Finite Group Databasebeta

← #10692597

Subgroups — #10692597

Order 768.

486 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 119 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#82475C211yesmaximal, characteristic
3#71637C211yesmaximal, characteristic
4#70778C212yesmaximal
5#67746C212yesmaximal
6#9402—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#944—21no—
9C2×#869C2211yescharacteristic
10C3×#446—24no—
11C3×#432—22no—
12C3×#431C2212yes—
13C2×#836—21no—
14#4981C414yes—
15#4973—21no—
16C4×C4⋊Dic3C2211yescharacteristic
17#4928—22no—
18#4924C2212yes—
19#4530—21no—
20#4486—24no—
21#2983D311yescharacteristic
22#2546—31no—
23#1801—31no—
24#1546—32no—
25#1303—32no—
26C22×C3×Q8D411yescharacteristic
27C22×Dic6—21no—
28C3×C4×Q8—22no—
29C3×C4×Q8—22no—
30C2×C3×C4⋊C4C2×C412yes—
31C2×C3×C4⋊C4D411yescharacteristic
32C2×C3×C42D412yes—
33C2×C3×C42C2311yescharacteristic
34#944—44no—
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3—21no—
37C2×C4⋊Dic3C2×C412yes—
38C2×Dic3⋊C4—24no—
39C2×C4×Dic3—21no—
40C2×#208—22no—
41#869—22no—
42#869—22no—
43C3×C4⋊C8—24no—
44C3×C4⋊C8D414yes—
45C3×C22⋊C8—22no—
46C3×#97—24no—
47C3×C4×C8—22no—
48#836—42no—
49C3⋊#95—24no—
50C3⋊#95C2×C412yes—
51C4⋊(C3⋊C8)—22no—
52C4⋊(C3⋊C8)—22no—
53C2×C4⋊Q8—31no—
54C2×C4⋊Q8—61no—
55C2×C4×Q8C2×D311yescharacteristic
56#446—24no—
57#432—22no—
58#431C2×D312yes—
59#423—32no—
60C2×#101—61no—
61#398—34no—
62#391—61no—
63#389—62no—
64C4×C4⋊C4—31no—
65#343—61no—
66#332—64no—
67C2×C3×Q8—22no—
68C2×C3×Q8—21no—
69C2×C3×Q8—22no—
70C2×C3×Q8—21no—
71C22×C3×C4C22⋊C412yes—
72C22×C3×C4C22⋊C412yes—
73C22×C3×C4C22×C411yescharacteristic
74C22×C3×C4C2×D411yescharacteristic
75C22×C3×C4C2×D411yescharacteristic
76C22×Dic3—22no—
77C22×Dic3—22no—
78C2×Dic6—42no—
79C2×Dic6—42no—
80C2×Dic6—41no—
81C2×Dic6—41no—
82C2×C3×C8—24no—
83C3×C4⋊C4—24no—
84C3×C4⋊C4—22no—
85C3×C42—22no—
86C3×C42—22no—
87C3×C42C2×D412yes—
88C3×C42#4112yes—
89C4⋊Dic3—24no—
90C4⋊Dic3—24no—
91C4⋊Dic3—41no—
92C4⋊Dic3—41no—
93Dic3⋊C4—48no—
94C4×Dic3—42no—
95#208—42no—
96#208—42no—
97C2×C3⋊C8—42no—
98C22×Q8—61no—
99C22×Q8D1211yescharacteristic
100C2×M16—62no—
101C4⋊Q8—62no—
102C4⋊Q8—62no—
103C4⋊Q8—124no—
104C4×Q8—22no—
105C4×Q8—22no—
106C2×C4⋊C4—32no—
107C2×C4⋊C4—61no—
108C2×C4⋊C4—64no—
109C2×C4⋊C4—61no—
110C2×C4⋊C4C4×D312yes—
111C2×C4⋊C4D1211yescharacteristic
112C2×C42—61no—
113C2×C42#6712yes—
114C2×C42C22×D311yescharacteristic
115C4⋊C8—24no—
116C4⋊C8—62no—
117C4⋊C8—62no—
118C4⋊C8#6714yes—
119#101—122no—
120C22⋊C8—22no—
121#97—24no—
122C4×C8—22no—
123#95—32no—
124#95—64no—
125C23×C3C2×C22⋊C411yescharacteristic
126C3×Q8—42no—
