Finite Group Databasebeta

← #10692705

Subgroups — #10692705

Order 768.

488 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3152C211yesmaximal, characteristic
2#83279C211yesmaximal, characteristic
3#71799C211yesmaximal, characteristic
4#71776C212yesmaximal, complemented
5#67670C212yesmaximal, complemented
6#9421—31nomaximal, Sylow
7C2×C3×#137—22no—
8C3×#532C2211yescharacteristic
9C2×C3×C4⋊Q8C2211yescharacteristic
10C3×#497—22no—
11C3×#494C412yes—
12C3×#492C412yes—
13C3×C4×M16C2211yescharacteristic
14#5234—22no—
15#5011—24no—
16#4926C2212yes—
17#4926C2212yes—
18#4547—24no—
19#4540—22no—
20#3152D311yescharacteristic
21#2980—31no—
22#1556—32no—
23#1536—31no—
24#1291—32no—
25C2×C3×#41D412yes—
26C22×C3×Q8—21no—
27C3×#137—44no—
28C3×#137—44no—
29C2×C3×Q16—24no—
30C2×C3×SD16—24no—
31C2×C3×M16—21no—
32C3×C4⋊Q8—22no—
33C3×C4⋊Q8—21no—
34C3×C4⋊Q8C2×C412yes—
35C3×C4⋊Q8C2×C411yescharacteristic
36C3×#127C2×C411yescharacteristic
37C3×#124D412yes—
38C3×#121—22no—
39C3×C4×D4—22no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C42C2311yescharacteristic
42C2×#208—22no—
43C22×C3⋊C8—22no—
44C3×#97—21no—
45C3×C4×C8C2×C412yes—
46#830—24no—
47#830—22no—
48#830—22no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)D414yes—
51#803—22no—
52#803—22no—
53C2×#137—22no—
54#532C2×D311yescharacteristic
55C2×C4⋊Q8C2×D311yescharacteristic
56#497—22no—
57#494Dic312yes—
58#492Dic312yes—
59#435—64no—
60#432—32no—
61#432—32no—
62#408—62no—
63C4×M16C2×D311yescharacteristic
64#332—64no—
65#329—62no—
66C3×#41—42no—
67C3×#41—42no—
68C2×C3×Q8—22no—
69C2×C3×Q8—42no—
70C2×C3×Q8—42no—
71C2×C3×Q8—41no—
72C2×C3×Q8C22⋊C412yes—
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4C22⋊C412yes—
76C22×C3×C4—22no—
77C22×C3×C4—41no—
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4#4112yes—
80C3×Q16—48no—
81C3×SD16—48no—
82C3×M16—24no—
83C2×C3×C8—22no—
84C3×C4⋊C4—22no—
85C3×C4⋊C4—24no—
86C3×C4⋊C4—42no—
87C3×C22⋊C4—24no—
88C3×C42C22×C411yescharacteristic
89C3×C42C2×D412yes—
90C3×C42C2×D411yescharacteristic
91#208—82no—
92C2×C3⋊C8—24no—
93C2×C3⋊C8—42no—
94C2×C3⋊C8—82no—
95C2×#41#6712yes—
96C22×Q8—21no—
97#137—44no—
98#137—44no—
99C2×Q16—24no—
100C2×SD16—24no—
101C2×M16—21no—
102C2×M16—62no—
103C22×C8—62no—
104C4⋊Q8—22no—
105C4⋊Q8—21no—
106C4⋊Q8C2×Dic312yes—
107C4⋊Q8C2×Dic311yescharacteristic
108#127C2×Dic311yescharacteristic
109#124#6712yes—
110#121—22no—
111C4×D4—22no—
112C2×C4⋊C4—22no—
113C2×C42C22×D311yescharacteristic
114C4⋊C8—34no—
115C4⋊C8—62no—
116C22⋊C8—64no—
117C22⋊C8—62no—
118C22⋊C8—62no—
119#97—21no—
120#97—62no—
121#97—62no—
122C4×C8C2×Dic312yes—
123C23×C3—22no—
124C23×C3#12311yescharacteristic
125C3×Q8—42no—
126C3×Q8—44no—
127C3×Q8—82no—
128C3×D4—42no—
129C3×D4—82no—
130C3×D4—82no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—42no—
137C2×C3×C4—81no—
138C2×C3×C4—81no—
139C2×C3×C4C2×C22⋊C411yescharacteristic
140C2×C3×C4C4×D412yes—
141C2×C3×C4#12011yescharacteristic
142C2×C3×C4#12112yes—
143C3×C8—24no—
144C3⋊C8—44no—
145C3⋊C8—82no—
146#41—42no—
147#41—42no—
148C2×Q8—22no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×Q8—41no—
152C2×Q8#21712yes—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4#21712yes—
156C22×C4—22no—
157C22×C4—41no—
158C22×C4#23712yes—
159C22×C4C2×#6711yescharacteristic
160Q16—48no—
161SD16—48no—
162M16—24no—
163M16—242no—
164C2×C8—22no—
165C2×C8—64no—
166C2×C8—122no—
167C2×C8—242no—
168C4⋊C4—22no—
169C4⋊C4—24no—
170C4⋊C4—42no—
171C22⋊C4—24no—
172C42D3×D412yes—
173C42C22×Dic311yescharacteristic
174C42C2×#6711yescharacteristic
175C22×C3—21no—
176C22×C3—22no—
177C22×C3—42no—
178C22×C3—44no—
179C22×C3#38611yescharacteristic
180C3×C4—21no—
181C3×C4—22no—
182C3×C4—22no—
183C3×C4—42no—
184C3×C4—82no—
185C3×C4#45312yes—
186C23—22no—
187C23#93511yescharacteristic
188Q8—42no—
189Q8—44no—
190Q8—82no—
191D4—42no—
192D4—82no—
193D4—82no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—81no—
201C2×C4—81no—
202C2×C4D4×Dic312yes—
203C2×C4#93711yescharacteristic
204C2×C4#93912yes—
205C2×C4C2×#21711yescharacteristic
206C8—24no—
207C8—124no—
208C8—242no—
209C2×C3—41no—
210C2×C3—82no—
211C2×C3#188711yescharacteristic
212C2×C3#190212yes—
213C22—21no—
214C22—22no—
215C22—42no—
216C22—44no—
217C22#522011yescharacteristic
218C4—21no—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—82no—
223C4#507712yes—
224C3#942111yescharacteristic, complemented, Sylow
225C2—41no—
226C2—82no—
227C2#7176311yescharacteristic
228C2#7210212yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3152C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3152D3
hypocenterC3#9421
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
4. #71776C2semidirect
5. #67670C2semidirect
224. C3#9421semidirect

Maximal subgroups

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

subgroupindex
C3×#31522
#832792
#717992
#717762
#676702
#94213