Finite Group Databasebeta

← #10696511

Subgroups — #10696511

Order 768.

600 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 97 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
2C2×#5055C212yesmaximal, complemented
3C2×#4540C212yesmaximal, complemented
4#69552C211yesmaximal, characteristic, complemented
5#67630C211yesmaximal, characteristic, complemented
6#10085—31nomaximal, Sylow
7C2×C3×#124C2212yes—
8C2×#869C2211yescharacteristic
9C3×#394—22no—
10C3×#393—22no—
11C3×#392—22no—
12C3×#382C2211yescharacteristic
13C2×#835—22no—
14C2×#832—22no—
15#5055—24no—
16#5055—24no—
17C2×#810—22no—
18C2×#803C2211yescharacteristic
19C2×#803C2212yes—
20#4540—24no—
21C2×#489—32no—
22#2819D311yescharacteristic
23#1960—31no—
24C2×#329—32no—
25#1295—31no—
26C22×C3×Q8D412yes—
27C22×C3×D4—22no—
28C22×C3×D4D412yes—
29C22×Dic6—21no—
30C3×#124—22no—
31C3×#124—22no—
32C3×#124—22no—
33C3×#124—22no—
34C2×C3×C4⋊C4D412yes—
35C2×C3×C22⋊C4—22no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—24no—
38C2×C3×C42C2311yescharacteristic
39C2×C4⋊Dic3—22no—
40C22×C3⋊C8—22no—
41C22×C3⋊C8—21no—
42#869—22no—
43#869—21no—
44#869D412yescomplemented
45C3×#95—22no—
46C3×#95—22no—
47#835—28no—
48#832—28no—
49#810—44no—
50#803—22no—
51#803—22no—
52#803D414yescomplemented
53C2×C4⋊Q8—31no—
54C2×#124C2×D312yes—
55#489—64no—
56#489—64no—
57C2×#103—62no—
58C2×#103—62no—
59C2×#102—62no—
60C2×#97—32no—
61C2×#97—31no—
62#394—22no—
63#393—22no—
64#392—22no—
65#382C2×D311yescharacteristic
66#329—64no—
67C24×C3—22no—
68C2×C3×Q8—22no—
69C2×C3×Q8—22no—
70C2×C3×Q8—44no—
71C2×C3×D4—22no—
72C2×C3×D4—22no—
73C2×C3×D4—44no—
74C2×C3×D4—82no—
75C2×C3×D4—84no—
76C22×C3×C4—22no—
77C22×C3×C4—22no—
78C22×C3×C4C2×D412yes—
79C22×C3×C4C2×D411yescharacteristic
80C22×Dic3—41no—
81C2×Dic6—22no—
82C2×Dic6—42no—
83C2×Dic6—42no—
84C2×Dic6—41no—
85C3×C4⋊C4—42no—
86C3×C22⋊C4—42no—
87C3×C22⋊C4—42no—
88C3×C22⋊C4—42no—
89C3×C22⋊C4—42no—
90C3×C22⋊C4—84no—
91C3×C42C2×D411yescharacteristic
92C3×C42C2×D412yes—
93C3×C42#4111yescharacteristic
94C4⋊Dic3—24no—
95C4⋊Dic3—42no—
96C2×C3⋊C8—22no—
97C2×C3⋊C8—24no—
98C2×C3⋊C8—41no—
99C2×C3⋊C8—44no—
100C2×C3⋊C8—41no—
101C22×Q8—61no—
102C22×Q8#6712yes—
103C22×D4—22no—
104C22×D4#6712yes—
105C22×C8—62no—
106C22×C8—61no—
107C4⋊Q8—32no—
108C4⋊Q8—62no—
109C4⋊Q8—61no—
110#124—22no—
111#124—22no—
112#124—22no—
113#124—22no—
114C2×C4⋊C4—62no—
115C2×C4⋊C4#6712yes—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—24no—
119C2×C42C22×D311yescharacteristic
120#103—68no—
121#103—124no—
122#102—68no—
123#97—34no—
124#97—62no—
125#97—62no—
126#95—22no—
127#95—22no—
128C23×C3—42no—
129C23×C3—42no—
130C23×C3—42no—
131C23×C3—44no—
132C23×C3—44no—
133C23×C3#12011yescharacteristic
134C3×Q8—44no—
135C3×Q8—82no—
136C3×D4—44no—
