Finite Group Databasebeta

← #10697418

Subgroups — #10697418

Order 768.

430 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 79 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#83327C212yesmaximal
2#79069C211yesmaximal, characteristic
3C3×#1995C211yesmaximal, characteristic
4#69433C212yesmaximal
5#67716C211yesmaximal, characteristic
6#10217—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8#5470C2212yes—
9C3×#423C2211yescharacteristic
10C3×#376C2211yescharacteristic
11#5137—22no—
12#5082—22no—
13#5081—22no—
14#5045—24no—
15#5021—22no—
16#5020—22no—
17#4926C2212yes—
18#4851—22no—
19#4849—22no—
20C3×#339—22no—
21#4452—24no—
22#3255—31no—
23#3237—32no—
24#1995D311yescharacteristic
25#1506—32no—
26#1307—31no—
27C2×C3×M16D412yes—
28C3×C4⋊Q8D412yes—
29C3×#125—21no—
30C3×C22⋊Q8—22no—
31C3×C4×Q8—22no—
32C3×#117—21no—
33C2×C3×C42—21no—
34C2×C3×C42C2311yescharacteristic
35#926—22no—
36#924—22no—
37#924—22no—
38C2×Dic12—22no—
39#870—22no—
40#869D412yes—
41#869D412yes—
42C4×Dic6—22no—
43C4×Dic6—22no—
44C3×C4⋊C8—22no—
45C3×#95—22no—
46#835—24no—
47#835—24no—
48#830D414yes—
49#817—22no—
50#816—24no—
51#808—24no—
52#807—24no—
53C4⋊(C3⋊C8)—22no—
54#803—22no—
55#533—32no—
56#533C2×D311yescharacteristic
57#501—62no—
58#491—62no—
59#485—64no—
60#477—62no—
61#475—62no—
62#475—62no—
63#470—62no—
64#462—62no—
65#432—32no—
66#423C2×D311yescharacteristic
67#376C2×D311yescharacteristic
68#339—22no—
69#328—64no—
70C2×C3×Q8—22no—
71C22×C3×C4—41no—
72C22×C3×C4C2×D412yes—
73C22×C3×C4C2×D411yescharacteristic
74C2×Dic6—22no—
75C2×Dic6—22no—
76C3×M16—42no—
77C2×C3×C8—22no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4—22no—
80C3×C4⋊C4—42no—
81C3×C22⋊C4—41no—
82C3×C42—22no—
83C3×C42—41no—
84C3×C42—41no—
85C3×C42C2×D411yescharacteristic
86C3×C42C2×D412yes—
87C3×C42#4111yescharacteristic
88#217—42no—
89C4⋊Dic3—22no—
90C4⋊Dic3—22no—
91C4⋊Dic3—22no—
92C4⋊Dic3—22no—
93Dic3⋊C4—42no—
94Dic3⋊C4—42no—
95C4×Dic3—42no—
96C2×C3⋊C8—24no—
97Dic12—82no—
98C2×Q16—62no—
99C2×M16#6712yes—
100C4⋊Q8—32no—
101C4⋊Q8—32no—
102C4⋊Q8D1212yes—
103#125—21no—
104#125—62no—
105C22⋊Q8—22no—
106C22⋊Q8—62no—
107C22⋊Q8—62no—
108C4×Q8—22no—
109C4×Q8—62no—
110C4×Q8—62no—
111#117—21no—
112#117—62no—
113C2×C42—21no—
114C2×C42C22×D311yescharacteristic
115#107—64no—
116#106—64no—
117#106—62no—
118C4⋊C8—22no—
119C4⋊C8—62no—
120#103—64no—
121#103—64no—
122#103—64no—
123C22⋊C8—34no—
124#97—62no—
125#95—22no—
126C23×C3#12711yescharacteristic
127C3×Q8—44no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—82no—
134C2×C3×C4#12012yes—
135C2×C3×C4#12011yescharacteristic
136C2×C3×C4#12112yes—
137C2×C3×C4#12111yescharacteristic
138C2×Dic3—42no—
139C2×Dic3—42no—
140Dic6—42no—
141Dic6—42no—
142Dic6—82no—
143C3×C8—42no—
144C3⋊C8—44no—
145C2×Q8—22no—
146C2×Q8—62no—
147C2×Q8—62no—
148C22×C4—41no—
149C22×C4D3×D412yes—
150C22×C4C2×#6711yescharacteristic
151Q16—242no—
152M16—42no—
153C2×C8—22no—
154C2×C8—64no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—42no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—122no—
163C4⋊C4—122no—
164C22⋊C4—41no—
165C22⋊C4—122no—
166C42—22no—
167C42—41no—
168C42—41no—
169C42—122no—
170C42C2×D1211yescharacteristic
171C42#23511yescharacteristic
172C42D3×D412yes—
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—44no—
187Q8—122no—
188Q8—122no—
189Q8—242no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—82no—
196C2×C4—122no—
197C2×C4—122no—
198C2×C4#88212yes—
199C2×C4#88412yes—
200C2×C4#93011yescharacteristic
201C2×C4#93711yescharacteristic
202C8—42no—
203C8—124no—
204C2×C3—41no—
205C2×C3#200711yescharacteristic
206C2×C3#200912yes—
207C22—22no—
208C22—21no—
209C22#496011yescharacteristic
210C4—22no—
211C4—22no—
212C4—21no—
213C4—82no—
214C4—82no—
215C4—242no—
216C4—242no—
217C4#506111yescharacteristic
218C4#509511yescharacteristic
219C3#1021711yescharacteristic, complemented, Sylow
220C2—41no—
221C2#6969411yescharacteristic
222C2#7041212yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1995C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1995D3
hypocenterC3#10217
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1995

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
#833272
#790692
C3×#19952
#694332
#677162
#102173