Finite Group Databasebeta

← #10692702

Subgroups — #10692702

Order 768.

488 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 95 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#83288C211yesmaximal, characteristic
2#83279C211yesmaximal, characteristic
3C3×#1536C211yesmaximal, characteristic
4#70822C212yesmaximal, complemented
5#67754C212yesmaximal, complemented
6#9421—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8C2×#951—22no—
9C2×#869C2211yescharacteristic
10C3×#432C2212yes—
11#5234—22no—
12#5092C412yes—
13#5076C412yes—
14#5060—22no—
15#5011—24no—
16#4926C2212yes—
17C4×#208C2211yescharacteristic
18C3×#329—22no—
19#4486—24no—
20#3152—31no—
21#2980—31no—
22#1556—32no—
23#1536D311yescharacteristic
24#1291—32no—
25C2×C3×#41D412yes—
26C22×Dic6—21no—
27C3×C4⋊Q8C2×C411yescharacteristic
28C3×#127C2×C411yescharacteristic
29C3×#124D412yes—
30C3×#121—22no—
31C3×C4×D4—22no—
32C2×C3×C42C2311yescharacteristic
33#951—44no—
34#951—44no—
35C2×#216—24no—
36C2×#214—24no—
37C2×C4⋊Dic3—22no—
38C2×#208—21no—
39C2×#208—22no—
40C22×C3⋊C8—22no—
41#869—22no—
42#869—21no—
43#869C2×C412yes—
44C3×C4⋊C8—22no—
45C3×C22⋊C8—22no—
46C3×#97—22no—
47#830—24no—
48#830—22no—
49C4⋊(C3⋊C8)D414yes—
50#803—21no—
51#803—22no—
52C4×C3⋊C8C2×C412yes—
53C2×#137—62no—
54#532C2×D311yescharacteristic
55C2×C4⋊Q8—31no—
56#497—62no—
57#494—32no—
58#492—32no—
59#435—64no—
60#432—32no—
61#432C2×D312yes—
62#408—62no—
63C4×M16—31no—
64#332—64no—
65#329—22no—
66C3×#41—42no—
67C3×#41—42no—
68C2×C3×Q8C22⋊C412yes—
69C2×C3×D4—22no—
70C2×C3×D4—22no—
71C2×C3×D4C22⋊C412yes—
72C22×C3×C4—22no—
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4#4112yes—
75C22×Dic3—41no—
76C2×Dic6—22no—
77C2×Dic6—42no—
78C2×Dic6—42no—
79C2×Dic6—41no—
80C2×C3×C8—42no—
81C3×C4⋊C4—22no—
82C3×C22⋊C4—24no—
83C3×C42C22×C411yescharacteristic
84C3×C42C2×D412yes—
85C3×C42C2×D411yescharacteristic
86#216—48no—
87#214—48no—
88C4⋊Dic3—24no—
89C4⋊Dic3—42no—
90#208—24no—
91#208—82no—
92C2×C3⋊C8—22no—
93C2×C3⋊C8—24no—
94C2×C3⋊C8—82no—
95C2×#41D1212yes—
96C22×Q8—61no—
97#137—124no—
98#137—124no—
99C2×Q16—64no—
100C2×SD16—64no—
101C2×M16—61no—
102C2×M16—62no—
103C22×C8—62no—
104C4⋊Q8—32no—
105C4⋊Q8—62no—
106C4⋊Q8—61no—
107C4⋊Q8C4×D311yescharacteristic
108#127C4×D311yescharacteristic
109#124#6712yes—
110#121—22no—
111C4×D4—22no—
112C2×C4⋊C4—62no—
113C2×C42C22×D311yescharacteristic
114C4⋊C8—22no—
115C4⋊C8—34no—
116C22⋊C8—22no—
117C22⋊C8—64no—
118C22⋊C8—62no—
119#97—22no—
120#97—62no—
121#97—61no—
122C4×C8—32no—
123C23×C3—22no—
124C23×C3#12311yescharacteristic
125C3×Q8—42no—
126C3×D4—42no—
127C3×D4—82no—
128C3×D4—82no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4C2×C22⋊C411yescharacteristic
135C2×C3×C4C4×D412yes—
136C2×C3×C4#12011yescharacteristic
137C2×C3×C4#12112yes—
138C2×Dic3—42no—
139C2×Dic3—81no—
140C2×Dic3—81no—
141Dic6—44no—
142Dic6—82no—
143C3×C8—82no—
144C3⋊C8—24no—
145C3⋊C8—44no—
146#41—42no—
147#41—42no—
148C2×Q8—62no—
149C2×Q8—122no—
150C2×Q8—122no—
151C2×Q8—121no—
152C2×Q8(C2×D3)⋊C412yes—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4(C2×D3)⋊C412yes—
156C22×C4—22no—
157C22×C4—121no—
158C22×C4C2×D1211yescharacteristic
159C22×C4#23712yes—
160Q16—128no—
161SD16—128no—
162M16—64no—
163M16—242no—
164C2×C8—42no—
165C2×C8—62no—
166C2×C8—64no—
167C2×C8—242no—
168C4⋊C4—22no—
169C4⋊C4—64no—
170C4⋊C4—122no—
171C22⋊C4—24no—
172C42C2×C4×D311yescharacteristic
173C42D3×D412yes—
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#45312yes—
185Dic3—82no—
186C23—22no—
187C23#88611yescharacteristic
188Q8—42no—
189Q8—124no—
190Q8—242no—
191D4—42no—
192D4—82no—
193D4—82no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—122no—
200C2×C4—241no—
201C2×C4—241no—
202C2×C4#67⋊C412yes—
203C2×C4#88211yescharacteristic
204C2×C4C2×(C2×D3)⋊C411yescharacteristic
205C2×C4#93912yes—
206C8—64no—
207C8—82no—
208C8—124no—
209C2×C3—41no—
210C2×C3—82no—
211C2×C3#188711yescharacteristic
212C2×C3#190212yes—
213C22—21no—
214C22—22no—
215C22—42no—
216C22—44no—
217C22#495811yescharacteristic
218C4—21no—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—242no—
223C4#474712yes—
224C3#942111yescharacteristic, complemented, Sylow
225C2—41no—
226C2—82no—
227C2#7007011yescharacteristic
228C2#7039912yes—
subgroupG/H
centerC22#4958
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1536C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1536D3
hypocenterC3#9421
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1536

Complements

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

NH ≅ G/Ntype
4. #70822C2semidirect
5. #67754C2semidirect
224. C3#9421semidirect

Maximal subgroups

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

subgroupindex
#832882
#832792
C3×#15362
#708222
#677542
#94213