Finite Group Databasebeta

← #10695836

Subgroups — #10695836

Order 768.

602 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 81 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×#2815C211yesmaximal, characteristic
2#82616C212yesmaximal, complemented
3#71674C212yesmaximal, complemented
4#69534C211yesmaximal, characteristic, complemented
5#67629C211yesmaximal, characteristic, complemented
6#9983—31nomaximal, Sylow
7C2×C3×#121C2212yes—
8C2×#926C2211yescharacteristic
9C3×#397—22no—
10C3×#392—22no—
11C3×#392—22no—
12C3×#381C2211yescharacteristic
13#5215—22no—
14#5211C2211yescharacteristic
15#5207C2212yes—
16#5035—24no—
17#5034—22no—
18#5033—22no—
19#4964—22no—
20#4537—24no—
21#4527—22no—
22#3094—32no—
23#2815D311yescharacteristic
24#1820—31no—
25#1510—32no—
26#1306—31no—
27C22×C3×D4—22no—
28C22×C3×D4—22no—
29C22×C3×D4D412yes—
30C23×C3×C4C2311yescharacteristic
31C3×#121—22no—
32C3×#121—22no—
33C3×#121—22no—
34C3×#121—22no—
35C2×C3×C4⋊C4D412yes—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—24no—
38C2×C3×C22⋊C4D412yes—
39C2×#217—21no—
40C2×#217—21no—
41#926—22no—
42#926—22no—
43C2×C4⋊Dic3D412yescomplemented
44C2×C4×Dic3—21no—
45C2×#208—22no—
46C3×#95—22no—
47C3×#95—22no—
48#832—24no—
49#832—24no—
50C3⋊#95—22no—
51#830—22no—
52#830D414yescomplemented
53#808—24no—
54#807—24no—
55C2×#121C2×D312yes—
56C2×#117—31no—
57#482—64no—
58#481—62no—
59#480—62no—
60#428—62no—
61#418—62no—
62#407—32no—
63#397—22no—
64#392—22no—
65#392—22no—
66#381—31no—
67#381C2×D311yescharacteristic
68#342—62no—
69#327—64no—
70C24×C3—22no—
71C24×C3C2×D411yescharacteristic
72C2×C3×D4—22no—
73C2×C3×D4—22no—
74C2×C3×D4—44no—
75C2×C3×D4—42no—
76C2×C3×D4—42no—
77C2×C3×D4—84no—
78C2×C3×D4—84no—
79C2×C3×D4—82no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—42no—
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D412yes—
85C22×C3×C4C2×D412yes—
86C22×C3×C4#4111yescharacteristic
87C22×Dic3—22no—
88C22×Dic3—41no—
89C3×C4⋊C4—22no—
90C3×C4⋊C4—22no—
91C3×C22⋊C4—42no—
92C3×C22⋊C4—42no—
93C3×C22⋊C4—42no—
94C3×C22⋊C4—84no—
95#217—41no—
96#217—41no—
97#217—42no—
98C4⋊Dic3—22no—
99C4⋊Dic3—22no—
100C4×Dic3—42no—
101#208—44no—
102C2×C3⋊C8—24no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4#6712yes—
106C23×C4C22×D311yescharacteristic
107C2×M16—62no—
108#121—22no—
109#121—22no—
110#121—22no—
111#121—22no—
112#117—62no—
113#117—62no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4#6712yes—
116C2×C22⋊C4—22no—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—61no—
119C2×C22⋊C4—61no—
120C2×C22⋊C4#6712yes—
121C2×C42—61no—
122#107—64no—
123#106—64no—
124#102—64no—
125#102—64no—
126C22⋊C8—34no—
127C22⋊C8—62no—
128#95—22no—
129#95—22no—
130#95—62no—
131C23×C3—22no—
132C23×C3—42no—
133C23×C3—42no—
134C23×C3—42no—
135C23×C3—44no—
136C23×C3—42no—
137C23×C3—44no—
138C23×C3#12011yescharacteristic
139C23×C3#12111yescharacteristic
140C23×C3#12711yescharacteristic
141C3×D4—44no—
142C3×D4—82no—
143C3×D4—84no—
144C3×D4—84no—
145C3×D4—88no—
146C2×C3×C4—24no—
147C2×C3×C4—41no—
148C2×C3×C4—42no—
149C2×C3×C4—41no—
150C2×C3×C4—44no—
151C2×C3×C4—44no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4#12012yes—
155C2×C3×C4#12112yes—
156C2×Dic3—42no—
157C2×Dic3—82no—
158C2×Dic3—81no—
159C2×Dic3—82no—
160C3⋊C8—44no—
161C24—22no—
162C24C2×#6711yescharacteristic
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—44no—
166C2×D4—42no—
167C2×D4—42no—
168C2×D4—84no—
169C2×D4—84no—
170C2×D4—82no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—42no—
174C22×C4—62no—
175C22×C4—121no—
176C22×C4D3×D411yescharacteristic
177C22×C4D3×D412yes—
178C22×C4#23711yescharacteristic
179C22×C4C2×#6712yes—
180M16—124no—
181C2×C8—64no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—62no—
185C4⋊C4—62no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C22⋊C4—84no—
190C22⋊C4—121no—
191C22⋊C4—121no—
192C22⋊C4—122no—
193C42—122no—
194C22×C3—22no—
195C22×C3—22no—
196C22×C3—22no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—82no—
200C22×C3—84no—
201C22×C3—84no—
202C22×C3—82no—
203C22×C3—82no—
204C22×C3—82no—
205C22×C3#39211yescharacteristic
206C22×C3#45812yes—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4—82no—
210C3×C4—82no—
211Dic3—82no—
212Dic3—161no—
213C23—22no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—44no—
218C23—42no—
219C23—44no—
220C23#93811yescharacteristic
221C23#94011yescharacteristic
222C23#95311yescharacteristic
223D4—44no—
224D4—82no—
225D4—84no—
226D4—84no—
227D4—88no—
228C2×C4—24no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—122no—
237C2×C4—242no—
238C2×C4—241no—
239C2×C4—242no—
240C2×C4#93712yes—
241C2×C4#93912yes—
242C8—124no—
243C2×C3—22no—
244C2×C3—41no—
245C2×C3—41no—
246C2×C3—162no—
247C2×C3#200812yes—
248C2×C3#202211yescharacteristic
249C22—22no—
250C22—21no—
251C22—21no—
252C22—22no—
253C22—22no—
254C22—82no—
255C22—84no—
256C22—84no—
257C22—82no—
258C22—82no—
259C22—82no—
260C22#522311yescharacteristic
261C22#524312yes—
262C4—22no—
263C4—22no—
264C4—82no—
265C4—82no—
266C4—242no—
267C4—481no—
268C3#998311yescharacteristic, complemented, Sylow
269C2—22no—
270C2—41no—
271C2—41no—
272C2—162no—
273C2#7184911yescharacteristic
274C2#7185812yes—
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2815C2
socleC22×C3#392
hypercenter#2815D3
hypocenterC3#9983
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2815

Complements

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

NH ≅ G/Ntype
2. #82616C2semidirect
3. #71674C2semidirect
4. #69534C2semidirect
5. #67629C2semidirect
43. C2×C4⋊Dic3D4semidirect
52. #830D4semidirect
268. C3#9983semidirect

Maximal subgroups

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

subgroupindex
C3×#28152
#826162
#716742
#695342
#676292
#99833