Finite Group Databasebeta

← #10698084

Subgroups — #10698084

Order 768.

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

normal subgroups only

This group has 806 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#84066C212yesmaximal, complemented
2#78641C211yesmaximal, characteristic
3C3×#1784C211yesmaximal, characteristic, complemented
4#69977C212yesmaximal, complemented
5#67683C211yesmaximal, characteristic, complemented
6#10327—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C2×#947—22no—
9C2×#930C2212yes—
10C2×C3×#101—22no—
11C3×#407C2211yescharacteristic
12C3×#407C2211yescharacteristic
13#5231—24no—
14#5229—22no—
15#5213—24no—
16C4×#217C2212yes—
17#5135—22no—
18#4735—22no—
19#4732—22no—
20#4470—24no—
21#3192—31no—
22#2436—32no—
23#1784D311yescharacteristic, complemented
24#1507—32no—
25#1297—31no—
26C22×C3×Q8—22no—
27C22×C3×Q8—21no—
28C22×C3×Q8—22no—
29C22×C3×Q8—21no—
30C22×C3×Q8—44no—
31C23×C3×C4D412yes—
32C23×C3×C4C2311yescharacteristic
33C22×#67—22no—
34C22×D12D412yes—
35C2×C3×M16—21no—
36C2×C3×M16D412yes—
37C2×#217—22no—
38#947—48no—
39#930—22no—
40#930—22no—
41#930—22no—
42#930—22no—
43C2×(C2×D3)⋊C4—28no—
44C2×(C2×D3)⋊C4—22no—
45C2×C4⋊Dic3D412yes—
46C2×C4×Dic3—22no—
47C2×D24—22no—
48C3×#101—44no—
49C3×C22⋊C8—22no—
50C3×C22⋊C8—22no—
51C3⋊#95—28no—
52C3⋊#95D414yes—
53#821—24no—
54#817—22no—
55C23×Q8C2×D311yescharacteristic
56C2×#124—62no—
57C2×#121—32no—
58#481—62no—
59#469—62no—
60#447—62no—
61C2×#101—22no—
62#407C2×D311yescharacteristic
63#407C2×D311yescharacteristic
64#393—64no—
65#391—62no—
66#386—64no—
67C4×C22⋊C4—32no—
68#327—64no—
69C24×C3C2×D411yescharacteristic
70C23×D3—22no—
71C2×C3×Q8—24no—
72C2×C3×Q8—28no—
73C2×C3×Q8—44no—
74C2×C3×Q8—42no—
75C2×C3×Q8—48no—
76C2×C3×Q8—44no—
77C2×C3×Q8—41no—
78C2×C3×Q8—42no—
79C2×C3×Q8—41no—
80C22×C3×C4—24no—
81C22×C3×C4—22no—
82C22×C3×C4—44no—
83C22×C3×C4—42no—
84C22×C3×C4C2×D412yes—
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4#4111yescharacteristic
88C2×#67—48no—
89C2×#67—42no—
90C2×#67—42no—
91C22×Dic3—22no—
92C22×Dic3—24no—
93C2×D12—22no—
94C2×D12—22no—
95C2×D12—44no—
96C3×M16—42no—
97C3×M16—42no—
98C2×C3×C8—22no—
99C2×C3×C8—41no—
100#217—44no—
101(C2×D3)⋊C4—416no—
102(C2×D3)⋊C4—42no—
103(C2×D3)⋊C4—42no—
104C4⋊Dic3—22no—
105C4⋊Dic3—22no—
106C4×Dic3—44no—
107D24—82no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—44no—
113C22×D4—32no—
114C22×D4—62no—
115C23×C4D1212yes—
116C23×C4C22×D311yescharacteristic
117C2×D8—62no—
118C2×M16—21no—
119C2×M16#6712yes—
120#124—128no—
121#121—62no—
122#121—62no—
123#121—62no—
124#121—62no—
125C2×C4⋊C4—32no—
126C2×C22⋊C4—68no—
127C2×C22⋊C4—62no—
128C2×C22⋊C4—62no—
129C2×C42—62no—
130#106—62no—
131#102—64no—
132#101—44no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135#95—34no—
136#95—68no—
137C23×C3—24no—
138C23×C3—41no—
139C23×C3#12012yes—
140C23×C3#12111yescharacteristic
141C22×D3—44no—
142C22×D3—44no—
143C22×D3—42no—
