Finite Group Databasebeta

← #10691206

Subgroups — #10691206

Order 768.

458 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2628C211yesmaximal, characteristic
2#82209C211yesmaximal, characteristic
3#71775C212yesmaximal, complemented
4#71571C211yesmaximal, characteristic
5#67631C212yesmaximal, complemented
6#9194—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C3×#400C412yes—
10C3×#399C412yes—
11C3×#395—21no—
12C3×#393—21no—
13C3×#389—21no—
14C3×#382C2211yescharacteristic
15C3×#380—21no—
16#5207—21no—
17#5000—24no—
18#4925C2212yes—
19#4924C2212yes—
20C4×#208—21no—
21#4547—24no—
22#4528—21no—
23#4525—21no—
24#2921—31no—
25#2628D311yescharacteristic
26#1549—32no—
27#1458—31no—
28#1286—32no—
29C22×C3×Q8—21no—
30C23×C3×C4D411yescharacteristic
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—22no—
33C3×#117—22no—
34C3×#117—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4C2×C412yes—
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4D411yescharacteristic
44C2×C3×C42D411yescharacteristic
45C2×C3×C42C2311yescharacteristic
46C2×#208—22no—
47C3×#95—22no—
48C3×#95—22no—
49C3×#95—21no—
50C3×#95C2×C412yes—
51#830—24no—
52#830—22no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)D414yes—
56#803—22no—
57C4×C3⋊C8—22no—
58C4×C3⋊C8—22no—
59C2×C4⋊Q8C2×D311yescharacteristic
60C2×#117C2×D311yescharacteristic
61#433—64no—
62#431—32no—
63#423—32no—
64#407—61no—
65C4×M16—61no—
66#400Dic312yes—
67#399Dic312yes—
68#395—21no—
69#393—21no—
70#389—21no—
71#382C2×D311yescharacteristic
72#380—21no—
73#332—64no—
74#324—61no—
75#323—61no—
76C24×C3C22⋊C411yescharacteristic
77C2×C3×Q8—42no—
78C2×C3×Q8—82no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C3×C4⋊C4—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—24no—
94C3×C4⋊C4—41no—
95C3×C4⋊C4—41no—
96C3×C4⋊C4—42no—
97C3×C4⋊C4—82no—
98C3×C22⋊C4—22no—
99C3×C22⋊C4—22no—
100C3×C22⋊C4—82no—
101C3×C42—21no—
102C3×C42—21no—
103C3×C42C2×D412yes—
104C3×C42#4112yes—
105#208—42no—
106#208—42no—
107C2×C3⋊C8—24no—
108C2×C3⋊C8—42no—
109C22×Q8—21no—
110C23×C4#6711yescharacteristic
111C2×M16—62no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114#117—22no—
115#117—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4C2×Dic312yes—
120C2×C4⋊C4C2×Dic311yescharacteristic
121C2×C4⋊C4#6711yescharacteristic
122C2×C22⋊C4—22no—
123C2×C22⋊C4C2×Dic311yescharacteristic
124C2×C22⋊C4#6711yescharacteristic
125C2×C42#6711yescharacteristic
126C2×C42C22×D311yescharacteristic
127C4⋊C8—34no—
128C4⋊C8—62no—
129C4⋊C8—62no—
130C22⋊C8—64no—
131C22⋊C8—62no—
132#97—62no—
133C4×C8—62no—
134C4×C8—62no—
135#95—22no—
136#95—22no—
137#95—21no—
138#95C2×Dic312yes—
139C23×C3—21no—
140C23×C3—21no—
141C23×C3—21no—
142C23×C3—41no—
143C23×C3—41no—
144C23×C3C2×C22⋊C411yescharacteristic
145C3×Q8—84no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4C4×D412yes—
164C2×C3×C4#12011yescharacteristic
165C2×C3×C4#12112yes—
166C2×C3×C4#12311yescharacteristic
167C2×C3×C4#13512yes—
168C2×C3×C4#13712yes—
169C3⋊C8—44no—
170C3⋊C8—82no—
171C24#21711yescharacteristic
172C2×Q8—42no—
173C2×Q8—82no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4#21711yescharacteristic
181C22×C4#21711yescharacteristic
182C22×C4#21711yescharacteristic
183C22×C4C22×Dic311yescharacteristic
184C22×C4C2×#6711yescharacteristic
185C22×C4C2×#6711yescharacteristic
186M16—122no—
187M16—122no—
188C2×C8—64no—
189C2×C8—122no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—82no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—82no—
200C42—21no—
201C42—21no—
202C42D3×D412yes—
203C42#23712yes—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—42no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—81no—
211C22×C3#38611yescharacteristic
212C22×C3#41611yescharacteristic
213C22×C3#41911yescharacteristic
214C3×C4—22no—
215C3×C4—22no—
216C3×C4—41no—
217C3×C4—41no—
218C3×C4—41no—
219C3×C4—82no—
220C3×C4—82no—
221C3×C4#45212yes—
222C3×C4#48512yes—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23C2×#21711yescharacteristic
229Q8—84no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4D4×Dic312yes—
248C2×C4#93511yescharacteristic
249C2×C4#93711yescharacteristic
250C2×C4#93912yes—
251C2×C4#94212yes—
252C2×C4#95012yes—
253C8—124no—
254C8—242no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—81no—
258C2×C3#187311yescharacteristic
259C2×C3#191011yescharacteristic
260C2×C3#191511yescharacteristic
261C22—21no—
262C22—21no—
263C22—42no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22#522011yescharacteristic
269C22#522511yescharacteristic
270C22#523311yescharacteristic
271C4—22no—
272C4—22no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—82no—
277C4—82no—
278C4#504512yes—
279C4#504712yes—
280C3#919411yescharacteristic, complemented, Sylow
281C2—21no—
282C2—21no—
283C2—81no—
284C2#7169311yescharacteristic
285C2#7185411yescharacteristic
286C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2628C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2628D3
hypocenterC3#9194
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2628

Complements

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

NH ≅ G/Ntype
3. #71775C2semidirect
5. #67631C2semidirect
280. C3#9194semidirect

Maximal subgroups

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

subgroupindex
C3×#26282
#822092
#717752
#715712
#676312
#91943