Finite Group Databasebeta

← #10697942

Subgroups — #10697942

Order 768.

554 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 87 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#83366C212yesmaximal, complemented
2#79064C211yesmaximal, characteristic
3C3×#1994C211yesmaximal, characteristic, complemented
4#69395C212yesmaximal, complemented
5C3⋊#1309C211yesmaximal, characteristic, complemented
6#10301—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8#5477C2212yes—
9C3×#423C2211yescharacteristic
10C3×C43C2211yescharacteristic
11#5135—22no—
12#5087—22no—
13#5083—22no—
14#5049—24no—
15#5027—22no—
16#5021—22no—
17C2×#805—24no—
18#4924C2212yes—
19#4843—22no—
20#4842—22no—
21C3×#339—22no—
22#3254—32no—
23#3253—31no—
24#1994D311yescharacteristic, complemented
25#1962—32no—
26#1309—31no—
27C2×#235—22no—
28C2×C3×M16D412yes—
29C3×C4⋊Q8D412yes—
30C3×#125—21no—
31C3×C22⋊Q8—22no—
32C3×C4×Q8—22no—
33C3×#117—21no—
34C2×C3×C42—22no—
35C2×C3×C42—21no—
36C2×C3×C42C2311yescharacteristic
37C2×#215—24no—
38#930—22no—
39#930—22no—
40C2×#208D414yes—
41C2×D24—22no—
42#875—22no—
43#874D412yes—
44C4×D12—22no—
45C4×D12—22no—
46#869D412yes—
47C3×C4⋊C8—22no—
48#835—24no—
49#821—24no—
50#817—22no—
51#809—24no—
52#807—24no—
53#805—48no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56#533C2×D311yescharacteristic
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#423—32no—
67#423C2×D311yescharacteristic
68C2×#104—64no—
69C43C2×D311yescharacteristic
70#339—22no—
71C2×C3×Q8—22no—
72C22×C3×C4—41no—
73C22×C3×C4C2×D412yes—
74C22×C3×C4C2×D411yescharacteristic
75C2×#67—42no—
76#235—44no—
77#235—44no—
78C2×D12—22no—
79C2×D12—22no—
80C2×D12—22no—
81C2×C4×D3—42no—
82C2×Dic6—22no—
83C3×M16—42no—
84C2×C3×C8—22no—
85C3×C4⋊C4—22no—
86C3×C4⋊C4—22no—
87C3×C4⋊C4—42no—
88C3×C22⋊C4—41no—
89C3×C42—24no—
90C3×C42—22no—
91C3×C42—42no—
92C3×C42—41no—
93C3×C42—41no—
94C3×C42C2×D411yescharacteristic
95C3×C42C2×D412yes—
96C3×C42#4111yescharacteristic
97#215—84no—
98(C2×D3)⋊C4—42no—
99(C2×D3)⋊C4—42no—
100C4⋊Dic3—22no—
101C4⋊Dic3—22no—
102#208—44no—
103C2×C3⋊C8—24no—
104D24—82no—
105C2×#41—62no—
106C2×SD16—64no—
107C2×D8—62no—
108C2×M16—34no—
109C2×M16#6712yes—
110C4⋊Q8—32no—
111C4⋊Q8D1212yes—
112#127—32no—
113#125—21no—
114#124—62no—
115C22⋊Q8—22no—
116#121—62no—
117#121—62no—
118C4×Q8—22no—
119C4×D4—62no—
120C4×D4—62no—
121#117—21no—
122C2×C42—22no—
123C2×C42—21no—
124C2×C42C22×D311yescharacteristic
125#107—64no—
126#106—62no—
127C4⋊C8—22no—
128C4⋊C8—62no—
129C4⋊C8—62no—
130#104—128no—
131#103—64no—
132#102—64no—
133#102—64no—
134C23×C3#12711yescharacteristic
135C22×D3—42no—
136C3×Q8—42no—
137C3×Q8—42no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—44no—
143C2×C3×C4—42no—
144C2×C3×C4—42no—
145C2×C3×C4#12012yes—
146C2×C3×C4#12011yescharacteristic
147C2×C3×C4#12112yes—
148C2×C3×C4#12111yescharacteristic
149#67—84no—
150C2×Dic3—42no—
151D12—44no—
152D12—42no—
153D12—42no—
154D12—42no—
155D12—42no—
156C4×D3—82no—
157C4×D3—82no—
158Dic6—44no—
159C3×C8—42no—
160C3⋊C8—44no—
161#41—124no—
162#41—124no—
163C2×Q8—22no—
164C2×Q8—62no—
165C2×D4—62no—
166C2×D4—62no—
167C2×D4—62no—
168C2×D4—122no—
169C22×C4—41no—
170C22×C4—122no—
171C22×C4D3×D412yes—
172C22×C4C2×#6711yescharacteristic
173SD16—244no—
174D8—242no—
175M16—42no—
176M16—124no—
177C2×C8—22no—
178C2×C8—64no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—42no—
182C4⋊C4—62no—
183C4⋊C4—62no—
184C22⋊C4—41no—
185C22⋊C4—122no—
186C22⋊C4—122no—
187C42—24no—
188C42—22no—
189C42—41no—
190C42—42no—
191C42—41no—
192C42C2×D1211yescharacteristic
193C42#23511yescharacteristic
194C42D3×D412yes—
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—44no—
204C3×C4—82no—
205C3×C4#45312yes—
206C3×C4#45412yes—
207C3×C4#45511yescharacteristic
208C3×C4#45611yescharacteristic
209Dic3—82no—
210C23—122no—
211C23#94011yescharacteristic
212Q8—42no—
213Q8—42no—
214Q8—124no—
215D4—124no—
216D4—122no—
217D4—122no—
218D4—122no—
219D4—122no—
220D4—244no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—122no—
229C2×C4—242no—
230C2×C4—242no—
231C2×C4#88212yes—
232C2×C4#88412yes—
233C2×C4#93011yescharacteristic
234C2×C4#93711yescharacteristic
235C8—42no—
236C8—124no—
237C2×C3—41no—
238C2×C3#200612yes—
239C2×C3#200711yescharacteristic
240D3—162no—
241C22—21no—
242C22—22no—
243C22—242no—
244C22—242no—
245C22—242no—
246C22#496011yescharacteristic
247C4—21no—
248C4—22no—
249C4—44no—
250C4—82no—
251C4—242no—
252C4#482212yes—
253C4#482712yes—
254C4#506111yescharacteristic
255C4#509511yescharacteristic
256C3#1030111yescharacteristic, complemented, Sylow
257C2—41no—
258C2—482no—
259C2#6969411yescharacteristic
260C2#7039212yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1994C2
socleC22×C3#392
prime radical#79064C2
hypercenter#1994D3
hypocenterC3#10301
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
1. #83366C2semidirect
3. C3×#1994C2semidirect
4. #69395C2semidirect
5. C3⋊#1309C2semidirect
24. #1994D3semidirect
256. C3#10301semidirect

Maximal subgroups

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

subgroupindex
#833662
#790642
C3×#19942
#693952
C3⋊#13092
#103013