Finite Group Databasebeta

← #10696030

Subgroups — #10696030

Order 768.

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

normal subgroups only

This group has 734 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
1#79297C212yesmaximal, complemented
2#73418C211yesmaximal, characteristic, complemented
3#70762C212yesmaximal, complemented
4#69440C211yesmaximal, characteristic, complemented
5C3×#1274C211yesmaximal, characteristic, complemented
6#10012—31nomaximal, Sylow
7#5477C2211yescharacteristic
8C2×C4×D12—22no—
9C2×C4×D12C2212yes—
10C3×#431C2212yes—
11C3×#376C2211yescharacteristic
12C2×#805—22no—
13#4924C2211yescharacteristic
14#4846—22no—
15#4843—22no—
16#4736—22no—
17#4732—22no—
18C4×D24—24no—
19#4677—22no—
20#4671—22no—
21#4459—24no—
22#3100—32no—
23#2397—31no—
24#1963—31no—
25#1486—32no—
26#1274D311yescharacteristic, complemented
27C2×#235—21no—
28C22×D12—22no—
29C22×D12D412yes—
30C22×C4×D3—22no—
31C2×C3×C42D412yes—
32C2×C3×C42C2311yescharacteristic
33#930—22no—
34C2×(C2×D3)⋊C4—24no—
35C2×(C2×D3)⋊C4—22no—
36C2×C4⋊Dic3—22no—
37C2×C4⋊Dic3D412yes—
38C2×#208—21no—
39C2×D24—24no—
40#875—21no—
41#874D411yescharacteristic, complemented
42C4×D12—22no—
43C4×D12—22no—
44C4×D12—22no—
45C4×D12—22no—
46C4×D12—22no—
47C4×D12—44no—
48C4×D12—44no—
49#869D411yescharacteristic, complemented
50C3×C4⋊C8—22no—
51C3×C22⋊C8—22no—
52C3×C4×C8D414yescomplemented
53C3×#95—22no—
54#821—24no—
55#821—24no—
56#818—24no—
57#805—44no—
58C4⋊(C3⋊C8)—22no—
59#532—31no—
60C2×C4×D4—32no—
61C2×C4×D4—62no—
62#480—62no—
63#474—62no—
64#459—62no—
65#447—62no—
66C4×D8—64no—
67#431C2×D312yes—
68#423—31no—
69C2×#104—62no—
70#399—62no—
71#392—62no—
72#376C2×D311yescharacteristic
73D4⋊C8—64no—
74C23×D3—22no—
75C22×C3×C4—22no—
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4C2×D412yes—
78C2×#67—41no—
79C22×Dic3—22no—
80#235—42no—
81#235—42no—
82C2×D12—22no—
83C2×D12—22no—
84C2×D12—22no—
85C2×D12—21no—
86C2×D12—44no—
87C2×D12—44no—
88C2×D12—48no—
89C2×C4×D3—42no—
90C2×C4×D3—42no—
91C2×C4×D3—41no—
92C2×C4×D3—44no—
93C2×C4×D3—44no—
94C2×Dic6—21no—
95C2×C3×C8—24no—
96C3×C42—22no—
97C3×C42D814yescomplemented
98C3×C42C2×D412yes—
99C3×C42C2×D411yescharacteristic
100C3×C42#4111yescharacteristic
101(C2×D3)⋊C4—48no—
102(C2×D3)⋊C4—42no—
103(C2×D3)⋊C4—42no—
104(C2×D3)⋊C4—42no—
105C4⋊Dic3—22no—
106C4⋊Dic3—22no—
107C4⋊Dic3—22no—
108C4⋊Dic3—44no—
109#208—42no—
110C2×C3⋊C8—41no—
111D24—48no—
112C2×#41—61no—
113C22×D4—32no—
114C22×D4—62no—
115C23×C4—62no—
116C2×D8—64no—
117C2×M16—61no—
118C4⋊Q8—31no—
119#127—31no—
120#124—61no—
121#121—62no—
122C4×D4—62no—
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×C42D1212yes—
134C2×C42C22×D311yescharacteristic
135#107—64no—
136C4⋊C8—22no—
137C4⋊C8—62no—
138#104—124no—
139#102—64no—
140#102—64no—
141C22⋊C8—22no—
142C4×C8D1214yescomplemented
143#95—22no—
144C23×C3#12011yescharacteristic
145C22×D3—24no—
146C22×D3—44no—
147C22×D3—42no—
148C22×D3—42no—
149C22×D3—44no—
150C22×D3—41no—
151C2×C3×C4—24no—
152C2×C3×C4—42no—
153C2×C3×C4—44no—
154C2×C3×C4#12012yes—
155C2×C3×C4#12112yes—
156C2×C3×C4#12111yescharacteristic
157C2×C3×C4#12711yescharacteristic
158C2×C3×C4C2×D812yes—
