Finite Group Databasebeta

← #10711739

Subgroups — #10711739

Order 768.

420 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#108C211yesmaximal, characteristic
2#71970C211yesmaximal, characteristic
3#71963C211yesmaximal, characteristic
4C2×#4494C212yesmaximal, complemented
5#67730C212yesmaximal, complemented
6#12928—31nomaximal, Sylow
7C22×C3×M16—21no—
8C23×C3×C8C2211yescharacteristic
9C2×C3×#108—21no—
10C2×C3×#108—24no—
11C2×C3×#108C2212yes—
12#5212—21no—
13C2×#817C2211yescharacteristic
14C2×#817C2211yescharacteristic
15C2×#812C2212yes—
16#4551—21no—
17#4549C412yescomplemented
18#4549C412yescomplemented
19#4494—24no—
20C22×#108D311yescharacteristic
21C2×#365—32no—
22#2112—31no—
23#1807—31no—
24#1383—32no—
25C23×C3×C4D411yescharacteristic
26C2×C3×M16—24no—
27C2×C3×M16—22no—
28C2×C3×M16—42no—
29C2×C3×M16—41no—
30C22×C3×C8—22no—
31C22×C3×C8—22no—
32C22×C3×C8—41no—
33C22×C3×C8C2×C411yescharacteristic
34C22×C3×C8C2×C411yescharacteristic
35C22×C3×C8D411yescharacteristic
36C22×C3×C8Q811yescharacteristic
37C22×C3×C8Q811yescharacteristic
38C22×C3×C8C2311yescharacteristic
39C2×#217—41no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C3×#108—24no—
43C3×#108—22no—
44C3×#108D414yes—
45C3⋊#95—41no—
46#817—22no—
47#817—22no—
48#812—22no—
49#812—22no—
50C2×C3⋊C16C2×C414yes—
51C22×M16—21no—
52C23×C8C2×D311yescharacteristic
53C2×M32—32no—
54C2×#108—21no—
55C2×#108—24no—
56C2×#108C2×D312yes—
57C2×#106—31no—
58C2×#106—31no—
59#380—61no—
60#365—64no—
61C22⋊C16—32no—
62C22⋊C16—32no—
63#340—61no—
64C24×C3SD1611yescharacteristic
65C22×C3×C4—21no—
66C22×C3×C4—21no—
67C22×C3×C4—21no—
68C22×C3×C4—21no—
69C22×C3×C4C4⋊C411yescharacteristic
70C22×C3×C4C4⋊C411yescharacteristic
71C22×C3×C4SD1611yescharacteristic
72C22×C3×C4SD1611yescharacteristic
73C22×C3×C4SD1611yescharacteristic
74C22×C3×C4C2×D411yescharacteristic
75C22×Dic3—41no—
76C22×Dic3—41no—
77C3×M16—24no—
78C3×M16—44no—
79C3×M16—42no—
80C2×C3×C8—21no—
81C2×C3×C8—21no—
82C2×C3×C8—22no—
83C2×C3×C8—22no—
84C2×C3×C8—22no—
85C2×C3×C8—22no—
86C2×C3×C8—44no—
87C2×C3×C8—42no—
88C2×C3×C8C4⋊C411yescharacteristic
89C2×C3×C8C4⋊C411yescharacteristic
90C2×C3×C8C22×C411yescharacteristic
91C2×C3×C8C2×D412yes—
92C2×C3×C8C2×Q811yescharacteristic
93C2×C3×C8#4112yes—
94#217—82no—
95C4⋊Dic3—41no—
96C4⋊Dic3—41no—
97C4⋊Dic3—41no—
98C4⋊Dic3—41no—
99C3⋊C16—24no—
100C23×C4#6711yescharacteristic
101C2×M16—24no—
102C2×M16—22no—
103C2×M16—42no—
104C2×M16—41no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—41no—
108C22×C8Dic611yescharacteristic
109C22×C8Dic611yescharacteristic
110C22×C8C4×D311yescharacteristic
111C22×C8C4×D311yescharacteristic
112C22×C8#6711yescharacteristic
113C22×C8C22×D311yescharacteristic
114C2×C4⋊C4—61no—
115C2×C4⋊C4—61no—
116C2×C22⋊C4—121no—
117M32—62no—
118M32—62no—
119C2×C16—34no—
120#108—24no—
121#108—22no—
122#108#6714yes—
123#106—62no—
124#106—62no—
125#95—121no—
126C23×C3—22no—
127C23×C3—21no—
128C23×C3—21no—
129C23×C3—41no—
130C23×C3#10611yescharacteristic
131C23×C3#10611yescharacteristic
132C23×C3C2×SD1611yescharacteristic
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4#10611yescharacteristic
144C2×C3×C4#10611yescharacteristic
145C2×C3×C4C2×C4⋊C411yescharacteristic
146C2×C3×C4#12011yescharacteristic
147C2×C3×C4#12311yescharacteristic
148C2×C3×C4C2×SD1611yescharacteristic
149C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
