Finite Group Databasebeta

← #10708208

Subgroups — #10708208

Order 768.

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

normal subgroups only

This group has 342 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#74217C211yesmaximal, characteristic
2C3×#2061C211yesmaximal, characteristic
3#71898C212yesmaximal
4#67632C211yesmaximal, characteristic
5#67495C212yesmaximal
6#12251—31nomaximal, Sylow
7C3×#544C2211yescharacteristic
8C2×#870—21no—
9C3×#428—21no—
10C3×#382C2211yescharacteristic
11#5043—22no—
12#4981—21no—
13#4981C2212yes—
14#4661—22no—
15#4659—22no—
16C3×#343C2211yescharacteristic
17C3×#339—21no—
18#4538—22no—
19#4528C2212yes—
20#2822—31no—
21#2061D311yescharacteristic
22#1611—32no—
23#1346—32no—
24#1324—31no—
25C2×C3×M16—21no—
26C3×C4⋊Q8—41no—
27C3×#125—41no—
28C3×C22⋊Q8—22no—
29C3×#117—21no—
30C2×C3×C4⋊C4D411yescharacteristic
31C2×C3×C4⋊C4C2311yescharacteristic
32C2×C3×C42D411yescharacteristic
33C2×C4⋊Dic3—21no—
34C2×C4⋊Dic3—21no—
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3D412yes—
37C2×Dic3⋊C4—22no—
38C2×Dic3⋊C4—22no—
39#870—42no—
40#870—42no—
41C3×#107—41no—
42C3×#106—41no—
43C3×#95—22no—
44#835—42no—
45C3⋊#95—21no—
46C3⋊#95—22no—
47C3⋊#95—22no—
48C3⋊#95D412yes—
49#830—22no—
50#808—42no—
51#544C2×D311yescharacteristic
52C2×#125—61no—
53#483—62no—
54#428—21no—
55#400—62no—
56#398—32no—
57#398—62no—
58#398—61no—
59#382C2×D311yescharacteristic
60#343C2×D311yescharacteristic
61#339—21no—
62#328—62no—
63#324—32no—
64C2×C3×Q8—22no—
65C22×C3×C4—21no—
66C22×C3×C4—21no—
67C22×C3×C4C2×D412yes—
68C22×C3×C4C2×D411yescharacteristic
69C22×Dic3—22no—
70C22×Dic3—22no—
71C3×M16—22no—
72C3×M16—41no—
73C2×C3×C8—41no—
74C3×C4⋊C4—22no—
75C3×C4⋊C4—21no—
76C3×C4⋊C4—21no—
77C3×C4⋊C4—41no—
78C3×C4⋊C4—41no—
79C3×C4⋊C4—42no—
80C3×C22⋊C4—41no—
81C3×C42—21no—
82C3×C42—41no—
83C3×C42SD1612yes—
84C4⋊Dic3—42no—
85C4⋊Dic3—42no—
86C4⋊Dic3—42no—
87C4⋊Dic3—81no—
88Dic3⋊C4—44no—
89Dic3⋊C4—42no—
90Dic3⋊C4—42no—
91C2×C3⋊C8—42no—
92C2×M16—21no—
93C4⋊Q8—41no—
94#125—41no—
95#125—122no—
96#125—122no—
97C22⋊Q8—22no—
98#117—21no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—61no—
101C2×C4⋊C4—62no—
102C2×C4⋊C4—61no—
103C2×C4⋊C4—62no—
104C2×C4⋊C4—61no—
105C2×C4⋊C4#6711yescharacteristic
106C2×C4⋊C4C22×D311yescharacteristic
107C2×C42#6711yescharacteristic
108#107—41no—
109#106—41no—
110#106—122no—
111#103—122no—
112C22⋊C8—62no—
113#95—22no—
114#95—32no—
115#95—62no—
116#95—61no—
117#95—62no—
118C23×C3#12011yescharacteristic
119C3×Q8—82no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—41no—
125C2×C3×C4—41no—
126C2×C3×C4—42no—
127C2×C3×C4—42no—
128C2×C3×C4—41no—
129C2×C3×C4—41no—
130C2×C3×C4—41no—
131C2×C3×C4C2×SD1611yescharacteristic
132C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
133C2×Dic3—42no—
134C2×Dic3—42no—
135C2×Dic3—42no—
136C2×Dic3—42no—
137C2×Dic3—42no—
138C2×Dic3—42no—
139C3×C8—42no—
140C3⋊C8—82no—
141C2×Q8—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—62no—
145C22×C4—62no—
146C22×C4D3×D412yes—
147C22×C4C2×#6711yescharacteristic
148M16—22no—
149M16—41no—
150C2×C8—41no—
151C2×C8—122no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—122no—
159C4⋊C4—124no—
160C4⋊C4—122no—
161C4⋊C4—122no—
162C4⋊C4—122no—
163C4⋊C4—122no—
164C4⋊C4—241no—
165C22⋊C4—41no—
166C42—21no—
167C42—41no—
168C42#21512yes—
169C22×C3—21no—
170C22×C3—21no—
171C22×C3#45011yescharacteristic
172C22×C3#45611yescharacteristic
173C22×C3B(4,𝔽2)11yescharacteristic
174C3×C4—21no—
175C3×C4—21no—
176C3×C4—42no—
177C3×C4—42no—
178C3×C4—81no—
179C3×C4—81no—
180C3×C4—81no—
181Dic3—82no—
182Dic3—82no—
183C23#93711yescharacteristic
184Q8—82no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—122no—
197C2×C4—122no—
198C2×C4—122no—
199C2×C4—122no—
200C2×C4—122no—
201C2×C4—122no—
202C2×C4C2×#21511yescharacteristic
203C2×C4#94911yescharacteristic
204C8—42no—
205C8—242no—
206C2×C3—21no—
207C2×C3—21no—
208C2×C3#160411yescharacteristic
209C2×C3#220111yescharacteristic
210C2×C3#220311yescharacteristic
211C22—21no—
212C22—21no—
213C22#503211yescharacteristic
214C22#504611yescharacteristic
215C22#509511yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—42no—
220C4—81no—
221C4—81no—
222C4—81no—
223C4—242no—
224C4—242no—
225C3#1225111yescharacteristic, complemented, Sylow
226C2—21no—
227C2—21no—
228C2#6970611yescharacteristic
229C2#7098911yescharacteristic
230C2#7099211yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2061C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#2061D3
hypocenterC3#12251
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
225. C3#12251semidirect

Maximal subgroups

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

subgroupindex
#742172
C3×#20612
#718982
#676322
#674952
#122513