Finite Group Databasebeta

← #10704831

Subgroups — #10704831

Order 768.

394 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 87 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
2C3×#1467C211yesmaximal, characteristic
3#71624C211yesmaximal, characteristic
4#67455C214yesmaximal
5#11568—31nomaximal, Sylow
6C2×C3×C4×Q8C2211yescharacteristic
7C2×#944C412yes—
8C2×#869C411yescharacteristic
9C3×#432—21no—
10#4981—21no—
11#4981—21no—
12#4973C2212yes—
13C4×C4⋊Dic3C411yescharacteristic
14#4928—21no—
15#4928—21no—
16#4925—21no—
17C3×#324—21no—
18C3×#324C2212yes—
19#4528—21no—
20#4528C2212yes—
21#2546—31no—
22#1467D311yescharacteristic
23#1461—31no—
24#1319—34no—
25C22×C3×Q8C2×C411yescharacteristic
26C22×Dic6C2×C412yes—
27C3×C4×Q8—41no—
28C3×C4×Q8—41no—
29C2×C3×C4⋊C4D411yescharacteristic
30C2×C3×C4⋊C4D411yescharacteristic
31C2×C3×C4⋊C4C2311yescharacteristic
32C2×C3×C42C2×C411yescharacteristic
33C2×C3×C42D411yescharacteristic
34C2×C3×C42D411yescharacteristic
35#944—44no—
36C2×C4⋊Dic3—21no—
37C2×C4⋊Dic3—21no—
38C2×C4⋊Dic3—21no—
39C2×Dic3⋊C4—22no—
40C2×Dic3⋊C4—22no—
41C2×C4×Dic3C2×C412yes—
42#869—41no—
43#869—41no—
44C3×C4⋊C8—41no—
45C3×C22⋊C8—22no—
46C3×#97—41no—
47C3⋊#95—21no—
48C3⋊#95—22no—
49C3⋊#95—22no—
50#830—22no—
51C4⋊(C3⋊C8)—41no—
52C4×C3⋊C8—41no—
53C2×C4⋊Q8—32no—
54C2×C4⋊Q8—31no—
55C2×C4×Q8C2×D311yescharacteristic
56#432—21no—
57#431—61no—
58#398—61no—
59#398—61no—
60#391—32no—
61#389—61no—
62#389—61no—
63C4×C4⋊C4—31no—
64#324—21no—
65#324—32no—
66#324—61no—
67#324C2×D312yes—
68C2×C3×Q8—22no—
69C2×C3×Q8—22no—
70C2×C3×Q8—41no—
71C22×C3×C4C22⋊C411yescharacteristic
72C22×C3×C4C22⋊C411yescharacteristic
73C22×C3×C4C22⋊C411yescharacteristic
74C22×C3×C4C22⋊C411yescharacteristic
75C22×C3×C4C22×C411yescharacteristic
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4C2×D411yescharacteristic
78C22×Dic3—22no—
79C22×Dic3—22no—
80C2×Dic6—24no—
81C2×Dic6—42no—
82C2×Dic6—42no—
83C2×C3×C8—42no—
84C3×C4⋊C4—41no—
85C3×C4⋊C4—41no—
86C3×C4⋊C4—41no—
87C3×C42—21no—
88C3×C42—21no—
89C3×C42—21no—
90C3×C42—21no—
91C3×C42—21no—
92C3×C42—21no—
93C4⋊Dic3—24no—
94C4⋊Dic3—42no—
95C4⋊Dic3—41no—
96C4⋊Dic3—41no—
97Dic3⋊C4—44no—
98Dic3⋊C4—44no—
99C4×Dic3—42no—
100C2×C3⋊C8—42no—
101C22×Q8—32no—
102C22×Q8C4×D311yescharacteristic
103C4⋊Q8—121no—
104C4⋊Q8—121no—
105C4⋊Q8—124no—
106C4×Q8—41no—
107C4×Q8—41no—
108C2×C4⋊C4—61no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4—61no—
112C2×C4⋊C4—61no—
113C2×C4⋊C4D1211yescharacteristic
114C2×C4⋊C4#6711yescharacteristic
115C2×C4⋊C4C22×D311yescharacteristic
116C2×C42—32no—
117C2×C42C4×D311yescharacteristic
118C2×C42D1211yescharacteristic
119C2×C42#6711yescharacteristic
120C4⋊C8—41no—
121C4⋊C8—121no—
