Finite Group Databasebeta

← #10692787

Subgroups — #10692787

Order 768.

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

normal subgroups only

This group has 504 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
1C3×#3152C211yesmaximal, characteristic
2#83278C211yesmaximal, characteristic
3#71810C211yesmaximal, characteristic
4#71775C212yesmaximal, complemented
5#67669C212yesmaximal, complemented
6#9435—31nomaximal, Sylow
7C2×C3×#137—22no—
8C3×#532C2211yescharacteristic
9C2×C3×C4⋊Q8C2211yescharacteristic
10C3×#497C414yes—
11C3×#494—21no—
12C3×#492—21no—
13C3×C4×M16C2211yescharacteristic
14#5234—22no—
15#5012—22no—
16D4×C3⋊C8—22no—
17#4925C2212yes—
18#4924C2212yes—
19#4547—22no—
20#4547—22no—
21#4546—21no—
22#4544—21no—
23#3152D311yescharacteristic
24#2979—31no—
25#1560—31no—
26#1549—32no—
27#1299—32no—
28C2×C3×#41D412yes—
29C22×C3×Q8—21no—
30C3×#137—44no—
31C3×#137—44no—
32C2×C3×Q16—24no—
33C2×C3×SD16—24no—
34C2×C3×M16—21no—
35C3×C4⋊Q8—22no—
36C3×C4⋊Q8—21no—
37C3×C4⋊Q8C2×C412yes—
38C3×C4⋊Q8D411yescharacteristic
39C3×#127D411yescharacteristic
40C3×#124C2×C412yes—
41C3×#121—22no—
42C3×C4×D4—22no—
43C2×C3×C4⋊C4—22no—
44C2×C3×C42C2311yescharacteristic
45C2×#208—22no—
46C2×#208—22no—
47C22×C3⋊C8—22no—
48C3×#97C2×C412yes—
49C3×C4×C8—21no—
50#830—24no—
51#830—22no—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)D412yes—
55C4⋊(C3⋊C8)D412yes—
56C4×C3⋊C8—22no—
57C2×#137—22no—
58#532C2×D311yescharacteristic
59C2×C4⋊Q8C2×D311yescharacteristic
60#497Dic314yes—
61#494—21no—
62#492—21no—
63#436—62no—
64C8×D4—62no—
65#431—32no—
66#423—32no—
67#408—62no—
68C4×M16C2×D311yescharacteristic
69#333—61no—
70#332—62no—
71#332—62no—
72#331—61no—
73C3×#41—42no—
74C3×#41—42no—
75C2×C3×Q8—22no—
76C2×C3×Q8—42no—
77C2×C3×Q8—42no—
78C2×C3×Q8—41no—
79C2×C3×Q8C22⋊C412yes—
80C2×C3×D4—22no—
81C2×C3×D4—22no—
82C2×C3×D4C22⋊C412yes—
83C22×C3×C4—22no—
84C22×C3×C4—41no—
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4#4112yes—
87C3×Q16—48no—
88C3×SD16—48no—
89C3×M16—22no—
90C3×M16—22no—
91C2×C3×C8—22no—
92C3×C4⋊C4—22no—
93C3×C4⋊C4—24no—
94C3×C4⋊C4—42no—
95C3×C22⋊C4—24no—
96C3×C42C22×C411yescharacteristic
97C3×C42C2×D412yes—
98C3×C42C2×D411yescharacteristic
99#208—82no—
100#208—82no—
101C2×C3⋊C8—22no—
102C2×C3⋊C8—22no—
103C2×C3⋊C8—42no—
104C2×C3⋊C8—82no—
105C2×#41#6712yes—
106C22×Q8—21no—
107#137—44no—
108#137—44no—
109C2×Q16—24no—
110C2×SD16—24no—
111C2×M16—21no—
112C2×M16—62no—
113C2×M16—62no—
114C22×C8—62no—
115C4⋊Q8—22no—
116C4⋊Q8—21no—
117C4⋊Q8C2×Dic312yes—
118C4⋊Q8#6711yescharacteristic
119#127#6711yescharacteristic
120#124C2×Dic312yes—
121#121—22no—
122C4×D4—22no—
123C2×C4⋊C4—22no—