127C3×Q8—42no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—41no—
134C2×C3×C4—42no—
135C2×C3×C4C4×D412yes—
136C2×C3×C4#12011yescharacteristic
137C2×C3×C4#12112yes—
138C2×C3×C4#12311yescharacteristic
139C2×C3×C4AGL(1,ℤ/8)12yes—
140C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
141C2×C3×C4#13711yescharacteristic
142C2×Dic3—24no—
143C2×Dic3—42no—
144C2×Dic3—42no—
145C2×Dic3—42no—
146C2×Dic3—42no—
147C2×Dic3—42no—
148Dic6—82no—
149Dic6—82no—
150C3×C8—44no—
151C3⋊C8—82no—
152C2×Q8—21no—
153C2×Q8—22no—
154C2×Q8—21no—
155C2×Q8—22no—
156C2×Q8—122no—
157C2×Q8—121no—
158C2×Q8—121no—
159C2×Q8—122no—
160C22×C4—62no—
161C22×C4—62no—
162C22×C4(C2×D3)⋊C412yes—
163C22×C4(C2×D3)⋊C412yes—
164C22×C4C2×C4×D311yescharacteristic
165C22×C4C2×D1211yescharacteristic
166C22×C4C2×#6711yescharacteristic
167M16—122no—
168M16—122no—
169C2×C8—24no—
170C2×C8—122no—
171C4⋊C4—24no—
172C4⋊C4—22no—
173C4⋊C4—64no—
174C4⋊C4—64no—
175C4⋊C4—128no—
176C4⋊C4—121no—
177C4⋊C4—121no—
178C42—22no—
179C42—22no—
180C42—122no—
181C42#23512yes—
182C42C2×#6712yes—
183C22×C3—21no—
184C22×C3—21no—
185C22×C3#38611yescharacteristic
186C22×C3#41711yescharacteristic
187C22×C3#41811yescharacteristic
188C3×C4—21no—
189C3×C4—22no—
190C3×C4—42no—
191C3×C4—42no—
192C3×C4#45612yes—
193C3×C4#46812yes—
194C3×C4#46912yes—
195Dic3—44no—
196Dic3—82no—
197C23C2×(C2×D3)⋊C411yescharacteristic
198Q8—42no—
199Q8—42no—
200Q8—242no—
201Q8—242no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—64no—
210C2×C4—122no—
211C2×C4—122no—
212C2×C4—122no—
213C2×C4—122no—
214C2×C4—122no—
215C2×C4#90812yes—
216C2×C4#90811yescharacteristic
217C2×C4#90911yescharacteristic
218C2×C4C4×#6712yes—
219C2×C4#92911yescharacteristic
220C2×C4#93012yes—
221C2×C4#95311yescharacteristic
222C8—44no—
223C8—242no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3#188611yescharacteristic
227C2×C3#189511yescharacteristic
228C2×C3#191511yescharacteristic
229C22—21no—
230C22—21no—
231C22#513111yescharacteristic
232C22#513311yescharacteristic
233C22#521311yescharacteristic
234C4—22no—
235C4—21no—
236C4—42no—
237C4—42no—
238C4—124no—
239C4—242no—
240C4#513412yes—
241C4#513512yes—
242C4#524612yes—
243C3#940211yescharacteristic, complemented, Sylow
244C2—21no—
245C2—21no—
246C2#7166011yescharacteristic
247C2#7199011yescharacteristic
248C2#7203111yescharacteristic
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#9402
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
243. C3#9402semidirect

Maximal subgroups

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

subgroupindex
C3×#29832
#824752
#716372
#707782
#677462
#94023