Finite Group Databasebeta

← #10692869

Subgroups — #10692869

Order 768.

468 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 103 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#82475C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic
3C3×#1467C211yesmaximal, characteristic
4#70824C212yesmaximal
5#67749C212yesmaximal
6#9452—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#944C412yes—
9C2×#869C2211yescharacteristic
10C3×#432C2212yes—
11#5024—24no—
12#4981—21no—
13#4981—21no—
14#4973C412yes—
15C4×C4⋊Dic3C2211yescharacteristic
16#4928—21no—
17#4928—21no—
18#4926—21no—
19#4926C2212yes—
20#4925—21no—
21C3×#324—21no—
22C3×#324—21no—
23#4481—24no—
24#2983—31no—
25#2546—31no—
26#1558—32no—
27#1467D311yescharacteristic
28#1285—32no—
29C22×C3×Q8C2×C411yescharacteristic
30C22×Dic6C2×C412yes—
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
39#944—44no—
40C2×C4⋊Dic3—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43C2×Dic3⋊C4—22no—
44C2×Dic3⋊C4—22no—
45C2×C4×Dic3C2×C412yes—
46#869—22no—
47#869—22no—
48C3×C4⋊C8—22no—
49C3×C22⋊C8—22no—
50C3×#97—22no—
51C3⋊#95—22no—
52C3⋊#95—22no—
53C3⋊#95—21no—
54#830—22no—
55C4⋊(C3⋊C8)—22no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)D414yes—
58#803—22no—
59#803—22no—
60C4×C3⋊C8—22no—
61C2×C4⋊Q8—32no—
62C2×C4⋊Q8—31no—
63C2×C4×Q8C2×D311yescharacteristic
64#446—64no—
65#432—32no—
66#432—61no—
67#432C2×D312yes—
68#431—61no—
69#398—61no—
70#398—61no—
71#391—32no—
72#389—61no—
73#389—61no—
74C4×C4⋊C4—31no—
75#333—64no—
76#324—21no—
77#324—21no—
78C2×C3×Q8—22no—
79C2×C3×Q8—22no—
80C2×C3×Q8—41no—
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C22⋊C411yescharacteristic
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×Dic3—22no—
89C22×Dic3—22no—
90C2×Dic6—24no—
91C2×Dic6—42no—
92C2×Dic6—42no—
93C2×C3×C8—42no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—22no—
97C3×C42—21no—
98C3×C42—21no—
99C3×C42—21no—
100C3×C42—21no—
101C3×C42C2×D412yes—
102C3×C42#4112yes—
103C4⋊Dic3—24no—
104C4⋊Dic3—41no—
105C4⋊Dic3—42no—
106C4⋊Dic3—41no—
107Dic3⋊C4—44no—
108Dic3⋊C4—44no—
109C4×Dic3—42no—
110C2×C3⋊C8—24no—
111C22×Q8—32no—
112C22×Q8C4×D311yescharacteristic
113C4⋊Q8—62no—
114C4⋊Q8—62no—
115C4⋊Q8—124no—
116C4×Q8—22no—
117C4×Q8—22no—
118C2×C4⋊C4—61no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4—62no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4—61no—
123C2×C4⋊C4C4×D311yescharacteristic
124C2×C4⋊C4#6711yescharacteristic
125C2×C4⋊C4#6711yescharacteristic
126C2×C42—32no—
127C2×C42D1211yescharacteristic
128C2×C42D1211yescharacteristic
129C2×C42C22×D311yescharacteristic
130C4⋊C8—22no—
131C4⋊C8—34no—
132C4⋊C8—62no—
133C4⋊C8—62no—
134C22⋊C8—22no—
135C22⋊C8—62no—
136#97—22no—
137#97—62no—
138#97—62no—
139C4×C8—62no—
140#95—62no—
141#95—62no—
142#95—61no—
143C23×C3C2×C22⋊C411yescharacteristic
144C3×Q8—44no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×C3×C4#10012yes—
159C2×C3×C4C4×D412yes—
160C2×C3×C4#12011yescharacteristic
161C2×C3×C4#12112yes—
162C2×C3×C4#12311yescharacteristic
163C2×Dic3—24no—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—42no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169Dic6—84no—
170C3×C8—82no—
171C3⋊C8—44no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—41no—
175C2×Q8—64no—
176C2×Q8—122no—
177C2×Q8—122no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4(C2×D3)⋊C411yescharacteristic
181C22×C4(C2×D3)⋊C411yescharacteristic
182C22×C4(C2×D3)⋊C411yescharacteristic
183C22×C4(C2×D3)⋊C411yescharacteristic
184C22×C4C2×C4×D311yescharacteristic
185C22×C4C2×D1211yescharacteristic
186C22×C4C2×#6711yescharacteristic
187C2×C8—42no—
188C2×C8—64no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—64no—
193C4⋊C4—124no—
194C4⋊C4—124no—
195C4⋊C4—121no—
196C4⋊C4—121no—
197C4⋊C4—122no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—122no—
203C42D3×D412yes—
204C42#23712yes—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3#38611yescharacteristic
208C22×C3C2×#10011yescharacteristic
209C22×C3#41311yescharacteristic
210C3×C4—21no—
211C3×C4—22no—
212C3×C4—22no—
213C3×C4—22no—
214C3×C4—41no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4#45612yes—
218Dic3—44no—
219Dic3—82no—
220C23C2×(C2×D3)⋊C411yescharacteristic
221Q8—44no—
222Q8—244no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—64no—
237C2×C4—122no—
238C2×C4—122no—
239C2×C4—122no—
240C2×C4—122no—
241C2×C4—122no—
242C2×C4#82312yes—
243C2×C4#67⋊C412yes—
244C2×C4#88211yescharacteristic
245C2×C4#88611yescharacteristic
246C2×C4#93912yes—
247C8—82no—
248C8—124no—
249C2×C3—21no—
250C2×C3—21no—
251C2×C3#188611yescharacteristic
252C2×C3#190011yescharacteristic
253C2×C3#190911yescharacteristic
254C22—21no—
255C22—21no—
256C22#495811yescharacteristic
257C22C2×#82311yescharacteristic
258C22#513211yescharacteristic
259C4—22no—
260C4—22no—
261C4—22no—
262C4—21no—
263C4—41no—
264C4—41no—
265C4—41no—
266C4—124no—
267C4—242no—
268C4#475212yes—
269C3#945211yescharacteristic, complemented, Sylow
270C2—21no—
271C2—21no—
272C2#6998511yescharacteristic
273C2#7004611yescharacteristic
274C2#7010311yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1467C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1467D3
hypocenterC3#9452
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
269. C3#9452semidirect

Maximal subgroups

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

subgroupindex
#824752
#824362
C3×#14672
#708242
#677492
#94523