Finite Group Databasebeta

← #10690697

Subgroups — #10690697

Order 768.

546 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 139 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×#2982C211yesmaximal, characteristic, complemented
2#82455C211yesmaximal, characteristic
3#71624C211yesmaximal, characteristic, complemented
4#70757C212yesmaximal
5#67737C212yesmaximal, complemented
6#9102—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#947C411yescharacteristic
9C2×C4×D12C2211yescharacteristic
10C3×C8×Q8—24no—
11C3×#431—22no—
12C3×#431C2212yes—
13#5229C411yescharacteristic
14#4989—22no—
15#4988C411yescharacteristic
16#4975C411yescharacteristic
17#4940—22no—
18#4932C2211yescharacteristic
19#4925C2212yes—
20#4528—22no—
21#4459—24no—
22#2982D311yescharacteristic, complemented
23#2326—31no—
24#1478—32no—
25#1461—31no—
26#1274—32no—
27C22×C3×Q8C2×C411yescharacteristic
28C22×D12C2×C411yescharacteristic
29C22×C4×D3—21no—
30C3×C4×Q8—22no—
31C3×C4×Q8—22no—
32C2×C3×C4⋊C4C2×C411yescharacteristic
33C2×C3×C4⋊C4D412yes—
34C2×C3×C42D412yes—
35C2×C3×C42C2311yescharacteristic
36#947—22no—
37#947C814yescomplemented
38C2×(C2×D3)⋊C4—22no—
39C2×(C2×D3)⋊C4—21no—
40C2×(C2×D3)⋊C4—22no—
41C2×C4⋊Dic3C2×C411yescharacteristic
42C2×Dic3⋊C4—22no—
43C2×C4×Dic3C2×C411yescharacteristic
44C2×C4×Dic3C2×C411yescharacteristic
45C4×D12—22no—
46C4×D12—22no—
47C3×C4⋊C8—24no—
48C3×C4⋊C8—22no—
49C3×C22⋊C8—22no—
50C3×C4×C8—24no—
51C3×C4×C8D414yes—
52C3⋊#95—22no—
53C3⋊#95—21no—
54C3⋊#95—21no—
55#830—22no—
56C4⋊(C3⋊C8)—22no—
57C4×C3⋊C8—22no—
58C2×#124—31no—
59C2×C4×Q8C2×D311yescharacteristic
60C2×C4×D4—31no—
61C8×Q8—24no—
62#431—22no—
63#431—32no—
64#431C2×D312yes—
65#391—31no—
66#390—31no—
67#388—62no—
68#385—62no—
69#384—31no—
70#376—31no—
71D4⋊C8—64no—
72#324—62no—
73C23×D3—21no—
74C2×C3×Q8—22no—
75C2×C3×Q8—22no—
76C2×C3×Q8C2×C812yes—
77C2×C3×Q8M1612yes—
78C22×C3×C4C22⋊C412yes—
79C22×C3×C4C22⋊C412yes—
80C22×C3×C4C22×C411yescharacteristic
81C22×C3×C4C2×D411yescharacteristic
82C22×C3×C4C2×D411yescharacteristic
83C22×Dic3—21no—
84C22×Dic3—21no—
85C22×Dic3—21no—
86C2×D12—22no—
87C2×D12—22no—
88C2×D12—21no—
89C2×D12C2×C812yes—
90C2×C4×D3—42no—
91C2×C4×D3—82no—
92C2×C3×C8—24no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—24no—
95C3×C42—22no—
96C3×C42—22no—
97C3×C42C2×D412yes—
98C3×C42#4112yes—
99(C2×D3)⋊C4—28no—
100(C2×D3)⋊C4—41no—
101(C2×D3)⋊C4—41no—
102(C2×D3)⋊C4—82no—
103C4⋊Dic3—22no—
104Dic3⋊C4—82no—
105C4×Dic3—21no—
106C4×Dic3—22no—
107C4×Dic3C2×C812yes—
108C2×C3⋊C8—42no—
109C22×Q8C4×D311yescharacteristic
110C22×D4—31no—
111C23×C4—61no—
112#124—34no—
113#124—62no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×D4—62no—
117C4×D4—62no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4C4×D311yescharacteristic
121C2×C4⋊C4D1212yes—
122C2×C22⋊C4—62no—
123C2×C22⋊C4—61no—
124C2×C22⋊C4—62no—
125C2×C42—31no—
126C2×C42—31no—
127C2×C42#6712yes—
128C2×C42C22×D311yescharacteristic
129C4⋊C8—24no—
130C4⋊C8—22no—
131C4⋊C8—62no—
132C22⋊C8—22no—
133C22⋊C8—62no—
134C4×C8—24no—
135C4×C8—62no—
136C4×C8#6714yes—
137#95—62no—
138#95—61no—
139#95—61no—
140C23×C3C2×C22⋊C411yescharacteristic
141C22×D3—22no—
142C22×D3—42no—
143C22×D3—42no—
144C22×D3—41no—
145C22×D3—41no—
146C3×Q8—24no—
147C3×Q8—24no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—42no—
