Finite Group Databasebeta

← #10698121

Subgroups — #10698121

Order 768.

914 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 89 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×#3195C211yesmaximal, characteristic, complemented
2#82761C212yesmaximal
3#71688C212yesmaximal, complemented
4#69988C211yesmaximal, characteristic
5#67683C211yesmaximal, characteristic, complemented
6#10333—31nomaximal, Sylow
7C23×D12C2211yescharacteristic
8C2×C3×C22⋊Q8C2212yes—
9C2×#947—22no—
10C3×#483—22no—
11C3×#470—22no—
12C3×#451—22no—
13C3×#407C2211yescharacteristic
14C4×#217C2212yes—
15#5207C2211yescharacteristic
16C2×#823—22no—
17#4991—24no—
18#4988—22no—
19#4955—24no—
20#4538—24no—
21#3195D311yescharacteristic, complemented
22#2435—32no—
23#1783—31no—
24#1508—32no—
25#1297—31no—
26C24×D3D412yes—
27C22×C3×Q8D412yes—
28C23×C3×C4C2311yescharacteristic
29C22×D12—21no—
30C22×D12—22no—
31C22×D12—22no—
32C22×D12—21no—
33C22×D12—44no—
34C2×C3×Q16—22no—
35C2×C3×M16D412yes—
36C3×C22⋊Q8—22no—
37C3×C22⋊Q8—22no—
38C3×C22⋊Q8—22no—
39C3×C22⋊Q8—22no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4D412yes—
42C2×C3×C22⋊C4—22no—
43C2×#217—22no—
44#947—48no—
45C2×(C2×D3)⋊C4—28no—
46C2×(C2×D3)⋊C4—28no—
47C2×C4×Dic3—22no—
48C2×#208—21no—
49C3×#106—22no—
50C3×#103—24no—
51C3×C22⋊C8—22no—
52C3⋊#95D414yes—
53#830—22no—
54#823—44no—
55C23×D4—31no—
56C2×#124—62no—
57C2×C22⋊Q8C2×D312yes—
58#483—22no—
59#470—22no—
60#451—22no—
61C2×#100—62no—
62#407—31no—
63#407C2×D311yescharacteristic
64#394—64no—
65#390—62no—
66#379—64no—
67C4×C22⋊C4—32no—
68#328—64no—
69C24×C3C2×D411yescharacteristic
70C23×D3—24no—
71C23×D3—22no—
72C23×D3—44no—
73C23×D3—48no—
74C2×C3×Q8—22no—
75C2×C3×Q8—22no—
76C2×C3×Q8—44no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—42no—
80C22×C3×C4C2×D412yes—
81C22×C3×C4C2×D412yes—
82C22×C3×C4C2×D411yescharacteristic
83C22×C3×C4#4111yescharacteristic
84C22×Dic3—24no—
85C2×D12—24no—
86C2×D12—28no—
87C2×D12—42no—
88C2×D12—44no—
89C2×D12—48no—
90C2×D12—44no—
91C2×D12—41no—
92C2×D12—42no—
93C2×D12—41no—
94C3×Q16—82no—
95C3×M16—42no—
96C2×C3×C8—22no—
97C3×C4⋊C4—22no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—42no—
100C3×C4⋊C4—42no—
101C3×C22⋊C4—42no—
102C3×C22⋊C4—42no—
103#217—44no—
104(C2×D3)⋊C4—416no—
105(C2×D3)⋊C4—88no—
106C4×Dic3—44no—
107#208—42no—
108C2×C3⋊C8—41no—
109C25—32no—
110C22×Q8#6712yes—
111C22×D4—62no—
112C22×D4—61no—
113C22×D4—62no—
114C22×D4—61no—
115C22×D4—124no—
116C23×C4C22×D311yescharacteristic
117C2×Q16—22no—
118C2×M16—61no—
119C2×M16#6712yes—
120#124—128no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4#6712yes—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—68no—
129C2×C22⋊C4—62no—
130C2×C22⋊C4—68no—
131C2×C42—62no—
132#106—22no—
133#103—24no—
134#100—124no—
135C22⋊C8—22no—
136C22⋊C8—62no—
137#95—34no—
138C23×C3—24no—
139C23×C3—41no—
140C23×C3#12012yes—
141C23×C3#12111yescharacteristic
142C22×D3—28no—
143C22×D3—24no—
