Finite Group Databasebeta

← #10697723

Subgroups — #10697723

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×#4849C212yesmaximal
2C2×#4682C211yesmaximal, characteristic
3C2×#4482C212yesmaximal
4#69788C211yesmaximal, characteristic
5C3×#1277C211yesmaximal, characteristic
6#10265—31nomaximal, Sylow
7C22×Dic12—22no—
8C2×#869C2212yes—
9C2×C4×Dic6C2211yescharacteristic
10C2×C3×C4⋊C8C2212yes—
11C2×C3×C4×C8C2211yescharacteristic
12C2×#818—22no—
13C2×#816—22no—
14C2×C4⋊(C3⋊C8)C2211yescharacteristic
15#4849—24no—
16#4849—24no—
17#4682—24no—
18#4550—22no—
19#4482—24no—
20C2×#500—31no—
21C2×#462—32no—
22#1984—31no—
23C2×#332—32no—
24#1277D311yescharacteristic
25C22×Dic6—22no—
26C22×Dic6D411yescharacteristic
27C22×C3×C8—22no—
28C22×C3×C8D412yes—
29C2×C3×C42C2311yescharacteristic
30C2×C4⋊Dic3—22no—
31C2×C4⋊Dic3—22no—
32C2×C4⋊Dic3D411yescharacteristic
33C2×Dic3⋊C4—21no—
34C2×C4×Dic3—21no—
35C22×C3⋊C8—21no—
36C2×Dic12—28no—
37C2×Dic12—44no—
38C2×Dic12—44no—
39#869—22no—
40#869—22no—
41#869—22no—
42#869D414yes—
43C4×Dic6—22no—
44C4×Dic6—22no—
45C3×C4⋊C8—22no—
46C3×C4⋊C8D414yes—
47C3×C4×C8—22no—
48#818—28no—
49#816—28no—
50C4⋊(C3⋊C8)—22no—
51C22×Q16—62no—
52C2×C4⋊Q8—32no—
53C2×C4×Q8—31no—
54#500—64no—
55#462—64no—
56#462—64no—
57C2×#107—62no—
58C2×C4⋊C8—31no—
59C2×C4⋊C8C2×D312yes—
60C2×#103—62no—
61C2×C4×C8C2×D311yescharacteristic
62#336—62no—
63#332—64no—
64C22×C3×C4C2×D411yescharacteristic
65C22×C3×C4C2×D411yescharacteristic
66C22×C3×C4#4111yescharacteristic
67C22×Dic3—21no—
68C22×Dic3—21no—
69C22×Dic3—42no—
70C2×Dic6—22no—
71C2×Dic6—22no—
72C2×Dic6—21no—
73C2×Dic6—24no—
74C2×Dic6—42no—
75C2×Dic6—42no—
76C2×Dic6—42no—
77C2×Dic6—42no—
78C2×Dic6—42no—
79C2×Dic6D812yes—
80C2×C3×C8—22no—
81C2×C3×C8—22no—
82C2×C3×C8—24no—
83C2×C3×C8—42no—
84C2×C3×C8—42no—
85C2×C3×C8Q1614yescomplemented
86C3×C42C2×D412yes—
87C3×C42C2×D412yes—
88C4⋊Dic3—22no—
89C4⋊Dic3—24no—
90C4⋊Dic3—24no—
91C4⋊Dic3—42no—
92C4⋊Dic3—42no—
93Dic3⋊C4—42no—
94C4×Dic3—41no—
95C4×Dic3—41no—
96C2×C3⋊C8—41no—
97C2×C3⋊C8—42no—
98Dic12—416no—
99C22×Q8—31no—
100C22×Q8—62no—
101C2×Q16—68no—
102C2×Q16—124no—
103C2×Q16—124no—
104C22×C8—22no—
105C22×C8—61no—
106C22×C8D1212yes—
107C4⋊Q8—34no—
108C4⋊Q8—62no—
109C4⋊Q8—62no—
110C4⋊Q8—62no—
111C4×Q8—62no—
112C4×Q8—62no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—62no—
115C2×C4⋊C4—62no—
116C2×C4⋊C4—61no—
117C2×C42—61no—
118C2×C42C22×D311yescharacteristic
119#107—68no—
120C4⋊C8—22no—
121C4⋊C8—62no—
122C4⋊C8D1214yes—
123#103—68no—
124C4×C8—22no—
125C23×C3#12111yescharacteristic
126C2×C3×C4—22no—
127C2×C3×C4#12012yes—
128C2×C3×C4#12011yescharacteristic
129C2×C3×C4#12112yes—
130C2×C3×C4#12711yescharacteristic
131C2×C3×C4C2×D811yescharacteristic
132C2×C3×C4C2×Q1612yes—
133C2×C3×C4#13511yescharacteristic
134C2×C3×C4AGL(1,ℤ/8)12yes—
135C2×C3×C4#13712yes—
136C2×Dic3—22no—
137C2×Dic3—41no—
