Finite Group Databasebeta

← #10690950

Subgroups — #10690950

Order 768.

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

normal subgroups only

This group has 586 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
1#82075C211yesmaximal, characteristic
2C2×#4937C211yesmaximal, characteristic
3C3×#1817C211yesmaximal, characteristic
4C2×#4467C212yesmaximal
5#67675C212yesmaximal
6#9153—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#924C2211yescharacteristic
9C2×C3×C22⋊C8C2212yes—
10#5212C2211yescharacteristic
11#5211—21no—
12#5210—21no—
13C2×C3⋊#95—21no—
14C2×C3⋊#95—21no—
15C2×C3⋊#95C2212yes—
16#4937—24no—
17#4933—22no—
18#4929C412yescomplemented
19#4928C412yescomplemented
20C3×#336—22no—
21#4467—24no—
22#2855—31no—
23C2×#383—31no—
24#1817D311yescharacteristic
25C2×#328—32no—
26#1294—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C23×Dic3—22no—
31C22×Dic6C2×C411yescharacteristic
32C22×C3×C8—22no—
33C2×C3×C42—22no—
34C2×C3×C42—22no—
35C2×C3×C42C2×C412yes—
36C2×C3×C42D412yes—
37C2×#217—21no—
38C2×#217—21no—
39C2×C4⋊Dic3C2×C411yescharacteristic
40C2×C4⋊Dic3C2×C412yes—
41#924—22no—
42#924—22no—
43C2×Dic3⋊C4—22no—
44C2×Dic3⋊C4—21no—
45C3×C22⋊C8—22no—
46C3×C22⋊C8—22no—
47C3⋊#95—24no—
48C3⋊#95—24no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51C3⋊#95—22no—
52C3⋊#95—21no—
53C3⋊#95—22no—
54C3⋊#95D414yes—
55C2×C22⋊Q8—31no—
56C22×C42C2×D311yescharacteristic
57C2×C22⋊C8C2×D312yes—
58#391—32no—
59#389—32no—
60#387—62no—
61#385—61no—
62#383—64no—
63#381—61no—
64#380—31no—
65C2×#95—32no—
66C2×#95—61no—
67C2×#95—61no—
68#336—22no—
69#328—64no—
70C24×C3C2×D411yescharacteristic
71C22×C3×C4—21no—
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—42no—
78C22×C3×C4—42no—
79C22×C3×C4C22⋊C412yes—
80C22×C3×C4C22⋊C412yes—
81C22×C3×C4C22×C411yescharacteristic
82C22×C3×C4C2×D412yes—
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4#4112yes—
85C22×Dic3—22no—
86C22×Dic3—21no—
87C22×Dic3—21no—
88C22×Dic3—24no—
89C22×Dic3—44no—
90C22×Dic3—44no—
91C22×Dic3—42no—
92C22×Dic3—42no—
93C2×Dic6—22no—
94C2×Dic6—42no—
95C2×C3×C8—44no—
96C2×C3×C8—42no—
97C3×C42—24no—
98C3×C42—24no—
99#217—41no—
100#217—41no—
101#217—42no—
102C4⋊Dic3—22no—
103C4⋊Dic3—42no—
104Dic3⋊C4—44no—
105Dic3⋊C4—42no—
106C22×Q8—31no—
107C23×C4—62no—
108C23×C4D1211yescharacteristic
109C23×C4D1211yescharacteristic
110C23×C4C22×D311yescharacteristic
111C22×C8—22no—
112C22⋊Q8—62no—
113C22⋊Q8—62no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—62no—
117C2×C4⋊C4—61no—
118C2×C22⋊C4—61no—
119C2×C22⋊C4—61no—
120C2×C42—22no—
121C2×C42—22no—
122C2×C42C4×D312yes—
123C2×C42#6712yes—
124C22⋊C8—22no—
125C22⋊C8—22no—
126#95—34no—
127#95—64no—
128#95—64no—
129#95—62no—
130#95—62no—
131#95—62no—
132#95—62no—
133#95—61no—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3C2×C22⋊C411yescharacteristic
137C23×C3#12011yescharacteristic
138C23×C3#12311yescharacteristic
139C23×C3#13512yes—
140C23×C3#13712yes—
141C2×C3×C4—22no—
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—42no—
150C2×C3×C4—42no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4C4×D412yes—
156C2×C3×C4#12112yes—
157C2×Dic3—44no—
158C2×Dic3—44no—
159C2×Dic3—41no—
160C2×Dic3—41no—
161C2×Dic3—42no—
162C2×Dic3—42no—
163C2×Dic3—42no—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—44no—
167C2×Dic3—84no—
168C2×Dic3—84no—
169Dic6—82no—
170C3×C8—82no—
171C24C2×D1211yescharacteristic
172C2×Q8—62no—
173C2×Q8—122no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—64no—
183C22×C4—62no—
184C22×C4—61no—
185C22×C4—61no—
186C22×C4—124no—
187C22×C4—124no—
188C22×C4—122no—
189C22×C4—122no—
190C22×C4(C2×D3)⋊C412yes—
191C22×C4(C2×D3)⋊C412yes—
192C22×C4C2×C4×D311yescharacteristic
193C22×C4D3×D412yes—
194C22×C4#23712yes—
195C22×C4C2×#6711yescharacteristic
196C2×C8—44no—
197C2×C8—42no—
198C4⋊C4—62no—
199C4⋊C4—122no—
200C4⋊C4—124no—
201C4⋊C4—122no—
202C22⋊C4—121no—
203C22⋊C4—121no—
204C22⋊C4—122no—
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—
224Dic3—84no—
225Dic3—81no—
226Dic3—81no—
227Dic3—82no—
228C23—22no—
229C23—22no—
230C23#88211yescharacteristic
231C23#88611yescharacteristic
232C23#90412yes—
233C23#90912yes—
234C23C2×(C2×D3)⋊C411yescharacteristic
235Q8—242no—
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—124no—
251C2×C4—124no—
252C2×C4—122no—
253C2×C4—122no—
254C2×C4—121no—
255C2×C4—121no—
256C2×C4—124no—
257C2×C4—122no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4—244no—
261C2×C4—244no—
262C2×C4#67⋊C412yes—
263C2×C4#93912yes—
264C8—82no—
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—24no—
275C22—21no—
276C22—21no—
277C22—24no—
278C22#472312yes—
279C22#473312yes—
280C22#495811yescharacteristic
281C22#500612yes—
282C22#511311yescharacteristic
283C22#512011yescharacteristic
284C22#513912yes—
285C4—22no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—244no—
290C4—241no—
291C4—242no—
292C4—241no—
293C3#915311yescharacteristic, complemented, Sylow
294C2—22no—
295C2—22no—
296C2#6953511yescharacteristic
297C2#6993111yescharacteristic
298C2#6996611yescharacteristic
299C2#7079112yes—
300C2#7083712yes—
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1817C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1817D3
hypocenterC3#9153
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
18. #4929C4semidirect
19. #4928C4semidirect
293. C3#9153semidirect

Maximal subgroups

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

subgroupindex
#820752
C2×#49372
C3×#18172
C2×#44672
#676752
#91533