Finite Group Databasebeta

← #10697467

Subgroups — #10697467

Order 768.

574 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3150C211yesmaximal, characteristic
2#83270C212yesmaximal, complemented
3#71814C212yesmaximal, complemented
4#69822C211yesmaximal, characteristic, complemented
5#67661C211yesmaximal, characteristic, complemented
6#10225—31nomaximal, Sylow
7C2×C3×#135—22no—
8C3×#532C2212yes—
9C2×#869C2211yescharacteristic
10C3×#497—22no—
11C3×#496—22no—
12C2×C3×#97C2211yescharacteristic
13C2×#816—22no—
14#5078—22no—
15#5070—22no—
16#5033—24no—
17#5007—24no—
18#4936C2211yescharacteristic
19#4925C2212yes—
20#4537—24no—
21#3226—32no—
22#3150D311yescharacteristic
23#1960—31no—
24#1517—32no—
25#1281—31no—
26C2×C3×#41—22no—
27C22×Dic6—21no—
28C3×#135—44no—
29C3×#135—44no—
30C2×C3×Q16—22no—
31C2×C3×SD16—24no—
32C2×C3×D8—22no—
33C22×C3×C8D412yes—
34C3×C4⋊Q8D412yes—
35C3×#127D412yes—
36C3×#124—22no—
37C3×#121—22no—
38C3×#121—22no—
39C3×C4×D4—22no—
40C3×C4×D4—22no—
41C2×C3×C42C2311yescharacteristic
42C2×C4⋊Dic3—22no—
43C2×C4⋊Dic3—21no—
44#869—22no—
45#869—21no—
46#869D412yes—
47C3×#97—22no—
48#832—28no—
49C3⋊#95—22no—
50#830D414yes—
51#816—44no—
52#807—28no—
53C4⋊(C3⋊C8)—22no—
54C4×C3⋊C8—22no—
55C2×#135—22no—
56#532C2×D312yes—
57C2×C4⋊Q8—31no—
58#500—62no—
59#497—22no—
60#496—22no—
61#486—62no—
62#480—64no—
63#474—64no—
64#431—32no—
65C2×#103—62no—
66C2×#97C2×D311yescharacteristic
67#382—31no—
68#327—64no—
69C3×#41—44no—
70C3×#41—44no—
71C2×C3×Q8—22no—
72C2×C3×D4—22no—
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—42no—
76C22×C3×C4—42no—
77C22×C3×C4C2×D412yes—
78C22×C3×C4C2×D411yescharacteristic
79C22×Dic3—41no—
80C22×Dic3—41no—
81C2×Dic6—22no—
82C2×Dic6—42no—
83C2×Dic6—42no—
84C2×Dic6—41no—
85C3×Q16—44no—
86C3×SD16—48no—
87C3×D8—44no—
88C2×C3×C8—22no—
89C2×C3×C8—22no—
90C2×C3×C8D814yes—
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C22⋊C4—42no—
94C3×C22⋊C4—42no—
95C3×C42C2×D411yescharacteristic
96C3×C42C2×D412yes—
97C3×C42#4111yescharacteristic
98C4⋊Dic3—24no—
99C4⋊Dic3—42no—
100C4⋊Dic3—42no—
101C2×C3⋊C8—24no—
102C2×#41—22no—
103C22×Q8—61no—
104#135—44no—
105#135—44no—
106C2×Q16—22no—
107C2×SD16—24no—
108C2×D8—22no—
109C22×C8#6712yes—
110C4⋊Q8—32no—
111C4⋊Q8—61no—
112C4⋊Q8—62no—
113C4⋊Q8#6712yes—
114#127#6712yes—
115#124—22no—
116#121—22no—
117#121—22no—
118C4×D4—22no—
119C4×D4—22no—
120C2×C4⋊C4—62no—
121C2×C4⋊C4—61no—
122C2×C42C22×D311yescharacteristic
123#107—68no—
124C4⋊C8—62no—
125#103—124no—
126#102—68no—
127C22⋊C8—34no—
128#97—22no—
129C4×C8—62no—
130#95—62no—
131C23×C3—42no—
132C23×C3#12711yescharacteristic
133C3×Q8—44no—
134C3×D4—44no—
135C3×D4—44no—
136C3×D4—82no—
137C3×D4—82no—
138C3×D4—82no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4—84no—
142C2×C3×C4#12012yes—
143C2×C3×C4#12011yescharacteristic
144C2×C3×C4#12111yescharacteristic
145C2×C3×C4#12112yes—
146C2×C3×C4C2×D812yes—
147C2×C3×C4AGL(1,ℤ/8)12yes—
148C2×Dic3—42no—
149C2×Dic3—81no—
150C2×Dic3—81no—
151C2×Dic3—82no—
152C2×Dic3—81no—
153Dic6—44no—
154Dic6—82no—
155C3×C8—24no—
156C3⋊C8—44no—
157#41—44no—
158#41—44no—
159C2×Q8—22no—
160C2×Q8—62no—
161C2×Q8—121no—
162C2×Q8—122no—
163C2×Q8—122no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—42no—
168C22×C4—42no—
169C22×C4—121no—
170C22×C4—121no—
171C22×C4D3×D412yes—
172C22×C4C2×#6711yescharacteristic
173Q16—44no—
174SD16—48no—
175D8—44no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—64no—
179C2×C8#21314yes—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—64no—
183C4⋊C4—122no—
184C4⋊C4—122no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C42D3×D411yescharacteristic
188C42#23711yescharacteristic
189C42C2×#6712yes—
190C22×C3—22no—
191C22×C3—82no—
192C22×C3—82no—
193C22×C3—82no—
194C22×C3#39211yescharacteristic
195C22×C3#49311yescharacteristic
196C22×C3#49611yescharacteristic
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—82no—
200C3×C4#44712yes—
201C3×C4#45912yes—
202Dic3—82no—
203Dic3—161no—
204C23—42no—
205C23#94011yescharacteristic
206Q8—44no—
207Q8—124no—
208Q8—242no—
209D4—44no—
210D4—44no—
211D4—82no—
212D4—82no—
213D4—82no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—84no—
217C2×C4—122no—
218C2×C4—241no—
219C2×C4—241no—
220C2×C4—242no—
221C2×C4—241no—
222C2×C4C2×#21312yes—
223C2×C4#93212yes—
224C2×C4#93712yes—
225C2×C4#93811yescharacteristic
226C2×C4#93912yes—
227C2×C4#95311yescharacteristic
228C8—24no—
229C8—124no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—162no—
233C2×C3#200912yes—
234C2×C3#201111yescharacteristic
235C22—22no—
236C22—82no—
237C22—82no—
238C22—82no—
239C22#507311yescharacteristic
240C22#507411yescharacteristic
241C22#522311yescharacteristic
242C4—22no—
243C4—22no—
244C4—82no—
245C4—242no—
246C4—481no—
247C4#507212yes—
248C4#521712yes—
249C3#1022511yescharacteristic, complemented, Sylow
250C2—21no—
251C2—21no—
252C2—162no—
253C2#7176711yescharacteristic
254C2#7212012yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3150C2
socleC22×C3#392
hypercenter#3150D3
hypocenterC3#10225
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3150

Complements

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

NH ≅ G/Ntype
2. #83270C2semidirect
3. #71814C2semidirect
4. #69822C2semidirect
5. #67661C2semidirect
249. C3#10225semidirect

Maximal subgroups

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

subgroupindex
C3×#31502
#832702
#718142
#698222
#676612
#102253