Finite Group Databasebeta

← #10693027

Subgroups — #10693027

Order 768.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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#82301C211yesmaximal, characteristic
3#71622C212yesmaximal, complemented
4#71617C211yesmaximal, characteristic
5#67631C212yesmaximal, complemented
6#9483—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—
16C2×#830—22no—
17D4×C3⋊C8—24no—
18#4925C2212yes—
19#4924C2212yes—
20#4535—24no—
21#4530—21no—
22#2972—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—22no—
45C22×C3⋊C8—24no—
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
58C8×D4—64no—
59#431—32no—
60#423—32no—
61C2×#100—61no—
62C2×C22⋊C8—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—22no—
76C2×C3×D4SD1612yes—
77C2×C3×D4Q1612yes—
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—42no—
93C3×C4⋊C4—82no—
94C3×C22⋊C4—24no—
95C3×C22⋊C4—44no—
96C3×C22⋊C4—82no—
97C3×C42—42no—
98C3×C42C2×D412yes—
99C3×C42#4112yes—
100#208—42no—
101#208—42no—
102C2×C3⋊C8—24no—
103C2×C3⋊C8—42no—
104C2×C3⋊C8—84no—
105C22×Q8C2×Dic311yescharacteristic
106C22×D4#6711yescharacteristic
107C23×C4#6712yes—
108C2×M16—62no—
109C22×C8—64no—
110C4×Q8—22no—
111C4×Q8—22no—
112C4×D4—22no—
113C4×D4—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4C2×Dic311yescharacteristic
118C2×C4⋊C4#6711yescharacteristic
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4C2×Dic312yes—
122C2×C42—21no—
123C2×C42C22×D311yescharacteristic
124C4⋊C8—62no—
125C4⋊C8—62no—
126C4⋊C8—62no—
127#100—122no—
128C22⋊C8—64no—
129C22⋊C8—62no—
130C4×C8—34no—
131#95—22no—
132#95—21no—
133#95C2×Dic312yes—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3—42no—
137C23×C3—42no—
138C23×C3#10314yes—
139C23×C3C2×C22⋊C411yescharacteristic
140C3×Q8—24no—
141C3×Q8—42no—
142C3×D4—24no—
143C3×D4—24no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—41no—
147C2×C3×C4—41no—
148C2×C3×C4—44no—
149C2×C3×C4—41no—
150C2×C3×C4—41no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4—41no—
154C2×C3×C4—42no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—82no—
159C2×C3×C4C4×D412yes—
160C2×C3×C4#12011yescharacteristic
161C2×C3×C4#12112yes—
162C2×C3×C4#12311yescharacteristic
163C2×C3×C4C2×SD1611yescharacteristic
164C2×C3×C4C2×Q1611yescharacteristic
165C2×C3×C4#13512yes—
166C3⋊C8—44no—
167C3⋊C8—82no—
168C24#21712yes—
169C2×Q8—22no—
170C2×Q8—42no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4#21512yes—
174C2×D4#21612yes—
175C22×C4—21no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—82no—
181C22×C4#21712yes—
182C22×C4C22×Dic311yescharacteristic
183C22×C4C2×#6711yescharacteristic
184C22×C4C2×#6711yescharacteristic
185M16—122no—
186M16—122no—
187C2×C8—64no—
188C2×C8—122no—
189C2×C8—244no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—82no—
196C22⋊C4—24no—
197C22⋊C4—44no—
198C22⋊C4—82no—
199C42—42no—
200C42D3×D412yes—
201C42#23712yes—
202C22×C3—21no—
203C22×C3—21no—
204C22×C3—24no—
205C22×C3—44no—
206C22×C3—42no—
207C22×C3—42no—
208C22×C3—82no—
209C22×C3#38611yescharacteristic
210C22×C3C2×#10311yescharacteristic
211C22×C3#41611yescharacteristic
212C3×C4—22no—
213C3×C4—21no—
214C3×C4—21no—
215C3×C4—42no—
216C3×C4—42no—
217C3×C4—82no—
218C3×C4—81no—
219C3×C4#46512yes—
220C3×C4#46712yes—
221C23—22no—
222C23—22no—
223C23—42no—
224C23—42no—
225C23#83514yes—
226C23C2×#21711yescharacteristic
227Q8—24no—
228Q8—42no—
229D4—24no—
230D4—24no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—44no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—82no—
246C2×C4D4×Dic312yes—
247C2×C4#93511yescharacteristic
248C2×C4#93711yescharacteristic
249C2×C4#93912yes—
250C2×C4C2×#21511yescharacteristic
251C2×C4C2×#21611yescharacteristic
252C2×C4#95012yes—
253C8—124no—
254C8—242no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—44no—
258C2×C3#189111yescharacteristic
259C2×C3#189211yescharacteristic
260C2×C3#191111yescharacteristic
261C22—21no—
262C22—21no—
263C22—24no—
264C22—44no—
265C22—42no—
266C22—42no—
267C22—82no—
268C22#522011yescharacteristic
269C22C2×#83511yescharacteristic
270C22#523311yescharacteristic
271C4—22no—
272C4—21no—
273C4—21no—
274C4—42no—
275C4—42no—
276C4—82no—
277C4—81no—
278C4#504812yes—
279C4#505112yes—
280C3#948311yescharacteristic, complemented, Sylow
281C2—21no—
282C2—21no—
283C2—44no—
284C2#7169611yescharacteristic
285C2#7171411yescharacteristic
286C2#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#9483
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
280. C3#9483semidirect

Maximal subgroups

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

subgroupindex
C3×#25802
#823012
#716222
#716172
#676312
#94833