Finite Group Databasebeta

← #10697720

Subgroups — #10697720

Order 768.

574 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C2×C3×#500C211yesmaximal, characteristic
2C2×#5090C212yesmaximal
3C2×#4547C212yesmaximal
4#69788C211yesmaximal, characteristic
5#67654C211yesmaximal, characteristic
6#10265—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2212yes—
8C22×#216—22no—
9C2×C4×Dic6C2211yescharacteristic
10C3×#500—24no—
11C2×C3×#107—22no—
12C2×C3×C4×C8C2211yescharacteristic
13C2×#835—22no—
14#5090—24no—
15#5090—24no—
16C2×C4⋊(C3⋊C8)C2211yescharacteristic
17C2×C4⋊(C3⋊C8)C2212yes—
18#4550—22no—
19#4547—24no—
20C2×#500D311yescharacteristic
21C2×#462—32no—
22#1984—31no—
23C2×#332—32no—
24#1277—31no—
25C22×C3×Q8—22no—
26C22×Dic6D411yescharacteristic
27C22×C3×C8D412yes—
28C3×C4⋊Q8—22no—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8—22no—
31C3×C4⋊Q8D414yes—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C42C2311yescharacteristic
35C2×#216—28no—
36C2×#216—44no—
37C2×#216—44no—
38C2×C4⋊Dic3D411yescharacteristic
39C2×Dic3⋊C4—21no—
40C2×C4×Dic3—21no—
41C22×C3⋊C8—22no—
42C22×C3⋊C8—21no—
43C4×Dic6—22no—
44C4×Dic6—22no—
45C3×#107—28no—
46C3×C4×C8—22no—
47#835—28no—
48C4⋊(C3⋊C8)—22no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)D414yes—
51C22×Q16—62no—
52C2×C4⋊Q8C2×D312yes—
53C2×C4×Q8—31no—
54#500—24no—
55#462—64no—
56#462—64no—
57C2×#107—22no—
58C2×C4⋊C8—32no—
59C2×C4⋊C8—31no—
60C2×#103—62no—
61C2×C4×C8C2×D311yescharacteristic
62#336—62no—
63#332—64no—
64C2×C3×Q8—24no—
65C2×C3×Q8—42no—
66C2×C3×Q8—42no—
67C2×C3×Q8—42no—
68C2×C3×Q8—42no—
69C2×C3×Q8—42no—
70C22×C3×C4—42no—
71C22×C3×C4C2×D411yescharacteristic
72C22×C3×C4C2×D411yescharacteristic
73C22×C3×C4#4111yescharacteristic
74C22×Dic3—21no—
75C22×Dic3—21no—
76C2×Dic6—22no—
77C2×Dic6—22no—
78C2×Dic6—21no—
79C2×Dic6D812yes—
80C2×C3×C8—22no—
81C2×C3×C8—22no—
82C2×C3×C8Q1614yescomplemented
83C3×C4⋊C4—24no—
84C3×C4⋊C4—24no—
85C3×C4⋊C4—42no—
86C3×C4⋊C4—42no—
87C3×C42C2×D412yes—
88C3×C42C2×D412yes—
89#216—416no—
90C4⋊Dic3—22no—
91Dic3⋊C4—42no—
92C4×Dic3—41no—
93C4×Dic3—41no—
94C2×C3⋊C8—24no—
95C2×C3⋊C8—42no—
96C2×C3⋊C8—42no—
97C2×C3⋊C8—41no—
98C2×C3⋊C8—42no—
99C22×Q8—22no—
100C22×Q8—31no—
101C2×Q16—68no—
102C2×Q16—124no—
103C2×Q16—124no—
104C22×C8—62no—
105C22×C8—61no—
106C22×C8#6712yes—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8#6714yes—
111C4×Q8—62no—
112C4×Q8—62no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—61no—
117C2×C42—61no—
118C2×C42C22×D311yescharacteristic
119#107—28no—
120C4⋊C8—34no—
121C4⋊C8—62no—
122C4⋊C8—62no—
123#103—68no—
124C4×C8—22no—
125C23×C3#12111yescharacteristic
126C3×Q8—44no—
127C3×Q8—44no—
128C3×Q8—82no—
129C3×Q8—82no—
130C2×C3×C4—22no—
131C2×C3×C4—44no—
132C2×C3×C4—82no—
133C2×C3×C4—82no—
134C2×C3×C4#12012yes—
135C2×C3×C4#12011yescharacteristic
136C2×C3×C4#12112yes—
