊C8—42no—
89C2×C3⋊C8—42no—
90C22×Q8#6712yes—
91C22×C8—21no—
92C22×C8—62no—
93C4⋊Q8—22no—
94C4⋊Q8Dic612yes—
95C4⋊Q8C4×D312yes—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—62no—
102C2×C42—62no—
103C2×C42—62no—
104C2×C42C22×D311yescharacteristic
105#107—124no—
106C4⋊C8—34no—
107C4⋊C8—62no—
108#103—44no—
109C4×C8—22no—
110#95—64no—
111#95—62no—
112C23×C3#12011yescharacteristic
113C3×Q8—44no—
114C3×Q8—82no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—24no—
118C2×C3×C4—42no—
119C2×C3×C4—42no—
120C2×C3×C4—82no—
121C2×C3×C4—81no—
122C2×C3×C4C2×C4⋊C411yescharacteristic
123C2×C3×C4C4×D412yes—
124C2×C3×C4C22⋊Q812yes—
125C2×C3×C4#12311yescharacteristic
126C2×C3×C4C2×Q1612yes—
127C2×C3×C4#13512yes—
128C2×C3×C4AGL(1,ℤ/8)12yes—
129C2×C3×C4#13712yes—
130C2×Dic3—24no—
131C2×Dic3—42no—
132C2×Dic3—42no—
133C2×Dic3—42no—
134C2×Dic3—42no—
135C2×Dic3—42no—
136C3×C8—42no—
137C3⋊C8—44no—
138C2×Q8—22no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8Dic3⋊C414yes—
142C22×C4—22no—
143C22×C4—41no—
144C22×C4—62no—
145C22×C4—62no—
146C22×C4D3×D412yes—
147C22×C4C2×#6711yescharacteristic
148C2×C8—22no—
149C2×C8—41no—
150C2×C8—41no—
151C2×C8—64no—
152C2×C8—122no—
153C2×C8—122no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—34no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—242no—
164C42—122no—
165C42—122no—
166C42—242no—
167C42C2×Dic611yescharacteristic
168C42C2×C4×D311yescharacteristic
169C42#23712yes—
170C22×C3#38511yescharacteristic
171C22×C3#44912yes—
172C22×C3#45112yes—
173C22×C3#45311yescharacteristic
174C22×C3#45511yescharacteristic
175C3×C4—21no—
176C3×C4—21no—
177C3×C4—44no—
178C3×C4—161no—
179C3×C4C4×Q1612yes—
180C3×C4#44012yes—
181C3×C4#47712yes—
182C3×C4#47912yes—
183Dic3—44no—
184Dic3—82no—
185C23#93711yescharacteristic
186Q8—44no—
187Q8—82no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—24no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—64no—
194C2×C4—82no—
195C2×C4—81no—
196C2×C4—122no—
197C2×C4—122no—
198C2×C4—122no—
199C2×C4—122no—
200C2×C4—122no—
201C2×C4#87812yes—
202C2×C4#67⋊C412yes—
203C2×C4#91412yes—
204C2×C4#91612yes—
205C2×C4D3×Q1612yes—
206C2×C4#91812yes—
207C2×C4C2×Dic3⋊C411yescharacteristic
208C2×C4#93511yescharacteristic
209C8—42no—
210C8—124no—
211C2×C3#162411yescharacteristic
212C2×C3#163311yescharacteristic
213C2×C3#164412yes—
214C2×C3#185611yescharacteristic
215C2×C3#186312yes—
216C22#494611yescharacteristic
217C22#506111yescharacteristic
218C22#507711yescharacteristic
219C22#517212yes—
220C22#518612yes—
221C4—21no—
222C4—21no—
223C4—44no—
224C4—124no—
225C4—161no—
226C4—242no—
227C4#479012yes—
228C4#479312yes—
229C4#479912yes—
230C4#481712yes—
231C3#892911yescharacteristic, complemented, Sylow
232C2#6964211yescharacteristic
233C2#7023512yes—
234C2#7123812yes—
235C2#7126511yescharacteristic
236C2#7130711yescharacteristic
subgroupG/H
centerC23#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1956C2
socle, prime radicalC23×C3#120
cosocleC2×C3×C42C23
hypercenter#1956D3
hypocenterC3#8929
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
17. #4546C4semidirect
231. C3#8929semidirect

Maximal subgroups

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

subgroupindex
#832302
C3×#19562
C2×#45462
#695672
#676552
#89293