Finite Group Databasebeta

← #10692806

Subgroups — #10692806

Order 768.

490 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2372C211yesmaximal, characteristic
2#82939C211yesmaximal, characteristic
3#71719C211yesmaximal, characteristic
4#69433C212yesmaximal, complemented
5#67615C212yesmaximal, complemented
6#9439—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8#5470C2211yescharacteristic
9C3×#391—21no—
10C3×#388—22no—
11C3×#386—21no—
12C3×C4×C22⋊C4—22no—
13C3×#376C2212yes—
14#5225—21no—
15#5215—21no—
16#5056—22no—
17#5055C414yescomplemented
18#4926C2212yes—
19C4×#208C2211yescharacteristic
20#4540—22no—
21#4452—22no—
22#4452—22no—
23#3140—31no—
24#2372D311yescharacteristic
25#1533—31no—
26#1506—32no—
27#1300—32no—
28C22×C3×Q8D411yescharacteristic
29C22×C3×D4D411yescharacteristic
30C23×C3×C4—21no—
31C3×#124—22no—
32C3×#124C2×C412yes—
33C3×#124D412yes—
34C2×C3×C4⋊C4—21no—
35C2×C3×C22⋊C4—22no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—21no—
38C2×C3×C42—22no—
39C2×C3×C42—21no—
40C2×C3×C42C2311yescharacteristic
41#926—21no—
42#924—22no—
43C2×#208—21no—
44#870—21no—
45#869C2×C412yescomplemented
46C4×Dic6—22no—
47C3×#95—22no—
48C3×#95—22no—
49C3×#95D412yes—
50C3×#95D412yes—
51#835—24no—
52#832—24no—
53#830—22no—
54C4⋊(C3⋊C8)—22no—
55#803—22no—
56#803C2×C412yescomplemented
57C4×C3⋊C8—21no—
58#533—31no—
59C2×#124C2×D311yescharacteristic
60#489—34no—
61#488—62no—
62#432—32no—
63#419—61no—
64#418—61no—
65C4×M16—31no—
66#391—21no—
67#388—22no—
68#386—21no—
69C4×C22⋊C4—22no—
70#376C2×D312yes—
71#329—62no—
72#328—62no—
73#328—62no—
74C24×C3—21no—
75C2×C3×Q8—21no—
76C2×C3×Q8—41no—
77C2×C3×Q8—41no—
78C2×C3×Q8C22⋊C412yes—
79C2×C3×D4—21no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C2×C3×D4C22⋊C412yes—
83C22×C3×C4—22no—
84C22×C3×C4—21no—
85C22×C3×C4—22no—
86C22×C3×C4—44no—
87C22×C3×C4—81no—
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D412yes—
90C2×Dic6—22no—
91C3×C4⋊C4—81no—
92C3×C22⋊C4—24no—
93C3×C22⋊C4—24no—
94C3×C22⋊C4—44no—
95C3×C22⋊C4—81no—
96C3×C42—44no—
97C3×C42—81no—
98C3×C42C22×C411yescharacteristic
99C3×C42C2×D411yescharacteristic
100C3×C42#4112yes—
101#217—41no—
102C4⋊Dic3—22no—
103C4⋊Dic3—22no—
104Dic3⋊C4—42no—
105C4×Dic3—41no—
106#208—41no—
107#208—41no—
108C2×C3⋊C8—22no—
109C2×C3⋊C8—42no—
110C22×Q8#6711yescharacteristic
111C22×D4#6711yescharacteristic
112C23×C4—21no—
113C2×M16—61no—
114C4⋊Q8—32no—
115#125—61no—
116#124—22no—
117#124C4×D312yes—
118#124D1212yes—
119C22⋊Q8—62no—
120C4×Q8—62no—
121#117—61no—
122C2×C4⋊C4—21no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—21no—
126C2×C42—22no—
127C2×C42—21no—
128C2×C42C22×D311yescharacteristic
129C4⋊C8—62no—
130#103—64no—
131#102—64no—
132C22⋊C8—62no—
133#97—32no—
134#97—62no—
135C4×C8—61no—
136#95—22no—
137#95—22no—
138#95#6712yes—
139#95#6712yes—
140C23×C3—22no—
141C23×C3—42no—
