Finite Group Databasebeta

← #10698703

Subgroups — #10698703

Order 768.

630 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 127 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#74368C211yesmaximal, characteristic
2C3×#1844C211yesmaximal, characteristic, complemented
3C2×#4459C212yesmaximal, complemented
4#69584C212yesmaximal, complemented
5#67634C211yesmaximal, characteristic, complemented
6#10432—31nomaximal, Sylow
7C2×#895—22no—
8C2×C4×D12C2212yes—
9C2×C3×#106—22no—
10C2×C3×C4×C8C2211yescharacteristic
11C3×#384C2211yescharacteristic
12C3×#384C2211yescharacteristic
13#4991—22no—
14#4987—22no—
15C2×#809—22no—
16C2×C4⋊(C3⋊C8)C2212yes—
17#4529—22no—
18#4459—24no—
19#2537—31no—
20#1844D311yescharacteristic, complemented
21#1802—32no—
22C2×D4⋊C8—32no—
23#1296—31no—
24C22×D12—22no—
25C22×D12D412yes—
26C22×C4×D3—22no—
27C22×C3×C8—21no—
28C2×C3×C4⋊C4—22no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4—21no—
31C2×C3×C4⋊C4D412yes—
32C2×C3×C42C2311yescharacteristic
33C2×(C2×D3)⋊C4—24no—
34C2×(C2×D3)⋊C4—22no—
35C2×C4⋊Dic3D412yes—
36C22×C3⋊C8—22no—
37#895—44no—
38#895—44no—
39C4×D12—22no—
40C4×D12—22no—
41C4×D12—22no—
42C4×D12—22no—
43C3×#106—44no—
44C3×C4×C8—21no—
45C3×C4×C8D412yescomplemented
46C3⋊#95—22no—
47#809—44no—
48C4⋊(C3⋊C8)—22no—
49C4⋊(C3⋊C8)—22no—
50C2×#121—62no—
51C2×C4×D4—32no—
52C2×#106—22no—
53C2×C4⋊C8—32no—
54C2×#102—62no—
55C2×C4×C8C2×D311yescharacteristic
56#394—62no—
57#384C2×D311yescharacteristic
58#384C2×D311yescharacteristic
59#380—62no—
60#340—62no—
61D4⋊C8—64no—
62C23×D3—22no—
63C22×C3×C4—22no—
64C22×C3×C4—41no—
65C22×C3×C4C2×D412yes—
66C22×C3×C4C2×D411yescharacteristic
67C22×Dic3—22no—
68C2×D12—22no—
69C2×D12—22no—
70C2×D12—22no—
71C2×D12—22no—
72C2×D12—22no—
73C2×D12—48no—
74C2×D12—82no—
75C2×D12SD1614yes—
76C2×C4×D3—42no—
77C2×C4×D3—42no—
78C2×C4×D3—42no—
79C2×C4×D3—42no—
80C2×C4×D3—42no—
81C2×C4×D3—42no—
82C2×C3×C8—22no—
83C2×C3×C8—22no—
84C2×C3×C8—22no—
85C3×C4⋊C4—42no—
86C3×C4⋊C4—42no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4—42no—
89C3×C4⋊C4—42no—
90C3×C4⋊C4—42no—
91C3×C42C2×D411yescharacteristic
92C3×C42#4111yescharacteristic
93C3×C42#4112yes—
94(C2×D3)⋊C4—48no—
95(C2×D3)⋊C4—42no—
96(C2×D3)⋊C4—42no—
97C4⋊Dic3—22no—
98C4⋊Dic3—22no—
99C2×C3⋊C8—42no—
100C2×C3⋊C8—42no—
101C2×C3⋊C8—42no—
102C22×D4—32no—
103C22×D4—62no—
104C23×C4—62no—
105C22×C8—21no—
106C22×C8—62no—
107#121—124no—
108#121—124no—
109C4×D4—62no—
110C4×D4—62no—
111C4×D4—62no—
112C4×D4—62no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4#6712yes—
118C2×C22⋊C4—64no—
119C2×C22⋊C4—62no—
120C2×C42C22×D311yescharacteristic
121#106—44no—
122C4⋊C8—62no—
123C4⋊C8—62no—
124#102—124no—
125C4×C8—21no—
126C4×C8D1212yescomplemented
127#95—62no—
128C23×C3#12011yescharacteristic
129C22×D3—24no—
130C22×D3—42no—
131C22×D3—42no—
132C22×D3—42no—
133C22×D3—42no—
134C22×D3—42no—
135C22×D3—42no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—44no—
139C2×C3×C4—42no—
140C2×C3×C4—82no—
141C2×C3×C4—81no—
142C2×C3×C4#12112yes—
143C2×C3×C4#12111yescharacteristic
144C2×C3×C4#12312yes—
145C2×C3×C4#12411yescharacteristic
146C2×C3×C4C2×SD1612yes—
147C2×C3×C4#13512yes—
148C2×C3×C4AGL(1,ℤ/8)12yes—
