Finite Group Databasebeta

← #10697939

Subgroups — #10697939

Order 768.

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

normal subgroups only

This group has 592 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#83313C212yesmaximal, complemented
2#79064C211yesmaximal, characteristic, complemented
3C3×#1962C211yesmaximal, characteristic, complemented
4#69395C212yesmaximal, complemented
5C3⋊#1309C211yesmaximal, characteristic, complemented
6#10300—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8#5477C2212yes—
9C3×#423C2211yescharacteristic
10C2×C3×#104—22no—
11C3×C43C2211yescharacteristic
12#5135—22no—
13#5084—22no—
14#5072—22no—
15#5039—24no—
16#5015—22no—
17#5008—22no—
18C2×#805—24no—
19#4924C2212yes—
20#4843—22no—
21#4842—22no—
22#3253—31no—
23#3253—32no—
24#1962—32no—
25#1962D311yescharacteristic, complemented
26#1309—31no—
27C2×C3×#41—21no—
28C2×#235—22no—
29C2×C3×M16D412yes—
30C3×C4⋊Q8D411yescharacteristic
31C3×#127D411yescharacteristic
32C3×#124—21no—
33C3×#121—22no—
34C3×C4×D4—22no—
35C2×C3×C42—22no—
36C2×C3×C42—21no—
37C2×C3×C42C2311yescharacteristic
38C2×#213—24no—
39#930—22no—
40#930—22no—
41C2×#208D414yes—
42C2×D24—22no—
43#875—22no—
44#874D412yes—
45C4×D12—22no—
46C4×D12—22no—
47#869D412yes—
48C3×C4⋊C8—22no—
49C3×#104—44no—
50#832—24no—
51#821—24no—
52#817—22no—
53#809—24no—
54#808—24no—
55#805—48no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)—22no—
58#532—32no—
59#532C2×D311yescharacteristic
60#476—62no—
61#476—62no—
62#476—62no—
63#469—64no—
64#469—62no—
65#459—62no—
66#459—62no—
67#459—62no—
68#423—32no—
69#423C2×D311yescharacteristic
70C2×#104—22no—
71C2×#104—64no—
72C43C2×D311yescharacteristic
73C3×#41—42no—
74C3×#41—42no—
75C2×C3×Q8—21no—
76C2×C3×D4—22no—
77C2×C3×D4—21no—
78C2×C3×D4—41no—
79C22×C3×C4—41no—
80C22×C3×C4—41no—
81C22×C3×C4C2×D412yes—
82C22×C3×C4C2×D411yescharacteristic
83C2×#67—42no—
84#235—44no—
85#235—44no—
86C2×D12—22no—
87C2×D12—22no—
88C2×D12—22no—
89C2×C4×D3—42no—
90C2×Dic6—22no—
91C3×M16—42no—
92C2×C3×C8—22no—
93C3×C4⋊C4—22no—
94C3×C22⋊C4—42no—
95C3×C42—24no—
96C3×C42—22no—
97C3×C42—42no—
98C3×C42—41no—
99C3×C42C2×D411yescharacteristic
100C3×C42C2×D412yes—
101C3×C42#4111yescharacteristic
102#213—84no—
103(C2×D3)⋊C4—42no—
104(C2×D3)⋊C4—42no—
105C4⋊Dic3—22no—
106C4⋊Dic3—22no—
107#208—44no—
108C2×C3⋊C8—24no—
109D24—82no—
110C2×#41—21no—
111C2×#41—62no—
112C2×D8—62no—
113C2×D8—64no—
114C2×M16—34no—
115C2×M16#6712yes—
116C4⋊Q8—32no—
117C4⋊Q8D1211yescharacteristic
118#127—32no—
119#127D1211yescharacteristic
120#124—21no—
121#124—62no—
122#121—22no—
123#121—62no—
124#121—62no—
125C4×D4—22no—
126C4×D4—62no—
127C4×D4—62no—
128C2×C42—22no—
129C2×C42—21no—
130C2×C42C22×D311yescharacteristic
131#106—64no—
132#106—62no—
133C4⋊C8—22no—
134C4⋊C8—62no—
135C4⋊C8—62no—
136#104—44no—
137#104—128no—
138#102—64no—
139#102—64no—
140#102—64no—
141C23×C3—41no—
142C23×C3#12711yescharacteristic
143C22×D3—42no—
144C3×Q8—42no—
