Finite Group Databasebeta

← #10693051

Subgroups — #10693051

Order 768.

454 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2983C211yesmaximal, characteristic
2#82481C211yesmaximal, characteristic
3#71637C211yesmaximal, characteristic
4#70770C212yesmaximal
5#67738C212yesmaximal
6#9488—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×C4×Dic6C2211yescharacteristic
9C3×#446—24no—
10C3×#432—22no—
11C3×#431C2212yes—
12#5229—21no—
13C2×#836—21no—
14#4986C412yes—
15#4979C412yes—
16#4972—21no—
17#4936C2211yescharacteristic
18#4933—22no—
19#4924C2212yes—
20#4530—21no—
21#4456—24no—
22#2983D311yescharacteristic
23#2581—31no—
24#1801—31no—
25#1487—32no—
26#1286—32no—
27C22×C3×Q8D411yescharacteristic
28C22×Dic6C2×C411yescharacteristic
29C3×C4×Q8—22no—
30C3×C4×Q8—22no—
31C2×C3×C4⋊C4C2×C412yes—
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C42D412yes—
34C2×C3×C42C2311yescharacteristic
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3C2×C411yescharacteristic
37C2×Dic3⋊C4—22no—
38C2×Dic3⋊C4—21no—
39C2×C4×Dic3—21no—
40C2×#208—22no—
41C4×Dic6—22no—
42C4×Dic6—22no—
43C3×C4⋊C8—24no—
44C3×C4⋊C8—22no—
45C3×C22⋊C8—22no—
46C3×#97—24no—
47C3×C4×C8D414yes—
48#836—42no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51C3⋊#95—22no—
52C3⋊#95—21no—
53C3⋊#95C2×C412yes—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56C2×C4×Q8—31no—
57C2×C4×Q8C2×D311yescharacteristic
58#446—24no—
59#432—22no—
60#431C2×D312yes—
61#423—32no—
62C2×#101—61no—
63#400—32no—
64#395—32no—
65#391—61no—
66#389—61no—
67#387—62no—
68#382—31no—
69#343—61no—
70Q8⋊C8—64no—
71C2×C3×Q8—22no—
72C2×C3×Q8—21no—
73C2×C3×Q8—21no—
74C2×C3×Q8—22no—
75C22×C3×C4C22⋊C412yes—
76C22×C3×C4C22⋊C412yes—
77C22×C3×C4C22×C411yescharacteristic
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4C2×D411yescharacteristic
80C22×Dic3—21no—
81C22×Dic3—22no—
82C22×Dic3—21no—
83C2×Dic6—22no—
84C2×Dic6—42no—
85C2×C3×C8—24no—
86C3×C4⋊C4—24no—
87C3×C4⋊C4—22no—
88C3×C42—22no—
89C3×C42—22no—
90C3×C42C2×D412yes—
91C3×C42#4112yes—
92C4⋊Dic3—22no—
93C4⋊Dic3—41no—
94C4⋊Dic3—41no—
95Dic3⋊C4—44no—
96Dic3⋊C4—41no—
97Dic3⋊C4—41no—
98C4×Dic3—42no—
99#208—42no—
100#208—42no—
101C2×C3⋊C8—42no—
102C22×Q8—31no—
103C22×Q8D1211yescharacteristic
104C2×M16—62no—
105C4×Q8—22no—
106C4×Q8—22no—
107C4×Q8—62no—
108C4×Q8—62no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4—61no—
112C2×C4⋊C4—61no—
113C2×C4⋊C4C4×D312yes—
114C2×C4⋊C4D1211yescharacteristic
115C2×C42—61no—
116C2×C42#6712yes—
117C2×C42C22×D311yescharacteristic
118C4⋊C8—24no—
119C4⋊C8—22no—
120C4⋊C8—62no—
121C4⋊C8—62no—
122#101—122no—
123C22⋊C8—22no—
124#97—24no—
125C4×C8#6714yes—
126#95—32no—
127#95—62no—
128#95—62no—
129#95—62no—
130#95—61no—
131C23×C3C2×C22⋊C411yescharacteristic
132C3×Q8—42no—
