Finite Group Databasebeta

← #10697665

Subgroups — #10697665

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#79067C212yesmaximal, complemented
2#79026C211yesmaximal, characteristic
3#70498C212yesmaximal, complemented
4#70472C211yesmaximal, characteristic, complemented
5C3×#1292C211yesmaximal, characteristic, complemented
6#10255—31nomaximal, Sylow
7C2×#909—22no—
8#5477C2212yes—
9C2×C4×Dic6C2211yescharacteristic
10C3×#423C2212yes—
11C2×C3×C4⋊C8C2211yescharacteristic
12#5133—22no—
13#5115—22no—
14C2×C4×C3⋊C8C2211yescharacteristic
15#4849—22no—
16#4848—22no—
17#4846—22no—
18#4845—24no—
19#4843—22no—
20#4550—22no—
21#4483—24no—
22#3245—31no—
23#3037—32no—
24#1984—31no—
25#1492—32no—
26#1292D311yescharacteristic, complemented
27C2×#235—22no—
28C22×Dic6D411yescharacteristic
29C2×C3×M16—22no—
30C22×C3×C8D412yes—
31C2×C3×C42C2311yescharacteristic
32#930—22no—
33#930—22no—
34C2×C4⋊Dic3D411yescharacteristic
35C2×Dic3⋊C4—21no—
36C2×C4×Dic3—21no—
37C22×C3⋊C8—21no—
38#909—44no—
39#909—44no—
40C2×Dic12—24no—
41C2×D24—22no—
42C2×#204—24no—
43#875—22no—
44#874D412yes—
45C4×D12—22no—
46C4×D12—22no—
47#869D412yes—
48C4×Dic6—22no—
49C4×Dic6—22no—
50C3×C4⋊C8—22no—
51C3×C4⋊C8—22no—
52C3×C4⋊C8D414yes—
53#821—24no—
54#821—24no—
55#818—22no—
56#816—24no—
57C4×C3⋊C8—22no—
58C2×#137—62no—
59#532—32no—
60C2×C4×Q8—31no—
61#474—62no—
62#466—62no—
63#464—64no—
64#462—62no—
65#460—62no—
66#459—62no—
67#423C2×D312yes—
68C2×C4⋊C8C2×D311yescharacteristic
69#417—62no—
70C2×C4×C8—31no—
71#336—62no—
72D4⋊C8—64no—
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D411yescharacteristic
75C22×C3×C4#4111yescharacteristic
76C2×#67—42no—
77C22×Dic3—21no—
78C22×Dic3—21no—
79#235—42no—
80#235—42no—
81#235—42no—
82#235—42no—
83C2×D12—22no—
84C2×D12—22no—
85C2×D12—22no—
86C2×C4×D3—42no—
87C2×Dic6—22no—
88C2×Dic6—22no—
89C2×Dic6—21no—
90C2×Dic6—22no—
91C2×Dic6D812yescomplemented
92C3×M16—42no—
93C3×M16—42no—
94C2×C3×C8—24no—
95C2×C3×C8—22no—
96C2×C3×C8—42no—
97C3×C42C2×D412yes—
98C3×C42C2×D412yes—
99(C2×D3)⋊C4—42no—
100(C2×D3)⋊C4—42no—
101C4⋊Dic3—22no—
102C4⋊Dic3—22no—
103C4⋊Dic3—22no—
104Dic3⋊C4—42no—
105C4×Dic3—41no—
106C4×Dic3—41no—
107C2×C3⋊C8—22no—
108C2×C3⋊C8—41no—
109C2×C3⋊C8—41no—
110Dic12—48no—
111D24—82no—
112#204—48no—
113C2×#41—62no—
114C22×Q8—31no—
115#137—124no—
116#137—124no—
117C2×Q16—64no—
118C2×SD16—64no—
119C2×D8—62no—
120C2×M16—22no—
121C22×C8—61no—
122C22×C8D1212yes—
123C4⋊Q8—32no—
124#127—32no—
125#124—62no—
126#121—62no—
127#121—62no—
128C4×Q8—62no—
129C4×Q8—62no—
130C4×D4—62no—
131C4×D4—62no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—61no—
134C2×C42—61no—
135C2×C42C22×D311yescharacteristic
136#107—62no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8D1214yes—
140#103—64no—
141#102—64no—
142#102—64no—
143C4×C8—62no—
144C23×C3#12111yescharacteristic
