Finite Group Databasebeta

← #10697664

Subgroups — #10697664

Order 768.

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

normal subgroups only

This group has 582 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#82441C212yesmaximal, complemented
2#79026C211yesmaximal, characteristic
3C3×#1984C211yesmaximal, characteristic, complemented
4#69735C212yesmaximal, complemented
5#67705C211yesmaximal, characteristic, complemented
6#10255—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#942—22no—
9#5477C2212yes—
10C2×C3×C4⋊C8C2211yescharacteristic
11C2×C3×C4×C8C2211yescharacteristic
12#5115—22no—
13#5031—22no—
14#5028—24no—
15#5027—22no—
16#4966—22no—
17#4924C2212yes—
18#4846—22no—
19#4843—22no—
20C3×#336—22no—
21#4459—24no—
22#3245—31no—
23#3037—32no—
24#1984D311yescharacteristic, complemented
25#1492—32no—
26#1292—31no—
27C22×C3×Q8D411yescharacteristic
28C2×#235—22no—
29C22×C3×C8—21no—
30C22×C3×C8D412yes—
31C3×C4×Q8—22no—
32C3×C4×Q8—22no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C42—21no—
36C2×C3×C42C2311yescharacteristic
37C2×#216—24no—
38#942—44no—
39#942—44no—
40C2×#215—24no—
41#930—22no—
42#930—22no—
43C2×#208—22no—
44C2×D24—22no—
45#875—22no—
46#874D412yes—
47C4×D12—22no—
48C4×D12—22no—
49#869D412yes—
50C3×C4⋊C8—22no—
51C3×C4×C8—22no—
52#821—24no—
53#818—22no—
54#810—24no—
55#809—24no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)D414yes—
58C2×#137—62no—
59#532—32no—
60C2×C4×Q8C2×D311yescharacteristic
61#474—62no—
62#466—62no—
63#464—64no—
64#462—62no—
65#460—62no—
66#459—62no—
67#423—32no—
68C2×C4⋊C8C2×D311yescharacteristic
69#417—62no—
70C2×C4×C8C2×D311yescharacteristic
71#336—22no—
72D4⋊C8—64no—
73C2×C3×Q8—21no—
74C2×C3×Q8—22no—
75C2×C3×Q8—22no—
76C2×C3×Q8D812yescomplemented
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4C2×D411yescharacteristic
80C22×C3×C4C2×D411yescharacteristic
81C22×C3×C4#4111yescharacteristic
82C2×#67—42no—
83#235—42no—
84#235—42no—
85#235—42no—
86#235—42no—
87C2×D12—22no—
88C2×D12—22no—
89C2×D12—22no—
90C2×C4×D3—42no—
91C2×Dic6—22no—
92C2×C3×C8—22no—
93C2×C3×C8—22no—
94C2×C3×C8—41no—
95C2×C3×C8—41no—
96C2×C3×C8—42no—
97C3×C4⋊C4—22no—
98C3×C4⋊C4—42no—
99C3×C42—41no—
100C3×C42—41no—
101C3×C42C2×D412yes—
102C3×C42C2×D412yes—
103#216—48no—
104#215—48no—
105(C2×D3)⋊C4—42no—
106(C2×D3)⋊C4—42no—
107C4⋊Dic3—22no—
108C4⋊Dic3—22no—
109#208—42no—
110#208—42no—
111C2×C3⋊C8—24no—
112D24—82no—
113C2×#41—62no—
114C22×Q8D1211yescharacteristic
115#137—124no—
116#137—124no—
117C2×Q16—64no—
118C2×SD16—64no—
119C2×D8—62no—
120C2×M16—62no—
121C22×C8—21no—
122C22×C8#6712yes—
123C4⋊Q8—32no—
124#127—32no—
125#124—62no—
126#121—62no—
127#121—62no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×D4—62no—
131C4×D4—62no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4D1211yescharacteristic
134C2×C42—21no—
135C2×C42C22×D311yescharacteristic
136#107—62no—
137C4⋊C8—22no—
138C4⋊C8—34no—
139C4⋊C8—62no—
140#103—64no—
141#102—64no—
142#102—64no—
143C4×C8—22no—
144C23×C3#12111yescharacteristic
145C22×D3—42no—
146C3×Q8—24no—
147C3×Q8—42no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—22no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—41no—
154C2×C3×C4—42no—
155C2×C3×C4—41no—