133C3×Q8—42no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—41no—
139C2×C3×C4—41no—
140C2×C3×C4—42no—
141C2×C3×C4C4×D412yes—
142C2×C3×C4#12011yescharacteristic
143C2×C3×C4#12112yes—
144C2×C3×C4#12311yescharacteristic
145C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
146C2×C3×C4#13712yes—
147C2×C3×C4#13711yescharacteristic
148C2×Dic3—22no—
149C2×Dic3—41no—
150C2×Dic3—41no—
151C2×Dic3—42no—
152C2×Dic3—42no—
153C2×Dic3—42no—
154C2×Dic3—41no—
155C2×Dic3—42no—
156Dic6—24no—
157Dic6—42no—
158C3×C8—44no—
159C3⋊C8—82no—
160C2×Q8—21no—
161C2×Q8—22no—
162C2×Q8—21no—
163C2×Q8—22no—
164C2×Q8—62no—
165C2×Q8—122no—
166C22×C4—61no—
167C22×C4—62no—
168C22×C4—61no—
169C22×C4(C2×D3)⋊C412yes—
170C22×C4(C2×D3)⋊C412yes—
171C22×C4C2×C4×D311yescharacteristic
172C22×C4C2×D1211yescharacteristic
173C22×C4C2×#6711yescharacteristic
174M16—122no—
175M16—122no—
176C2×C8—24no—
177C2×C8—122no—
178C4⋊C4—24no—
179C4⋊C4—22no—
180C4⋊C4—62no—
181C4⋊C4—124no—
182C4⋊C4—121no—
183C4⋊C4—121no—
184C4⋊C4—121no—
185C4⋊C4—121no—
186C42—22no—
187C42—22no—
188C42—122no—
189C42#23512yes—
190C42C2×#6712yes—
191C22×C3—21no—
192C22×C3—21no—
193C22×C3#38611yescharacteristic
194C22×C3#41711yescharacteristic
195C22×C3#41911yescharacteristic
196C3×C4—22no—
197C3×C4—21no—
198C3×C4—21no—
199C3×C4—42no—
200C3×C4—42no—
201C3×C4#46812yes—
202C3×C4#47012yes—
203Dic3—42no—
204Dic3—82no—
205Dic3—81no—
206C23C2×(C2×D3)⋊C411yescharacteristic
207Q8—42no—
208Q8—42no—
209Q8—64no—
210Q8—122no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—62no—
219C2×C4—121no—
220C2×C4—121no—
221C2×C4—122no—
222C2×C4—122no—
223C2×C4—122no—
224C2×C4—122no—
225C2×C4—121no—
226C2×C4#90811yescharacteristic
227C2×C4#90912yes—
228C2×C4#90911yescharacteristic
229C2×C4C4×#6712yes—
230C2×C4#92911yescharacteristic
231C2×C4#93012yes—
232C2×C4#95311yescharacteristic
233C8—44no—
234C8—242no—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3#189111yescharacteristic
238C2×C3#189411yescharacteristic
239C2×C3#191211yescharacteristic
240C22—21no—
241C22—21no—
242C22#512011yescharacteristic
243C22#513311yescharacteristic
244C22#521311yescharacteristic
245C4—22no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—42no—
250C4—122no—
251C4—242no—
252C4—241no—
253C4#513412yes—
254C4#513712yes—
255C3#948811yescharacteristic, complemented, Sylow
256C2—21no—
257C2—21no—
258C2#7166311yescharacteristic
259C2#7198911yescharacteristic
260C2#7202911yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2983C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2983D3
hypocenterC3#9488
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
255. C3#9488semidirect

Maximal subgroups

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

subgroupindex
C3×#29832
#824812
#716372
#707702
#677382
#94883