Finite Group Databasebeta

← #10690718

Subgroups — #10690718

Order 768.

522 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 127 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×#2326C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic, complemented
3#71624C211yesmaximal, characteristic, complemented
4#71597C212yesmaximal
5#67614C212yesmaximal, complemented
6#9106—31nomaximal, Sylow
7C2×C3×#124C411yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C2×C3×C4×D4C2211yescharacteristic
10C3×#391C411yescharacteristic
11C3×#390C411yescharacteristic
12C3×#388—22no—
13C3×#385—22no—
14C3×#384C411yescharacteristic
15C3×#376C2211yescharacteristic
16#5024—24no—
17#4926—22no—
18#4925C2212yes—
19#4925C2212yes—
20D4⋊(C3⋊C8)—24no—
21#4528—22no—
22#2983—31no—
23#2326D311yescharacteristic
24#1478—32no—
25#1461—31no—
26#1274—32no—
27C22×C3×Q8C2×C411yescharacteristic
28C22×C3×D4C2×C411yescharacteristic
29C23×C3×C4—21no—
30C3×#124—22no—
31C3×#124—22no—
32C3×C4×Q8—22no—
33C3×C4×Q8—22no—
34C3×C4×D4—22no—
35C3×C4×D4—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4D412yes—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C42C2×C411yescharacteristic
44C2×C3×C42C2×C411yescharacteristic
45C2×C3×C42D412yes—
46C2×C3×C42C2311yescharacteristic
47C3×#95—22no—
48C3×#95—21no—
49C3×#95—21no—
50#830—22no—
51#830—22no—
52C4⋊(C3⋊C8)—24no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55#803—24no—
56C4×C3⋊C8—22no—
57C4×C3⋊C8D414yes—
58C2×#124Dic311yescharacteristic
59C2×C4×Q8C2×D311yescharacteristic
60C2×C4×D4C2×D311yescharacteristic
61#446—64no—
62#432—62no—
63#431—32no—
64#431—32no—
65#391Dic311yescharacteristic
66#390Dic311yescharacteristic
67#388—22no—
68#385—22no—
69#384Dic311yescharacteristic
70#376C2×D311yescharacteristic
71D4⋊C8—64no—
72#324—62no—
73C24×C3—21no—
74C2×C3×Q8—22no—
75C2×C3×Q8—21no—
76C2×C3×Q8—21no—
77C2×C3×Q8—22no—
78C2×C3×D4—22no—
79C2×C3×D4—21no—
80C2×C3×D4—22no—
81C2×C3×D4M1612yes—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—42no—
86C22×C3×C4—82no—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22⋊C412yes—
89C22×C3×C4C22×C411yescharacteristic
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C3×C4⋊C4—22no—
93C3×C4⋊C4—24no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—82no—
96C3×C22⋊C4—28no—
97C3×C22⋊C4—41no—
98C3×C22⋊C4—41no—
99C3×C22⋊C4—82no—
100C3×C42—22no—
101C3×C42—22no—
102C3×C42—21no—
103C3×C42—22no—
104C3×C42M1612yes—
105C3×C42C2×D412yes—
106C3×C42#4112yes—
107C2×C3⋊C8—24no—
108C2×C3⋊C8—42no—
109C22×Q8C2×Dic311yescharacteristic
110C22×D4C2×Dic311yescharacteristic
111C23×C4—21no—
112#124—22no—
113#124—22no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×D4—22no—
117C4×D4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4C2×Dic311yescharacteristic
120C2×C4⋊C4C2×Dic311yescharacteristic
121C2×C4⋊C4#6712yes—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—22no—
125C2×C42C2×Dic311yescharacteristic
126C2×C42C2×Dic311yescharacteristic
127C2×C42#6712yes—
128C2×C42C22×D311yescharacteristic
129C4⋊C8—64no—
130C4⋊C8—62no—
131C4⋊C8—62no—
132C22⋊C8—62no—