144C22×D3—44no—
145C3×Q8—44no—
146C3×Q8—48no—
147C3×Q8—44no—
148C2×C3×C4—24no—
149C2×C3×C4—22no—
150C2×C3×C4—28no—
151C2×C3×C4—24no—
152C2×C3×C4—42no—
153C2×C3×C4—41no—
154C2×C3×C4—44no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4#12011yescharacteristic
158C2×C3×C4#12112yes—
159C2×C3×C4#12711yescharacteristic
160#67—84no—
161#67—84no—
162C2×Dic3—44no—
163C2×Dic3—44no—
164C2×Dic3—44no—
165C2×Dic3—42no—
166C2×Dic3—44no—
167D12—44no—
168D12—82no—
169C3×C8—42no—
170C3×C8—81no—
171C24—62no—
172C24C2×D1211yescharacteristic
173C2×Q8—24no—
174C2×Q8—28no—
175C2×Q8—44no—
176C2×Q8—42no—
177C2×Q8—48no—
178C2×Q8—44no—
179C2×Q8—42no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×D4—62no—
183C2×D4—62no—
184C2×D4—124no—
185C2×D4—128no—
186C2×D4—122no—
187C2×D4—122no—
188C22×C4—24no—
189C22×C4—22no—
190C22×C4—44no—
191C22×C4—42no—
192C22×C4—64no—
193C22×C4—62no—
194C22×C4#23511yescharacteristic
195C22×C4D3×D412yes—
196C22×C4D3×D412yes—
197C22×C4C2×#6711yescharacteristic
198D8—242no—
199M16—42no—
200M16—42no—
201C2×C8—22no—
202C2×C8—41no—
203C4⋊C4—62no—
204C4⋊C4—62no—
205C22⋊C4—1216no—
206C22⋊C4—124no—
207C22⋊C4—122no—
208C22⋊C4—122no—
209C42—124no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—21no—
214C22×C3—44no—
215C22×C3#39211yescharacteristic
216C22×C3#45414yes—
217C2×D3—88no—
218C2×D3—82no—
219C2×D3—82no—
220C2×D3—82no—
221C3×C4—21no—
222C3×C4—22no—
223C3×C4—44no—
224C3×C4B(4,𝔽2)12yes—
225Dic3—84no—
226Dic3—82no—
227C23—24no—
228C23—41no—
229C23—124no—
230C23—124no—
231C23—122no—
232C23—124no—
233C23#88212yes—
234C23#93011yescharacteristic
235Q8—44no—
236Q8—48no—
237Q8—44no—
238D4—124no—
239D4—242no—
240D4—244no—
241D4—244no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—28no—
245C2×C4—24no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—124no—
252C2×C4—124no—
253C2×C4—124no—
254C2×C4—122no—
255C2×C4—124no—
256C2×C4#88412yes—
257C2×C4#93711yescharacteristic
258C2×C4#94011yescharacteristic
259C8—42no—
260C8—81no—
261C2×C3—24no—
262C2×C3—41no—
263C2×C3#200312yes—
264C2×C3#202111yescharacteristic
265D3—162no—
266C22—22no—
267C22—22no—
268C22—21no—
269C22—22no—
270C22—44no—
271C22—248no—
272C22—242no—
273C22—242no—
274C22—242no—
275C22#475314yes—
276C22#496011yescharacteristic
277C4—21no—
278C4—22no—
279C4—44no—
280C4—244no—
281C4—242no—
282C4#503212yes—
283C3#1032711yescharacteristic, complemented, Sylow
284C2—24no—
285C2—41no—
286C2—482no—
287C2#6950011yescharacteristic
288C2#7011612yes—
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1784C2
socleC22×C3#392
prime radical#78641C2
hypercenter#1784D3
hypocenterC3#10327
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
1. #84066C2semidirect
3. C3×#1784C2semidirect
4. #69977C2semidirect
5. #67683C2semidirect
23. #1784D3semidirect
283. C3#10327semidirect

Maximal subgroups

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

subgroupindex
#840662
#786412
C3×#17842
#699772
#676832
#103273