Finite Group Databasebeta

← #11235193

Subgroups — #11235193

Order 768.

1514 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2399C211yesmaximal, characteristic
2#82479C214yesmaximal
3#77892C218yesmaximal
4#77200C218yesmaximal
5#73711C214yesmaximal
6#73643C214yesmaximal
7#73388C212yesmaximal
8#45436—31nomaximal, Sylow
9C2×C3×C4×Q8C2212yes—
10C2×#869C2212yes—
11C2×#869C2214yes—
12C2×C4×Dic6C2214yes—
13C3×#398C2214yes—
14C3×#395C2214yes—
15C3×C4×C4⋊C4C2214yes—
16C3×#376C2211yescharacteristic
17#5229C2218yes—
18#4986C2218yes—
19#4986C2218yes—
20#4984C22116yes—
21#4978C22116yes—
22#4977C2218yes—
23#4936C2212yes—
24C4×C4⋊Dic3C2214yes—
25#4929C2214yes—
26#4929C2218yes—
27#4661C2218yes—
28#4661C2218yes—
29#4653C22116yes—
30#4651C2218yes—
31#4648C2218yes—
32#2769—38no—
33#2612—38no—
34#2568—34no—
35#2546—34no—
36#2399D311yescharacteristic
37#2356—34no—
38#2327—32no—
39C22×C3×Q8C2312yes—
40C22×Dic6C2314yes—
41C3×C4×Q8—44no—
42C2×C3×C4⋊C4C2318yes—
43C2×C3×C4⋊C4C2314yes—
44C2×C3×C4⋊C4C2312yes—
45C2×C3×C42C2311yescharacteristic
46C2×C3×C42C2314yes—
47C2×C3×C42C2312yes—
48C2×C4⋊Dic3C2318yes—
49C2×C4⋊Dic3C2314yes—
50C2×C4⋊Dic3C2318yes—
51C2×C4⋊Dic3C2318yes—
52C2×Dic3⋊C4C2318yes—
53C2×C4×Dic3C2318yes—
54C2×C4×Dic3C2318yes—
55#869—28no—
56#869—48no—
57C4×Dic6—216no—
58C3×#95C2314yes—
59C3×#95C2314yes—
60C3×#95C2314yes—
61C3⋊#95C2318yes—
62C3⋊#95C23116yes—
63C3⋊#95C2318yes—
64C3⋊#95C23116yes—
65C3⋊#95C23116yes—
66C2×C4⋊Q8—32no—
67C2×C4⋊Q8—34no—
68C2×C4×Q8—34no—
69C2×C4×Q8C2×D312yes—
70#400—38no—
71#400—38no—
72#398—38no—
73#398—38no—
74#398C2×D314yes—
75#395C2×D314yes—
76#391—38no—
77#391—34no—
78#391—38no—
79#389—316no—
80#387—316no—
81#387—316no—
82#383—38no—
83#382—38no—
84#382—32no—
85C4×C4⋊C4—34no—
86C4×C4⋊C4C2×D314yes—
87#376—38no—
88#376C2×D311yescharacteristic
89C2×C3×Q8—28no—
90C2×C3×Q8—42no—
91C22×C3×C4C2412yes—
92C22×C3×C4C2414yes—
93C22×C3×C4C2414yes—
94C22×C3×C4C2414yes—
95C22×C3×C4C2411yescharacteristic
96C22×Dic3C2418yes—
97C22×Dic3C2418yes—
98C2×Dic6—28no—
99C2×Dic6—28no—
100C2×Dic6—28no—
101C3×C4⋊C4—216no—
102C3×C4⋊C4—44no—
103C3×C4⋊C4—42no—
104C3×C42—28no—
105C3×C42—22no—
106C3×C42#4118yes—
107C4⋊Dic3—28no—
108C4⋊Dic3—216no—
109C4⋊Dic3—48no—
110C4⋊Dic3—48no—
111Dic3⋊C4—216no—
112C4×Dic3—216no—
113C4×Dic3—216no—
114C22×Q8—34no—
115C22×Q8C22×D312yes—
116C4⋊Q8—68no—
117C4⋊Q8—128no—
118C4×Q8—44no—
119C4×Q8—616no—
120C2×C4⋊C4—34no—
121C2×C4⋊C4—38no—
122C2×C4⋊C4—38no—
123C2×C4⋊C4—38no—
124C2×C4⋊C4—38no—
