Finite Group Databasebeta

← #10696466

Subgroups — #10696466

Order 768.

780 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 97 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×#2819C211yesmaximal, characteristic, complemented
2C2×#5058C212yesmaximal, complemented
3C2×#4540C212yesmaximal, complemented
4#69553C211yesmaximal, characteristic, complemented
5#67630C211yesmaximal, characteristic, complemented
6#10078—31nomaximal, Sylow
7C2×C3×#124C2212yes—
8C22×#215—22no—
9C22×#213—22no—
10C2×#874C2211yescharacteristic
11C3×#394—22no—
12C3×#393—22no—
13C3×#392—22no—
14C3×#382C2211yescharacteristic
15#5058—24no—
16#5058—24no—
17C2×#809—22no—
18C2×#803C2211yescharacteristic
19C2×#803C2212yes—
20#4540—24no—
21C2×#496—32no—
22#2819D311yescharacteristic, complemented
23#1961—31no—
24C2×#329—32no—
25#1295—31no—
26C22×C3×Q8D412yes—
27C22×C3×D4—22no—
28C22×C3×D4D412yes—
29C22×D12—21no—
30C22×D12—22no—
31C3×#124—22no—
32C3×#124—22no—
33C3×#124—22no—
34C3×#124—22no—
35C2×C3×C4⋊C4D412yes—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—24no—
39C2×C3×C42C2311yescharacteristic
40C2×#215—28no—
41C2×#215—48no—
42C2×#213—28no—
43C2×#213—48no—
44C22×C3⋊C8—22no—
45C22×C3⋊C8—21no—
46#874—22no—
47#874—21no—
48#874D412yescomplemented
49C3×#95—22no—
50C3×#95—22no—
51#809—44no—
52#803—22no—
53#803—22no—
54#803D414yescomplemented
55C22×SD16—62no—
56C22×D8—62no—
57C2×#127—31no—
58C2×#124C2×D312yes—
59#496—64no—
60#496—64no—
61C2×#102—62no—
62C2×#97—32no—
63C2×#97—31no—
64#394—22no—
65#393—22no—
66#392—22no—
67#382C2×D311yescharacteristic
68#329—64no—
69C24×C3—22no—
70C23×D3—41no—
71C2×C3×Q8—22no—
72C2×C3×Q8—22no—
73C2×C3×Q8—44no—
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—44no—
77C2×C3×D4—84no—
78C2×C3×D4—82no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4C2×D412yes—
82C22×C3×C4C2×D411yescharacteristic
83C2×D12—24no—
84C2×D12—22no—
85C2×D12—42no—
86C2×D12—44no—
87C2×D12—44no—
88C2×D12—42no—
89C2×D12—42no—
90C2×D12—41no—
91C3×C4⋊C4—42no—
92C3×C22⋊C4—42no—
93C3×C22⋊C4—42no—
94C3×C22⋊C4—42no—
95C3×C22⋊C4—42no—
96C3×C22⋊C4—84no—
97C3×C42C2×D411yescharacteristic
98C3×C42C2×D412yes—
99C3×C42#4111yescharacteristic
100#215—416no—
101#213—416no—
102C2×C3⋊C8—28no—
103C2×C3⋊C8—24no—
104C2×C3⋊C8—22no—
105C2×C3⋊C8—41no—
106C2×C3⋊C8—41no—
107C22×Q8#6712yes—
108C22×D4—22no—
109C22×D4—62no—
110C22×D4—61no—
111C22×D4#6712yes—
112C2×SD16—68no—
113C2×SD16—128no—
114C2×D8—68no—
115C2×D8—128no—
116C22×C8—62no—
117C22×C8—61no—
118#127—32no—
119#127—62no—
120#127—61no—
121#124—22no—
122#124—22no—
123#124—22no—
124#124—22no—
125C2×C4⋊C4#6712yes—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—24no—
129C2×C42C22×D311yescharacteristic
130#102—124no—
131#97—34no—
132#97—62no—
133#97—62no—
134#95—22no—
135#95—22no—
136C23×C3—42no—
137C23×C3—44no—
138C23×C3—44no—
139C23×C3—42no—
140C23×C3—42no—
141C23×C3#12011yescharacteristic
142C22×D3—42no—
143C22×D3—84no—
144C22×D3—81no—
145C22×D3—81no—
146C3×Q8—44no—
147C3×Q8—82no—
148C3×D4—44no—
149C3×D4—82no—
150C3×D4—88no—
