Finite Group Databasebeta

← #10697848

Subgroups — #10697848

Order 768.

542 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 111 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#83271C212yesmaximal
2#73308C211yesmaximal, characteristic, complemented
3C3×#1962C211yesmaximal, characteristic
4#70508C212yesmaximal, complemented
5#67611C211yesmaximal, characteristic, complemented
6#10285—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8C2×#869C2212yes—
9C3×#423C2211yescharacteristic
10C2×C3×#104—22no—
11C3×C43C2211yescharacteristic
12#5080—22no—
13#5070—22no—
14#5034—24no—
15#5008—24no—
16#4936—22no—
17#4929—24no—
18#4925C2212yes—
19#4486—24no—
20#3227—32no—
21#2862—31no—
22#1962D311yescharacteristic
23#1549—32no—
24#1276—31no—
25C2×C3×#41—21no—
26C22×Dic6—22no—
27C2×C3×M16—21no—
28C3×C4⋊Q8D411yescharacteristic
29C3×#127D411yescharacteristic
30C3×#124—21no—
31C3×#121—22no—
32C3×C4×D4—22no—
33C2×C3×C42—22no—
34C2×C3×C42D412yes—
35C2×C3×C42C2311yescharacteristic
36C2×C4⋊Dic3—22no—
37C2×C4⋊Dic3D414yes—
38#869—22no—
39#869—22no—
40#869—22no—
41#869—22no—
42C3×C4⋊C8—22no—
43C3×#104—44no—
44#832—28no—
45C3⋊#95—28no—
46#830—22no—
47#808—28no—
48C4⋊(C3⋊C8)D414yes—
49C4×C3⋊C8—22no—
50#532C2×D311yescharacteristic
51C2×C4⋊Q8—32no—
52#498—62no—
53#486—62no—
54#481—64no—
55#476—64no—
56#431—32no—
57#423C2×D311yescharacteristic
58C2×#104—22no—
59#391—64no—
60#382—62no—
61C43C2×D311yescharacteristic
62#332—64no—
63C3×#41—42no—
64C3×#41—42no—
65C2×C3×Q8—21no—
66C2×C3×D4—22no—
67C2×C3×D4—21no—
68C2×C3×D4—41no—
69C22×C3×C4—22no—
70C22×C3×C4—41no—
71C22×C3×C4C2×D412yes—
72C22×C3×C4C2×D411yescharacteristic
73C22×Dic3—24no—
74C2×Dic6—48no—
75C2×Dic6—42no—
76C2×Dic6—42no—
77C3×M16—42no—
78C2×C3×C8—41no—
79C3×C4⋊C4—22no—
80C3×C22⋊C4—42no—
81C3×C42—22no—
82C3×C42—44no—
83C3×C42SD1614yes—
84C3×C42C2×D411yescharacteristic
85C3×C42C2×D412yes—
86C3×C42#4111yescharacteristic
87C4⋊Dic3—24no—
88C4⋊Dic3—24no—
89C4⋊Dic3—42no—
90C4⋊Dic3—42no—
91C2×C3⋊C8—24no—
92C2×#41—21no—
93C22×Q8—62no—
94C2×M16—21no—
95C4⋊Q8—62no—
96C4⋊Q8—62no—
97C4⋊Q8—62no—
98C4⋊Q8—62no—
99C4⋊Q8D1211yescharacteristic
100#127D1211yescharacteristic
101#124—21no—
102#121—22no—
103C4×D4—22no—
104C2×C4⋊C4—34no—
105C2×C4⋊C4—62no—
106C2×C42—22no—
107C2×C42#6712yes—
108C2×C42C22×D311yescharacteristic
109#106—68no—
110C4⋊C8—22no—
111C4⋊C8—34no—
112#104—44no—
113#102—68no—
114C22⋊C8—62no—
115C4×C8—62no—
116#95—68no—
117C23×C3—41no—
118C23×C3#12011yescharacteristic
119C3×Q8—42no—
120C3×D4—44no—
121C3×D4—42no—
122C3×D4—82no—
123C2×C3×C4—24no—
124C2×C3×C4—24no—
125C2×C3×C4—44no—
126C2×C3×C4—41no—
127C2×C3×C4—81no—
128C2×C3×C4—81no—
129C2×C3×C4#12012yes—
130C2×C3×C4#12112yes—
131C2×C3×C4#12111yescharacteristic
132C2×C3×C4#12711yescharacteristic
133C2×C3×C4C2×SD1612yes—
134C2×C3×C4#13712yes—
135C2×Dic3—48no—
136C2×Dic3—44no—
137Dic6—84no—
138Dic6—84no—
139C3×C8—81no—
140C3⋊C8—44no—
141#41—42no—
142#41—42no—
143C2×Q8—21no—
144C2×Q8—128no—
145C2×Q8—122no—
146C2×Q8—122no—
147C2×D4—22no—
148C2×D4—21no—
149C2×D4—41no—
150C22×C4—22no—
151C22×C4—41no—
152C22×C4—64no—
153C22×C4D3×D412yes—
154C22×C4C2×#6711yescharacteristic
155M16—42no—
156C2×C8—41no—
157C2×C8—64no—
158C4⋊C4—22no—
159C4⋊C4—64no—
160C4⋊C4—64no—
161C4⋊C4—122no—
162C4⋊C4—122no—
163C22⋊C4—42no—
164C42—22no—
165C42—44no—
166C42#21414yes—
167C42C2×D1211yescharacteristic
168C42#23511yescharacteristic
169C42D3×D412yes—
170C22×C3—22no—
171C22×C3—82no—
172C22×C3—81no—
173C22×C3#39211yescharacteristic
174C22×C3#44812yes—
175C3×C4—22no—
176C3×C4—44no—
177C3×C4—81no—
178C3×C4#45414yes—
179C3×C4#46112yes—
180C3×C4#49211yescharacteristic
181C3×C4#49711yescharacteristic
182Dic3—84no—
183C23—41no—
184C23#93711yescharacteristic
185Q8—42no—
186Q8—244no—
187Q8—244no—
188D4—42no—
189D4—44no—
190D4—82no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—44no—
194C2×C4—41no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—128no—
198C2×C4—124no—
199C2×C4#88212yes—
200C2×C4#88412yes—
201C2×C4#93011yescharacteristic
202C2×C4C2×#21412yes—
203C2×C4#94011yescharacteristic
204C2×C4#95112yes—
205C8—81no—
206C8—124no—
207C2×C3—22no—
208C2×C3—161no—
209C2×C3#200312yes—
210C2×C3#201311yescharacteristic
211C22—22no—
212C22—81no—
213C22—82no—
214C22#496011yescharacteristic
215C22#504012yes—
216C4—22no—
217C4—44no—
218C4—81no—
219C4—244no—
220C4#482714yes—
221C4#501912yes—
222C4#506011yescharacteristic
223C4#507611yescharacteristic
224C3#1028511yescharacteristic, complemented, Sylow
225C2—22no—
226C2—161no—
227C2#6966111yescharacteristic
228C2#7038812yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1962C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1962D3
hypocenterC3#10285
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
2. #73308C2semidirect
4. #70508C2semidirect
5. #67611C2semidirect
224. C3#10285semidirect

Maximal subgroups

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

subgroupindex
#832712
#733082
C3×#19622
#705082
#676112
#102853