Finite Group Databasebeta

← #10696027

Subgroups — #10696027

Order 768.

734 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 103 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#83298C212yesmaximal, complemented
2#73418C211yesmaximal, characteristic, complemented
3C3×#1963C211yesmaximal, characteristic, complemented
4#70496C212yesmaximal, complemented
5#67614C211yesmaximal, characteristic, complemented
6#10012—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8C2×C4×D12—22no—
9C2×C4×D12C2212yes—
10C3×#423C2211yescharacteristic
11C2×C3×#104—22no—
12C3×#376C2211yescharacteristic
13#5087—22no—
14#5072—22no—
15#5036—22no—
16#5033—22no—
17C4×#213—24no—
18#4925C2212yes—
19#4677—22no—
20#4671—22no—
21#4483—24no—
22#3100—32no—
23#2397—31no—
24#1963D311yescharacteristic, complemented
25#1486—32no—
26#1274—31no—
27C2×C3×#41—21no—
28C22×D12—22no—
29C22×D12D412yes—
30C22×C4×D3—22no—
31C2×C3×M16—21no—
32C3×C4⋊Q8D411yescharacteristic, complemented
33C3×#127D411yescharacteristic, complemented
34C3×#124—21no—
35C3×#121—22no—
36C3×C4×D4—22no—
37C2×C3×C42D412yes—
38C2×C3×C42C2311yescharacteristic
39C2×#213—24no—
40C2×(C2×D3)⋊C4—24no—
41C2×(C2×D3)⋊C4—22no—
42C2×C4⋊Dic3—22no—
43C2×C4⋊Dic3D412yes—
44C4×D12—22no—
45C4×D12—22no—
46C4×D12—22no—
47C4×D12—22no—
48C4×D12—44no—
49C4×D12—44no—
50C3×C4⋊C8—22no—
51C3×#104—44no—
52C3×#95—22no—
53#832—24no—
54#830—22no—
55#809—24no—
56#807—24no—
57C4⋊(C3⋊C8)—22no—
58C4×C3⋊C8D414yescomplemented
59#532C2×D311yescharacteristic
60C2×C4×D4—32no—
61C2×C4×D4—62no—
62#480—62no—
63#474—62no—
64#459—62no—
65#447—62no—
66C4×D8—64no—
67#431—32no—
68#423C2×D311yescharacteristic
69C2×#104—22no—
70#399—62no—
71#392—62no—
72#376C2×D311yescharacteristic
73D4⋊C8—64no—
74C23×D3—22no—
75C3×#41—42no—
76C3×#41—42no—
77C2×C3×Q8—21no—
78C2×C3×D4—22no—
79C2×C3×D4—21no—
80C2×C3×D4—41no—
81C22×C3×C4—22no—
82C22×C3×C4—41no—
83C22×C3×C4C2×D412yes—
84C22×C3×C4C2×D411yescharacteristic
85C22×Dic3—22no—
86C2×D12—22no—
87C2×D12—22no—
88C2×D12—44no—
89C2×D12—44no—
90C2×D12—48no—
91C2×C4×D3—42no—
92C2×C4×D3—42no—
93C2×C4×D3—44no—
94C2×C4×D3—44no—
95C3×M16—42no—
96C2×C3×C8—41no—
97C3×C4⋊C4—22no—
98C3×C22⋊C4—42no—
99C3×C42—22no—
100C3×C42D814yescomplemented
101C3×C42C2×D411yescharacteristic
102C3×C42C2×D412yes—
103C3×C42#4111yescharacteristic
104#213—48no—
105(C2×D3)⋊C4—48no—
106(C2×D3)⋊C4—42no—
107(C2×D3)⋊C4—42no—
108C4⋊Dic3—22no—
109C4⋊Dic3—22no—
110C4⋊Dic3—44no—
111C2×C3⋊C8—24no—
112C2×#41—21no—
113C22×D4—32no—
114C22×D4—62no—
115C23×C4—62no—
116C2×D8—64no—
117C2×M16—21no—
118C4⋊Q8D1211yescharacteristic, complemented
119#127D1211yescharacteristic, complemented
120#124—21no—
121#121—22no—
122C4×D4—22no—
123C4×D4—62no—
124C4×D4—62no—
125C4×D4—62no—
126C4×D4—62no—
127C4×D4—124no—
128C4×D4—124no—
129C2×C4⋊C4—32no—
130C2×C4⋊C4—62no—
131C2×C22⋊C4—64no—
132C2×C22⋊C4—62no—
133C2×C42#6712yes—
134C2×C42C22×D311yescharacteristic
135#107—64no—
136C4⋊C8—22no—
137C4⋊C8—62no—
138#104—44no—
139#102—64no—
140#102—64no—
141C22⋊C8—62no—
142C4×C8—34no—
143#95—22no—
144C23×C3—41no—
145C23×C3#12011yescharacteristic
146C22×D3—24no—
147C22×D3—44no—
148C22×D3—42no—
149C22×D3—42no—
150C22×D3—44no—
151C3×Q8—42no—
152C3×D4—44no—
153C3×D4—42no—
154C3×D4—82no—
