Finite Group Databasebeta

← #10697419

Subgroups — #10697419

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
1#79321C212yesmaximal
2#79069C211yesmaximal, characteristic
3#70836C212yesmaximal
4#70515C211yesmaximal, characteristic
5C3×#1307C211yesmaximal, characteristic
6#10217—31nomaximal, Sylow
7#5470C2211yescharacteristic
8#5470C2212yes—
9C3×#432C2212yes—
10C3×#423C2211yescharacteristic
11#5137—22no—
12#4932C2211yescharacteristic
13#4851—22no—
14#4851—22no—
15#4850—22no—
16#4849—22no—
17#4723—24no—
18#4703—22no—
19#4702—22no—
20#4555—22no—
21#4467—24no—
22#3255—31no—
23#3237—32no—
24#1995—31no—
25#1506—32no—
26#1307D311yescharacteristic
27C2×C3×M16D412yes—
28C2×C3×C42C2311yescharacteristic
29#926—22no—
30#926—21no—
31#924—22no—
32#924—22no—
33#924—22no—
34C2×C4×Dic3—21no—
35C2×Dic12—22no—
36#870—22no—
37#870—21no—
38#869D412yes—
39#869D412yes—
40#869D412yes—
41C4×Dic6—22no—
42C4×Dic6—22no—
43C4×Dic6—22no—
44C3×C4⋊C8—22no—
45C3×C4⋊C8—22no—
46C3×C22⋊C8D414yes—
47C3×#97—22no—
48C3⋊#95—22no—
49#818—24no—
50#817—22no—
51#817—24no—
52#816—24no—
53#816—24no—
54#816—24no—
55#533—31no—
56#533—32no—
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#432C2×D312yes—
66#423C2×D311yescharacteristic
67#376—31no—
68#339—62no—
69#328—64no—
70C22×C3×C4C2×D411yescharacteristic
71C22×C3×C4C2×D412yes—
72C22×Dic3—41no—
73C2×Dic6—22no—
74C2×Dic6—22no—
75C2×Dic6—22no—
76C3×M16—42no—
77C2×C3×C8—24no—
78C2×C3×C8—22no—
79C3×C42C2×D411yescharacteristic
80C3×C42C2×D412yes—
81C3×C42#4111yescharacteristic
82#217—42no—
83#217—41no—
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86C4⋊Dic3—22no—
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89C4⋊Dic3—22no—
90Dic3⋊C4—42no—
91Dic3⋊C4—42no—
92Dic3⋊C4—42no—
93C4×Dic3—22no—
94C4×Dic3—41no—
95C4×Dic3—41no—
96C4×Dic3—42no—
97Dic12—82no—
98C2×Q16—62no—
99C2×M16D1212yes—
100C4⋊Q8—32no—
101C4⋊Q8—32no—
102C4⋊Q8—32no—
103#125—62no—
104#125—61no—
105C22⋊Q8—62no—
106C22⋊Q8—62no—
107C22⋊Q8—62no—
108C4×Q8—62no—
109C4×Q8—62no—
110C4×Q8—62no—
111#117—61no—
112#117—62no—
113C2×C42—61no—
114C2×C42C22×D311yescharacteristic
115#107—64no—
116#106—62no—
117#106—64no—
118C4⋊C8—22no—
119C4⋊C8—22no—
120#103—64no—
121#103—64no—
122#103—64no—
123C22⋊C8D1214yes—
124#97—22no—
125#95—62no—
126C23×C3#12711yescharacteristic
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—42no—
130C2×C3×C4#12012yes—
131C2×C3×C4#12011yescharacteristic
132C2×C3×C4#12112yes—
133C2×C3×C4#12111yescharacteristic
134C2×Dic3—42no—
135C2×Dic3—42no—
136C2×Dic3—42no—
137C2×Dic3—42no—
138C2×Dic3—82no—
139Dic6—44no—
140Dic6—42no—
141Dic6—42no—
142Dic6—82no—
143C3×C8—44no—
144C3×C8—42no—
145C2×Q8—62no—
146C2×Q8—62no—
147C2×Q8—62no—
148C22×C4—121no—
149C22×C4C2×D1211yescharacteristic
150C22×C4C2×D1212yes—
151Q16—242no—
152M16—42no—
153C2×C8—24no—
154C2×C8—22no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—62no—
161C4⋊C4—122no—
162C4⋊C4—122no—
163C4⋊C4—122no—
164C22⋊C4—121no—
165C22⋊C4—122no—
166C42—62no—
167C42—121no—
168C42—121no—
169C42—122no—
170C42D3×D411yescharacteristic
171C42D3×D412yes—
172C42#23911yescharacteristic
173C22×C3—21no—
174C22×C3—22no—
175C22×C3#39211yescharacteristic
176C3×C4—21no—
177C3×C4—22no—
178C3×C4—22no—
179C3×C4#45511yescharacteristic
180C3×C4#45611yescharacteristic
181Dic3—82no—
182Dic3—82no—
183Dic3—82no—
184Dic3—82no—
185C23#87411yescharacteristic
186Q8—124no—
187Q8—122no—
188Q8—122no—
189Q8—242no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—42no—
193C2×C4—122no—
194C2×C4—122no—
195C2×C4—122no—
196C2×C4—122no—
197C2×C4—242no—
198C2×C4#88212yes—
199C2×C4#88211yescharacteristic
200C2×C4#89512yes—
201C2×C4#89511yescharacteristic
202C8—44no—
203C8—42no—
204C2×C3—41no—
205C2×C3#200711yescharacteristic
206C2×C3#200912yes—
207C22—21no—
208C22—22no—
209C22#467111yescharacteristic
210C4—21no—
211C4—22no—
212C4—22no—
213C4—242no—
214C4—242no—
215C4—242no—
216C4—242no—
217C4#474611yescharacteristic
218C4#475211yescharacteristic
219C3#1021711yescharacteristic, complemented, Sylow
220C2—41no—
221C2#6900911yescharacteristic
222C2#6903112yes—
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1307C2
socleC22×C3#392
prime radicalC23×C3#127
hypercenter#1307D3
hypocenterC3#10217
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1307

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
#793212
#790692
#708362
#705152
C3×#13072
#102173