154C2×C3×C4C22⋊C814yes—
155C2×C3×C4C4×D412yes—
156C2×C3×C4#12011yescharacteristic
157C2×C3×C4#12112yes—
158C2×C3×C4#12311yescharacteristic
159C2×C3×C4C22×C811yescharacteristic
160C2×C3×C4C2×M1611yescharacteristic
161C2×C3×C4#13112yes—
162C2×Dic3—22no—
163C2×Dic3—41no—
164C2×Dic3—41no—
165C2×Dic3—42no—
166C2×Dic3—41no—
167C2×Dic3—41no—
168C2×Dic3—41no—
169C2×Dic3—41no—
170D12—24no—
171D12—42no—
172C4×D3—84no—
173C3×C8—44no—
174C3⋊C8—82no—
175C24—61no—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8D3×C812yes—
179C2×Q8ΓL(1,𝔽25)12yes—
180C2×D4—32no—
181C2×D4—61no—
182C2×D4—62no—
183C2×D4—62no—
184C22×C4—61no—
185C22×C4—61no—
186C22×C4—61no—
187C22×C4—122no—
188C22×C4—242no—
189C22×C4(C2×D3)⋊C412yes—
190C22×C4(C2×D3)⋊C412yes—
191C22×C4C2×C4×D311yescharacteristic
192C22×C4C2×D1211yescharacteristic
193C22×C4C2×#6711yescharacteristic
194C2×C8—24no—
195C2×C8—122no—
196C4⋊C4—22no—
197C4⋊C4—24no—
198C4⋊C4—62no—
199C4⋊C4—242no—
200C22⋊C4—68no—
201C22⋊C4—121no—
202C22⋊C4—121no—
203C22⋊C4—242no—
204C42—22no—
205C42—22no—
206C42—32no—
207C42—61no—
208C42—62no—
209C42#23512yes—
210C42C2×#6712yes—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3#38611yescharacteristic
214C22×C3C2×C22⋊C811yescharacteristic
215C22×C3#40811yescharacteristic
216C2×D3—44no—
217C2×D3—42no—
218C2×D3—82no—
219C2×D3—81no—
220C2×D3—81no—
221C3×C4—24no—
222C3×C4—22no—
223C3×C4—21no—
224C3×C4—21no—
225C3×C4—42no—
226C3×C4C8×D412yes—
227C3×C4#43512yes—
228Dic3—42no—
229Dic3—81no—
230Dic3—81no—
231C23—62no—
232C23—122no—
233C23—122no—
234C23—121no—
235C23—121no—
236C23C2×(C2×D3)⋊C411yescharacteristic
237Q8—24no—
238Q8—24no—
239D4—64no—
240D4—122no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—62no—
248C2×C4—121no—
249C2×C4—121no—
250C2×C4—121no—
251C2×C4—122no—
252C2×C4—121no—
253C2×C4—121no—
254C2×C4—121no—
255C2×C4—244no—
256C2×C4(C2×D3)⋊C814yes—
257C2×C4C2×D3×C811yescharacteristic
258C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
259C2×C4#90112yes—
260C2×C4C4×#6712yes—
261C2×C4#92911yescharacteristic
262C2×C4#93012yes—
263C2×C4#95311yescharacteristic
264C8—44no—
265C8—242no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3#186911yescharacteristic
269C2×C3#191011yescharacteristic
270C2×C3#191211yescharacteristic
271D3—82no—
272C22—21no—
273C22—21no—
274C22—124no—
275C22—122no—
276C22—242no—
277C22—241no—
278C22—241no—
279C22C2×(C2×D3)⋊C811yescharacteristic
280C22#511011yescharacteristic
281C22#521311yescharacteristic
282C4—24no—
283C4—22no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—122no—
288C4—241no—
289C4—241no—
290C4C8×#6712yes—
291C4#511112yes—
292C3#910211yescharacteristic, complemented, Sylow
293C2—21no—
294C2—21no—
295C2—242no—
296C2#7166311yescharacteristic
297C2#7197511yescharacteristic
298C2#7202511yescharacteristic
subgroupG/H
centerC22#5213
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#2982C2
socleC22×C3#386
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#2982D3
hypocenterC3#9102
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
1. C3×#2982C2semidirect
3. #71624C2semidirect
5. #67737C2semidirect
22. #2982D3semidirect
37. #947C8semidirect
292. C3#9102semidirect

Maximal subgroups

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

subgroupindex
C3×#29822
#824552
#716242
#707572
#677372
#91023