Finite Group Databasebeta

← #10687352

Subgroups — #10687352

Order 768.

658 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 119 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#85803C211yesmaximal, characteristic
2C3×#1467C211yesmaximal, characteristic
3#71635C211yesmaximal, characteristic
4#70820C212yesmaximal
5#70813C212yesmaximal
6#8460—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8#5812C412yes—
9#5570—22no—
10#5569—24no—
11Q8×Dic6—22no—
12#5500—22no—
13#5500C412yes—
14#5470—22no—
15#5470C2212yes—
16C3×#432C2212yes—
17C2×C4⋊(C3⋊C8)C2211yescharacteristic
18C2×#803C2211yescharacteristic
19C3×#324—22no—
20#4524—22no—
21#4486—22no—
22#4482—22no—
23#4465—24no—
24#3502—31no—
25#1985—31no—
26#1552—32no—
27#1531—32no—
28#1467D311yescharacteristic
29C22×C3×Q8C2×C411yescharacteristic
30C3×C4×Q8—22no—
31C3×C4×Q8—22no—
32C2×C3×C4⋊C4C2×C411yescharacteristic
33C2×C3×C4⋊C4D412yes—
34C2×C3×C42D412yes—
35C2×C3×C42C2311yescharacteristic
36Q8×Dic3—24no—
37Q8×Dic3—24no—
38#926—24no—
39#926C2×C414yes—
40#924—24no—
41#924—22no—
42#924—22no—
43C22×C3⋊C8—21no—
44C22×C3⋊C8—21no—
45#890—28no—
46#890—24no—
47#888—28no—
48#888—24no—
49#870—24no—
50#870D414yes—
51#869—22no—
52#869—22no—
53#869D412yes—
54#869D412yes—
55C4×Dic6—24no—
56C4×Dic6—24no—
57C4×Dic6—22no—
58C4×Dic6—22no—
59C3×C4⋊C8—22no—
60C3×C22⋊C8—22no—
61C3×#97—22no—
62C4⋊(C3⋊C8)—22no—
63#803—22no—
64Q82—62no—
65#557—64no—
66#557—62no—
67#544—32no—
68#544—62no—
69#533—32no—
70#533—62no—
71#519—32no—
72C2×C4×Q8C2×D311yescharacteristic
73#432C2×D312yes—
74C2×C4⋊C8—31no—
75C2×#97—31no—
76#336—62no—
77#332—62no—
78#332—62no—
79#330—64no—
80#324—22no—
81C2×C3×Q8—24no—
82C2×C3×Q8—22no—
83C22×C3×C4C22⋊C412yes—
84C22×C3×C4C22⋊C412yes—
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C2×Dic6—22no—
89C2×Dic6—24no—
90C2×Dic6—22no—
91C2×C3×C8—42no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—24no—
94C3×C42—22no—
95C3×C42—22no—
96C3×C42C2×D412yes—
97C3×C42C2×D412yes—
98#217—24no—
99C4⋊Dic3—24no—
100C4⋊Dic3—24no—
101C4⋊Dic3—24no—
102C4⋊Dic3—22no—
103C4⋊Dic3—22no—
104C4⋊Dic3C22⋊C414yes—
105C4⋊Dic3C22⋊C414yes—
106Dic3⋊C4—28no—
107Dic3⋊C4—24no—
108Dic3⋊C4—24no—
109C4×Dic3—28no—
110C4×Dic3—24no—
111C2×C3⋊C8—22no—
112C2×C3⋊C8—41no—
113C2×C3⋊C8—41no—
114C2×C3⋊C8—42no—
115C2×C3⋊C8—41no—
116C22×Q8C4×D311yescharacteristic
117C22×C8—61no—
118C22×C8—61no—
119C4⋊Q8—32no—
120C4⋊Q8—32no—
121C4⋊Q8—62no—
122C4⋊Q8—62no—
123C4⋊Q8—64no—
124C4⋊Q8—68no—
125#125—34no—
126#125—64no—
127#125—64no—
128#125—68no—
129C22⋊Q8—64no—
130C22⋊Q8—62no—
131C22⋊Q8—62no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—64no—
135C4×Q8—64no—
136C4×Q8—64no—
137C4×Q8—64no—
138C4×Q8—62no—
139C4×Q8—62no—
140#117—34no—
141#117—64no—