145C22×D3—42no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—41no—
149C2×C3×C4#12012yes—
150C2×C3×C4#12011yescharacteristic
151C2×C3×C4#12112yes—
152C2×C3×C4#12711yescharacteristic
153C2×C3×C4C2×D811yescharacteristic
154C2×C3×C4#13511yescharacteristic
155C2×C3×C4AGL(1,ℤ/8)12yes—
156#67—82no—
157#67—82no—
158C2×Dic3—22no—
159C2×Dic3—41no—
160C2×Dic3—41no—
161C2×Dic3—42no—
162C2×Dic3—41no—
163C2×Dic3—42no—
164D12—44no—
165D12—42no—
166D12—42no—
167D12—42no—
168D12—42no—
169C4×D3—82no—
170C4×D3—82no—
171Dic6—24no—
172Dic6—42no—
173Dic6—42no—
174Dic6—42no—
175C3×C8—44no—
176C3×C8—42no—
177C3⋊C8—42no—
178#41—122no—
179#41—122no—
180#41—122no—
181#41—122no—
182C2×Q8—32no—
183C2×Q8—61no—
184C2×Q8—62no—
185C2×Q8—62no—
186C2×Q8—62no—
187C2×D4—62no—
188C2×D4—62no—
189C2×D4—62no—
190C2×D4—122no—
191C22×C4—61no—
192C22×C4—61no—
193C22×C4—122no—
194C22×C4C2×D1211yescharacteristic
195C22×C4D3×D411yescharacteristic
196C22×C4#23911yescharacteristic
197Q16—128no—
198SD16—128no—
199D8—242no—
200M16—42no—
201M16—42no—
202C2×C8—22no—
203C2×C8—24no—
204C2×C8—42no—
205C2×C8—62no—
206C2×C8—121no—
207C2×C8—121no—
208C4⋊C4—62no—
209C4⋊C4—62no—
210C4⋊C4—62no—
211C4⋊C4—122no—
212C22⋊C4—122no—
213C22⋊C4—122no—
214C42—121no—
215C42—121no—
216C42C2×D1212yes—
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#44712yes—
229C3×C4#44912yes—
230C3×C4#45511yescharacteristic
231C3×C4#45611yescharacteristic
232Dic3—42no—
233Dic3—82no—
234Dic3—81no—
235C23—122no—
236C23#89511yescharacteristic
237Q8—64no—
238Q8—122no—
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—41no—
251C2×C4—62no—
252C2×C4—121no—
253C2×C4—121no—
254C2×C4—122no—
255C2×C4—121no—
256C2×C4—122no—
257C2×C4—242no—
258C2×C4—242no—
259C2×C4#87411yescharacteristic
260C2×C4#88211yescharacteristic
261C2×C4#88212yes—
262C2×C4#89512yes—
263C2×C4D3×D811yescharacteristic
264C2×C4#91412yes—
265C2×C4#91911yescharacteristic
266C8—42no—
267C8—44no—
268C8—122no—
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#467111yescharacteristic
281C22#486511yescharacteristic
282C22#488311yescharacteristic
283C4—22no—
284C4—21no—
285C4—122no—
286C4—242no—
287C4—241no—
288C4#474611yescharacteristic
289C4#475211yescharacteristic
290C4#477312yes—
291C4#481012yes—
292C3#1025511yescharacteristic, complemented, Sylow
293C2—21no—
294C2—21no—
295C2—482no—
296C2#6872711yescharacteristic
297C2#6888611yescharacteristic
298C2#6900911yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1292C2
socleC22×C3#392
prime radical#79026C2
hypercenter#1292D3
hypocenterC3#10255
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
1. #79067C2semidirect
3. #70498C2semidirect
4. #70472C2semidirect
5. C3×#1292C2semidirect
26. #1292D3semidirect
91. C2×Dic6D8semidirect
292. C3#10255semidirect

Maximal subgroups

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

subgroupindex
#790672
#790262
#704982
#704722
C3×#12922
#102553