124C2×C42C22×D311yescharacteristic
125C4⋊C8—32no—
126C4⋊C8—32no—
127C4⋊C8—62no—
128C4⋊C8—62no—
129C22⋊C8—64no—
130C22⋊C8—62no—
131#97C2×Dic312yes—
132C4×C8—21no—
133C4×C8—62no—
134C23×C3—22no—
135C23×C3#12311yescharacteristic
136C3×Q8—42no—
137C3×Q8—44no—
138C3×Q8—81no—
139C3×Q8—81no—
140C3×D4—42no—
141C3×D4—42no—
142C3×D4—42no—
143C3×D4—82no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—42no—
150C2×C3×C4—42no—
151C2×C3×C4—42no—
152C2×C3×C4—82no—
153C2×C3×C4C2×C22⋊C411yescharacteristic
154C2×C3×C4C4×D412yes—
155C2×C3×C4#12011yescharacteristic
156C2×C3×C4#12112yes—
157C3×C8—24no—
158C3⋊C8—42no—
159C3⋊C8—42no—
160C3⋊C8—82no—
161#41—42no—
162#41—42no—
163C2×Q8—22no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—41no—
167C2×Q8#21712yes—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4#21712yes—
171C22×C4—22no—
172C22×C4—41no—
173C22×C4#23712yes—
174C22×C4C2×#6711yescharacteristic
175Q16—48no—
176SD16—48no—
177M16—22no—
178M16—22no—
179M16—242no—
180M16—242no—
181C2×C8—22no—
182C2×C8—62no—
183C2×C8—62no—
184C2×C8—122no—
185C2×C8—242no—
186C4⋊C4—22no—
187C4⋊C4—24no—
188C4⋊C4—42no—
189C22⋊C4—24no—
190C42D3×D412yes—
191C42C22×Dic311yescharacteristic
192C42C2×#6711yescharacteristic
193C22×C3—21no—
194C22×C3—22no—
195C22×C3—42no—
196C22×C3—44no—
197C22×C3#38611yescharacteristic
198C3×C4—21no—
199C3×C4—22no—
200C3×C4—42no—
201C3×C4—82no—
202C3×C4#45311yescharacteristic
203C3×C4#45311yescharacteristic
204C3×C4#47112yes—
205C3×C4#47312yes—
206C23—22no—
207C23#93511yescharacteristic
208Q8—42no—
209Q8—44no—
210Q8—81no—
211Q8—81no—
212D4—42no—
213D4—42no—
214D4—42no—
215D4—82no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—82no—
225C2×C4D4×Dic312yes—
226C2×C4#93711yescharacteristic
227C2×C4#93912yes—
228C2×C4C2×#21711yescharacteristic
229C8—24no—
230C8—122no—
231C8—122no—
232C8—242no—
233C2×C3—41no—
234C2×C3—82no—
235C2×C3#188711yescharacteristic
236C2×C3#190812yes—
237C22—21no—
238C22—22no—
239C22—42no—
240C22—44no—
241C22#522011yescharacteristic
242C4—21no—
243C4—22no—
244C4—42no—
245C4—82no—
246C4#507711yescharacteristic
247C4#507711yescharacteristic
248C4#520312yes—
249C4#520412yes—
250C3#943511yescharacteristic, complemented, Sylow
251C2—41no—
252C2—82no—
253C2#7176311yescharacteristic
254C2#7210312yes—
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3152C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#3152D3
hypocenterC3#9435
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. #71775C2semidirect
5. #67669C2semidirect
250. C3#9435semidirect

Maximal subgroups

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

subgroupindex
C3×#31522
#832782
#718102
#717752
#676692
#94353