155C2×C3×C4—24no—
156C2×C3×C4—42no—
157C2×C3×C4—44no—
158C2×C3×C4—41no—
159C2×C3×C4—81no—
160C2×C3×C4—81no—
161C2×C3×C4#12012yes—
162C2×C3×C4#12112yes—
163C2×C3×C4#12111yescharacteristic
164C2×C3×C4#12711yescharacteristic
165C2×C3×C4C2×D812yes—
166C2×C3×C4AGL(1,ℤ/8)12yes—
167C2×Dic3—42no—
168C2×Dic3—44no—
169D12—24no—
170D12—24no—
171D12—42no—
172D12—42no—
173D12—416no—
174C4×D3—84no—
175C4×D3—82no—
176C4×D3—82no—
177C3×C8—81no—
178C3⋊C8—44no—
179C24—62no—
180#41—42no—
181#41—42no—
182C2×Q8—21no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—41no—
186C2×D4—62no—
187C2×D4—62no—
188C2×D4—124no—
189C2×D4—124no—
190C2×D4—128no—
191C22×C4—22no—
192C22×C4—41no—
193C22×C4—62no—
194C22×C4—122no—
195C22×C4—124no—
196C22×C4—122no—
197C22×C4—124no—
198C22×C4D3×D412yes—
199C22×C4C2×#6711yescharacteristic
200D8—128no—
201M16—42no—
202C2×C8—41no—
203C2×C8—64no—
204C4⋊C4—22no—
205C4⋊C4—62no—
206C4⋊C4—62no—
207C4⋊C4—124no—
208C22⋊C4—42no—
209C22⋊C4—128no—
210C22⋊C4—122no—
211C22⋊C4—122no—
212C42—22no—
213C42#21314yescomplemented
214C42C2×D1211yescharacteristic
215C42#23511yescharacteristic
216C42D3×D412yes—
217C22×C3—22no—
218C22×C3—82no—
219C22×C3—81no—
220C22×C3#39211yescharacteristic
221C22×C3#44712yes—
222C2×D3—48no—
223C2×D3—44no—
224C2×D3—84no—
225C2×D3—82no—
226C2×D3—82no—
227C3×C4—22no—
228C3×C4—22no—
229C3×C4—44no—
230C3×C4—81no—
231C3×C4#45912yes—
232C3×C4#49311yescharacteristic
233C3×C4#49611yescharacteristic
234Dic3—82no—
235C23—41no—
236C23—64no—
237C23—124no—
238C23—122no—
239C23—122no—
240C23—124no—
241C23#93711yescharacteristic
242Q8—42no—
243D4—44no—
244D4—42no—
245D4—64no—
246D4—64no—
247D4—82no—
248D4—122no—
249D4—122no—
250D4—1216no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—41no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—122no—
258C2×C4—124no—
259C2×C4—244no—
260C2×C4—242no—
261C2×C4—242no—
262C2×C4#88212yes—
263C2×C4#88412yes—
264C2×C4#93011yescharacteristic
265C2×C4C2×#21312yes—
266C2×C4#94011yescharacteristic
267C2×C4#94912yes—
268C8—81no—
269C8—124no—
270C2×C3—22no—
271C2×C3—161no—
272C2×C3#201012yes—
273C2×C3#201111yescharacteristic
274D3—84no—
275C22—22no—
276C22—82no—
277C22—81no—
278C22—128no—
279C22—124no—
280C22—244no—
281C22—242no—
282C22—242no—
283C22#496011yescharacteristic
284C22#503612yes—
285C4—22no—
286C4—22no—
287C4—44no—
288C4—81no—
289C4—242no—
290C4#501512yes—
291C4#505811yescharacteristic
292C4#507311yescharacteristic
293C3#1001211yescharacteristic, complemented, Sylow
294C2—22no—
295C2—161no—
296C2—244no—
297C2#6965811yescharacteristic
298C2#7040612yes—
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1963C2
socleC22×C3#392
hypercenter#1963D3
hypocenterC3#10012
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
1. #83298C2semidirect
2. #73418C2semidirect
3. C3×#1963C2semidirect
4. #70496C2semidirect
5. #67614C2semidirect
24. #1963D3semidirect
32. C3×C4⋊Q8D4semidirect
33. C3×#127D4semidirect
58. C4×C3⋊C8D4semidirect
100. C3×C42D8semidirect
118. C4⋊Q8D12semidirect
119. #127D12semidirect
213. C42#213semidirect
293. C3#10012semidirect

Maximal subgroups

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

subgroupindex
#832982
#734182
C3×#19632
#704962
#676142
#100123