Finite Group Databasebeta

← #10693001

Subgroups — #10693001

Order 768.

610 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2274C211yesmaximal, characteristic
2#82176C211yesmaximal, characteristic
3#71552C211yesmaximal, characteristic
4#69442C212yesmaximal
5#67616C212yesmaximal
6#9478—31nomaximal, Sylow
7C2×C3×#117—21no—
8C22×C3×C4⋊C4C2211yescharacteristic
9C2×#926C2211yescharacteristic
10C3×C4×C4⋊C4—24no—
11C3×C4×C22⋊C4—21no—
12C3×C4×C22⋊C4C2212yes—
13#5211—21no—
14#5211C2211yescharacteristic
15#5210—21no—
16#5207C2212yes—
17#4986C412yescomplemented
18#4981C412yescomplemented
19#4978—22no—
20#4528—22no—
21#4523—24no—
22#2484—31no—
23#2274D311yescharacteristic
24#1826—32no—
25#1463—31no—
26#1284—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C3×#117—28no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4C2×C412yes—
35C2×C3×C4⋊C4D412yes—
36C2×C3×C22⋊C4—22no—
37C2×C3×C42—22no—
38C2×C3×C42—22no—
39C2×C3×C42D414yes—
40C2×#217—21no—
41C2×#217—21no—
42#926—22no—
43#926—22no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×Dic3⋊C4—22no—
47C2×C4×Dic3—21no—
48C2×#208—22no—
49C3×#95—22no—
50C3×#95—22no—
51C3⋊#95—22no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95—21no—
55C3⋊#95C2×C412yes—
56#830—22no—
57#830—22no—
58C2×#117—21no—
59C2×#117—31no—
60C22×C4⋊C4C2×D311yescharacteristic
61#407—32no—
62#400—32no—
63#398—32no—
64#387—62no—
65#385—61no—
66#381—31no—
67#381—61no—
68C4×C4⋊C4—24no—
69C4×C22⋊C4—21no—
70C4×C22⋊C4C2×D312yes—
71#339—64no—
72#324—62no—
73C24×C3C2×D411yescharacteristic
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—24no—
77C22×C3×C4—24no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4C22⋊C412yes—
83C22×C3×C4C22⋊C412yes—
84C22×C3×C4C22×C411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4C2×D412yes—
87C22×C3×C4#4112yes—
88C22×Dic3—22no—
89C22×Dic3—21no—
90C22×Dic3—21no—
91C3×C4⋊C4—28no—
92C3×C4⋊C4—24no—
93C3×C4⋊C4—24no—
94C3×C22⋊C4—28no—
95C3×C42—28no—
96C3×C42—44no—
97C3×C42—82no—
98#217—42no—
99#217—41no—
100#217—41no—
101C4⋊Dic3—22no—
102C4⋊Dic3—22no—
103Dic3⋊C4—44no—
104C4×Dic3—42no—
105#208—44no—
106C2×C3⋊C8—42no—
107C23×C4#6711yescharacteristic
108C23×C4#6711yescharacteristic
109C23×C4C22×D311yescharacteristic
110C2×M16—62no—
111#117—28no—
112#117—62no—
113#117—62no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4C4×D312yes—
121C2×C4⋊C4D1212yes—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—61no—
124C2×C22⋊C4—61no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—61no—
128C2×C42#6714yes—
129C22⋊C8—62no—
130C22⋊C8—62no—
131#95—22no—
132#95—22no—
133#95—32no—
134#95—62no—
135#95—62no—
136#95—62no—
137#95—61no—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—41no—
141C23×C3C2×C22⋊C411yescharacteristic
142C23×C3#12011yescharacteristic
143C23×C3#12311yescharacteristic
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—24no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—24no—
150C2×C3×C4—28no—
151C2×C3×C4—22no—
152C2×C3×C4—41no—
153C2×C3×C4—41no—
154C2×C3×C4—42no—
155C2×C3×C4—44no—
156C2×C3×C4—44no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—81no—
160C2×C3×C4#10414yes—
161C2×C3×C4C4×D412yes—
162C2×C3×C4#12112yes—
163C2×Dic3—42no—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—41no—
167C2×Dic3—41no—
168C2×Dic3—42no—
169C2×Dic3—42no—
170C3⋊C8—82no—
171C24C2×#6711yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—62no—
181C22×C4—61no—
182C22×C4—61no—
183C22×C4(C2×D3)⋊C412yes—
184C22×C4(C2×D3)⋊C412yes—
185C22×C4C2×C4×D311yescharacteristic
186C22×C4C2×D1211yescharacteristic
187C22×C4#23512yes—
188C22×C4C2×#6712yes—
189M16—124no—
190C2×C8—122no—
191C4⋊C4—28no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—62no—
195C4⋊C4—62no—
196C4⋊C4—124no—
197C22⋊C4—28no—
198C22⋊C4—122no—
199C22⋊C4—121no—
200C22⋊C4—121no—
201C42—28no—
202C42—44no—
203C42—82no—
204C42—122no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—44no—
213C22×C3#38611yescharacteristic
214C22×C3C2×#10412yes—
215C22×C3#45812yes—
216C3×C4—22no—
217C3×C4—22no—
218C3×C4—24no—
219C3×C4—48no—
220C3×C4—42no—
221Dic3—82no—
222Dic3—81no—
223Dic3—81no—
224C23—22no—
225C23—22no—
226C23—41no—
227C23C2×(C2×D3)⋊C411yescharacteristic
228C23#92911yescharacteristic
229C23#95311yescharacteristic
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—28no—
237C2×C4—22no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—81no—
246C2×C4—122no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—122no—
250C2×C4—122no—
251C2×C4—121no—
252C2×C4—121no—
253C2×C4#80514yes—
254C2×C4C4×#6712yes—
255C2×C4#93012yes—
256C8—242no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—41no—
260C2×C3#189011yescharacteristic
261C2×C3#189611yescharacteristic
262C2×C3#191211yescharacteristic
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—44no—
271C22C2×#80512yes—
272C22#521311yescharacteristic
273C22#524312yes—
274C4—24no—
275C4—22no—
276C4—22no—
277C4—48no—
278C4—42no—
279C4—242no—
280C4—241no—
281C4—241no—
282C3#947811yescharacteristic, complemented, Sylow
283C2—22no—
284C2—22no—
285C2—41no—
286C2#7153211yescharacteristic
287C2#7165511yescharacteristic
288C2#7166311yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2274C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2274D3
hypocenterC3#9478
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
17. #4986C4semidirect
18. #4981C4semidirect
282. C3#9478semidirect

Maximal subgroups

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

subgroupindex
C3×#22742
#821762
#715522
#694422
#676162
#94783