Finite Group Databasebeta

← #10693995

Subgroups — #10693995

Order 768.

558 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×#124C211yesmaximal, characteristic
2C2×#4535C212yesmaximal, complemented
3C2×D4⋊(C3⋊C8)C212yesmaximal
4#69392C211yesmaximal, characteristic, complemented
5#69388C211yesmaximal, characteristic, complemented
6#9645—31nomaximal, Sylow
7C2×C3×#124C411yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C2×C3×C4×D4C2211yescharacteristic
10C3×#391C411yescharacteristic
11C3×#390C411yescharacteristic
12C3×#388—22no—
13C3×#382C411yescharacteristic
14C3×C4×C22⋊C4—22no—
15C3×C43C2211yescharacteristic
16C2×C4⋊(C3⋊C8)C2212yes—
17C2×C4×C3⋊C8C2212yes—
18#4535—24no—
19D4⋊(C3⋊C8)—24no—
20#4524—22no—
21#4524—22no—
22C4×#124D311yescharacteristic
23#1831—31no—
24#1744—31no—
25C2×Q8⋊C8—32no—
26C2×D4⋊C8—32no—
27C22×C3×Q8C2×C411yescharacteristic
28C22×C3×D4C2×C411yescharacteristic
29C23×C3×C4—21no—
30C3×#124—22no—
31C3×#124—22no—
32C3×C4×Q8—22no—
33C3×C4×Q8—22no—
34C3×C4×D4—22no—
35C3×C4×D4—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C42—21no—
43C2×C3×C42C2×C411yescharacteristic
44C2×C3×C42C2×C411yescharacteristic
45C2×C3×C42D412yes—
46C2×C3×C42D412yes—
47C2×C3×C42C2311yescharacteristic
48C22×C3⋊C8—22no—
49C22×C3⋊C8—22no—
50C3×#95—22no—
51C3×#95—22no—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)—22no—
54C4×C3⋊C8—22no—
55C4×C3⋊C8D414yes—
56C2×#124Dic311yescharacteristic
57C2×C4×Q8C2×D311yescharacteristic
58C2×C4×D4C2×D311yescharacteristic
59C2×C4⋊C8—32no—
60C2×C4×C8—32no—
61#391Dic311yescharacteristic
62#390Dic311yescharacteristic
63#388—22no—
64#382Dic311yescharacteristic
65C4×C22⋊C4—22no—
66C43C2×D311yescharacteristic
67#336—62no—
68#336—62no—
69Q8⋊C8—64no—
70D4⋊C8—64no—
71C24×C3—21no—
72C2×C3×Q8—21no—
73C2×C3×Q8—22no—
74C2×C3×Q8—22no—
75C2×C3×Q8M1612yes—
76C2×C3×D4—22no—
77C2×C3×D4—21no—
78C2×C3×D4—22no—
79C2×C3×D4M1612yes—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—41no—
84C22×C3×C4—41no—
85C22×C3×C4—44no—
86C22×C3×C4C22⋊C412yes—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—42no—
94C3×C22⋊C4—28no—
95C3×C22⋊C4—44no—
96C3×C22⋊C4—42no—
97C3×C42—21no—
98C3×C42—21no—
99C3×C42—22no—
100C3×C42—24no—
101C3×C42—42no—
102C3×C42—41no—
103C3×C42—41no—
104C3×C42C2×D412yes—
105C3×C42#4112yes—
106C2×C3⋊C8—24no—
107C2×C3⋊C8—44no—
108C2×C3⋊C8—42no—
109C2×C3⋊C8—44no—
110C22×Q8C2×Dic311yescharacteristic
111C22×D4C2×Dic311yescharacteristic
112C23×C4—21no—
113C22×C8—62no—
114C22×C8—62no—
115#124—22no—
116#124—22no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×D4—22no—
120C4×D4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4C2×Dic311yescharacteristic
123C2×C4⋊C4C2×Dic311yescharacteristic
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—21no—
127C2×C42—21no—
128C2×C42C2×Dic311yescharacteristic
129C2×C42C2×Dic311yescharacteristic
130C2×C42#6712yes—
131C2×C42#6712yes—
132C2×C42C22×D311yescharacteristic
133C4⋊C8—62no—
134C4⋊C8—62no—
135C4×C8—34no—
136C4×C8—62no—
137#95—22no—
138#95—22no—
139C23×C3—22no—
140C23×C3—41no—
141C23×C3—41no—
142C23×C3—42no—
143C23×C3—42no—
144C23×C3C2×C22⋊C411yescharacteristic