142C23×C3—42no—
143C23×C3—41no—
144C23×C3—41no—
145C23×C3#12011yescharacteristic
146C3×Q8—42no—
147C3×Q8—81no—
148C3×D4—42no—
149C3×D4—81no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—24no—
153C2×C3×C4—22no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—84no—
163C2×C3×C4C2×C22⋊C411yescharacteristic
164C2×C3×C4C4×D412yes—
165C2×C3×C4#12112yes—
166C2×C3×C4#12311yescharacteristic
167C2×Dic3—42no—
168Dic6—82no—
169C3⋊C8—42no—
170C3⋊C8—82no—
171C24—21no—
172C2×Q8—21no—
173C2×Q8—41no—
174C2×Q8—41no—
175C2×Q8—62no—
176C2×Q8(C2×D3)⋊C412yes—
177C2×D4—21no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4(C2×D3)⋊C412yes—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—44no—
185C22×C4—81no—
186C22×C4C2×#6711yescharacteristic
187C22×C4C2×#6712yes—
188M16—121no—
189M16—121no—
190C2×C8—62no—
191C2×C8—122no—
192C4⋊C4—62no—
193C4⋊C4—62no—
194C4⋊C4—81no—
195C4⋊C4—122no—
196C22⋊C4—24no—
197C22⋊C4—24no—
198C22⋊C4—44no—
199C22⋊C4—81no—
200C22⋊C4—121no—
201C42—44no—
202C42—81no—
203C42—121no—
204C42C2×C4×D311yescharacteristic
205C42C2×D1211yescharacteristic
206C42#23512yes—
207C22×C3—22no—
208C22×C3—42no—
209C22×C3—44no—
210C22×C3—81no—
211C22×C3—81no—
212C22×C3—82no—
213C22×C3#38611yescharacteristic
214C22×C3#45511yescharacteristic
215C22×C3#45511yescharacteristic
216C3×C4—21no—
217C3×C4—21no—
218C3×C4—22no—
219C3×C4—22no—
220C3×C4—44no—
221C3×C4—42no—
222C3×C4—82no—
223Dic3—82no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—41no—
228C23—41no—
229C23#95311yescharacteristic
230Q8—42no—
231Q8—81no—
232Q8—242no—
233D4—42no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—84no—
248C2×C4—122no—
249C2×C4C2×(C2×D3)⋊C411yescharacteristic
250C2×C4C4×#6712yes—
251C2×C4#92911yescharacteristic
252C2×C4#93012yes—
253C8—122no—
254C8—242no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—82no—
258C2×C3#188511yescharacteristic
259C2×C3#190612yes—
260C22—22no—
261C22—44no—
262C22—42no—
263C22—81no—
264C22—81no—
265C22—82no—
266C22#521311yescharacteristic
267C22#524211yescharacteristic
268C22#524211yescharacteristic
269C4—22no—
270C4—22no—
271C4—21no—
272C4—21no—
273C4—44no—
274C4—42no—
275C4—82no—
276C4—242no—
277C3#943911yescharacteristic, complemented, Sylow
278C2—21no—
279C2—21no—
280C2—82no—
281C2#7164512yes—
282C2#7164911yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2372C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#2372D3
hypocenterC3#9439
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2372

Complements

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

NH ≅ G/Ntype
4. #69433C2semidirect
5. #67615C2semidirect
17. #5055C4semidirect
45. #869C2×C4semidirect
56. #803C2×C4semidirect
277. C3#9439semidirect

Maximal subgroups

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

subgroupindex
C3×#23722
#829392
#717192
#694332
#676152
#94393