159C2×C3×C4AGL(1,ℤ/8)12yes—
160#67—82no—
161C2×Dic3—41no—
162C2×Dic3—42no—
163C2×Dic3—44no—
164D12—24no—
165D12—24no—
166D12—42no—
167D12—42no—
168D12—44no—
169D12—416no—
170D12—42no—
171C4×D3—84no—
172C4×D3—82no—
173C4×D3—81no—
174C4×D3—82no—
175C4×D3—81no—
176Dic6—42no—
177C3×C8—44no—
178C3⋊C8—81no—
179C24—62no—
180#41—122no—
181#41—122no—
182C2×Q8—61no—
183C2×D4—62no—
184C2×D4—62no—
185C2×D4—62no—
186C2×D4—61no—
187C2×D4—124no—
188C2×D4—124no—
189C2×D4—128no—
190C2×D4—121no—
191C22×C4—22no—
192C22×C4—62no—
193C22×C4—122no—
194C22×C4—121no—
195C22×C4—122no—
196C22×C4—124no—
197C22×C4—124no—
198C22×C4C2×D1211yescharacteristic
199C22×C4D3×D412yes—
200D8—128no—
201M16—122no—
202C2×C8—24no—
203C2×C8—121no—
204C4⋊C4—62no—
205C4⋊C4—62no—
206C4⋊C4—62no—
207C4⋊C4—124no—
208C22⋊C4—128no—
209C22⋊C4—122no—
210C22⋊C4—122no—
211C22⋊C4—122no—
212C42—22no—
213C42D2414yescomplemented
214C42C2×D1212yes—
215C42D3×D411yescharacteristic
216C42#23911yescharacteristic
217C22×C3—22no—
218C22×C3#39211yescharacteristic
219C22×C3#44712yes—
220C2×D3—48no—
221C2×D3—44no—
222C2×D3—84no—
223C2×D3—82no—
224C2×D3—82no—
225C2×D3—82no—
226C2×D3—81no—
227C3×C4—22no—
228C3×C4—22no—
229C3×C4—44no—
230C3×C4#45912yes—
231C3×C4#49311yescharacteristic
232C3×C4#49611yescharacteristic
233Dic3—81no—
234Dic3—82no—
235C23—64no—
236C23—124no—
237C23—122no—
238C23—122no—
239C23—124no—
240C23—121no—
241C23#88211yescharacteristic
242Q8—122no—
243D4—64no—
244D4—64no—
245D4—122no—
246D4—122no—
247D4—124no—
248D4—1216no—
249D4—122no—
250D4—242no—
251C2×C4—24no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—122no—
255C2×C4—121no—
256C2×C4—124no—
257C2×C4—242no—
258C2×C4—241no—
259C2×C4—242no—
260C2×C4—241no—
261C2×C4—244no—
262C2×C4#87411yescharacteristic
263C2×C4#88212yes—
264C2×C4#89512yes—
265C2×C4#89511yescharacteristic
266C2×C4C2×D2412yes—
267C2×C4#90812yes—
268C8—44no—
269C8—241no—
270C2×C3—22no—
271C2×C3#201012yes—
272C2×C3#201111yescharacteristic
273D3—84no—
274D3—161no—
275C22—22no—
276C22—128no—
277C22—124no—
278C22—244no—
279C22—242no—
280C22—242no—
281C22—241no—
282C22—242no—
283C22#467111yescharacteristic
284C22#473212yes—
285C4—22no—
286C4—22no—
287C4—44no—
288C4—242no—
289C4—241no—
290C4#469511yescharacteristic
291C4#471211yescharacteristic
292C4#484312yes—
293C3#1001211yescharacteristic, complemented, Sylow
294C2—22no—
295C2—244no—
296C2—481no—
297C2#6867411yescharacteristic
298C2#6870312yes—
subgroupG/H
centerC2×C4#874
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1274C2
socleC22×C3#392
hypercenter#1274D3
hypocenterC3#10012
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1274

Complements

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

NH ≅ G/Ntype
1. #79297C2semidirect
2. #73418C2semidirect
3. #70762C2semidirect
4. #69440C2semidirect
5. C3×#1274C2semidirect
26. #1274D3semidirect
41. #874D4semidirect
49. #869D4semidirect
52. C3×C4×C8D4semidirect
97. C3×C42D8semidirect
142. C4×C8D12semidirect
213. C42D24semidirect
293. C3#10012semidirect

Maximal subgroups

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

subgroupindex
#792972
#734182
#707622
#694402
C3×#12742
#100123