142C2×C4⋊C4C4×D311yescharacteristic
143C2×C4⋊C4#6712yes—
144C2×C42D1212yes—
145C2×C42C22×D311yescharacteristic
146C4⋊C8—22no—
147C4⋊C8—62no—
148C22⋊C8—22no—
149#97—22no—
150#97—62no—
151C23×C3C2×C22⋊C411yescharacteristic
152C3×Q8—28no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—22no—
159C2×C3×C4—41no—
160C2×C3×C4—42no—
161C2×C3×C4#10114yes—
162C2×C3×C4C2×C22⋊C412yes—
163C2×C3×C4#12011yescharacteristic
164C2×C3×C4#12011yescharacteristic
165C2×C3×C4#12012yes—
166C2×Dic3—24no—
167C2×Dic3—24no—
168Dic6—42no—
169Dic6—42no—
170Dic6—82no—
171Dic6—84no—
172C3×C8—82no—
173C3⋊C8—42no—
174C3⋊C8—81no—
175C2×Q8—24no—
176C2×Q8—22no—
177C2×Q8—64no—
178C2×Q8—62no—
179C2×Q8—62no—
180C22×C4(C2×D3)⋊C412yes—
181C22×C4(C2×D3)⋊C412yes—
182C22×C4C2×C4×D311yescharacteristic
183C22×C4C2×D1211yescharacteristic
184C22×C4C2×#6711yescharacteristic
185C2×C8—42no—
186C2×C8—62no—
187C2×C8—121no—
188C2×C8—121no—
189C2×C8—122no—
190C2×C8—121no—
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—34no—
194C4⋊C4—34no—
195C4⋊C4—64no—
196C4⋊C4—64no—
197C4⋊C4—64no—
198C4⋊C4—68no—
199C4⋊C4—64no—
200C4⋊C4—62no—
201C4⋊C4—64no—
202C4⋊C4—62no—
203C22⋊C4—64no—
204C42—22no—
205C42—22no—
206C42—68no—
207C42—64no—
208C42D3×D412yes—
209C42D3×D412yes—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3#37911yescharacteristic
213C22×C3C2×#10112yes—
214C3×C4—21no—
215C3×C4—22no—
216C3×C4—24no—
217C3×C4—22no—
218C3×C4—42no—
219C3×C4#45411yescharacteristic
220C3×C4#45611yescharacteristic
221Dic3—44no—
222Dic3—44no—
223C23C2×(C2×D3)⋊C411yescharacteristic
224Q8—28no—
225Q8—122no—
226Q8—122no—
227Q8—242no—
228Q8—244no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—64no—
238C2×C4—64no—
239C2×C4#82414yes—
240C2×C4D3×C22⋊C412yes—
241C2×C4#88211yescharacteristic
242C2×C4#88211yescharacteristic
243C2×C4#93712yes—
244C8—82no—
245C8—122no—
246C8—241no—
247C2×C3—22no—
248C2×C3#178411yescharacteristic
249C2×C3#179311yescharacteristic
250C2×C3#179411yescharacteristic
251C22—21no—
252C22—21no—
253C22#495511yescharacteristic
254C22C2×#82412yes—
255C4—24no—
256C4—22no—
257C4—22no—
258C4—21no—
259C4—42no—
260C4—124no—
261C4—124no—
262C4#475211yescharacteristic
263C4#475311yescharacteristic
264C3#846011yescharacteristic, complemented, Sylow
265C2—22no—
266C2#6999011yescharacteristic
267C2#7005411yescharacteristic
268C2#7012111yescharacteristic
subgroupG/H
centerC22#4955
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1467C2
socleC22×C3#379
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1467D3
hypocenterC3#8460
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
264. C3#8460semidirect

Maximal subgroups

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

subgroupindex
#858032
C3×#14672
#716352
#708202
#708132
#84603