Finite Group Databasebeta

← #10697416

Subgroups — #10697416

Order 768.

430 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 81 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×#3255C211yesmaximal, characteristic
2#83372C212yesmaximal
3#71836C212yesmaximal
4#70515C211yesmaximal, characteristic
5#67716C211yesmaximal, characteristic
6#10217—31nomaximal, Sylow
7C3×#533C2212yes—
8#5470C2211yescharacteristic
9C3×#475—22no—
10C3×#470—22no—
11C3×#462—22no—
12C3×#423C2211yescharacteristic
13#5094—22no—
14#5091—22no—
15#5069—22no—
16#5064—22no—
17#4996—24no—
18#4932C2211yescharacteristic
19#4926C2212yes—
20#4555—22no—
21#4538—24no—
22#3255D311yescharacteristic
23#3237—32no—
24#1995—31no—
25#1506—32no—
26#1307—31no—
27C2×C3×Q16—22no—
28C2×C3×M16D412yes—
29C3×C4⋊Q8D412yes—
30C3×C4⋊Q8D412yes—
31C3×#125—22no—
32C3×C22⋊Q8—22no—
33C3×C22⋊Q8—22no—
34C3×C4×Q8—22no—
35C3×C4×Q8—22no—
36C3×#117—22no—
37C2×C3×C42C2311yescharacteristic
38#926—21no—
39#924—22no—
40C2×C4×Dic3—21no—
41#870—21no—
42#869D412yes—
43C4×Dic6—22no—
44C3×#106—22no—
45C3×C4⋊C8—22no—
46C3×#103—24no—
47C3⋊#95—22no—
48#830D414yes—
49#810—24no—
50#810—24no—
51#808—24no—
52#807—24no—
53C4⋊(C3⋊C8)—22no—
54#803—22no—
55#533—31no—
56#533C2×D312yes—
57#501—62no—
58#491—62no—
59#485—64no—
60#477—62no—
61#475—22no—
62#475—62no—
63#470—22no—
64#462—22no—
65#432—32no—
66#423C2×D311yescharacteristic
67#376—31no—
68#339—62no—
69#328—64no—
70C2×C3×Q8—22no—
71C2×C3×Q8—22no—
72C22×C3×C4C2×D412yes—
73C22×C3×C4C2×D411yescharacteristic
74C22×Dic3—41no—
75C2×Dic6—22no—
76C3×Q16—82no—
77C3×M16—42no—
78C2×C3×C8—22no—
79C3×C4⋊C4—22no—
80C3×C4⋊C4—22no—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—22no—
83C3×C4⋊C4—42no—
84C3×C4⋊C4—42no—
85C3×C22⋊C4—42no—
86C3×C42—42no—
87C3×C42C2×D411yescharacteristic
88C3×C42C2×D412yes—
89C3×C42#4111yescharacteristic
90#217—41no—
91C4⋊Dic3—22no—
92C4⋊Dic3—22no—
93Dic3⋊C4—42no—
94C4×Dic3—22no—
95C4×Dic3—41no—
96C4×Dic3—41no—
97C2×C3⋊C8—24no—
98C2×Q16—22no—
99C2×M16#6712yes—
100C4⋊Q8—32no—
101C4⋊Q8#6712yes—
102C4⋊Q8#6712yes—
103#125—22no—
104#125—61no—
105C22⋊Q8—22no—
106C22⋊Q8—22no—
107C22⋊Q8—62no—
108C4×Q8—22no—
109C4×Q8—22no—
110C4×Q8—62no—
111#117—22no—
112#117—61no—
113C2×C42—61no—
114C2×C42C22×D311yescharacteristic
115#107—64no—
116#106—22no—
117#106—64no—
118C4⋊C8—22no—
119C4⋊C8—62no—
120#103—24no—
121#103—64no—
122#103—64no—
123C22⋊C8—34no—
124#97—62no—
125#95—62no—
126C23×C3#12711yescharacteristic
127C3×Q8—42no—
128C3×Q8—42no—
129C3×Q8—82no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4#12012yes—
136C2×C3×C4#12011yescharacteristic
137C2×C3×C4#12111yescharacteristic
138C2×C3×C4#12112yes—
139C2×Dic3—42no—
140C2×Dic3—42no—
141C2×Dic3—82no—
142Dic6—44no—
143C3×C8—42no—
144C3⋊C8—44no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—62no—
148C22×C4—121no—
149C22×C4D3×D412yes—
150C22×C4C2×#6711yescharacteristic
151Q16—82no—
152M16—42no—
153C2×C8—22no—
154C2×C8—64no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—122no—
164C22⋊C4—42no—
165C22⋊C4—121no—
166C42—42no—
167C42—62no—
168C42—121no—
169C42—121no—
170C42D3×D411yescharacteristic
171C42#23711yescharacteristic
172C42C2×#6712yes—
173C22×C3—21no—
174C22×C3—22no—
175C22×C3#39211yescharacteristic
176C3×C4—21no—
177C3×C4—22no—
178C3×C4—22no—
179C3×C4—82no—
180C3×C4—82no—
181C3×C4#45511yescharacteristic
182C3×C4#45611yescharacteristic
183Dic3—82no—
184Dic3—82no—
185C23#94011yescharacteristic
186Q8—42no—
187Q8—42no—
188Q8—82no—
189Q8—124no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—122no—
196C2×C4—122no—
197C2×C4—242no—
198C2×C4#93712yes—
199C2×C4#93811yescharacteristic
200C2×C4#93912yes—
201C2×C4#95311yescharacteristic
202C8—42no—
203C8—124no—
204C2×C3—41no—
205C2×C3#200711yescharacteristic
206C2×C3#200912yes—
207C22—22no—
208C22—21no—
209C22#522311yescharacteristic
210C4—22no—
211C4—22no—
212C4—21no—
213C4—82no—
214C4—82no—
215C4—242no—
216C4—242no—
217C4#524211yescharacteristic
218C4#524611yescharacteristic
219C3#1021711yescharacteristic, complemented, Sylow
220C2—41no—
221C2#7188111yescharacteristic
222C2#7211112yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3255C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#3255D3
hypocenterC3#10217
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3255

Complements

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

NH ≅ G/Ntype
219. C3#10217semidirect

Maximal subgroups

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

subgroupindex
C3×#32552
#833722
#718362
#705152
#677162
#102173