Finite Group Databasebeta

← #10698124

Subgroups — #10698124

Order 768.

914 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 87 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#84050C212yesmaximal
2#78646C211yesmaximal, characteristic, complemented
3C3×#1783C211yesmaximal, characteristic
4#69980C212yesmaximal, complemented
5#67683C211yesmaximal, characteristic, complemented
6#10333—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C2×#936—22no—
9C2×#924C2212yes—
10C2×C3×#100—22no—
11C3×#407C2211yescharacteristic
12C3×#407C2211yescharacteristic
13#5247—24no—
14#5222—24no—
15#5221—22no—
16C4×#217C2212yes—
17#5137—22no—
18#4739—22no—
19#4721—22no—
20#4467—24no—
21#3195—31no—
22#2435—32no—
23#1783D311yescharacteristic
24#1508—32no—
25#1297—31no—
26C25×C3D412yes—
27C22×C3×D4—22no—
28C22×C3×D4—21no—
29C22×C3×D4—22no—
30C22×C3×D4—21no—
31C22×C3×D4—44no—
32C23×C3×C4C2311yescharacteristic
33C22×Dic6D412yes—
34C2×C3×M16—21no—
35C2×C3×M16D412yes—
36C2×#217—28no—
37C2×#217—22no—
38C2×#217—22no—
39C2×#217—28no—
40#936—48no—
41C2×C4⋊Dic3D412yes—
42#924—22no—
43#924—22no—
44#924—22no—
45#924—22no—
46C2×Dic3⋊C4—22no—
47C2×C4×Dic3—22no—
48C2×Dic12—22no—
49C3×#100—44no—
50C3×C22⋊C8—22no—
51C3×C22⋊C8—22no—
52C3⋊#95D414yes—
53#817—22no—
54#816—24no—
55C23×D4C2×D311yescharacteristic
56C2×#124—62no—
57C2×C22⋊Q8—32no—
58#483—62no—
59#470—62no—
60#451—62no—
61C2×#100—22no—
62#407C2×D311yescharacteristic
63#407C2×D311yescharacteristic
64#394—64no—
65#390—62no—
66#379—64no—
67C4×C22⋊C4—32no—
68#328—64no—
69C24×C3—24no—
70C24×C3—22no—
71C24×C3—44no—
72C24×C3—48no—
73C24×C3C2×D411yescharacteristic
74C2×C3×D4—24no—
75C2×C3×D4—28no—
76C2×C3×D4—44no—
77C2×C3×D4—42no—
78C2×C3×D4—48no—
79C2×C3×D4—44no—
80C2×C3×D4—41no—
81C2×C3×D4—42no—
82C2×C3×D4—41no—
83C22×C3×C4—42no—
84C22×C3×C4C2×D412yes—
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4#4111yescharacteristic
88C22×Dic3—24no—
89C22×Dic3—22no—
90C22×Dic3—22no—
91C2×Dic6—22no—
92C2×Dic6—22no—
93C2×Dic6—44no—
94C3×M16—42no—
95C3×M16—42no—
96C2×C3×C8—22no—
97C2×C3×C8—41no—
98#217—416no—
99#217—44no—
100#217—42no—
101#217—42no—
102#217—88no—
103C4⋊Dic3—22no—
104C4⋊Dic3—22no—
105Dic3⋊C4—42no—
106Dic3⋊C4—42no—
107C4×Dic3—44no—
108Dic12—82no—
109C25D1212yes—
110C22×Q8—32no—
111C22×D4—21no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—44no—
116C23×C4C22×D311yescharacteristic
117C2×Q16—62no—
118C2×M16—21no—
119C2×M16#6712yes—
120#124—128no—
121C22⋊Q8—62no—
122C22⋊Q8—62no—
123C22⋊Q8—62no—
124C22⋊Q8—62no—
125C2×C4⋊C4—32no—
126C2×C4⋊C4—62no—
127C2×C22⋊C4—68no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4—62no—
130C2×C22⋊C4—68no—
131C2×C42—62no—
132#106—62no—
133#103—64no—
134#100—44no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137#95—34no—
138C23×C3—24no—
139C23×C3—28no—
140C23×C3—24no—
141C23×C3—41no—
142C23×C3—44no—
143C23×C3—42no—
144C23×C3—48no—
145C23×C3—42no—
146C23×C3—48no—
147C23×C3—84no—
148C23×C3—816no—
149C23×C3#12012yes—
