Finite Group Databasebeta

← #10697949

Subgroups — #10697949

Order 768.

516 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 87 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#83321C212yesmaximal
2#79069C211yesmaximal, characteristic, complemented
3C3×#1962C211yesmaximal, characteristic
4#69396C212yesmaximal, complemented
5C3⋊#1309C211yesmaximal, characteristic, complemented
6#10302—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8#5470C2212yes—
9C3×#423C2211yescharacteristic
10C2×C3×#104—22no—
11C3×C43C2211yescharacteristic
12#5137—22no—
13#5091—22no—
14#5075—22no—
15#5042—24no—
16#5019—22no—
17#5007—22no—
18#4924C2212yes—
19#4851—22no—
20#4849—22no—
21#4523—24no—
22#3255—31no—
23#3254—32no—
24#1994—32no—
25#1962D311yescharacteristic
26#1309—31no—
27C2×C3×#41—21no—
28C2×C3×M16D412yes—
29C3×C4⋊Q8D411yescharacteristic
30C3×#127D411yescharacteristic
31C3×#124—21no—
32C3×#121—22no—
33C3×C4×D4—22no—
34C2×C3×C42—22no—
35C2×C3×C42—21no—
36C2×C3×C42C2311yescharacteristic
37C2×#214—24no—
38#926—22no—
39#924—22no—
40#924—22no—
41C2×#208D414yes—
42C2×Dic12—22no—
43#870—22no—
44#869D412yes—
45#869D412yes—
46C4×Dic6—22no—
47C4×Dic6—22no—
48C3×C4⋊C8—22no—
49C3×#104—44no—
50#832—24no—
51#817—22no—
52#816—24no—
53#810—24no—
54#807—24no—
55C4⋊(C3⋊C8)—22no—
56C4⋊(C3⋊C8)—22no—
57#533—32no—
58#532C2×D311yescharacteristic
59#477—62no—
60#475—62no—
61#474—62no—
62#470—62no—
63#468—64no—
64#462—62no—
65#461—62no—
66#460—62no—
67#423—32no—
68#423C2×D311yescharacteristic
69C2×#104—22no—
70C43C2×D311yescharacteristic
71#339—64no—
72C3×#41—42no—
73C3×#41—42no—
74C2×C3×Q8—21no—
75C2×C3×D4—22no—
76C2×C3×D4—21no—
77C2×C3×D4—41no—
78C22×C3×C4—41no—
79C22×C3×C4—41no—
80C22×C3×C4C2×D412yes—
81C22×C3×C4C2×D411yescharacteristic
82C2×Dic6—22no—
83C2×Dic6—22no—
84C3×M16—42no—
85C2×C3×C8—22no—
86C3×C4⋊C4—22no—
87C3×C22⋊C4—42no—
88C3×C42—22no—
89C3×C42—24no—
90C3×C42—41no—
91C3×C42—42no—
92C3×C42C2×D411yescharacteristic
93C3×C42C2×D412yes—
94C3×C42#4111yescharacteristic
95#217—42no—
96#214—84no—
97C4⋊Dic3—22no—
98C4⋊Dic3—22no—
99C4⋊Dic3—22no—
100C4⋊Dic3—22no—
101Dic3⋊C4—42no—
102Dic3⋊C4—42no—
103C4×Dic3—42no—
104#208—44no—
105C2×C3⋊C8—24no—
106Dic12—82no—
107C2×#41—21no—
108C2×Q16—62no—
109C2×SD16—64no—
110C2×M16—34no—
111C2×M16#6712yes—
112C4⋊Q8—32no—
113C4⋊Q8—32no—
114C4⋊Q8D1211yescharacteristic
115#127D1211yescharacteristic
116#125—62no—
117#124—21no—
118C22⋊Q8—62no—
119C22⋊Q8—62no—
120#121—22no—
121C4×Q8—62no—
122C4×Q8—62no—
123C4×D4—22no—
124#117—62no—
125C2×C42—22no—
126C2×C42—21no—
127C2×C42C22×D311yescharacteristic
128#107—64no—
129#106—62no—
130C4⋊C8—22no—
131C4⋊C8—62no—
132C4⋊C8—62no—
133#104—44no—
134#103—64no—
135#103—64no—
