Finite Group Databasebeta

← #10692635

Subgroups — #10692635

Order 768.

644 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#82182C211yesmaximal, characteristic
2#82144C211yesmaximal, characteristic, complemented
3C3×#1463C211yesmaximal, characteristic, complemented
4#69977C212yesmaximal
5#67683C212yesmaximal, complemented
6#9410—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C2×#926—21no—
10C3×#407C2212yes—
11#5213—21no—
12#5213—21no—
13C4×#217—21no—
14C4×#217C2211yescharacteristic
15C4×#217C2212yes—
16#4991C412yescomplemented
17#4990—22no—
18#4986C412yescomplemented
19Dic3×C4⋊C4—24no—
20C3×#324—22no—
21#4470—24no—
22#2591—31no—
23#2274—31no—
24#1507—32no—
25#1463D311yescharacteristic, complemented
26#1297—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C22×D12C2×C411yescharacteristic
32C2×C3×M16—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4C2×C412yes—
36C2×C3×C4⋊C4D412yes—
37C2×#217—22no—
38C2×#217—21no—
39#930—22no—
40#930—22no—
41C2×(C2×D3)⋊C4—22no—
42C2×(C2×D3)⋊C4—21no—
43C2×(C2×D3)⋊C4—22no—
44#926—28no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3C2×C411yescharacteristic
47C2×C4×Dic3—22no—
48C2×C4×Dic3—22no—
49C2×C4×Dic3—22no—
50C2×C4×Dic3—21no—
51C3×C22⋊C8—22no—
52C3×C22⋊C8—22no—
53C3⋊#95—22no—
54C3⋊#95—22no—
55C3⋊#95—22no—
56C3⋊#95C2×C412yes—
57C3⋊#95D414yes—
58C2×#121—31no—
59C2×#117—61no—
60C22×C4⋊C4C2×D311yescharacteristic
61#407C2×D312yes—
62#400—32no—
63#394—32no—
64#388—62no—
65#386—61no—
66#386—61no—
67C4×C4⋊C4—64no—
68C4×C22⋊C4—31no—
69C4×C22⋊C4—32no—
70C4×C22⋊C4—61no—
71#327—64no—
72#324—22no—
73C24×C3C2×D411yescharacteristic
74C23×D3—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—24no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—81no—
81C22×C3×C4—81no—
82C22×C3×C4C22⋊C412yes—
83C22×C3×C4C22⋊C412yes—
84C22×C3×C4C22×C411yescharacteristic
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4#4112yes—
88C2×#67—44no—
89C2×#67—41no—
90C2×#67—41no—
91C22×Dic3—24no—
92C22×Dic3—21no—
93C22×Dic3—22no—
94C2×D12—22no—
95C2×D12—81no—
96C3×M16—82no—
97C2×C3×C8—42no—
98C3×C4⋊C4—24no—
99C3×C4⋊C4—24no—
100#217—28no—
101#217—42no—
102(C2×D3)⋊C4—44no—
103(C2×D3)⋊C4—42no—
104(C2×D3)⋊C4—82no—
105C4⋊Dic3—28no—
106C4⋊Dic3—22no—
107C4×Dic3—28no—
108C4×Dic3—82no—
109C4×Dic3—82no—
110C4×Dic3—81no—
111C22×D4—31no—
112C22×D4—61no—
113C23×C4D1211yescharacteristic
114C23×C4D1211yescharacteristic
115C23×C4C22×D311yescharacteristic
116C2×M16—22no—
117#121—62no—
118#121—62no—
119#117—68no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—62no—
124C2×C4⋊C4C4×D312yes—
125C2×C4⋊C4#6712yes—
126C2×C22⋊C4—62no—
127C2×C22⋊C4—61no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4—61no—
130C2×C22⋊C4—62no—
131C2×C42—62no—
132C2×C42—62no—
133C2×C42—62no—
134C2×C42—61no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137#95—32no—
138#95—34no—
139#95—62no—
140#95—62no—
141#95—62no—
142C23×C3—22no—
143C23×C3—22no—
144C23×C3—41no—
145C23×C3C2×C22⋊C411yescharacteristic
146C23×C3#12011yescharacteristic
147C23×C3#12311yescharacteristic
148C22×D3—42no—
149C22×D3—42no—
150C22×D3—41no—