137C3×D4—82no—
138C3×D4—88no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—24no—
142C2×C3×C4—42no—
143C2×C3×C4—42no—
144C2×C3×C4—44no—
145C2×C3×C4—44no—
146C2×C3×C4#12012yes—
147C2×C3×C4#12111yescharacteristic
148C2×C3×C4#12112yes—
149C2×C3×C4#12711yescharacteristic
150C2×Dic3—42no—
151C2×Dic3—81no—
152C2×Dic3—81no—
153Dic6—44no—
154Dic6—82no—
155C3⋊C8—44no—
156C3⋊C8—42no—
157C24—22no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—44no—
161C2×Q8—62no—
162C2×Q8—122no—
163C2×Q8—122no—
164C2×Q8—121no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—44no—
168C2×D4—82no—
169C2×D4—84no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—121no—
173C22×C4C2×#6712yes—
174C22×C4C2×#6711yescharacteristic
175C2×C8—62no—
176C2×C8—64no—
177C2×C8—124no—
178C2×C8—121no—
179C2×C8—121no—
180C4⋊C4—42no—
181C4⋊C4—64no—
182C4⋊C4—122no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—84no—
188C42D3×D411yescharacteristic
189C42D3×D412yes—
190C42#23711yescharacteristic
191C22×C3—84no—
192C22×C3—84no—
193C22×C3—82no—
194C22×C3—82no—
195C22×C3—82no—
196C22×C3#39211yescharacteristic
197C22×C3#45311yescharacteristic
198C22×C3#45411yescharacteristic
199C22×C3#45514yes—
200C3×C4—22no—
201C3×C4—22no—
202C3×C4—21no—
203C3×C4—21no—
204C3×C4—82no—
205C3×C4—82no—
206Dic3—82no—
207C23—42no—
208C23—42no—
209C23—44no—
210C23—44no—
211C23—42no—
212C23#95311yescharacteristic
213Q8—44no—
214Q8—82no—
215Q8—124no—
216Q8—242no—
217D4—44no—
218D4—82no—
219D4—88no—
220C2×C4—21no—
221C2×C4—24no—
222C2×C4—21no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—122no—
228C2×C4—241no—
229C2×C4—241no—
230C2×C4#93712yes—
231C2×C4#93811yescharacteristic
232C2×C4#93912yes—
233C2×C4#94011yescharacteristic
234C8—124no—
235C8—122no—
236C2×C3—162no—
237C2×C3#165912yes—
238C2×C3#166212yes—
239C2×C3#200412yes—
240C2×C3#200611yescharacteristic
241C22—84no—
242C22—84no—
243C22—82no—
244C22—82no—
245C22—82no—
246C22#522311yescharacteristic
247C22#524111yescharacteristic
248C22#524214yes—
249C22#524511yescharacteristic
250C4—22no—
251C4—22no—
252C4—21no—
253C4—21no—
254C4—82no—
255C4—82no—
256C4—242no—
257C3#1008511yescharacteristic, complemented, Sylow
258C2—162no—
259C2#7185712yes—
260C2#7187911yescharacteristic
261C2#7194912yes—
262C2#7195412yes—
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2819C2
socleC23×C3#120
hypercenter#2819D3
hypocenterC3#10085
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. C2×#5055C2semidirect
3. C2×#4540C2semidirect
4. #69552C2semidirect
5. #67630C2semidirect
44. #869D4semidirect
52. #803D4semidirect
257. C3#10085semidirect

Maximal subgroups

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

subgroupindex
C3×#28192
C2×#50552
C2×#45402
#695522
#676302
#100853