133C22⋊C8—62no—
134#97—64no—
135C4×C8—34no—
136C4×C8—62no—
137#95—22no—
138#95—21no—
139#95—21no—
140C23×C3—22no—
141C23×C3—42no—
142C23×C3—42no—
143C23×C3—41no—
144C23×C3—41no—
145C23×C3C2×C22⋊C411yescharacteristic
146C3×Q8—42no—
147C3×Q8—42no—
148C3×D4—24no—
149C3×D4—42no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—42no—
160C2×C3×C4—41no—
161C2×C3×C4—42no—
162C2×C3×C4—41no—
163C2×C3×C4—41no—
164C2×C3×C4—41no—
165C2×C3×C4—84no—
166C2×C3×C4C4×D412yes—
167C2×C3×C4#12011yescharacteristic
168C2×C3×C4#12112yes—
169C2×C3×C4#12311yescharacteristic
170C2×C3×C4C2×M1611yescharacteristic
171C2×C3×C4C2×M1611yescharacteristic
172C2×C3×C4#13112yes—
173C3⋊C8—44no—
174C3⋊C8—82no—
175C24—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4#20812yes—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—82no—
189C22×C4#21712yes—
190C22×C4#21712yes—
191C22×C4C22×Dic311yescharacteristic
192C22×C4C2×#6711yescharacteristic
193C22×C4C2×#6711yescharacteristic
194C2×C8—64no—
195C2×C8—122no—
196C4⋊C4—22no—
197C4⋊C4—24no—
198C4⋊C4—22no—
199C4⋊C4—82no—
200C22⋊C4—28no—
201C22⋊C4—41no—
202C22⋊C4—41no—
203C22⋊C4—82no—
204C42—21no—
205C42—22no—
206C42—22no—
207C42—22no—
208C42#20812yes—
209C42D3×D412yes—
210C42#23712yes—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—44no—
214C22×C3—42no—
215C22×C3—82no—
216C22×C3—81no—
217C22×C3—81no—
218C22×C3#38611yescharacteristic
219C22×C3#40711yescharacteristic
220C22×C3#40811yescharacteristic
221C3×C4—22no—
222C3×C4—21no—
223C3×C4—21no—
224C3×C4—42no—
225C3×C4—42no—
226C3×C4—42no—
227C3×C4—81no—
228C3×C4—81no—
229C3×C4#43512yes—
230C3×C4#43612yes—
231C23—22no—
232C23—42no—
233C23—42no—
234C23—41no—
235C23—41no—
236C23C2×#21711yescharacteristic
237Q8—42no—
238Q8—42no—
239D4—24no—
240D4—42no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—84no—
257C2×C4C2×#20811yescharacteristic
258C2×C4C2×#20811yescharacteristic
259C2×C4D4×Dic312yes—
260C2×C4#93511yescharacteristic
261C2×C4#93711yescharacteristic
262C2×C4#93912yes—
263C2×C4#94812yes—
264C8—124no—
265C8—242no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—82no—
269C2×C3#187011yescharacteristic
270C2×C3#191011yescharacteristic
271C2×C3#191211yescharacteristic
272C22—21no—
273C22—21no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22#520711yescharacteristic
280C22#522011yescharacteristic
281C22#523411yescharacteristic
282C4—22no—
283C4—21no—
284C4—21no—
285C4—42no—
286C4—42no—
287C4—42no—
288C4—81no—
289C4—81no—
290C4#501112yes—
291C4#501212yes—
292C3#910611yescharacteristic, complemented, Sylow
293C2—21no—
294C2—21no—
295C2—82no—
296C2#7161111yescharacteristic
297C2#7185411yescharacteristic
298C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2326C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#2326D3
hypocenterC3#9106
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #82436C2semidirect
3. #71624C2semidirect
5. #67614C2semidirect
292. C3#9106semidirect

Maximal subgroups

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

subgroupindex
C3×#23262
#824362
#716242
#715972
#676142
#91063