Finite Group Databasebeta

← #10693047

Subgroups — #10693047

Order 768.

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

normal subgroups only

This group has 518 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
1C3×#2580C211yesmaximal, characteristic
2#82304C211yesmaximal, characteristic
3#71622C212yesmaximal, complemented
4#71617C211yesmaximal, characteristic
5#67631C212yesmaximal, complemented
6#9487—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#399C412yes—
10C3×#393C412yes—
11C3×#390—21no—
12C3×#389—21no—
13C3×#385—22no—
14C3×#382C2211yescharacteristic
15C2×#833—21no—
16#5207—22no—
17#5012—24no—
18#4925C2212yes—
19#4924C2212yes—
20#4535—24no—
21#4530—21no—
22#2975—31no—
23#2580D311yescharacteristic
24#1800—31no—
25#1487—32no—
26#1286—32no—
27C22×C3×Q8C2×C411yescharacteristic
28C22×C3×D4D411yescharacteristic
29C23×C3×C4D412yes—
30C3×C4×Q8—22no—
31C3×C4×Q8—22no—
32C3×C4×D4—22no—
33C3×C4×D4—22no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4C2×C412yes—
42C2×C3×C42—21no—
43C2×C3×C42C2311yescharacteristic
44C2×#208—24no—
45C2×#208—22no—
46C3×#95—22no—
47C3×#95—21no—
48C3×#95C2×C412yes—
49#833—42no—
50#830—24no—
51#830—22no—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55C4×C3⋊C8D414yes—
56C2×C4×Q8C2×D311yescharacteristic
57C2×C4×D4C2×D311yescharacteristic
58#436—64no—
59#431—32no—
60#423—32no—
61C2×#100—61no—
62#407—62no—
63#399Dic312yes—
64#393Dic312yes—
65#390—21no—
66#389—21no—
67#385—22no—
68#382C2×D311yescharacteristic
69#343—61no—
70Q8⋊C8—64no—
71C24×C3C22⋊C412yes—
72C2×C3×Q8—22no—
73C2×C3×Q8—42no—
74C2×C3×D4—22no—
75C2×C3×D4—21no—
76C2×C3×D4—22no—
77C2×C3×D4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—21no—
83C22×C3×C4—82no—
84C22×C3×C4C22⋊C412yes—
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C3×C4⋊C4—22no—
89C3×C4⋊C4—22no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4—41no—
92C3×C4⋊C4—41no—
93C3×C4⋊C4—41no—
94C3×C4⋊C4—82no—
95C3×C22⋊C4—24no—
96C3×C22⋊C4—44no—
97C3×C22⋊C4—82no—
98C3×C42—42no—
99C3×C42C2×D412yes—
100C3×C42#4112yes—
101#208—42no—
102#208—42no—
103#208—84no—
104C2×C3⋊C8—24no—
105C2×C3⋊C8—42no—
106C22×Q8C2×Dic311yescharacteristic
107C22×D4#6711yescharacteristic
108C23×C4#6712yes—
109C2×M16—64no—
110C2×M16—62no—
111C4×Q8—22no—
112C4×Q8—22no—
113C4×D4—22no—
114C4×D4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4C2×Dic311yescharacteristic
119C2×C4⋊C4#6711yescharacteristic
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4C2×Dic312yes—
123C2×C42—21no—
124C2×C42C22×D311yescharacteristic
125C4⋊C8—62no—
126C4⋊C8—62no—
127C4⋊C8—62no—
128#100—122no—
129C22⋊C8—64no—
130C22⋊C8—62no—
131C4×C8—34no—
132#95—22no—
133#95—21no—
134#95C2×Dic312yes—
135C23×C3—22no—
136C23×C3—22no—
137C23×C3—22no—
138C23×C3—42no—
139C23×C3—42no—
140C23×C3C2×C22⋊C411yescharacteristic
141C3×Q8—24no—
142C3×Q8—42no—
143C3×D4—42no—
144C3×D4—42no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—41no—
148C2×C3×C4—41no—
149C2×C3×C4—44no—
150C2×C3×C4—41no—
151C2×C3×C4—41no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×C3×C4—42no—
159C2×C3×C4—82no—
160C2×C3×C4C4×D412yes—
161C2×C3×C4#12011yescharacteristic
162C2×C3×C4#12112yes—
163C2×C3×C4#12311yescharacteristic
164C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
165C2×C3×C4#13712yes—
166C2×C3×C4#13711yescharacteristic
167C3⋊C8—44no—
168C3⋊C8—82no—
169C24#21712yes—
170C2×Q8—22no—
171C2×Q8—42no—
172C2×D4—22no—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—21no—
176C22×C4—21no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—82no—
182C22×C4#21712yes—
183C22×C4C22×Dic311yescharacteristic
184C22×C4C2×#6711yescharacteristic
185C22×C4C2×#6711yescharacteristic
186M16—122no—
187M16—122no—
188M16—244no—
189C2×C8—64no—
190C2×C8—122no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—82no—
198C22⋊C4—24no—
199C22⋊C4—44no—
200C22⋊C4—82no—
201C42—42no—
202C42D3×D412yes—
203C42#23712yes—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—44no—
207C22×C3—42no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—82no—
211C22×C3#38611yescharacteristic
212C22×C3#41711yescharacteristic
213C22×C3#41911yescharacteristic
214C3×C4—22no—
215C3×C4—21no—
216C3×C4—21no—
217C3×C4—42no—
218C3×C4—42no—
219C3×C4—82no—
220C3×C4—81no—
221C3×C4#46812yes—
222C3×C4#47012yes—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—42no—
227C23—42no—
228C23C2×#21711yescharacteristic
229Q8—24no—
230Q8—42no—
231D4—42no—
232D4—42no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—44no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—82no—
248C2×C4D4×Dic312yes—
249C2×C4#93511yescharacteristic
250C2×C4#93711yescharacteristic
251C2×C4#93912yes—
252C2×C4#94212yes—
253C2×C4#94911yescharacteristic
254C2×C4#95111yescharacteristic
255C8—124no—
256C8—242no—
257C2×C3—21no—
258C2×C3—21no—
259C2×C3—82no—
260C2×C3#189111yescharacteristic
261C2×C3#189411yescharacteristic
262C2×C3#191111yescharacteristic
263C22—21no—
264C22—21no—
265C22—44no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—82no—
270C22#522011yescharacteristic
271C22#522511yescharacteristic
272C22#523211yescharacteristic
273C4—22no—
274C4—21no—
275C4—21no—
276C4—42no—
277C4—42no—
278C4—82no—
279C4—81no—
280C4#504912yes—
281C4#505212yes—
282C3#948711yescharacteristic, complemented, Sylow
283C2—21no—
284C2—21no—
285C2—82no—
286C2#7169711yescharacteristic
287C2#7171411yescharacteristic
288C2#7185311yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2580C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#2580D3
hypocenterC3#9487
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2580

Complements

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

NH ≅ G/Ntype
3. #71622C2semidirect
5. #67631C2semidirect
282. C3#9487semidirect

Maximal subgroups

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

subgroupindex
C3×#25802
#823042
#716222
#716172
#676312
#94873