Finite Group Databasebeta

← #10697945

Subgroups — #10697945

Order 768.

554 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 89 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#79065C212yesmaximal, complemented
2#79064C211yesmaximal, characteristic
3#70514C211yesmaximal, characteristic, complemented
4#70512C212yesmaximal, complemented
5C3×#1309C211yesmaximal, characteristic, complemented
6#10301—31nomaximal, Sylow
7#5477C2212yes—
8#5470C2211yescharacteristic
9C3×#423C2212yes—
10C3×#423C2211yescharacteristic
11C2×#825—24no—
12#5135—22no—
13#5134—24no—
14C42×Dic3C2211yescharacteristic
15#4850—22no—
16#4848—22no—
17#4846—22no—
18#4843—22no—
19#4842—22no—
20#4841—22no—
21#4555—22no—
22#3254—32no—
23#3253—31no—
24#1994—31no—
25#1962—32no—
26#1309D311yescharacteristic, complemented
27C2×#235—22no—
28C2×C3×M16D414yes—
29C2×C3×M16D412yes—
30C2×C3×C42C2311yescharacteristic
31#930—22no—
32#930—22no—
33#926—21no—
34#924—22no—
35C2×C4×Dic3—22no—
36C2×C4×Dic3—21no—
37C2×D24—22no—
38C2×#204—24no—
39#875—22no—
40#874D412yes—
41C4×D12—22no—
42C4×D12—22no—
43#870—21no—
44#869D412yes—
45#869D412yes—
46C4×Dic6—22no—
47C3×C4⋊C8—22no—
48C3×C4⋊C8—22no—
49C3×C4⋊C8—22no—
50#825—48no—
51#821—24no—
52#821—24no—
53#818—24no—
54#817—22no—
55#816—24no—
56#533—31no—
57#532—32no—
58#477—62no—
59#476—62no—
60#474—62no—
61#469—62no—
62#468—64no—
63#461—62no—
64#460—62no—
65#459—62no—
66#423C2×D312yes—
67#423C2×D311yescharacteristic
68C2×#104—64no—
69C43—31no—
70#339—62no—
71C22×C3×C4C2×D412yes—
72C22×C3×C4C2×D411yescharacteristic
73C2×#67—42no—
74C22×Dic3—41no—
75#235—44no—
76#235—44no—
77C2×D12—22no—
78C2×D12—22no—
79C2×D12—22no—
80C2×C4×D3—42no—
81C2×Dic6—22no—
82C2×Dic6—22no—
83C3×M16—44no—
84C3×M16—42no—
85C2×C3×C8—24no—
86C2×C3×C8—22no—
87C3×C42C2×D412yes—
88C3×C42C2×D411yescharacteristic
89C3×C42#4111yescharacteristic
90#217—41no—
91(C2×D3)⋊C4—42no—
92(C2×D3)⋊C4—42no—
93C4⋊Dic3—22no—
94C4⋊Dic3—22no—
95C4⋊Dic3—22no—
96C4⋊Dic3—22no—
97Dic3⋊C4—42no—
98C4×Dic3—24no—
99C4×Dic3—22no—
100C4×Dic3—42no—
101C4×Dic3—41no—
102C4×Dic3—41no—
103D24—82no—
104#204—84no—
105C2×#41—62no—
106C2×SD16—64no—
107C2×D8—62no—
108C2×M16D1212yes—
109C2×M16D1214yes—
110C4⋊Q8—32no—
111C4⋊Q8—32no—
112#127—32no—
113#125—61no—
114#124—62no—
115C22⋊Q8—62no—
116#121—62no—
117#121—62no—
118C4×Q8—62no—
119C4×D4—62no—
120C4×D4—62no—
121#117—61no—
122C2×C42—61no—
123C2×C42—62no—
124C2×C42C22×D311yescharacteristic
125#107—64no—
126#106—62no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130#104—128no—
131#103—64no—
132#102—64no—
133#102—64no—
134C23×C3#12711yescharacteristic
135C22×D3—42no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4#12012yes—
141C2×C3×C4#12011yescharacteristic
142C2×C3×C4#12112yes—
143C2×C3×C4#12111yescharacteristic
144#67—84no—
145C2×Dic3—44no—
146C2×Dic3—42no—
147C2×Dic3—42no—
148C2×Dic3—42no—
149D12—44no—
150D12—42no—
151D12—42no—
152D12—42no—
153D12—42no—
154C4×D3—82no—
155C4×D3—82no—
156Dic6—44no—
157Dic6—42no—
158Dic6—42no—
159C3×C8—44no—
160C3×C8—42no—
161#41—124no—
162#41—124no—
163C2×Q8—62no—
164C2×Q8—62no—
165C2×D4—62no—
166C2×D4—62no—
167C2×D4—62no—
168C2×D4—122no—
169C22×C4—121no—
170C22×C4—122no—
171C22×C4C2×D1211yescharacteristic
172C22×C4C2×D1212yes—
173SD16—244no—
174D8—242no—
175M16—44no—
176M16—42no—
177C2×C8—22no—
178C2×C8—24no—
179C4⋊C4—62no—
180C4⋊C4—62no—
181C4⋊C4—62no—
182C4⋊C4—62no—
183C4⋊C4—122no—
184C22⋊C4—121no—
185C22⋊C4—122no—
186C22⋊C4—122no—
187C42—64no—
188C42—62no—
189C42—121no—
190C42—122no—
191C42—121no—
192C42D3×D411yescharacteristic
193C42D3×D412yes—
194C42#23911yescharacteristic
195C22×C3—22no—
196C22×C3—21no—
197C22×C3#39211yescharacteristic
198C2×D3—82no—
199C2×D3—82no—
200C2×D3—82no—
201C3×C4—22no—
202C3×C4—21no—
203C3×C4#45312yes—
204C3×C4#45412yes—
205C3×C4#45511yescharacteristic
206C3×C4#45611yescharacteristic
207Dic3—44no—
208Dic3—82no—
209Dic3—82no—
210C23—122no—
211C23#87411yescharacteristic
212Q8—124no—
213Q8—122no—
214Q8—122no—
215D4—124no—
216D4—122no—
217D4—122no—
218D4—122no—
219D4—122no—
220D4—244no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—124no—
226C2×C4—122no—
227C2×C4—122no—
228C2×C4—122no—
229C2×C4—242no—
230C2×C4—242no—
231C2×C4#88211yescharacteristic
232C2×C4#88212yes—
233C2×C4#89511yescharacteristic
234C2×C4#89512yes—
235C8—42no—
236C8—44no—
237C2×C3—41no—
238C2×C3#200612yes—
239C2×C3#200711yescharacteristic
240D3—162no—
241C22—22no—
242C22—21no—
243C22—242no—
244C22—242no—
245C22—242no—
246C22#467111yescharacteristic
247C4—22no—
248C4—21no—
249C4—124no—
250C4—242no—
251C4—242no—
252C4#474611yescharacteristic
253C4#474712yes—
254C4#475211yescharacteristic
255C4#475312yes—
256C3#1030111yescharacteristic, complemented, Sylow
257C2—41no—
258C2—482no—
259C2#6900812yes—
260C2#6900911yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1309C2
socleC22×C3#392
prime radical#79064C2
hypercenter#1309D3
hypocenterC3#10301
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1309

Complements

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

NH ≅ G/Ntype
1. #79065C2semidirect
3. #70514C2semidirect
4. #70512C2semidirect
5. C3×#1309C2semidirect
26. #1309D3semidirect
256. C3#10301semidirect

Maximal subgroups

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

subgroupindex
#790652
#790642
#705142
#705122
C3×#13092
#103013