Finite Group Databasebeta

← #10708249

Subgroups — #10708249

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#74342C211yesmaximal, characteristic
2#69573C211yesmaximal, characteristic
3#68141C212yesmaximal
4C3×#1325C211yesmaximal, characteristic
5#67459C212yesmaximal
6#12257—31nomaximal, Sylow
7#5500C2211yescharacteristic
8C3×#384C2211yescharacteristic
9#4986—21no—
10#4986C2212yes—
11#4984—22no—
12#4977—22no—
13#4964—21no—
14C4×C4⋊Dic3—21no—
15#4722—22no—
16C3×#324C2212yes—
17#4530C2211yescharacteristic
18#4523—21no—
19#4467—22no—
20#2571—31no—
21#2060—31no—
22#1617—32no—
23#1353—32no—
24#1325D311yescharacteristic
25C2×C3×C4⋊C4—22no—
26C2×C3×C4⋊C4D411yescharacteristic
27C2×C3×C4⋊C4C2311yescharacteristic
28C2×C3×C42D411yescharacteristic
29#926—21no—
30C2×C4⋊Dic3—21no—
31C2×C4⋊Dic3—21no—
32C2×C4⋊Dic3—21no—
33C2×C4⋊Dic3D412yes—
34#924—22no—
35C2×C4×Dic3—22no—
36C2×#208—21no—
37#890—41no—
38#888—41no—
39C3×C22⋊C8—22no—
40C3⋊#95—21no—
41C3⋊#95—22no—
42C3⋊#95—22no—
43C3⋊#95—22no—
44C3⋊#95D412yes—
45#818—42no—
46#816—42no—
47#808—41no—
48#807—41no—
49#544—31no—
50#484—62no—
51#428—61no—
52#400—32no—
53#400—61no—
54#389—62no—
55#384C2×D311yescharacteristic
56#382—62no—
57C4×C4⋊C4—61no—
58#343—31no—
59#339—61no—
60#328—62no—
61#324C2×D312yes—
62C22×C3×C4—21no—
63C22×C3×C4—21no—
64C22×C3×C4C2×D411yescharacteristic
65C22×C3×C4C2×D412yes—
66C22×Dic3—22no—
67C22×Dic3—22no—
68C2×Dic6—22no—
69C2×C3×C8—42no—
70C3×C4⋊C4—41no—
71C3×C4⋊C4—41no—
72C3×C4⋊C4—44no—
73C3×C42—21no—
74C3×C42Q1612yes—
75#217—41no—
76C4⋊Dic3—21no—
77C4⋊Dic3—21no—
78C4⋊Dic3—22no—
79C4⋊Dic3—42no—
80C4⋊Dic3—42no—
81C4⋊Dic3—42no—
82C4⋊Dic3—81no—
83Dic3⋊C4—42no—
84C4×Dic3—41no—
85C4×Dic3—44no—
86#208—22no—
87#208—41no—
88C2×C3⋊C8—41no—
89C2×M16—61no—
90C4⋊Q8—121no—
91#125—121no—
92C22⋊Q8—62no—
93#117—61no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—61no—
97C2×C4⋊C4—61no—
98C2×C4⋊C4—61no—
99C2×C4⋊C4D1211yescharacteristic
100C2×C4⋊C4C22×D311yescharacteristic
101C2×C42—62no—
102C2×C42D1211yescharacteristic
103#107—121no—
104#107—122no—
105#106—121no—
106#103—122no—
107C22⋊C8—22no—
108#95—32no—
109#95—61no—
110#95—62no—
111#95—62no—
112#95—62no—
113C23×C3#12011yescharacteristic
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4—22no—
120C2×C3×C4—42no—
121C2×C3×C4—42no—
122C2×C3×C4—41no—
123C2×C3×C4C2×Q1611yescharacteristic
124C2×C3×C4#13711yescharacteristic
125C2×Dic3—42no—
126C2×Dic3—42no—
127C2×Dic3—42no—
128C2×Dic3—42no—
129C2×Dic3—42no—
130C2×Dic3—41no—
131C2×Dic3—41no—
132C2×Dic3—42no—
133Dic6—82no—
134C3×C8—82no—
135C3⋊C8—42no—
136C2×Q8—62no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—62no—
140C22×C4—62no—
141C22×C4C2×D1211yescharacteristic
142C22×C4D3×D412yes—
143M16—62no—
144M16—121no—
145C2×C8—42no—
146C2×C8—121no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—44no—
150C4⋊C4—61no—
151C4⋊C4—61no—
152C4⋊C4—62no—
153C4⋊C4—122no—
154C4⋊C4—122no—
155C4⋊C4—122no—
156C4⋊C4—122no—
157C4⋊C4—241no—
158C22⋊C4—121no—
159C42—21no—
160C42—124no—
161C42—121no—
162C42Dic1212yes—
163C22×C3—21no—
164C22×C3—21no—
165C22×C3#45111yescharacteristic
166C22×C3#45611yescharacteristic
167C22×C3B(4,𝔽2)11yescharacteristic
168C3×C4—24no—
169C3×C4—21no—
170C3×C4—21no—
171C3×C4—42no—
172C3×C4—81no—
173Dic3—82no—
174Dic3—81no—
175Dic3—81no—
176Dic3—82no—
177C23#88211yescharacteristic
178Q8—242no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—122no—
189C2×C4—122no—
190C2×C4—122no—
191C2×C4—122no—
192C2×C4—121no—
193C2×C4—121no—
194C2×C4—122no—
195C2×C4—122no—
196C2×C4C2×Dic1211yescharacteristic
197C2×C4#90911yescharacteristic
198C8—82no—
199C8—122no—
200C2×C3—21no—
201C2×C3—21no—
202C2×C3#160511yescharacteristic
203C2×C3#219811yescharacteristic
204C2×C3#220411yescharacteristic
205C22—21no—
206C22—21no—
207C22#473911yescharacteristic
208C22#474111yescharacteristic
209C22#475211yescharacteristic
210C4—24no—
211C4—21no—
212C4—21no—
213C4—42no—
214C4—81no—
215C4—242no—
216C4—241no—
217C4—242no—
218C4—241no—
219C3#1225711yescharacteristic, complemented, Sylow
220C2—21no—
221C2—21no—
222C2#6813911yescharacteristic
223C2#6914811yescharacteristic
224C2#6916811yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1325C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1325D3
hypocenterC3#12257
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
219. C3#12257semidirect

Maximal subgroups

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

subgroupindex
#743422
#695732
#681412
C3×#13252
#674592
#122573