137C2×C3×C4#12711yescharacteristic
138C2×C3×C4C2×D811yescharacteristic
139C2×C3×C4C2×Q1612yes—
140C2×C3×C4#13511yescharacteristic
141C2×C3×C4AGL(1,ℤ/8)12yes—
142C2×C3×C4#13712yes—
143C2×Dic3—22no—
144C2×Dic3—41no—
145C2×Dic3—41no—
146C2×Dic3—41no—
147C2×Dic3—42no—
148Dic6—24no—
149Dic6—42no—
150C3×C8—24no—
151C3⋊C8—44no—
152C3⋊C8—81no—
153C2×Q8—24no—
154C2×Q8—32no—
155C2×Q8—42no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—61no—
161C2×Q8—62no—
162C2×Q8—62no—
163C22×C4—42no—
164C22×C4—61no—
165C22×C4—61no—
166C22×C4D3×D411yescharacteristic
167C22×C4#23711yescharacteristic
168C22×C4C2×#6711yescharacteristic
169Q16—1216no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—64no—
173C2×C8—122no—
174C2×C8—122no—
175C2×C8—121no—
176C2×C8—122no—
177C2×C8#21614yescomplemented
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—62no—
183C4⋊C4—122no—
184C42—121no—
185C42—121no—
186C42D3×D412yes—
187C42C2×#6712yes—
188C22×C3#39211yescharacteristic
189C22×C3#46212yes—
190C22×C3#46611yescharacteristic
191C22×C3#46911yescharacteristic
192C22×C3#47312yes—
193C3×C4—22no—
194C3×C4—84no—
195C3×C4#44712yes—
196C3×C4#44912yes—
197C3×C4#45112yes—
198C3×C4#49411yescharacteristic
199C3×C4#49711yescharacteristic
200Dic3—42no—
201Dic3—81no—
202C23#93811yescharacteristic
203Q8—44no—
204Q8—44no—
205Q8—64no—
206Q8—82no—
207Q8—82no—
208Q8—122no—
209C2×C4—22no—
210C2×C4—44no—
211C2×C4—62no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4—121no—
215C2×C4—121no—
216C2×C4—122no—
217C2×C4—121no—
218C2×C4D3×D811yescharacteristic
219C2×C4#91112yes—
220C2×C4#91211yescharacteristic
221C2×C4#93712yes—
222C2×C4#93912yes—
223C2×C4#94011yescharacteristic
224C2×C4#94212yes—
225C2×C4C2×#21612yes—
226C2×C4#95311yescharacteristic
227C8—24no—
228C8—124no—
229C8—241no—
230C2×C3#167412yes—
231C2×C3#169412yes—
232C2×C3#199911yescharacteristic
233C2×C3#201611yescharacteristic
234C2×C3#202011yescharacteristic
235C22#509012yes—
236C22#515711yescharacteristic
237C22#515911yescharacteristic
238C22#519212yes—
239C22#522311yescharacteristic
240C4—22no—
241C4—84no—
242C4—122no—
243C4—241no—
244C4#508611yescharacteristic
245C4#509211yescharacteristic
246C4#515612yes—
247C4#515812yes—
248C4#522812yes—
249C3#1026511yescharacteristic, complemented, Sylow
250C2#7137712yes—
251C2#7137912yes—
252C2#7179311yescharacteristic
253C2#7204311yescharacteristic
254C2#7207411yescharacteristic
subgroupG/H
centerC23#938
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC2×C3×#500C2
socle, prime radicalC23×C3#121
hypercenterC2×#500D3
hypocenterC3#10265
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#500

Complements

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

NH ≅ G/Ntype
82. C2×C3×C8Q16semidirect
177. C2×C8#216semidirect
249. C3#10265semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5002
C2×#50902
C2×#45472
#697882
#676542
#102653