Finite Group Databasebeta

← #10697479

Subgroups — #10697479

Order 768.

466 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3162C211yesmaximal, characteristic
2#83323C212yesmaximal
3#71838C212yesmaximal
4#69822C211yesmaximal, characteristic
5#67661C211yesmaximal, characteristic
6#10227—31nomaximal, Sylow
7C3×#533C2212yes—
8C2×#869C2211yescharacteristic
9C3×#501—22no—
10C3×#501—22no—
11C3×#427—22no—
12C2×C3×#97C2211yescharacteristic
13C2×#816—22no—
14#5080—22no—
15#5079—22no—
16#5043—24no—
17#5020—24no—
18#4936C2211yescharacteristic
19#4925C2212yes—
20#4538—24no—
21#3229—32no—
22#3162D311yescharacteristic
23#1960—31no—
24#1518—32no—
25#1281—31no—
26C22×Dic6—21no—
27C22×C3×C8D412yes—
28C3×C4⋊Q8D412yes—
29C3×C4⋊Q8D412yes—
30C3×#125—22no—
31C3×C22⋊Q8—22no—
32C3×C22⋊Q8—22no—
33C3×C4×Q8—22no—
34C3×C4×Q8—22no—
35C3×#117—22no—
36C2×C3×C42C2311yescharacteristic
37C2×C4⋊Dic3—22no—
38C2×C4⋊Dic3—21no—
39#869—22no—
40#869—21no—
41#869D412yes—
42C3×#107—24no—
43C3×#106—22no—
44C3×#106—22no—
45C3×#97—22no—
46#835—28no—
47C3⋊#95—22no—
48#830D414yes—
49#816—44no—
50#808—28no—
51C4⋊(C3⋊C8)—22no—
52C4×C3⋊C8—22no—
53#533C2×D312yes—
54C2×C4⋊Q8—31no—
55#501—22no—
56#501—22no—
57#498—62no—
58#487—62no—
59#483—64no—
60#475—64no—
61#431—32no—
62#427—22no—
63C2×#103—62no—
64C2×#97C2×D311yescharacteristic
65#382—31no—
66#328—64no—
67C2×C3×Q8—22no—
68C2×C3×Q8—22no—
69C22×C3×C4C2×D412yes—
70C22×C3×C4C2×D411yescharacteristic
71C22×Dic3—41no—
72C22×Dic3—41no—
73C2×Dic6—22no—
74C2×Dic6—42no—
75C2×Dic6—42no—
76C2×Dic6—41no—
77C2×C3×C8—22no—
78C2×C3×C8—22no—
79C2×C3×C8SD1614yes—
80C3×C4⋊C4—22no—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—22no—
83C3×C4⋊C4—22no—
84C3×C4⋊C4—42no—
85C3×C4⋊C4—42no—
86C3×C22⋊C4—42no—
87C3×C42—42no—
88C3×C42C2×D411yescharacteristic
89C3×C42C2×D412yes—
90C3×C42#4111yescharacteristic
91C4⋊Dic3—24no—
92C4⋊Dic3—42no—
93C4⋊Dic3—42no—
94C2×C3⋊C8—24no—
95C22×Q8—61no—
96C22×C8#6712yes—
97C4⋊Q8—32no—
98C4⋊Q8—61no—
99C4⋊Q8—62no—
100C4⋊Q8#6712yes—
101C4⋊Q8#6712yes—
102#125—22no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C4×Q8—22no—
106C4×Q8—22no—
107#117—22no—
108C2×C4⋊C4—62no—
109C2×C4⋊C4—61no—
110C2×C42C22×D311yescharacteristic
111#107—24no—
112#106—22no—
113#106—22no—
114#106—68no—
115C4⋊C8—62no—
116#103—68no—
117#103—124no—
118C22⋊C8—34no—
119#97—22no—
120C4×C8—62no—
121#95—62no—
122C23×C3#12711yescharacteristic
123C3×Q8—44no—
124C3×Q8—82no—
125C2×C3×C4—42no—
126C2×C3×C4—42no—
127C2×C3×C4—42no—
128C2×C3×C4#12012yes—
129C2×C3×C4#12011yescharacteristic
130C2×C3×C4#12111yescharacteristic
131C2×C3×C4#12112yes—
132C2×C3×C4C2×SD1612yes—
133C2×C3×C4#13712yes—
134C2×Dic3—42no—
135C2×Dic3—81no—
136C2×Dic3—81no—
137C2×Dic3—82no—
138C2×Dic3—81no—
139Dic6—44no—
140Dic6—82no—
141C3×C8—24no—
142C3⋊C8—44no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—62no—
146C2×Q8—121no—
147C2×Q8—122no—
148C2×Q8—122no—
149C22×C4—121no—
150C22×C4—121no—
151C22×C4D3×D412yes—
152C22×C4C2×#6711yescharacteristic
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—64no—
156C2×C8#21514yes—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—64no—
164C4⋊C4—122no—
165C4⋊C4—122no—
166C22⋊C4—42no—
167C42—42no—
168C42D3×D411yescharacteristic
169C42#23711yescharacteristic
170C42C2×#6712yes—
171C22×C3—22no—
172C22×C3#39211yescharacteristic
173C22×C3#49211yescharacteristic
174C22×C3#49711yescharacteristic
175C3×C4—22no—
176C3×C4—22no—
177C3×C4—82no—
178C3×C4—82no—
179C3×C4#44812yes—
180C3×C4#46112yes—
181Dic3—82no—
182Dic3—161no—
183C23#94011yescharacteristic
184Q8—44no—
185Q8—82no—
186Q8—124no—
187Q8—242no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—122no—
192C2×C4—241no—
193C2×C4—241no—
194C2×C4—242no—
195C2×C4—241no—
196C2×C4#93712yes—
197C2×C4#93811yescharacteristic
198C2×C4#93912yes—
199C2×C4C2×#21512yes—
200C2×C4#94212yes—
201C2×C4#95311yescharacteristic
202C8—24no—
203C8—124no—
204C2×C3—21no—
205C2×C3—21no—
206C2×C3#200912yes—
207C2×C3#201311yescharacteristic
208C22—22no—
209C22#508511yescharacteristic
210C22#508611yescharacteristic
211C22#522311yescharacteristic
212C4—22no—
213C4—22no—
214C4—82no—
215C4—82no—
216C4—242no—
217C4—481no—
218C4#508312yes—
219C4#522712yes—
220C3#1022711yescharacteristic, complemented, Sylow
221C2—21no—
222C2—21no—
223C2#7179111yescharacteristic
224C2#7212012yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3162C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#3162D3
hypocenterC3#10227
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
220. C3#10227semidirect

Maximal subgroups

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

subgroupindex
C3×#31622
#833232
#718382
#698222
#676612
#102273