145C3×D4—42no—
146C3×D4—42no—
147C3×D4—42no—
148C3×D4—82no—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4—22no—
152C2×C3×C4—21no—
153C2×C3×C4—44no—
154C2×C3×C4—42no—
155C2×C3×C4—41no—
156C2×C3×C4—81no—
157C2×C3×C4—81no—
158C2×C3×C4#12012yes—
159C2×C3×C4#12011yescharacteristic
160C2×C3×C4#12112yes—
161C2×C3×C4#12111yescharacteristic
162#67—84no—
163C2×Dic3—42no—
164D12—44no—
165D12—42no—
166D12—42no—
167D12—42no—
168D12—42no—
169C4×D3—82no—
170C4×D3—82no—
171Dic6—44no—
172C3×C8—42no—
173C3⋊C8—44no—
174#41—42no—
175#41—42no—
176#41—124no—
177#41—124no—
178C2×Q8—21no—
179C2×Q8—62no—
180C2×D4—22no—
181C2×D4—21no—
182C2×D4—41no—
183C2×D4—62no—
184C2×D4—62no—
185C2×D4—62no—
186C2×D4—122no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—122no—
190C22×C4D3×D412yes—
191C22×C4C2×#6711yescharacteristic
192D8—242no—
193D8—244no—
194M16—42no—
195M16—124no—
196C2×C8—22no—
197C2×C8—64no—
198C4⋊C4—22no—
199C4⋊C4—62no—
200C4⋊C4—62no—
201C22⋊C4—42no—
202C22⋊C4—122no—
203C22⋊C4—122no—
204C42—22no—
205C42—24no—
206C42—41no—
207C42—42no—
208C42C2×D1211yescharacteristic
209C42#23511yescharacteristic
210C42D3×D412yes—
211C22×C3—22no—
212C22×C3—21no—
213C22×C3—81no—
214C22×C3—82no—
215C22×C3#39211yescharacteristic
216C2×D3—82no—
217C2×D3—82no—
218C2×D3—82no—
219C3×C4—22no—
220C3×C4—21no—
221C3×C4—44no—
222C3×C4—81no—
223C3×C4#45311yescharacteristic
224C3×C4#45312yes—
225C3×C4#45411yescharacteristic
226C3×C4#45412yes—
227Dic3—82no—
228C23—41no—
229C23—122no—
230C23#94011yescharacteristic
231Q8—42no—
232Q8—124no—
233D4—42no—
234D4—42no—
235D4—42no—
236D4—82no—
237D4—124no—
238D4—122no—
239D4—122no—
240D4—122no—
241D4—122no—
242D4—244no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—41no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—122no—
253C2×C4—242no—
254C2×C4—242no—
255C2×C4#88212yes—
256C2×C4#88412yes—
257C2×C4#93011yescharacteristic
258C2×C4#93711yescharacteristic
259C8—42no—
260C8—124no—
261C2×C3—41no—
262C2×C3—161no—
263C2×C3#200611yescharacteristic
264C2×C3#200612yes—
265D3—162no—
266C22—21no—
267C22—22no—
268C22—81no—
269C22—82no—
270C22—242no—
271C22—242no—
272C22—242no—
273C22#496011yescharacteristic
274C4—21no—
275C4—22no—
276C4—44no—
277C4—81no—
278C4—242no—
279C4#482212yes—
280C4#482712yes—
281C4#506211yescharacteristic
282C4#507711yescharacteristic
283C3#1030011yescharacteristic, complemented, Sylow
284C2—41no—
285C2—161no—
286C2—482no—
287C2#6966411yescharacteristic
288C2#7039212yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1962C2
socleC22×C3#392
hypercenter#1962D3
hypocenterC3#10300
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
1. #83313C2semidirect
2. #79064C2semidirect
3. C3×#1962C2semidirect
4. #69395C2semidirect
5. C3⋊#1309C2semidirect
25. #1962D3semidirect
283. C3#10300semidirect

Maximal subgroups

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

subgroupindex
#833132
#790642
C3×#19622
#693952
C3⋊#13092
#103003