150C23×C3#12111yescharacteristic
151C3×D4—44no—
152C3×D4—48no—
153C3×D4—44no—
154C2×C3×C4—24no—
155C2×C3×C4—22no—
156C2×C3×C4—42no—
157C2×C3×C4—41no—
158C2×C3×C4#12011yescharacteristic
159C2×C3×C4#12112yes—
160C2×C3×C4#12711yescharacteristic
161C2×Dic3—44no—
162C2×Dic3—44no—
163C2×Dic3—44no—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—44no—
167C2×Dic3—44no—
168Dic6—44no—
169Dic6—82no—
170C3×C8—42no—
171C3×C8—81no—
172C24—24no—
173C24—22no—
174C24—44no—
175C24—48no—
176C24C2×D1211yescharacteristic
177C2×Q8—62no—
178C2×Q8—62no—
179C2×Q8—124no—
180C2×D4—24no—
181C2×D4—28no—
182C2×D4—44no—
183C2×D4—42no—
184C2×D4—44no—
185C2×D4—48no—
186C2×D4—42no—
187C2×D4—41no—
188C2×D4—41no—
189C22×C4—42no—
190C22×C4—64no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4#23511yescharacteristic
194C22×C4D3×D412yes—
195C22×C4D3×D412yes—
196C22×C4C2×#6711yescharacteristic
197Q16—242no—
198M16—42no—
199M16—42no—
200C2×C8—22no—
201C2×C8—41no—
202C4⋊C4—62no—
203C4⋊C4—62no—
204C4⋊C4—122no—
205C4⋊C4—122no—
206C22⋊C4—1216no—
207C22⋊C4—124no—
208C22⋊C4—122no—
209C22⋊C4—122no—
210C22⋊C4—248no—
211C42—124no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—21no—
216C22×C3—44no—
217C22×C3—416no—
218C22×C3—44no—
219C22×C3—88no—
220C22×C3—84no—
221C22×C3—84no—
222C22×C3—84no—
223C22×C3#39211yescharacteristic
224C22×C3#45514yes—
225C3×C4—21no—
226C3×C4—22no—
227C3×C4B(4,𝔽2)12yes—
228Dic3—84no—
229Dic3—82no—
230Dic3—82no—
231C23—24no—
232C23—28no—
233C23—24no—
234C23—41no—
235C23—44no—
236C23—42no—
237C23—48no—
238C23—42no—
239C23—48no—
240C23—84no—
241C23—816no—
242C23#88212yes—
243C23#93011yescharacteristic
244Q8—124no—
245Q8—242no—
246D4—44no—
247D4—48no—
248D4—44no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—124no—
254C2×C4—124no—
255C2×C4—124no—
256C2×C4—122no—
257C2×C4—122no—
258C2×C4—124no—
259C2×C4—124no—
260C2×C4#88412yes—
261C2×C4#93711yescharacteristic
262C2×C4#94011yescharacteristic
263C8—42no—
264C8—81no—
265C2×C3—24no—
266C2×C3—41no—
267C2×C3—84no—
268C2×C3#200412yes—
269C2×C3#202111yescharacteristic
270C22—22no—
271C22—22no—
272C22—21no—
273C22—22no—
274C22—44no—
275C22—416no—
276C22—44no—
277C22—88no—
278C22—84no—
279C22—84no—
280C22—84no—
281C22#474614yes—
282C22#496011yescharacteristic
283C4—21no—
284C4—22no—
285C4—244no—
286C4—242no—
287C4—242no—
288C4#503212yes—
289C3#1033311yescharacteristic, complemented, Sylow
290C2—24no—
291C2—41no—
292C2—84no—
293C2#6950011yescharacteristic
294C2#7008412yes—
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1783C2
socleC22×C3#392
prime radicalC23×C3×D4C22
hypercenter#1783D3
hypocenterC3#10333
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1783

Complements

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

NH ≅ G/Ntype
2. #78646C2semidirect
4. #69980C2semidirect
5. #67683C2semidirect
289. C3#10333semidirect

Maximal subgroups

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

subgroupindex
#840502
#786462
C3×#17832
#699802
#676832
#103333