Finite Group Databasebeta

← #10698659

Subgroups — #10698659

Order 768.

518 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C3×#2819C211yesmaximal, characteristic
2#71726C212yesmaximal, complemented
3C2×#4540C212yesmaximal, complemented
4#69548C211yesmaximal, characteristic, complemented
5#67630C211yesmaximal, characteristic, complemented
6#10425—31nomaximal, Sylow
7C2×C3×#124C2212yes—
8C3×#394—22no—
9C3×#393—22no—
10C3×#392—22no—
11C3×#382C2211yescharacteristic
12C2×#835—22no—
13C2×#832—22no—
14#4935C2211yescharacteristic
15C2×#803C2212yes—
16C2×#803C2211yescharacteristic
17#4540—24no—
18#4529—22no—
19#2819D311yescharacteristic
20#1959—32no—
21#1836—31no—
22C2×#329—32no—
23#1295—31no—
24C22×C3×Q8D412yes—
25C22×C3×D4—22no—
26C22×C3×D4D412yes—
27C3×#124—22no—
28C3×#124—22no—
29C3×#124—22no—
30C3×#124—22no—
31C2×C3×C4⋊C4D412yes—
32C2×C3×C22⋊C4—22no—
33C2×C3×C22⋊C4—22no—
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4—22no—
36C2×C3×C42C2311yescharacteristic
37C2×C4⋊Dic3—22no—
38C2×C4⋊Dic3—21no—
39C22×C3⋊C8—22no—
40C22×C3⋊C8—21no—
41C3×#95—22no—
42C3×#95—22no—
43#835—44no—
44#832—44no—
45#803—21no—
46#803—22no—
47#803—22no—
48#803D412yescomplemented
49C2×#124C2×D312yes—
50C2×#103—62no—
51C2×#102—62no—
52C2×#97—31no—
53C2×#97—32no—
54#394—22no—
55#393—22no—
56#392—22no—
57#384—31no—
58#382C2×D311yescharacteristic
59#340—62no—
60#329—64no—
61C24×C3—22no—
62C2×C3×Q8—22no—
63C2×C3×Q8—22no—
64C2×C3×Q8—42no—
65C2×C3×Q8—42no—
66C2×C3×D4—22no—
67C2×C3×D4—22no—
68C2×C3×D4—42no—
69C2×C3×D4—42no—
70C2×C3×D4—82no—
71C2×C3×D4—82no—
72C2×C3×D4—82no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4C2×D412yes—
76C22×C3×C4C2×D411yescharacteristic
77C22×Dic3—41no—
78C3×C4⋊C4—42no—
79C3×C22⋊C4—42no—
80C3×C22⋊C4—42no—
81C3×C22⋊C4—42no—
82C3×C22⋊C4—42no—
83C3×C22⋊C4—82no—
84C3×C22⋊C4—82no—
85C3×C42C2×D411yescharacteristic
86C3×C42#4111yescharacteristic
87C3×C42#4112yes—
88C4⋊Dic3—42no—
89C4⋊Dic3—42no—
90C4⋊Dic3—42no—
91C2×C3⋊C8—24no—
92C2×C3⋊C8—22no—
93C2×C3⋊C8—41no—
94C2×C3⋊C8—41no—
95C2×C3⋊C8—42no—
96C2×C3⋊C8—42no—
97C22×Q8#6712yes—
98C22×D4—22no—
99C22×D4#6712yes—
100C22×C8—61no—
101C22×C8—62no—
102#124—22no—
103#124—22no—
104#124—22no—
105#124—22no—
106C2×C4⋊C4—61no—
107C2×C4⋊C4—62no—
108C2×C4⋊C4#6712yes—
109C2×C22⋊C4—22no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C42C22×D311yescharacteristic
114#103—124no—
115#102—124no—
116#97—32no—
117#97—62no—
118#97—61no—
119#97—62no—
120#95—22no—
121#95—22no—
122C23×C3—42no—
123C23×C3—42no—
124C23×C3—42no—
125C23×C3—42no—
126C23×C3—42no—
127C23×C3—42no—
128C23×C3—42no—
129C23×C3#12011yescharacteristic
130C3×Q8—44no—
131C3×Q8—82no—
132C3×D4—44no—
133C3×D4—82no—
134C3×D4—88no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4—42no—
144C2×C3×C4—42no—
145C2×C3×C4#12111yescharacteristic
146C2×C3×C4#12112yes—
147C2×C3×C4#12312yes—
148C2×C3×C4#12411yescharacteristic
149C2×Dic3—82no—
150C2×Dic3—81no—
151C3⋊C8—44no—
152C3⋊C8—42no—
153C24—22no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—82no—
163C2×D4—82no—
164C2×D4—82no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—121no—
168C22×C4C2×#6712yes—
169C22×C4C2×#6711yescharacteristic
170C2×C8—64no—
171C2×C8—62no—
172C2×C8—121no—
173C2×C8—122no—
174C2×C8—121no—
175C2×C8—122no—
176C4⋊C4—42no—
177C4⋊C4—122no—
178C4⋊C4—122no—
179C4⋊C4—122no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—82no—
185C22⋊C4—82no—
186C42D3×D411yescharacteristic
187C42#23711yescharacteristic
188C42#23712yes—
189C22×C3—82no—
190C22×C3—82no—
191C22×C3—82no—
192C22×C3—82no—
193C22×C3—82no—
194C22×C3—82no—
195C22×C3—82no—
196C22×C3#39411yescharacteristic
197C22×C3#45312yes—
198C22×C3#45411yescharacteristic
199C22×C3#45511yescharacteristic
200C22×C3#45512yes—
201C3×C4—22no—
202C3×C4—22no—
203C3×C4—21no—
204C3×C4—21no—
205C3×C4—82no—
206C3×C4—82no—
207Dic3—161no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23#95311yescharacteristic
216Q8—44no—
217Q8—82no—
218D4—44no—
219D4—82no—
220D4—88no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—242no—
232C2×C4—241no—
233C2×C4#93512yes—
234C2×C4#93611yescharacteristic
235C2×C4#93811yescharacteristic
236C2×C4#93912yes—
237C8—124no—
238C8—122no—
239C2×C3—162no—
240C2×C3#165611yescharacteristic
241C2×C3#165912yes—
242C2×C3#166211yescharacteristic
243C2×C3#203712yes—
244C2×C3#204011yescharacteristic
245C22—82no—
246C22—82no—
247C22—82no—
248C22—82no—
249C22—82no—
250C22—82no—
251C22—82no—
252C22#522211yescharacteristic
253C22#524112yes—
254C22#524211yescharacteristic
255C22#524212yes—
256C22#524511yescharacteristic
257C4—22no—
258C4—22no—
259C4—21no—
260C4—21no—
261C4—82no—
262C4—82no—
263C4—481no—
264C3#1042511yescharacteristic, complemented, Sylow
265C2—162no—
266C2#7185212yes—
267C2#7187711yescharacteristic
268C2#7194912yes—
269C2#7195211yescharacteristic
270C2#7195411yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2819C2
socleC23×C3#120
hypercenter#2819D3
hypocenterC3#10425
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2819

Complements

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

NH ≅ G/Ntype
2. #71726C2semidirect
3. C2×#4540C2semidirect
4. #69548C2semidirect
5. #67630C2semidirect
48. #803D4semidirect
264. C3#10425semidirect

Maximal subgroups

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

subgroupindex
C3×#28192
#717262
C2×#45402
#695482
#676302
#104253