156C2×C3×C4#12012yes—
157C2×C3×C4#12011yescharacteristic
158C2×C3×C4#12112yes—
159C2×C3×C4#12711yescharacteristic
160C2×C3×C4C2×D811yescharacteristic
161C2×C3×C4#13511yescharacteristic
162C2×C3×C4AGL(1,ℤ/8)12yes—
163#67—82no—
164#67—82no—
165C2×Dic3—42no—
166D12—44no—
167D12—42no—
168D12—42no—
169D12—42no—
170D12—42no—
171C4×D3—82no—
172C4×D3—82no—
173Dic6—42no—
174Dic6—42no—
175C3×C8—42no—
176C3×C8—42no—
177C3⋊C8—44no—
178#41—122no—
179#41—122no—
180#41—122no—
181#41—122no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—62no—
186C2×Q8D2412yescomplemented
187C2×D4—62no—
188C2×D4—62no—
189C2×D4—62no—
190C2×D4—122no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—122no—
194C22×C4C2×D1211yescharacteristic
195C22×C4#23511yescharacteristic
196C22×C4C2×#6711yescharacteristic
197Q16—128no—
198SD16—128no—
199D8—242no—
200M16—122no—
201M16—122no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—42no—
207C2×C8—64no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—62no—
211C4⋊C4—62no—
212C22⋊C4—122no—
213C22⋊C4—122no—
214C42—41no—
215C42—41no—
216C42D3×D412yes—
217C42D3×D412yes—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3#39211yescharacteristic
221C22×C3#46611yescharacteristic
222C22×C3#46911yescharacteristic
223C2×D3—82no—
224C2×D3—82no—
225C2×D3—82no—
226C3×C4—21no—
227C3×C4—22no—
228C3×C4—42no—
229C3×C4—81no—
230C3×C4#44712yes—
231C3×C4#44912yes—
232C3×C4#45511yescharacteristic
233C3×C4#45611yescharacteristic
234Dic3—82no—
235C23—122no—
236C23#93011yescharacteristic
237Q8—24no—
238Q8—42no—
239Q8—122no—
240Q8—122no—
241D4—124no—
242D4—122no—
243D4—122no—
244D4—122no—
245D4—122no—
246D4—242no—
247D4—242no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—122no—
257C2×C4—242no—
258C2×C4—242no—
259C2×C4#88212yes—
260C2×C4#88412yes—
261C2×C4C2×D2411yescharacteristic
262C2×C4#90411yescharacteristic
263C2×C4#90812yes—
264C2×C4#93711yescharacteristic
265C2×C4#94011yescharacteristic
266C8—42no—
267C8—42no—
268C8—124no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3#199911yescharacteristic
272C2×C3#200711yescharacteristic
273C2×C3#201711yescharacteristic
274D3—162no—
275C22—21no—
276C22—21no—
277C22—242no—
278C22—242no—
279C22—242no—
280C22#496011yescharacteristic
281C22#511511yescharacteristic
282C22#513511yescharacteristic
283C4—21no—
284C4—22no—
285C4—42no—
286C4—81no—
287C4—242no—
288C4#473212yes—
289C4#473412yes—
290C4#506111yescharacteristic
291C4#509511yescharacteristic
292C3#1025511yescharacteristic, complemented, Sylow
293C2—21no—
294C2—21no—
295C2—482no—
296C2#6969411yescharacteristic
297C2#6996111yescharacteristic
298C2#7001311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1984C2
socleC22×C3#392
prime radical#79026C2
hypercenter#1984D3
hypocenterC3#10255
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1984

Complements

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

NH ≅ G/Ntype
1. #82441C2semidirect
3. C3×#1984C2semidirect
4. #69735C2semidirect
5. #67705C2semidirect
24. #1984D3semidirect
76. C2×C3×Q8D8semidirect
186. C2×Q8D24semidirect
292. C3#10255semidirect

Maximal subgroups

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

subgroupindex
#824412
#790262
C3×#19842
#697352
#677052
#102553