151C2×C3×C4—21no—
152C2×C3×C4—24no—
153C2×C3×C4—21no—
154C2×C3×C4—42no—
155C2×C3×C4—44no—
156C2×C3×C4—44no—
157C2×C3×C4—42no—
158C2×C3×C4#12012yes—
159C2×C3×C4#12111yescharacteristic
160C2×C3×C4#12112yes—
161C2×C3×C4#12711yescharacteristic
162D12—44no—
163D12—44no—
164D12—44no—
165D12—84no—
166D12—82no—
167C3⋊C8—28no—
168C3⋊C8—42no—
169C24—22no—
170C24—121no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—44no—
174C2×D4—22no—
175C2×D4—22no—
176C2×D4—44no—
177C2×D4—64no—
178C2×D4—62no—
179C2×D4—84no—
180C2×D4—82no—
181C2×D4—122no—
182C2×D4—124no—
183C2×D4—124no—
184C2×D4—122no—
185C2×D4—122no—
186C2×D4—121no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4C2×#6712yes—
190C22×C4C2×#6711yescharacteristic
191SD16—1216no—
192D8—1216no—
193C2×C8—68no—
194C2×C8—64no—
195C2×C8—62no—
196C2×C8—121no—
197C2×C8—121no—
198C4⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—84no—
204C42D3×D411yescharacteristic
205C42D3×D412yes—
206C42#23711yescharacteristic
207C22×C3—84no—
208C22×C3—84no—
209C22×C3—82no—
210C22×C3—82no—
211C22×C3—82no—
212C22×C3#39211yescharacteristic
213C22×C3#45311yescharacteristic
214C22×C3#45314yes—
215C22×C3#45411yescharacteristic
216C2×D3—84no—
217C2×D3—82no—
218C2×D3—162no—
219C2×D3—162no—
220C3×C4—22no—
221C3×C4—22no—
222C3×C4—21no—
223C3×C4—21no—
224C3×C4—82no—
225C3×C4—82no—
226C23—42no—
227C23—44no—
228C23—44no—
229C23—42no—
230C23—42no—
231C23—122no—
232C23—244no—
233C23—241no—
234C23—241no—
235C23#95311yescharacteristic
236Q8—44no—
237Q8—82no—
238D4—44no—
239D4—82no—
240D4—88no—
241D4—124no—
242D4—124no—
243D4—124no—
244D4—244no—
245D4—242no—
246C2×C4—21no—
247C2×C4—24no—
248C2×C4—21no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4#93712yes—
254C2×C4#93811yescharacteristic
255C2×C4#93912yes—
256C2×C4#94011yescharacteristic
257C8—68no—
258C8—122no—
259C2×C3—162no—
260C2×C3#165512yes—
261C2×C3#165612yes—
262C2×C3#200212yes—
263C2×C3#200611yescharacteristic
264D3—162no—
265C22—84no—
266C22—84no—
267C22—82no—
268C22—82no—
269C22—82no—
270C22—244no—
271C22—242no—
272C22—482no—
273C22—482no—
274C22#522311yescharacteristic
275C22#524111yescharacteristic
276C22#524114yes—
277C22#524511yescharacteristic
278C4—22no—
279C4—22no—
280C4—21no—
281C4—21no—
282C4—82no—
283C4—82no—
284C3#1007811yescharacteristic, complemented, Sylow
285C2—162no—
286C2—482no—
287C2#7185612yes—
288C2#7187911yescharacteristic
289C2#7194812yes—
290C2#7195212yes—
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2819C2
socleC23×C3#120
hypercenter#2819D3
hypocenterC3#10078
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2819

Complements

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

NH ≅ G/Ntype
1. C3×#2819C2semidirect
2. C2×#5058C2semidirect
3. C2×#4540C2semidirect
4. #69553C2semidirect
5. #67630C2semidirect
22. #2819D3semidirect
48. #874D4semidirect
54. #803D4semidirect
284. C3#10078semidirect

Maximal subgroups

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

subgroupindex
C3×#28192
C2×#50582
C2×#45402
#695532
#676302
#100783