151C22×D3—42no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—24no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4—42no—
159C2×C3×C4—41no—
160C2×C3×C4—42no—
161C2×C3×C4—41no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—81no—
165C2×C3×C4#10414yes—
166C2×C3×C4C4×D412yes—
167C2×C3×C4#12112yes—
168#67—82no—
169#67—82no—
170C2×Dic3—28no—
171C2×Dic3—44no—
172C2×Dic3—44no—
173C2×Dic3—42no—
174C2×Dic3—42no—
175C2×Dic3—42no—
176C2×Dic3—41no—
177C2×Dic3—42no—
178D12—82no—
179C3×C8—82no—
180C24—61no—
181C24C2×D1211yescharacteristic
182C2×D4—62no—
183C2×D4—124no—
184C2×D4—121no—
185C2×D4—121no—
186C2×D4—241no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—24no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—64no—
193C22×C4—62no—
194C22×C4—61no—
195C22×C4—81no—
196C22×C4—81no—
197C22×C4(C2×D3)⋊C412yes—
198C22×C4(C2×D3)⋊C412yes—
199C22×C4C2×C4×D311yescharacteristic
200C22×C4D3×D412yes—
201C22×C4#23712yes—
202C22×C4C2×#6711yescharacteristic
203M16—82no—
204C2×C8—42no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—68no—
208C4⋊C4—62no—
209C22⋊C4—68no—
210C22⋊C4—122no—
211C22⋊C4—124no—
212C22⋊C4—122no—
213C22⋊C4—242no—
214C42—68no—
215C42—242no—
216C42—241no—
217C42—242no—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—44no—
226C22×C3#38611yescharacteristic
227C22×C3C2×#10412yes—
228C22×C3#45412yes—
229C2×D3—84no—
230C2×D3—81no—
231C2×D3—81no—
232C2×D3—81no—
233C3×C4—24no—
234C3×C4—22no—
235C3×C4—22no—
236C3×C4—42no—
237Dic3—48no—
238Dic3—82no—
239Dic3—81no—
240C23—22no—
241C23—22no—
242C23—41no—
243C23—122no—
244C23—122no—
245C23—122no—
246C23—121no—
247C23#88211yescharacteristic
248C23#88611yescharacteristic
249C23C2×(C2×D3)⋊C411yescharacteristic
250D4—242no—
251D4—242no—
252D4—242no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—68no—
266C2×C4—81no—
267C2×C4—124no—
268C2×C4—124no—
269C2×C4—122no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4—122no—
273C2×C4—121no—
274C2×C4#82514yes—
275C2×C4#67⋊C412yes—
276C2×C4#93912yes—
277C8—82no—
278C2×C3—22no—
279C2×C3—22no—
280C2×C3—41no—
281C2×C3#188811yescharacteristic
282C2×C3#189611yescharacteristic
283C2×C3#191311yescharacteristic
284D3—161no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—44no—
293C22—244no—
294C22—241no—
295C22—241no—
296C22—241no—
297C22#475312yes—
298C22#495811yescharacteristic
299C22C2×#82512yes—
300C4—24no—
301C4—22no—
302C4—22no—
303C4—42no—
304C4—128no—
305C4—242no—
306C4—241no—
307C3#941011yescharacteristic, complemented, Sylow
308C2—22no—
309C2—22no—
310C2—41no—
311C2—481no—
312C2#6996511yescharacteristic
313C2#7002411yescharacteristic
314C2#7010411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1463C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1463D3
hypocenterC3#9410
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #82144C2semidirect
3. C3×#1463C2semidirect
5. #67683C2semidirect
16. #4991C4semidirect
18. #4986C4semidirect
25. #1463D3semidirect
307. C3#9410semidirect

Maximal subgroups

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

subgroupindex
#821822
#821442
C3×#14632
#699772
#676832
#94103