125C2×C4⋊C4C22×D318yes—
126C2×C4⋊C4C22×D314yes—
127C2×C4⋊C4C22×D312yes—
128C2×C42—38no—
129C2×C42—38no—
130C2×C42C22×D311yescharacteristic
131C2×C42C22×D314yes—
132C2×C42C22×D312yes—
133#95—38no—
134#95—316no—
135#95—38no—
136#95—316no—
137#95—316no—
138#95C22×D314yes—
139#95C22×D314yes—
140#95C22×D314yes—
141C23×C3C2511yescharacteristic
142C3×Q8—48no—
143C2×C3×C4—24no—
144C2×C3×C4—28no—
145C2×C3×C4—24no—
146C2×C3×C4—24no—
147C2×C3×C4—22no—
148C2×C3×C4—28no—
149C2×C3×C4—24no—
150C2×C3×C4C2×#4118yes—
151C2×C3×C4C2×#4114yes—
152C2×C3×C4H(5,𝔽2)12yes—
153C2×C3×C42−1+418yes—
154C2×Dic3—28no—
155C2×Dic3—28no—
156C2×Dic3—28no—
157C2×Dic3—28no—
158C2×Dic3—216no—
159Dic6—416no—
160C2×Q8—28no—
161C2×Q8—42no—
162C2×Q8—68no—
163C2×Q8—68no—
164C2×Q8—68no—
165C22×C4—38no—
166C22×C4—38no—
167C22×C4C23×D312yes—
168C22×C4C23×D314yes—
169C22×C4C23×D314yes—
170C22×C4C23×D314yes—
171C22×C4C23×D311yescharacteristic
172C4⋊C4—216no—
173C4⋊C4—44no—
174C4⋊C4—42no—
175C4⋊C4—616no—
176C4⋊C4—68no—
177C4⋊C4—616no—
178C4⋊C4—128no—
179C4⋊C4—128no—
180C42—22no—
181C42—28no—
182C42—616no—
183C42—616no—
184C42#23918yes—
185C22×C3C22×#4112yes—
186C22×C3C2×H(5,𝔽2)11yescharacteristic
187C22×C3C2×2−1+414yes—
188C3×C4—28no—
189C3×C4—28no—
190C3×C4—44no—
191C3×C4—42no—
192C3×C4#55514yes—
193Dic3—48no—
194Dic3—48no—
195C23C24×D311yescharacteristic
196Q8—48no—
197Q8—1216no—
198C2×C4—24no—
199C2×C4—28no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—28no—
204C2×C4—24no—
205C2×C4—68no—
206C2×C4—68no—
207C2×C4—68no—
208C2×C4—68no—
209C2×C4—616no—
210C2×C4#100412yes—
211C2×C4C2×#23918yes—
212C2×C4C2×#23914yes—
213C2×C4#101018yes—
214C2×C3C2×#55511yescharacteristic
215C2×C3#350914yes—
216C2×C3#353812yes—
217C22C2×#100411yescharacteristic
218C22C22×#23912yes—
219C22C2×#101014yes—
220C4—28no—
221C4—28no—
222C4—42no—
223C4—44no—
224C4—128no—
225C4—128no—
226C4#573414yes—
227C3#4543611yescharacteristic, complemented, Sylow
228C2#8603112yes—
229C2C2×#573411yescharacteristic
230C2#8626214yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, prime radical, cosocleC23×C3C25
FittingC3×#2399C2
hypercenter#2399D3
hypocenterC3#45436
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2399

Complements

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

NH ≅ G/Ntype
227. C3#45436semidirect

Maximal subgroups

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

subgroupindex
C3×#23992
#824792
#778922
#772002
#737112
#736432
#733882
#454363