Finite Group Databasebeta

← #10690913

Subgroups — #10690913

Order 768.

694 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 143 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×#2855C211yesmaximal, characteristic
2C2×#4932C211yesmaximal, characteristic
3C2×#4538C212yesmaximal
4#71531C211yesmaximal, characteristic
5#67624C212yesmaximal
6#9145—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C22×C3×C42C2211yescharacteristic
9C22×C4×Dic3—21no—
10C3×#391C412yescomplemented
11C3×#389C412yescomplemented
12C3×#387—22no—
13C3×#385—21no—
14C3×#381—21no—
15C3×#380C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2212yes—
18C2×#830C2212yes—
19#4932—24no—
20#4538—24no—
21#4524—22no—
22#2855D311yescharacteristic
23C2×#376—31no—
24#1817—31no—
25C2×#328—32no—
26#1294—32no—
27C22×C3×Q8C2×C411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C23×Dic3—22no—
32C3×C22⋊Q8—22no—
33C3×C22⋊Q8—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C22⋊C4—21no—
39C2×C3×C22⋊C4—21no—
40C2×C3×C42—22no—
41C2×C3×C42—22no—
42C2×C3×C42C2×C412yes—
43C2×C3×C42D412yes—
44C2×C4×Dic3—24no—
45C2×C4×Dic3—28no—
46C22×C3⋊C8—22no—
47C3×#95—22no—
48C3×#95—22no—
49C3×#95—22no—
50C3×#95—21no—
51C3⋊#95—24no—
52C3⋊#95—22no—
53C3⋊#95D414yes—
54#830—22no—
55#830—22no—
56C2×C22⋊Q8C2×D311yescharacteristic
57C22×C42—61no—
58C22×C42C2×D311yescharacteristic
59C2×C22⋊C8—32no—
60#391Dic312yescomplemented
61#389Dic312yescomplemented
62#387—22no—
63#385—21no—
64#381—21no—
65#380C2×D311yescharacteristic
66#376—64no—
67C2×#95—32no—
68C2×#95—61no—
69#336—62no—
70#328—64no—
71C24×C3C2×D411yescharacteristic
72C2×C3×Q8—22no—
73C2×C3×Q8—42no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—24no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—42no—
85C22×C3×C4C22⋊C412yes—
86C22×C3×C4C22⋊C412yes—
87C22×C3×C4C22×C411yescharacteristic
88C22×C3×C4C2×D412yes—
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4#4112yes—
91C22×Dic3—28no—
92C22×Dic3—24no—
93C22×Dic3—44no—
94C22×Dic3—42no—
95C22×Dic3—42no—
96C3×C4⋊C4—22no—
97C3×C4⋊C4—42no—
98C3×C4⋊C4—44no—
99C3×C4⋊C4—42no—
100C3×C22⋊C4—41no—
101C3×C22⋊C4—41no—
102C3×C22⋊C4—42no—
103C3×C42—24no—
104C3×C42—24no—
105C4×Dic3—216no—
106C2×C3⋊C8—44no—
107C2×C3⋊C8—42no—
108C22×Q8C2×Dic311yescharacteristic
109C23×C4—62no—
110C23×C4#6711yescharacteristic
111C23×C4#6711yescharacteristic
112C23×C4C22×D311yescharacteristic
113C22×C8—62no—
114C22⋊Q8—22no—
115C22⋊Q8—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4C2×Dic311yescharacteristic
119C2×C4⋊C4C2×Dic312yes—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—64no—
125C2×C42—68no—
126C2×C42C2×Dic312yes—
127C2×C42#6712yes—
128C22⋊C8—62no—
129C22⋊C8—62no—
130#95—22no—
131#95—22no—
132#95—22no—
133#95—21no—
134#95—34no—
135#95—64no—
136#95—62no—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3C2×C22⋊C411yescharacteristic
140C23×C3#12011yescharacteristic
141C23×C3#12311yescharacteristic
142C23×C3#13512yes—
143C23×C3#13712yes—
144C3×Q8—82no—
145C2×C3×C4—24no—
146C2×C3×C4—24no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4#10414yes—
166C2×C3×C4C4×D412yes—
167C2×C3×C4#12112yes—
168C2×Dic3—28no—
169C2×Dic3—44no—
170C2×Dic3—48no—
171C2×Dic3—48no—
172C2×Dic3—44no—
173C3⋊C8—82no—
174C24C2×#6711yescharacteristic
175C2×Q8—22no—
176C2×Q8—42no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—24no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—42no—
188C22×C4—68no—
189C22×C4—64no—
190C22×C4—124no—
191C22×C4—122no—
192C22×C4—122no—
193C22×C4#21712yes—
194C22×C4#21712yes—
195C22×C4D3×D412yes—
196C22×C4#23712yes—
197C22×C4C22×Dic311yescharacteristic
198C22×C4C2×#6711yescharacteristic
199C2×C8—124no—
200C2×C8—122no—
201C4⋊C4—22no—
202C4⋊C4—42no—
203C4⋊C4—44no—
204C4⋊C4—42no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—42no—
208C42—24no—
209C42—24no—
210C42—616no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—24no—
216C22×C3—24no—
217C22×C3#38611yescharacteristic
218C22×C3#41012yes—
219C22×C3#41611yescharacteristic
220C22×C3#41911yescharacteristic
221C22×C3C2×#10412yes—
222C22×C3#45212yes—
223C22×C3#48512yes—
224C3×C4—24no—
225C3×C4—22no—
226C3×C4—22no—
227C3×C4—42no—
228C3×C4—81no—
229C3×C4—81no—
230C3×C4—82no—
231Dic3—48no—
232C23—22no—
233C23—22no—
234C23#93511yescharacteristic
235C23#93711yescharacteristic
236C23#94212yes—
237C23#95012yes—
238C23C2×#21711yescharacteristic
239Q8—82no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—68no—
261C2×C4—124no—
262C2×C4—128no—
263C2×C4—128no—
264C2×C4—124no—
265C2×C4#83714yes—
266C2×C4D4×Dic312yes—
267C2×C4#93912yes—
268C8—242no—
269C2×C3—22no—
270C2×C3—22no—
271C2×C3#150012yes—
272C2×C3#150612yes—
273C2×C3#187311yescharacteristic
274C2×C3#189611yescharacteristic
275C2×C3#189911yescharacteristic
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—24no—
281C22—24no—
282C22#504512yes—
283C22#504712yes—
284C22#522011yescharacteristic
285C22#522511yescharacteristic
286C22#523311yescharacteristic
287C22C2×#83712yes—
288C22#524012yes—
289C4—22no—
290C4—22no—
291C4—24no—
292C4—42no—
293C4—81no—
294C4—81no—
295C4—82no—
296C4—128no—
297C3#914511yescharacteristic, complemented, Sylow
298C2—22no—
299C2—22no—
300C2#7169311yescharacteristic
301C2#7170511yescharacteristic
302C2#7171111yescharacteristic
303C2#7175412yes—
304C2#7183612yes—
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2855C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2855D3
hypocenterC3#9145
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
10. C3×#391C4semidirect
11. C3×#389C4semidirect
60. #391Dic3semidirect
61. #389Dic3semidirect
297. C3#9145semidirect

Maximal subgroups

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

subgroupindex
C3×#28552
C2×#49322
C2×#45382
#715312
#676242
#91453