Finite Group Databasebeta

← #10804178

Subgroups — #10804178

Order 768.

1254 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 235 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
1C2×#5816C211yesmaximal, characteristic
2#78630C212yesmaximal
3C3×#1784C211yesmaximal, characteristic, complemented
4#72133C211yesmaximal, characteristic, complemented
5#70115C218yesmaximal, complemented
6#69974C212yesmaximal, complemented
7#21756—31nomaximal, Sylow
8C23×C3×Q8C2211yescharacteristic
9C22×#235C2211yescharacteristic
10#5816—28no—
11C2×#947C412yes—
12C2×#946C2214yes—
13C2×#944C412yes—
14C2×#929C414yes—
15C2×#926C2211yescharacteristic
16C2×D3×M16C2214yes—
17C2×C3×#101C2214yes—
18C3×#407C2212yes—
19C2×#836C2214yes—
20#5207C2212yes—
21#5126C2214yes—
22#4727—28no—
23#4554C2218yes—
24C2×#536—31no—
25#2879—32no—
26#1901—38no—
27#1822—32no—
28#1784—31no—
29#1784D311yescharacteristic, complemented
30C22×C3×Q8—24no—
31C22×C3×Q8—44no—
32C22×C3×Q8C2×C412yes—
33C22×C3×Q8C2312yes—
34C23×C3×C4C2×C412yes—
35C23×C3×C4C2311yescharacteristic
36C22×#67C2×C412yes—
37C2×#235—44no—
38C2×#235D418yes—
39C22×D12C2×C411yescharacteristic
40C22×C4×D3C2312yes—
41C22×Dic6C2×C411yescharacteristic
42C2×C3×M16C2314yes—
43C2×#217C2×C412yes—
44#947—28no—
45#946—28no—
46#946—28no—
47#944—28no—
48#929—216no—
49C2×(C2×D3)⋊C4C2×C418yes—
50#926—24no—
51C2×C4⋊Dic3C2312yes—
52C2×Dic3⋊C4C2×C418yes—
53C2×C4×Dic3C2×C412yes—
54C2×#208C2314yes—
55D3×M16—48no—
56C2×ΓL(1,𝔽25)—24no—
57C2×D3×C8—24no—
58C3×#101—44no—
59C3×C22⋊C8—24no—
60#836—44no—
61#830—24no—
62(C2×D3)⋊C8—28no—
63C22×#41—31no—
64C23×Q8C2×D311yescharacteristic
65C22×M16—34no—
66#536—68no—
67C2×C4⋊Q8—32no—
68C2×#124—32no—
69C2×#123—34no—
70C2×C22⋊Q8—34no—
71C2×#117—31no—
72C2×#101—34no—
73C2×#101C2×D314yes—
74#408—68no—
75#407—32no—
76#407—34no—
77#407C2×D312yes—
78#343—38no—
79C24×C3C22×C411yescharacteristic
80C23×D3C22×C412yes—
81C2×C3×Q8—24no—
82C2×C3×Q8—24no—
83C2×C3×Q8—24no—
84C2×C3×Q8—24no—
85C2×C3×Q8—44no—
86C2×C3×Q8—416no—
87C22×C3×C4—24no—
88C22×C3×C4—24no—
89C22×C3×C4—44no—
90C22×C3×C4C22×C414yes—
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D414yes—
93C22×C3×C4C2411yescharacteristic
94C2×#67—44no—
95C2×#67C22⋊C418yes—
96C22×Dic3C22×C412yes—
97C22×Dic3C22×C414yes—
98#235—48no—
99#235—48no—
100C2×D12—42no—
101C2×D12C22⋊C414yes—
102C2×C4×D3—28no—
103C2×C4×D3C2×D414yes—
104C2×C4×D3C2×D414yes—
105C2×Dic6—42no—
106C2×Dic6C22⋊C414yes—
107C3×M16—28no—
108C2×C3×C8—24no—
109#217—24no—
110(C2×D3)⋊C4—216no—
111C4⋊Dic3—24no—
112Dic3⋊C4—216no—
113C4×Dic3—24no—
114#208—28no—
115C2×C3⋊C8—24no—
116ΓL(1,𝔽25)—48no—
117D3×C8—48no—
118C2×#41—38no—
119C2×#41—124no—
120C22×Q8—24no—
121C22×Q8—31no—
122C22×Q8—44no—
123C22×Q8C4×D312yes—
124C22×Q8C22×D312yes—
125C22×D4—32no—
126C22×D4—31no—
127C23×C4—32no—
128C23×C4C4×D312yes—
129C23×C4C22×D311yescharacteristic
130C2×M16—34no—
131C2×M16—64no—
132C2×M16—128no—
133C2×M16C22×D314yes—
134C22×C8—64no—
135C4⋊Q8—68no—
136#124—68no—
137#123—616no—
138C22⋊Q8—68no—
139C22⋊Q8—68no—
140#117—64no—
141C2×C4⋊C4—38no—
142C2×C4⋊C4—32no—