150C2×C3×C4#13711yescharacteristic
151C2×Dic3—81no—
152C2×Dic3—81no—
153C2×Dic3—81no—
154C2×Dic3—81no—
155C2×Dic3—81no—
156C2×Dic3—81no—
157C3×C8—21no—
158C3×C8—21no—
159C3×C8—44no—
160C3×C8C4×D412yes—
161C3×C8C22⋊Q812yes—
162C24#21511yescharacteristic
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—121no—
168C22×C4—121no—
169C22×C4Dic3⋊C411yescharacteristic
170C22×C4Dic3⋊C411yescharacteristic
171C22×C4#21411yescharacteristic
172C22×C4#21411yescharacteristic
173C22×C4#21511yescharacteristic
174C22×C4C2×#6711yescharacteristic
175M16—24no—
176M16—44no—
177M16—42no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C2×C8—22no—
184C2×C8—44no—
185C2×C8—42no—
186C2×C8Dic3⋊C411yescharacteristic
187C2×C8Dic3⋊C411yescharacteristic
188C2×C8C2×Dic611yescharacteristic
189C2×C8C2×C4×D311yescharacteristic
190C2×C8#23512yes—
191C2×C8C2×#6712yes—
192C4⋊C4—121no—
193C4⋊C4—121no—
194C4⋊C4—121no—
195C4⋊C4—121no—
196C22⋊C4—242no—
197C16—64no—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—42no—
206C22×C3—42no—
207C22×C3#36512yescomplemented
208C22×C3C2×#10611yescharacteristic
209C22×C3#44811yescharacteristic
210C22×C3#48111yescharacteristic
211C3×C4—21no—
212C3×C4—21no—
213C3×C4#38511yescharacteristic
214C3×C4#42811yescharacteristic
215C3×C4#45011yescharacteristic
216C3×C4#48311yescharacteristic
217Dic3—161no—
218Dic3—161no—
219C23—21no—
220C23—21no—
221C23—22no—
222C23—41no—
223C23#80811yescharacteristic
224C23#80811yescharacteristic
225C23C2×#21511yescharacteristic
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—241no—
237C2×C4—241no—
238C2×C4—241no—
239C2×C4—241no—
240C2×C4—241no—
241C2×C4—241no—
242C2×C4#80811yescharacteristic
243C2×C4#80811yescharacteristic
244C2×C4C2×Dic3⋊C411yescharacteristic
245C2×C4#92911yescharacteristic
246C2×C4#93211yescharacteristic
247C2×C4C2×#21411yescharacteristic
248C2×C4#94211yescharacteristic
249C2×C4#95311yescharacteristic
250C8—21no—
251C8—21no—
252C8—44no—
253C8#92412yes—
254C8C4×#6712yes—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—22no—
258C2×C3—41no—
259C2×C3#185311yescharacteristic
260C2×C3C2×#36511yescharacteristic
261C2×C3#215511yescharacteristic
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—42no—
270C22—42no—
271C22#449412yescomplemented
272C22C2×#80811yescharacteristic
273C22#496811yescharacteristic
274C22#522711yescharacteristic
275C4—21no—
276C4—21no—
277C4—481no—
278C4—481no—
279C4#496411yescharacteristic
280C4#497111yescharacteristic
281C4#521011yescharacteristic
282C4#521811yescharacteristic
283C3#1292811yescharacteristic, complemented, Sylow
284C2—21no—
285C2—21no—
286C2—22no—
287C2—41no—
288C2C2×#449411yescharacteristic
289C2#7068511yescharacteristic
290C2#7155611yescharacteristic
subgroupG/H
centerC22C2×#808
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC22×C3×#108C2
socleC22×C3C2×#106
prime radicalC24×C3SD16
cosocleC22×C3×C8C23
hypercenterC22×#108D3
hypocenterC3#12928
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C22×#108

Complements

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

NH ≅ G/Ntype
4. C2×#4494C2semidirect
5. #67730C2semidirect
17. #4549C4semidirect
18. #4549C4semidirect
207. C22×C3#365semidirect
271. C22#4494semidirect
283. C3#12928semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1082
#719702
#719632
C2×#44942
#677302
#129283