145C3×Q8—24no—
146C3×Q8—42no—
147C3×D4—24no—
148C3×D4—42no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4—41no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—82no—
164C2×C3×C4—82no—
165C2×C3×C4C4×D412yes—
166C2×C3×C4#12112yes—
167C2×C3×C4#12411yescharacteristic
168C2×C3×C4#12711yescharacteristic
169C2×C3×C4C2×M1611yescharacteristic
170C2×C3×C4C2×M1611yescharacteristic
171C2×C3×C4#13112yes—
172C2×C3×C4#13514yes—
173C3⋊C8—44no—
174C3⋊C8—82no—
175C24—21no—
176C2×Q8—21no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8#20812yes—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4#20812yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—44no—
190C22×C4#21712yes—
191C22×C4#21712yes—
192C22×C4C22×Dic311yescharacteristic
193C22×C4C2×#6711yescharacteristic
194C22×C4C2×#6711yescharacteristic
195C2×C8—64no—
196C2×C8—124no—
197C2×C8—122no—
198C2×C8—124no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—42no—
202C22⋊C4—28no—
203C22⋊C4—44no—
204C22⋊C4—42no—
205C42—21no—
206C42—21no—
207C42—22no—
208C42—24no—
209C42—42no—
210C42—41no—
211C42—41no—
212C42D3×D412yes—
213C42#23712yes—
214C22×C3—42no—
215C22×C3—44no—
216C22×C3—81no—
217C22×C3—81no—
218C22×C3—82no—
219C22×C3#39011yescharacteristic
220C22×C3#40711yescharacteristic
221C22×C3#40811yescharacteristic
222C22×C3#41612yes—
223C22×C3#42112yes—
224C3×C4—22no—
225C3×C4—42no—
226C3×C4—42no—
227C3×C4—42no—
228C3×C4—81no—
229C3×C4—81no—
230C3×C4#43512yes—
231C3×C4#43512yes—
232C3×C4#48812yes—
233C3×C4#49512yes—
234C23—22no—
235C23—41no—
236C23—42no—
237C23—41no—
238C23—42no—
239C23C2×#21711yescharacteristic
240Q8—24no—
241Q8—42no—
242D4—24no—
243D4—42no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4C2×#20811yescharacteristic
261C2×C4C2×#20811yescharacteristic
262C2×C4D4×Dic312yes—
263C2×C4#93611yescharacteristic
264C2×C4#93812yes—
265C2×C4#94011yescharacteristic
266C2×C4#94812yes—
267C2×C4#95014yes—
268C8—124no—
269C8—242no—
270C2×C3—82no—
271C2×C3#148212yes—
272C2×C3#148812yes—
273C2×C3#195511yescharacteristic
274C2×C3#195811yescharacteristic
275C2×C3#196311yescharacteristic
276C22—42no—
277C22—44no—
278C22—81no—
279C22—82no—
280C22—81no—
281C22#520711yescharacteristic
282C22#522111yescharacteristic
283C22#523312yes—
284C22#523411yescharacteristic
285C22#523612yes—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—42no—
290C4—81no—
291C4—81no—
292C4#501112yes—
293C4#501112yes—
294C4#505612yes—
295C4#505912yes—
296C3#964511yescharacteristic, complemented, Sylow
297C2—82no—
298C2#7161511yescharacteristic
299C2#7164012yes—
300C2#7172511yescharacteristic
301C2#7175012yes—
302C2#7181611yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×#124C2
socleC23×C3C2×C22⋊C4
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenterC4×#124D3
hypocenterC3#9645
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×#124

Complements

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

NH ≅ G/Ntype
2. C2×#4535C2semidirect
4. #69392C2semidirect
5. #69388C2semidirect
296. C3#9645semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1242
C2×#45352
C2×D4⋊(C3⋊C8)2
#693922
#693882
#96453