Finite Group Databasebeta

← #10692881

Subgroups — #10692881

Order 768.

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

normal subgroups only

This group has 448 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
1C3×#2983C211yesmaximal, characteristic
2#82478C211yesmaximal, characteristic
3#71625C211yesmaximal, characteristic
4#70822C212yesmaximal
5#67750C212yesmaximal
6#9454—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#869C2211yescharacteristic
9C3×#446—24no—
10C3×#432—21no—
11C3×#432C2212yes—
12C3×#431—21no—
13#5229C412yes—
14#4979—21no—
15#4979—21no—
16#4975C412yes—
17C4×C4⋊Dic3C2211yescharacteristic
18#4928—21no—
19#4928—21no—
20#4926C2212yes—
21#4528—21no—
22#4528—21no—
23#4486—24no—
24#2983D311yescharacteristic
25#2567—31no—
26#1556—32no—
27#1467—31no—
28#1285—32no—
29C22×C3×Q8C2×C411yescharacteristic
30C22×Dic6—21no—
31C3×C4×Q8—22no—
32C3×C4×Q8—22no—
33C2×C3×C4⋊C4C2×C411yescharacteristic
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C42D411yescharacteristic
37C2×C3×C42D411yescharacteristic
38C2×C3×C42C2311yescharacteristic
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3C2×C412yes—
42C2×Dic3⋊C4—22no—
43C2×Dic3⋊C4—22no—
44C2×C4×Dic3C2×C412yes—
45#869—22no—
46#869—22no—
47C3×C4⋊C8—22no—
48C3×C4⋊C8—22no—
49C3×C4⋊C8D414yes—
50C3×C22⋊C8—22no—
51C3×#97—22no—
52C3×#97—22no—
53C3×C4×C8—22no—
54C3⋊#95—22no—
55C3⋊#95—22no—
56C3⋊#95—21no—
57#830—22no—
58C4⋊(C3⋊C8)—22no—
59#803—22no—
60C2×C4⋊Q8—31no—
61C2×C4×Q8C2×D311yescharacteristic
62#446—24no—
63#432—21no—
64#432—32no—
65#432C2×D312yes—
66#431—21no—
67#395—61no—
68#395—61no—
69#391—32no—
70#389—61no—
71#389—61no—
72#384—32no—
73C4×C4⋊C4—31no—
74#332—64no—
75#324—61no—
76#324—61no—
77C2×C3×Q8—22no—
78C2×C3×Q8—22no—
79C2×C3×Q8—41no—
80C22×C3×C4C22⋊C411yescharacteristic
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C22⋊C411yescharacteristic
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22×C411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×Dic3—22no—
88C22×Dic3—22no—
89C2×Dic6—42no—
90C2×Dic6—82no—
91C2×C3×C8—24no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C42—21no—
96C3×C42—21no—
97C3×C42—21no—
98C3×C42—21no—
99C3×C42C2×D412yes—
100C3×C42#4112yes—
101C4⋊Dic3—24no—
102C4⋊Dic3—24no—
103C4⋊Dic3—41no—
104C4⋊Dic3—41no—
105Dic3⋊C4—82no—
106Dic3⋊C4—82no—
107C4×Dic3—42no—
108C2×C3⋊C8—42no—
109C22×Q8—61no—
110C22×Q8C4×D311yescharacteristic
111C4⋊Q8—62no—
112C4⋊Q8—62no—
113C4×Q8—22no—
114C4×Q8—22no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—61no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4C4×D311yescharacteristic
121C2×C4⋊C4D1211yescharacteristic
122C2×C4⋊C4D1211yescharacteristic
123C2×C42—32no—
124C2×C42#6711yescharacteristic
125C2×C42#6711yescharacteristic
126C2×C42C22×D311yescharacteristic
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—62no—
130C4⋊C8#6714yes—
131C22⋊C8—22no—
132C22⋊C8—62no—
133#97—22no—
134#97—22no—
135#97—62no—
136C4×C8—22no—
137#95—62no—
138#95—62no—
139#95—61no—
140C23×C3C2×C22⋊C411yescharacteristic
141C3×Q8—44no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—41no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4#10112yes—
156C2×C3×C4C4×D412yes—
157C2×C3×C4#12011yescharacteristic
158C2×C3×C4#12112yes—
159C2×C3×C4#12311yescharacteristic
160C2×Dic3—24no—
161C2×Dic3—42no—
162C2×Dic3—42no—
163C2×Dic3—42no—
164C2×Dic3—42no—
165C2×Dic3—42no—
166Dic6—84no—
167C3×C8—44no—
168C3⋊C8—82no—
169C2×Q8—22no—
170C2×Q8—22no—
171C2×Q8—41no—
172C2×Q8—122no—
173C2×Q8—242no—
174C22×C4—62no—
175C22×C4—62no—
176C22×C4(C2×D3)⋊C411yescharacteristic
177C22×C4(C2×D3)⋊C411yescharacteristic
178C22×C4(C2×D3)⋊C411yescharacteristic
179C22×C4(C2×D3)⋊C411yescharacteristic
180C22×C4C2×C4×D311yescharacteristic
181C22×C4C2×D1211yescharacteristic
182C22×C4C2×#6711yescharacteristic
183C2×C8—24no—
184C2×C8—122no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—64no—
189C4⋊C4—64no—
190C4⋊C4—121no—
191C4⋊C4—121no—
192C4⋊C4—242no—
193C4⋊C4—242no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—122no—
199C42#23512yes—
200C42C2×#6712yes—
201C22×C3—21no—
202C22×C3—21no—
203C22×C3#38611yescharacteristic
204C22×C3C2×#10111yescharacteristic
205C22×C3#41311yescharacteristic
206C3×C4—21no—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4—22no—
210C3×C4—41no—
211C3×C4—41no—
212C3×C4—41no—
213C3×C4#45612yes—
214Dic3—44no—
215Dic3—82no—
216C23C2×(C2×D3)⋊C411yescharacteristic
217Q8—44no—
218Q8—244no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—64no—
233C2×C4—122no—
234C2×C4—122no—
235C2×C4—122no—
236C2×C4—122no—
237C2×C4—122no—
238C2×C4#82412yes—
239C2×C4C4×#6712yes—
240C2×C4#92911yescharacteristic
241C2×C4#93012yes—
242C2×C4#95311yescharacteristic
243C8—44no—
244C8—242no—
245C2×C3—21no—
246C2×C3—21no—
247C2×C3#188611yescharacteristic
248C2×C3#190111yescharacteristic
249C2×C3#191011yescharacteristic
250C22—21no—
251C22—21no—
252C22#513211yescharacteristic
253C22C2×#82411yescharacteristic
254C22#521311yescharacteristic
255C4—22no—
256C4—22no—
257C4—22no—
258C4—21no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—124no—
263C4—242no—
264C4#524612yes—
265C3#945411yescharacteristic, complemented, Sylow
266C2—21no—
267C2—21no—
268C2#7166011yescharacteristic
269C2#7200311yescharacteristic
270C2#7202511yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2983C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2983D3
hypocenterC3#9454
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
265. C3#9454semidirect

Maximal subgroups

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

subgroupindex
C3×#29832
#824782
#716252
#708222
#677502
#94543