136#102—64no—
137C23×C3—41no—
138C23×C3#12711yescharacteristic
139C3×Q8—42no—
140C3×D4—42no—
141C3×D4—42no—
142C3×D4—42no—
143C3×D4—82no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—42no—
149C2×C3×C4—44no—
150C2×C3×C4—41no—
151C2×C3×C4—81no—
152C2×C3×C4—81no—
153C2×C3×C4#12012yes—
154C2×C3×C4#12011yescharacteristic
155C2×C3×C4#12112yes—
156C2×C3×C4#12111yescharacteristic
157C2×Dic3—42no—
158C2×Dic3—42no—
159Dic6—42no—
160Dic6—42no—
161Dic6—42no—
162Dic6—42no—
163C3×C8—42no—
164C3⋊C8—44no—
165#41—42no—
166#41—42no—
167C2×Q8—21no—
168C2×Q8—62no—
169C2×Q8—62no—
170C2×D4—22no—
171C2×D4—21no—
172C2×D4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4D3×D412yes—
176C22×C4C2×#6711yescharacteristic
177Q16—242no—
178SD16—244no—
179M16—42no—
180M16—124no—
181C2×C8—22no—
182C2×C8—64no—
183C4⋊C4—22no—
184C4⋊C4—62no—
185C4⋊C4—62no—
186C4⋊C4—62no—
187C4⋊C4—62no—
188C4⋊C4—122no—
189C4⋊C4—122no—
190C22⋊C4—42no—
191C22⋊C4—122no—
192C42—22no—
193C42—24no—
194C42—41no—
195C42—42no—
196C42—122no—
197C42C2×D1211yescharacteristic
198C42#23511yescharacteristic
199C42D3×D412yes—
200C22×C3—21no—
201C22×C3—22no—
202C22×C3—81no—
203C22×C3—82no—
204C22×C3#39211yescharacteristic
205C3×C4—21no—
206C3×C4—22no—
207C3×C4—44no—
208C3×C4—81no—
209C3×C4#45311yescharacteristic
210C3×C4#45411yescharacteristic
211C3×C4#45512yes—
212C3×C4#45612yes—
213Dic3—82no—
214Dic3—82no—
215C23—41no—
216C23#94011yescharacteristic
217Q8—42no—
218Q8—122no—
219Q8—122no—
220Q8—122no—
221Q8—122no—
222D4—42no—
223D4—42no—
224D4—42no—
225D4—82no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—41no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—122no—
236C2×C4—122no—
237C2×C4#88212yes—
238C2×C4#88412yes—
239C2×C4#93011yescharacteristic
240C2×C4#93711yescharacteristic
241C8—42no—
242C8—124no—
243C2×C3—41no—
244C2×C3—161no—
245C2×C3#200611yescharacteristic
246C2×C3#200712yes—
247C22—21no—
248C22—22no—
249C22—81no—
250C22—82no—
251C22#496011yescharacteristic
252C4—21no—
253C4—22no—
254C4—44no—
255C4—81no—
256C4—242no—
257C4—242no—
258C4#482512yes—
259C4#482612yes—
260C4#506211yescharacteristic
261C4#507711yescharacteristic
262C3#1030211yescharacteristic, complemented, Sylow
263C2—41no—
264C2—161no—
265C2#6966411yescharacteristic
266C2#7039312yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1962C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1962D3
hypocenterC3#10302
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
2. #79069C2semidirect
4. #69396C2semidirect
5. C3⋊#1309C2semidirect
262. C3#10302semidirect

Maximal subgroups

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

subgroupindex
#833212
#790692
C3×#19622
#693962
C3⋊#13092
#103023