Finite Group Databasebeta

← #10690953

Subgroups — #10690953

Order 768.

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

normal subgroups only

This group has 586 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×#2855C211yesmaximal, characteristic
2C2×#4937C211yesmaximal, characteristic
3C2×#4538C212yesmaximal
4#71531C211yesmaximal, characteristic
5#67624C212yesmaximal
6#9153—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C22×C3×C42C2211yescharacteristic
9C3×#391C412yescomplemented
10C3×#389C412yescomplemented
11C3×#387—22no—
12C3×#385—21no—
13C3×#381—21no—
14C3×#380C2211yescharacteristic
15C2×C3⋊#95—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2212yes—
18C2×#830C2212yes—
19#4937—24no—
20#4538—24no—
21#4524—22no—
22#2855D311yescharacteristic
23C2×#383—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—
44C22×C3⋊C8—22no—
45C3×#95—22no—
46C3×#95—22no—
47C3×#95—22no—
48C3×#95—21no—
49C3⋊#95—24no—
50C3⋊#95—24no—
51C3⋊#95—22no—
52C3⋊#95D414yes—
53#830—22no—
54#830—22no—
55C2×C22⋊Q8C2×D311yescharacteristic
56C22×C42C2×D311yescharacteristic
57C2×C22⋊C8—32no—
58#391Dic312yescomplemented
59#389Dic312yescomplemented
60#387—22no—
61#385—21no—
62#383—64no—
63#381—21no—
64#380C2×D311yescharacteristic
65C2×#95—32no—
66C2×#95—61no—
67C2×#95—61no—
68#336—62no—
69#328—64no—
70C24×C3C2×D411yescharacteristic
71C2×C3×Q8—22no—
72C2×C3×Q8—42no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—42no—
83C22×C3×C4—42no—
84C22×C3×C4C22⋊C412yes—
85C22×C3×C4C22⋊C412yes—
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C2×D412yes—
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4#4112yes—
90C22×Dic3—24no—
91C22×Dic3—44no—
92C22×Dic3—44no—
93C22×Dic3—42no—
94C22×Dic3—42no—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—42no—
97C3×C4⋊C4—44no—
98C3×C4⋊C4—42no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—41no—
101C3×C22⋊C4—42no—
102C3×C42—24no—
103C3×C42—24no—
104C2×C3⋊C8—44no—
105C2×C3⋊C8—42no—
106C22×Q8C2×Dic311yescharacteristic
107C23×C4—62no—
108C23×C4#6711yescharacteristic
109C23×C4#6711yescharacteristic
110C23×C4C22×D311yescharacteristic
111C22×C8—62no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C2×Dic311yescharacteristic
117C2×C4⋊C4C2×Dic312yes—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42C2×Dic312yes—
123C2×C42#6712yes—
124C22⋊C8—62no—
125C22⋊C8—62no—
126#95—22no—
127#95—22no—
128#95—22no—
129#95—21no—
130#95—34no—
131#95—64no—
132#95—64no—
133#95—62no—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3C2×C22⋊C411yescharacteristic
137C23×C3#12011yescharacteristic
138C23×C3#12311yescharacteristic
139C23×C3#13512yes—
140C23×C3#13712yes—
141C3×Q8—82no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—42no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4C4×D412yes—
164C2×C3×C4#12112yes—
165C2×Dic3—44no—
166C2×Dic3—44no—
167C2×Dic3—44no—
168C2×Dic3—84no—
169C2×Dic3—84no—
170C3⋊C8—82no—
171C24C2×#6711yescharacteristic
172C2×Q8—22no—
173C2×Q8—42no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—64no—
186C22×C4—124no—
187C22×C4—124no—
188C22×C4—122no—
189C22×C4—122no—
190C22×C4#21712yes—
191C22×C4#21712yes—
192C22×C4D3×D412yes—
193C22×C4#23712yes—
194C22×C4C22×Dic311yescharacteristic
195C22×C4C2×#6711yescharacteristic
196C2×C8—124no—
197C2×C8—122no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—44no—
201C4⋊C4—42no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—42no—
205C42—24no—
206C42—24no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—24no—
212C22×C3—24no—
213C22×C3#38611yescharacteristic
214C22×C3#41012yes—
215C22×C3#41611yescharacteristic
216C22×C3#41911yescharacteristic
217C22×C3#42112yes—
218C22×C3#45212yes—
219C22×C3#48512yes—
220C3×C4—22no—
221C3×C4—22no—
222C3×C4—42no—
223C3×C4—42no—
224C3×C4—81no—
225C3×C4—81no—
226C3×C4—82no—
227Dic3—84no—
228C23—22no—
229C23—22no—
230C23#93511yescharacteristic
231C23#93711yescharacteristic
232C23#94212yes—
233C23#95012yes—
234C23C2×#21711yescharacteristic
235Q8—82no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—124no—
258C2×C4—124no—
259C2×C4—124no—
260C2×C4—244no—
261C2×C4—244no—
262C2×C4D4×Dic312yes—
263C2×C4#93912yes—
264C8—242no—
265C2×C3—22no—
266C2×C3—22no—
267C2×C3#151512yes—
268C2×C3#152112yes—
269C2×C3#187311yescharacteristic
270C2×C3#189711yescharacteristic
271C2×C3#189911yescharacteristic
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—24no—
277C22—24no—
278C22#504512yes—
279C22#504712yes—
280C22#522011yescharacteristic
281C22#522511yescharacteristic
282C22#523311yescharacteristic
283C22#523612yes—
284C22#524012yes—
285C4—22no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—81no—
290C4—81no—
291C4—82no—
292C4—244no—
293C3#915311yescharacteristic, complemented, Sylow
294C2—22no—
295C2—22no—
296C2#7169311yescharacteristic
297C2#7170511yescharacteristic
298C2#7171211yescharacteristic
299C2#7175712yes—
300C2#7183712yes—
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#9153
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
9. C3×#391C4semidirect
10. C3×#389C4semidirect
58. #391Dic3semidirect
59. #389Dic3semidirect
293. C3#9153semidirect

Maximal subgroups

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

subgroupindex
C3×#28552
C2×#49372
C2×#45382
#715312
#676242
#91533