Finite Group Databasebeta

← #10708271

Subgroups — #10708271

Order 768.

330 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2844C211yesmaximal, characteristic
2#69719C212yesmaximal
3#69543C211yesmaximal, characteristic
4#67652C212yesmaximal
5#67632C211yesmaximal, characteristic
6#12261—31nomaximal, Sylow
7#5500C2211yescharacteristic
8C3×#400—22no—
9C3×#400C2212yes—
10C3×#395—22no—
11C3×#395—21no—
12C3×#382—21no—
13C3×#382C2211yescharacteristic
14#4970—22no—
15#4964—21no—
16#4530C2211yescharacteristic
17#4528C2212yes—
18#4523—21no—
19#4452—22no—
20#2844D311yescharacteristic
21#2061—31no—
22#1617—32no—
23#1353—32no—
24#1324—31no—
25C22×C3×Q8—21no—
26C2×C3×C4⋊C4—21no—
27C2×C3×C4⋊C4—21no—
28C2×C3×C4⋊C4—22no—
29C2×C3×C4⋊C4D411yescharacteristic
30C2×C3×C4⋊C4D412yes—
31C2×C3×C4⋊C4C2311yescharacteristic
32C2×C3×C42D411yescharacteristic
33#926—21no—
34#924—22no—
35C2×#208—21no—
36#890—41no—
37#888—41no—
38C3×#95—22no—
39C3×#95—21no—
40C3×#95—22no—
41C3×#95—22no—
42C3×#95—22no—
43C3×#95D412yes—
44#830—22no—
45#810—42no—
46#808—41no—
47#807—41no—
48#807—42no—
49#544—31no—
50#484—62no—
51#428—61no—
52#400—22no—
53#400C2×D312yes—
54#395—22no—
55#395—21no—
56#382—21no—
57#382C2×D311yescharacteristic
58#343—31no—
59#339—61no—
60#328—62no—
61#324—32no—
62C2×C3×Q8—82no—
63C2×C3×Q8—81no—
64C22×C3×C4—21no—
65C22×C3×C4—22no—
66C22×C3×C4—22no—
67C22×C3×C4—21no—
68C22×C3×C4C2×D411yescharacteristic
69C22×C3×C4C2×D412yes—
70C2×Dic6—22no—
71C3×C4⋊C4—41no—
72C3×C4⋊C4—41no—
73C3×C4⋊C4—42no—
74C3×C4⋊C4—81no—
75C3×C4⋊C4—81no—
76C3×C4⋊C4—82no—
77C3×C42—21no—
78C3×C42Q1612yes—
79#217—41no—
80C4⋊Dic3—22no—
81C4⋊Dic3—21no—
82C4⋊Dic3—21no—
83Dic3⋊C4—42no—
84C4×Dic3—41no—
85#208—22no—
86#208—41no—
87C2×C3⋊C8—41no—
88C2×C3⋊C8—42no—
89C22×Q8—21no—
90C2×M16—61no—
91C4⋊Q8—121no—
92#125—121no—
93C22⋊Q8—62no—
94#117—61no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4#6711yescharacteristic
99C2×C4⋊C4#6712yes—
100C2×C4⋊C4C22×D311yescharacteristic
101C2×C42#6711yescharacteristic
102#107—121no—
103#107—122no—
104#106—121no—
105#103—122no—
106C22⋊C8—62no—
107#95—22no—
108#95—21no—
109#95—22no—
110#95—22no—
111#95—22no—
112#95#6712yes—
113C23×C3#12011yescharacteristic
114C3×Q8—84no—
115C2×C3×C4—21no—
116C2×C3×C4—21no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—41no—
120C2×C3×C4—41no—
121C2×C3×C4—42no—
122C2×C3×C4—42no—
123C2×C3×C4—42no—
124C2×C3×C4—42no—
125C2×C3×C4—42no—
126C2×C3×C4—42no—
127C2×C3×C4—42no—
128C2×C3×C4—41no—
129C2×C3×C4—42no—
130C2×C3×C4C2×Q1611yescharacteristic
131C2×C3×C4#13711yescharacteristic
132C2×Dic3—41no—
133C2×Dic3—41no—
134Dic6—82no—
135C3⋊C8—42no—
136C3⋊C8—82no—
137C2×Q8—62no—
138C2×Q8—82no—
139C2×Q8—81no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4C2×#6711yescharacteristic
145C22×C4C2×#6712yes—
146M16—62no—
147M16—121no—
148C2×C8—121no—
149C2×C8—122no—
150C4⋊C4—41no—
151C4⋊C4—42no—
152C4⋊C4—41no—
153C4⋊C4—61no—
154C4⋊C4—61no—
155C4⋊C4—62no—
156C4⋊C4—81no—
157C4⋊C4—81no—
158C4⋊C4—82no—
159C4⋊C4—122no—
160C22⋊C4—121no—
161C42—21no—
162C42—121no—
163C42#21612yes—
164C22×C3—21no—
165C22×C3—21no—
166C22×C3#45111yescharacteristic
167C22×C3#45611yescharacteristic
168C22×C3B(4,𝔽2)11yescharacteristic
169C3×C4—21no—
170C3×C4—21no—
171C3×C4—42no—
172C3×C4—42no—
173C3×C4—82no—
174C3×C4—81no—
175C3×C4—82no—
176Dic3—81no—
177Dic3—81no—
178C23#95311yescharacteristic
179Q8—84no—
180Q8—242no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—121no—
197C2×C4—121no—
198C2×C4#94211yescharacteristic
199C2×C4C2×#21611yescharacteristic
200C8—122no—
201C8—242no—
202C2×C3—21no—
203C2×C3—21no—
204C2×C3#160511yescharacteristic
205C2×C3#220111yescharacteristic
206C2×C3#220211yescharacteristic
207C22—21no—
208C22—21no—
209C22#522811yescharacteristic
210C22#524411yescharacteristic
211C22#524611yescharacteristic
212C4—21no—
213C4—21no—
214C4—42no—
215C4—42no—
216C4—82no—
217C4—82no—
218C4—81no—
219C4—241no—
220C4—241no—
221C3#1226111yescharacteristic, complemented, Sylow
222C2—21no—
223C2—21no—
224C2#7190411yescharacteristic
225C2#7195111yescharacteristic
226C2#7195811yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2844C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#2844D3
hypocenterC3#12261
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2844

Complements

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

NH ≅ G/Ntype
221. C3#12261semidirect

Maximal subgroups

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

subgroupindex
C3×#28442
#697192
#695432
#676522
#676322
#122613