143C2×C22⋊C4—32no—
144C2×C22⋊C4—38no—
145C2×C42—32no—
146#101—44no—
147#101—124no—
148C22⋊C8—24no—
149C22⋊C8—68no—
150C22⋊C8—64no—
151C23×C3—22no—
152C23×C3—42no—
153C23×C3C2×C22⋊C412yes—
154C23×C3C23×C411yescharacteristic
155C22×D3—24no—
156C22×D3—44no—
157C22×D3C2×C22⋊C414yes—
158C3×Q8—48no—
159C3×Q8—48no—
160C2×C3×C4—22no—
161C2×C3×C4—28no—
162C2×C3×C4—24no—
163C2×C3×C4—24no—
164C2×C3×C4—24no—
165C2×C3×C4—48no—
166C2×C3×C4C2×C22⋊C412yes—
167C2×C3×C4C22×D412yes—
168C2×C3×C4H(5,𝔽2)14yes—
169#67—48no—
170C2×Dic3—24no—
171C2×Dic3—24no—
172C2×Dic3—44no—
173C2×Dic3C2×C22⋊C414yes—
174D12—44no—
175C4×D3—24no—
176C4×D3—24no—
177C4×D3—28no—
178Dic6—44no—
179C3×C8—44no—
180C3⋊C8—44no—
181C24—32no—
182C24C2×C4×D311yescharacteristic
183#41—128no—
184#41—128no—
185C2×Q8—24no—
186C2×Q8—24no—
187C2×Q8—24no—
188C2×Q8—24no—
189C2×Q8—34no—
190C2×Q8—44no—
191C2×Q8—416no—
192C2×Q8—122no—
193C2×D4—38no—
194C2×D4—34no—
195C2×D4—122no—
196C2×D4—124no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—32no—
200C22×C4—34no—
201C22×C4—34no—
202C22×C4—34no—
203C22×C4—44no—
204C22×C4—68no—
205C22×C4C2×C4×D314yes—
206C22×C4C2×C4×D311yescharacteristic
207C22×C4D3×D414yes—
208C22×C4C23×D311yescharacteristic
209M16—28no—
210M16—68no—
211M16—128no—
212C2×C8—24no—
213C2×C8—64no—
214C2×C8—128no—
215C4⋊C4—616no—
216C4⋊C4—64no—
217C22⋊C4—616no—
218C22⋊C4—64no—
219C42—64no—
220C22×C3—22no—
221C22×C3—22no—
222C22×C3—24no—
223C22×C3—44no—
224C22×C3C22×C22⋊C411yescharacteristic
225C22×C3#51812yes—
226C2×D3—24no—
227C2×D3—28no—
228C2×D3—84no—
229C3×C4—22no—
230C3×C4—44no—
231C3×C4#53414yes—
232Dic3—24no—
233Dic3—44no—
234C23—22no—
235C23—34no—
236C23—42no—
237C23—64no—
238C23—124no—
239C23D3×C22⋊C412yes—
240C23C22×C4×D311yescharacteristic
241Q8—48no—
242Q8—48no—
243Q8—124no—
244D4—124no—
245D4—128no—
246C2×C4—22no—
247C2×C4—28no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—34no—
252C2×C4—48no—
253C2×C4—64no—
254C2×C4—64no—
255C2×C4—64no—
256C2×C4—68no—
257C2×C4—64no—
258C2×C4—124no—
259C2×C4D3×C22⋊C412yes—
260C2×C4C2×D3×D412yes—
261C2×C4#100414yes—
262C8—44no—
263C8—124no—
264C2×C3—22no—
265C2×C3—42no—
266C2×C3#230811yescharacteristic
267C2×C3#288912yes—
268D3—44no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—44no—
273C22—68no—
274C22—64no—
275C22—244no—
276C22C2×D3×C22⋊C411yescharacteristic
277C22#548612yes—
278C4—22no—
279C4—44no—
280C4—64no—
281C4—124no—
282C4#558914yes—
283C3#2175611yescharacteristic, complemented, Sylow
284C2—22no—
285C2—42no—
286C2—124no—
287C2#7384911yescharacteristic
288C2#7870612yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1784C2
socleC22×C3C22×C22⋊C4
prime radicalC22×#235C22
cosocleC22×C3×C4C24
hypercenter#1784D3
hypocenterC3#21756
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
3. C3×#1784C2semidirect
4. #72133C2semidirect
5. #70115C2semidirect
6. #69974C2semidirect
29. #1784D3semidirect
283. C3#21756semidirect

Maximal subgroups

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

subgroupindex
C2×#58162
#786302
C3×#17842
#721332
#701152
#699742
#217563