138C2×Dic3—41no—
139C2×Dic3—41no—
140C2×Dic3—44no—
141C2×Dic3—42no—
142C2×Dic3—82no—
143C2×Dic3—82no—
144Dic6—24no—
145Dic6—42no—
146Dic6—44no—
147Dic6—44no—
148Dic6—82no—
149Dic6—82no—
150C3×C8—24no—
151C3×C8—44no—
152C3⋊C8—81no—
153C2×Q8—32no—
154C2×Q8—61no—
155C2×Q8—62no—
156C2×Q8—64no—
157C2×Q8—62no—
158C2×Q8—122no—
159C2×Q8—122no—
160C2×Q8—122no—
161C2×Q8—122no—
162C2×Q8—122no—
163C22×C4—61no—
164C22×C4—61no—
165C22×C4—122no—
166C22×C4C2×D1211yescharacteristic
167C22×C4D3×D411yescharacteristic
168C22×C4#23911yescharacteristic
169Q16—1216no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—24no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—121no—
176C2×C8—122no—
177C2×C8Dic1214yescomplemented
178C4⋊C4—62no—
179C4⋊C4—64no—
180C4⋊C4—64no—
181C4⋊C4—122no—
182C4⋊C4—122no—
183C4⋊C4—122no—
184C42—121no—
185C42—121no—
186C42C2×D1212yes—
187C42D3×D412yes—
188C22×C3#39211yescharacteristic
189C22×C3#46212yes—
190C22×C3#46611yescharacteristic
191C22×C3#46911yescharacteristic
192C22×C3#47312yes—
193C3×C4—22no—
194C3×C4#44712yes—
195C3×C4#44912yes—
196C3×C4#45112yes—
197C3×C4#49411yescharacteristic
198C3×C4#49711yescharacteristic
199Dic3—42no—
200Dic3—84no—
201Dic3—81no—
202C23#89511yescharacteristic
203Q8—64no—
204Q8—122no—
205Q8—124no—
206Q8—124no—
207Q8—242no—
208Q8—242no—
209C2×C4—22no—
210C2×C4—62no—
211C2×C4—121no—
212C2×C4—121no—
213C2×C4—122no—
214C2×C4—121no—
215C2×C4—124no—
216C2×C4—242no—
217C2×C4—242no—
218C2×C4#87411yescharacteristic
219C2×C4#88211yescharacteristic
220C2×C4#88212yes—
221C2×C4#89512yes—
222C2×C4C2×Dic1212yes—
223C2×C4#90912yes—
224C2×C4D3×D811yescharacteristic
225C2×C4#91412yes—
226C2×C4#91911yescharacteristic
227C8—24no—
228C8—44no—
229C8—241no—
230C2×C3#167412yes—
231C2×C3#169412yes—
232C2×C3#199911yescharacteristic
233C2×C3#201611yescharacteristic
234C2×C3#202011yescharacteristic
235C22#467111yescharacteristic
236C22#484912yes—
237C22#486511yescharacteristic
238C22#488311yescharacteristic
239C22#489612yes—
240C4—22no—
241C4—122no—
242C4—244no—
243C4—241no—
244C4#469911yescharacteristic
245C4#471511yescharacteristic
246C4#473912yes—
247C4#477312yes—
248C4#481012yes—
249C3#1026511yescharacteristic, complemented, Sylow
250C2#6855212yes—
251C2#6858812yes—
252C2#6868811yescharacteristic
253C2#6888611yescharacteristic
254C2#6894411yescharacteristic
subgroupG/H
centerC23#895
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1277C2
socle, prime radicalC23×C3#121
hypercenter#1277D3
hypocenterC3#10265
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1277

Complements

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

NH ≅ G/Ntype
85. C2×C3×C8Q16semidirect
177. C2×C8Dic12semidirect
249. C3#10265semidirect

Maximal subgroups

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

subgroupindex
C2×#48492
C2×#46822
C2×#44822
#697882
C3×#12772
#102653