Finite Group Databasebeta

← #10708029

Subgroups — #10708029

Order 768.

380 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 59 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#76604C211yesmaximal, characteristic
2C3×#2028C211yesmaximal, characteristic
3#71904C212yesmaximal
4#67643C211yesmaximal, characteristic
5#67494C212yesmaximal
6#12224—31nomaximal, Sylow
7C3×#544C2211yescharacteristic
8C3×#419—21no—
9C3×#391C2211yescharacteristic
10#5229—21no—
11#5050—22no—
12#4979C2212yes—
13#4929—21no—
14#4656—22no—
15#4646—22no—
16C3×#343C2211yescharacteristic
17C3×#339—21no—
18#4538—22no—
19#4528C2212yes—
20#2811—31no—
21#2028D311yescharacteristic
22#1605—32no—
23#1354—32no—
24#1319—31no—
25C22×C3×Q8D411yescharacteristic
26C22×Dic6—21no—
27C22×Dic6D412yes—
28C2×C3×M16—21no—
29C3×C4⋊Q8—41no—
30C3×#125—41no—
31C3×C22⋊Q8—22no—
32C3×#117—21no—
33C2×C3×C4⋊C4C2311yescharacteristic
34C2×C3×C42D411yescharacteristic
35C2×#216—42no—
36C2×C4⋊Dic3—21no—
37C2×C4⋊Dic3—21no—
38C2×Dic3⋊C4—22no—
39C2×C4×Dic3—21no—
40C3×#103—41no—
41C3×#103—41no—
42C3×#95—22no—
43C3⋊#95—22no—
44C3⋊#95—22no—
45C3⋊#95—22no—
46C3⋊#95D412yes—
47#830—22no—
48#810—42no—
49#544C2×D311yescharacteristic
50#451—62no—
51#419—21no—
52#395—32no—
53#395—62no—
54#391—62no—
55#391—61no—
56#391—61no—
57#391C2×D311yescharacteristic
58#343C2×D311yescharacteristic
59#339—21no—
60#328—62no—
61#324—32no—
62C2×C3×Q8—21no—
63C2×C3×Q8—22no—
64C2×C3×Q8—41no—
65C2×C3×Q8—41no—
66C2×C3×Q8Q1612yescomplemented
67C22×C3×C4—21no—
68C22×C3×C4—21no—
69C22×C3×C4C2×D412yes—
70C22×C3×C4C2×D411yescharacteristic
71C22×Dic3—22no—
72C22×Dic3—22no—
73C2×Dic6—24no—
74C2×Dic6—42no—
75C2×Dic6—82no—
76C2×Dic6—81no—
77C2×Dic6—81no—
78C3×M16—41no—
79C3×M16—41no—
80C2×C3×C8—41no—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—21no—
83C3×C4⋊C4—21no—
84C3×C4⋊C4—41no—
85C3×C4⋊C4—42no—
86C3×C22⋊C4—41no—
87C3×C42—41no—
88C3×C42—41no—
89#216—84no—
90C4⋊Dic3—81no—
91C4⋊Dic3—81no—
92Dic3⋊C4—82no—
93C4×Dic3—81no—
94C2×C3⋊C8—42no—
95C22×Q8—32no—
96C22×Q8—61no—
97C22×Q8#6711yescharacteristic
98C2×Q16—122no—
99C2×M16—21no—
100C4⋊Q8—41no—
101#125—41no—
102C22⋊Q8—22no—
103#117—21no—
104C2×C4⋊C4—61no—
105C2×C4⋊C4—61no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4C22×D311yescharacteristic
108C2×C42—61no—
109C2×C42#6711yescharacteristic
110#103—41no—
111#103—41no—
112#103—122no—
113C22⋊C8—62no—
114#95—22no—
115#95—32no—
116#95—62no—
117#95—62no—
118#95—62no—
119C23×C3#12011yescharacteristic
120C3×Q8—42no—
121C3×Q8—81no—
122C3×Q8—82no—
123C2×C3×C4—21no—
124C2×C3×C4—21no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—41no—
129C2×C3×C4—41no—
130C2×C3×C4—41no—
131C2×C3×C4—41no—
132C2×C3×C4—42no—
133C2×C3×C4—41no—
134C2×C3×C4—41no—
135C2×C3×C4C2×Q1611yescharacteristic
136C2×C3×C4#13711yescharacteristic
137C2×Dic3—24no—
138C2×Dic3—42no—
139C2×Dic3—42no—
140C2×Dic3—42no—
141C2×Dic3—42no—
142C2×Dic3—42no—
143Dic6—44no—
144Dic6—84no—
145Dic6—82no—
146C3×C8—81no—
147C3⋊C8—82no—
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—41no—
151C2×Q8—41no—
152C2×Q8—64no—
153C2×Q8—122no—
154C2×Q8—242no—
155C2×Q8—241no—
156C2×Q8—241no—
157C2×Q8#21612yescomplemented
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—62no—
161C22×C4—62no—
162C22×C4D3×D412yes—
163C22×C4C2×#6711yescharacteristic
164Q16—244no—
165M16—41no—
166M16—41no—
167C2×C8—41no—
168C2×C8—122no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—42no—
174C4⋊C4—242no—
175C4⋊C4—241no—
176C4⋊C4—241no—
177C22⋊C4—41no—
178C42—41no—
179C42—41no—
180C42—241no—
181C22×C3—21no—
182C22×C3—21no—
183C22×C3#45111yescharacteristic
184C22×C3#45511yescharacteristic
185C22×C3B(4,𝔽2)11yescharacteristic
186C3×C4—21no—
187C3×C4—21no—
188C3×C4—42no—
189C3×C4—42no—
190C3×C4—42no—
191C3×C4—81no—
192C3×C4—81no—
193Dic3—44no—
194Dic3—82no—
195C23#93711yescharacteristic
196Q8—42no—
197Q8—81no—
198Q8—82no—
199Q8—124no—
200Q8—244no—
201Q8—242no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—64no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4C2×#21611yescharacteristic
221C2×C4#95111yescharacteristic
222C8—81no—
223C8—242no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3#160211yescharacteristic
227C2×C3#219911yescharacteristic
228C2×C3#220511yescharacteristic
229C22—21no—
230C22—21no—
231C22#503211yescharacteristic
232C22#505011yescharacteristic
233C22#506111yescharacteristic
234C4—21no—
235C4—21no—
236C4—42no—
237C4—42no—
238C4—42no—
239C4—81no—
240C4—81no—
241C4—124no—
242C4—242no—
243C3#1222411yescharacteristic, complemented, Sylow
244C2—21no—
245C2—21no—
246C2#6970111yescharacteristic
247C2#7099111yescharacteristic
248C2#7099511yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2028C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#2028D3
hypocenterC3#12224
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2028

Complements

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

NH ≅ G/Ntype
66. C2×C3×Q8Q16semidirect
157. C2×Q8#216semidirect
243. C3#12224semidirect

Maximal subgroups

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

subgroupindex
#766042
C3×#20282
#719042
#676432
#674942
#122243