144C22×D3—44no—
145C22×D3—42no—
146C22×D3—48no—
147C22×D3—42no—
148C22×D3—48no—
149C22×D3—816no—
150C22×D3—84no—
151C3×Q8—44no—
152C3×Q8—82no—
153C2×C3×C4—22no—
154C2×C3×C4—24no—
155C2×C3×C4—42no—
156C2×C3×C4—41no—
157C2×C3×C4—44no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—44no—
161C2×C3×C4#12011yescharacteristic
162C2×C3×C4#12112yes—
163C2×C3×C4#12711yescharacteristic
164C2×Dic3—44no—
165C2×Dic3—44no—
166C2×Dic3—44no—
167D12—44no—
168D12—48no—
169D12—44no—
170C3×C8—42no—
171C3⋊C8—81no—
172C24—64no—
173C24—62no—
174C24—128no—
175C24—124no—
176C24D3×D411yescharacteristic
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8—44no—
180C2×D4—64no—
181C2×D4—68no—
182C2×D4—124no—
183C2×D4—122no—
184C2×D4—128no—
185C2×D4—124no—
186C2×D4—122no—
187C2×D4—121no—
188C2×D4—121no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—42no—
192C22×C4—64no—
193C22×C4D3×D412yes—
194C22×C4#23711yescharacteristic
195C22×C4C2×#6712yes—
196C22×C4C2×#6711yescharacteristic
197Q16—82no—
198M16—42no—
199M16—122no—
200C2×C8—22no—
201C2×C8—121no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—1216no—
209C22⋊C4—124no—
210C22⋊C4—248no—
211C42—124no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—21no—
216C22×C3—44no—
217C22×C3#39211yescharacteristic
218C22×C3#45514yes—
219C2×D3—416no—
220C2×D3—44no—
221C2×D3—88no—
222C2×D3—84no—
223C2×D3—84no—
224C2×D3—84no—
225C3×C4—21no—
226C3×C4—22no—
227C3×C4—82no—
228C3×C4—82no—
229C3×C4B(4,𝔽2)12yes—
230Dic3—84no—
231C23—24no—
232C23—41no—
233C23—68no—
234C23—64no—
235C23—124no—
236C23—122no—
237C23—128no—
238C23—128no—
239C23—122no—
240C23—244no—
241C23—2416no—
242C23#93712yes—
243C23#93811yescharacteristic
244Q8—44no—
245Q8—82no—
246D4—124no—
247D4—128no—
248D4—124no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—124no—
258C2×C4—124no—
259C2×C4—124no—
260C2×C4#93912yes—
261C2×C4#94011yescharacteristic
262C2×C4#95311yescharacteristic
263C8—42no—
264C8—241no—
265C2×C3—24no—
266C2×C3—41no—
267C2×C3#200412yes—
268C2×C3#202111yescharacteristic
269D3—84no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—21no—
274C22—44no—
275C22—1216no—
276C22—124no—
277C22—248no—
278C22—244no—
279C22—244no—
280C22—244no—
281C22#520214yes—
282C22#522311yescharacteristic
283C4—21no—
284C4—22no—
285C4—82no—
286C4—82no—
287C4—244no—
288C4#524412yes—
289C3#1033311yescharacteristic, complemented, Sylow
290C2—24no—
291C2—41no—
292C2—244no—
293C2#7185111yescharacteristic
294C2#7210612yes—
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3195C2
socleC22×C3#392
prime radicalC23×D12C22
hypercenter#3195D3
hypocenterC3#10333
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
1. C3×#3195C2semidirect
3. #71688C2semidirect
5. #67683C2semidirect
21. #3195D3semidirect
289. C3#10333semidirect

Maximal subgroups

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

subgroupindex
C3×#31952
#827612
#716882
#699882
#676832
#103333