122C22⋊C8—22no—
123C22⋊C8—62no—
124#97—41no—
125C4×C8—121no—
126#95—61no—
127#95—62no—
128#95—62no—
129C23×C3C2×C22⋊C411yescharacteristic
130C3×Q8—44no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—41no—
143C2×C3×C4—41no—
144C2×C3×C4—41no—
145C2×C3×C4#9911yescharacteristic
146C2×C3×C4#9911yescharacteristic
147C2×C3×C4#10012yes—
148C2×C3×C4#13112yes—
149C2×Dic3—24no—
150C2×Dic3—42no—
151C2×Dic3—42no—
152C2×Dic3—42no—
153C2×Dic3—42no—
154C2×Dic3—42no—
155Dic6—84no—
156C3×C8—82no—
157C3⋊C8—82no—
158C2×Q8—22no—
159C2×Q8—22no—
160C2×Q8—41no—
161C2×Q8—64no—
162C2×Q8—122no—
163C2×Q8—122no—
164C22×C4—62no—
165C22×C4—62no—
166C22×C4(C2×D3)⋊C411yescharacteristic
167C22×C4(C2×D3)⋊C411yescharacteristic
168C22×C4(C2×D3)⋊C411yescharacteristic
169C22×C4(C2×D3)⋊C411yescharacteristic
170C22×C4C2×C4×D311yescharacteristic
171C22×C4C2×D1211yescharacteristic
172C22×C4C2×#6711yescharacteristic
173C2×C8—42no—
174C2×C8—122no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—64no—
179C4⋊C4—124no—
180C4⋊C4—122no—
181C4⋊C4—121no—
182C4⋊C4—121no—
183C4⋊C4—124no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—122no—
191C22×C3—21no—
192C22×C3—21no—
193C22×C3#40811yescharacteristic
194C22×C3C2×#9911yescharacteristic
195C22×C3C2×#10011yescharacteristic
196C3×C4—21no—
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—41no—
200C3×C4—41no—
201C3×C4—41no—
202C3×C4—41no—
203C3×C4#35612yes—
204Dic3—44no—
205Dic3—82no—
206C23C2×(C2×D3)⋊C411yescharacteristic
207Q8—44no—
208Q8—244no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—64no—
224C2×C4—122no—
225C2×C4—122no—
226C2×C4—122no—
227C2×C4—122no—
228C2×C4—122no—
229C2×C4#80611yescharacteristic
230C2×C4#80611yescharacteristic
231C2×C4#82312yes—
232C2×C4#90712yes—
233C8—82no—
234C8—242no—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3#147111yescharacteristic
238C2×C3#212911yescharacteristic
239C2×C3C2×#35611yescharacteristic
240C22—21no—
241C22—21no—
242C22C2×#80611yescharacteristic
243C22#512911yescharacteristic
244C22C2×#82311yescharacteristic
245C4—22no—
246C4—22no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—41no—
251C4—41no—
252C4—124no—
253C4—242no—
254C4#447812yes—
255C3#1156811yescharacteristic, complemented, Sylow
256C2—21no—
257C2—21no—
258C2#6997011yescharacteristic
259C2#7005911yescharacteristic
260C2C2×#447811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1467C2
socleC22×C3C2×#99
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1467D3
hypocenterC3#11568
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
255. C3#11568semidirect

Maximal subgroups

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

subgroupindex
#824752
C3×#14672
#716242
#674552
#115683