149C2×C3×C4AGL(1,ℤ/8)12yes—
150C2×Dic3—42no—
151C2×Dic3—42no—
152C2×Dic3—42no—
153D12—24no—
154D12—24no—
155D12—42no—
156D12—42no—
157D12—88no—
158C4×D3—82no—
159C4×D3—82no—
160C4×D3—82no—
161C4×D3—82no—
162C3×C8—24no—
163C3⋊C8—82no—
164C24—62no—
165C2×D4—34no—
166C2×D4—62no—
167C2×D4—62no—
168C2×D4—62no—
169C2×D4—62no—
170C2×D4—62no—
171C2×D4—128no—
172C2×D4—242no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4—62no—
176C22×C4—122no—
177C22×C4—122no—
178C22×C4—122no—
179C22×C4—122no—
180C22×C4—122no—
181C22×C4—122no—
182C22×C4D3×D412yes—
183C22×C4C2×#6711yescharacteristic
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—122no—
188C2×C8—122no—
189C2×C8—122no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—62no—
197C4⋊C4—62no—
198C22⋊C4—128no—
199C22⋊C4—122no—
200C22⋊C4—122no—
201C42C2×D1211yescharacteristic
202C42#23511yescharacteristic
203C42#23912yes—
204C22×C3#39411yescharacteristic
205C22×C3#44912yes—
206C22×C3#45012yes—
207C22×C3#45311yescharacteristic
208C22×C3#45611yescharacteristic
209C2×D3—44no—
210C2×D3—44no—
211C2×D3—44no—
212C2×D3—82no—
213C2×D3—82no—
214C2×D3—82no—
215C2×D3—82no—
216C3×C4—21no—
217C3×C4—21no—
218C3×C4—82no—
219C3×C4—161no—
220C3×C4#46512yes—
221C3×C4#46912yes—
222C3×C4#48112yes—
223C3×C4#48212yes—
224Dic3—82no—
225C23—64no—
226C23—122no—
227C23—122no—
228C23—122no—
229C23—122no—
230C23—122no—
231C23—122no—
232C23#93711yescharacteristic
233D4—64no—
234D4—64no—
235D4—122no—
236D4—122no—
237D4—248no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—82no—
243C2×C4—81no—
244C2×C4—122no—
245C2×C4—122no—
246C2×C4—122no—
247C2×C4—242no—
248C2×C4—242no—
249C2×C4—242no—
250C2×C4—242no—
251C2×C4#89412yes—
252C2×C4#89512yes—
253C2×C4D3×SD1612yes—
254C2×C4#91412yes—
255C2×C4#91412yes—
256C2×C4#91612yes—
257C2×C4#93011yescharacteristic
258C2×C4#94711yescharacteristic
259C8—24no—
260C8—242no—
261C2×C3#162411yescharacteristic
262C2×C3#162911yescharacteristic
263C2×C3#165412yes—
264C2×C3#203412yes—
265C2×C3#203911yescharacteristic
266D3—84no—
267C22—124no—
268C22—124no—
269C22—124no—
270C22—242no—
271C22—242no—
272C22—242no—
273C22—242no—
274C22#499111yescharacteristic
275C22#507711yescharacteristic
276C22#509511yescharacteristic
277C22#516912yes—
278C22#517212yes—
279C4—21no—
280C4—21no—
281C4—82no—
282C4—161no—
283C4—242no—
284C4#486312yes—
285C4#486412yes—
286C4#486512yes—
287C4#486612yes—
288C3#1043211yescharacteristic, complemented, Sylow
289C2—244no—
290C2#6959311yescharacteristic
291C2#7058112yes—
292C2#7126511yescharacteristic
293C2#7126611yescharacteristic
294C2#7134312yes—
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1844C2
socleC23×C3#120
prime radical#74368C2
hypercenter#1844D3
hypocenterC3#10432
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1844

Complements

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

NH ≅ G/Ntype
2. C3×#1844C2semidirect
3. C2×#4459C2semidirect
4. #69584C2semidirect
5. #67634C2semidirect
20. #1844D3semidirect
45. C3×C4×C8D4semidirect
126. C4×C8D12semidirect
288. C3#10432semidirect

Maximal subgroups

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

subgroupindex
#743682
C3×#18442
C2×#44592
#695842
#676342
#104323