Finite Group Databasebeta

← #10697943

Subgroups — #10697943

Order 768.

554 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 89 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×#3253C211yesmaximal, characteristic
2#83321C212yesmaximal, complemented
3#71815C212yesmaximal, complemented
4#70514C211yesmaximal, characteristic, complemented
5C3⋊#1309C211yesmaximal, characteristic, complemented
6#10301—31nomaximal, Sylow
7C3×#532C2212yes—
8#5470C2211yescharacteristic
9C3×#476—22no—
10C3×#469—22no—
11C3×#459—22no—
12C3×#423C2211yescharacteristic
13C2×#837—24no—
14#5091—22no—
15#5075—22no—
16#5042—24no—
17#5019—22no—
18#5007—22no—
19C42×Dic3C2211yescharacteristic
20#4924C2212yes—
21#4555—22no—
22#3254—32no—
23#3253D311yescharacteristic
24#1994—31no—
25#1962—32no—
26#1309—31no—
27C2×C3×#41—22no—
28C2×C3×D8—22no—
29C2×C3×M16D412yes—
30C3×C4⋊Q8D412yes—
31C3×#127D412yes—
32C3×#124—22no—
33C3×#121—22no—
34C3×#121—22no—
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C2×C3×C42C2311yescharacteristic
38C2×#214—24no—
39#926—21no—
40#924—22no—
41C2×C4×Dic3—22no—
42C2×C4×Dic3—21no—
43C2×#208D414yes—
44#870—21no—
45#869D412yes—
46C4×Dic6—22no—
47C3×#106—22no—
48C3×C4⋊C8—22no—
49C3×#102—24no—
50#837—48no—
51#832—24no—
52#810—24no—
53#807—24no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56#533—31no—
57#532C2×D312yes—
58#477—62no—
59#476—22no—
60#474—62no—
61#469—22no—
62#468—64no—
63#461—62no—
64#460—62no—
65#459—22no—
66#423—32no—
67#423C2×D311yescharacteristic
68C2×#104—64no—
69C43—31no—
70#339—62no—
71C3×#41—44no—
72C3×#41—44no—
73C2×C3×Q8—22no—
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4—42no—
78C22×C3×C4—42no—
79C22×C3×C4C2×D412yes—
80C22×C3×C4C2×D411yescharacteristic
81C22×Dic3—41no—
82C2×Dic6—22no—
83C3×D8—82no—
84C3×M16—42no—
85C2×C3×C8—22no—
86C3×C4⋊C4—22no—
87C3×C4⋊C4—22no—
88C3×C22⋊C4—42no—
89C3×C22⋊C4—42no—
90C3×C42C2×D411yescharacteristic
91C3×C42C2×D412yes—
92C3×C42#4111yescharacteristic
93#217—41no—
94#214—84no—
95C4⋊Dic3—22no—
96C4⋊Dic3—22no—
97Dic3⋊C4—42no—
98C4×Dic3—24no—
99C4×Dic3—22no—
100C4×Dic3—42no—
101C4×Dic3—41no—
102C4×Dic3—41no—
103#208—44no—
104C2×C3⋊C8—24no—
105C2×#41—22no—
106C2×SD16—64no—
107C2×D8—22no—
108C2×M16—34no—
109C2×M16#6712yes—
110C4⋊Q8—32no—
111C4⋊Q8#6712yes—
112#127#6712yes—
113#125—61no—
114#124—22no—
115C22⋊Q8—62no—
116#121—22no—
117#121—22no—
118C4×Q8—62no—
119C4×D4—22no—
120C4×D4—22no—
121#117—61no—
122C2×C42—62no—
123C2×C42—61no—
124C2×C42C22×D311yescharacteristic
125#107—64no—
126#106—22no—
127C4⋊C8—22no—
128C4⋊C8—62no—
129C4⋊C8—62no—
130#104—128no—
131#103—64no—
132#102—24no—
133#102—64no—
134C23×C3—42no—
135C23×C3#12711yescharacteristic
136C3×Q8—44no—
137C3×D4—44no—
138C3×D4—42no—
139C3×D4—42no—
140C3×D4—42no—
141C3×D4—42no—
142C3×D4—84no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—42no—
148C2×C3×C4—82no—
149C2×C3×C4—82no—
150C2×C3×C4#12012yes—
151C2×C3×C4#12011yescharacteristic
152C2×C3×C4#12111yescharacteristic
153C2×C3×C4#12112yes—
154C2×Dic3—44no—
155C2×Dic3—42no—
156C2×Dic3—42no—
157Dic6—42no—
158Dic6—42no—
159C3×C8—42no—
160C3⋊C8—44no—
161#41—44no—
162#41—44no—
163C2×Q8—22no—
164C2×Q8—62no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—42no—
169C22×C4—42no—
170C22×C4—121no—
171C22×C4D3×D412yes—
172C22×C4C2×#6711yescharacteristic
173SD16—244no—
174D8—82no—
175M16—42no—
176M16—124no—
177C2×C8—22no—
178C2×C8—64no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—62no—
182C4⋊C4—62no—
183C4⋊C4—122no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—121no—
187C42—64no—
188C42—62no—
189C42—122no—
190C42—121no—
191C42—121no—
192C42D3×D411yescharacteristic
193C42#23711yescharacteristic
194C42C2×#6712yes—
195C22×C3—22no—
196C22×C3—21no—
197C22×C3—82no—
198C22×C3—82no—
199C22×C3—82no—
200C22×C3#39211yescharacteristic
201C3×C4—22no—
202C3×C4—21no—
203C3×C4—82no—
204C3×C4#45312yes—
205C3×C4#45412yes—
206C3×C4#45511yescharacteristic
207C3×C4#45611yescharacteristic
208Dic3—44no—
209Dic3—82no—
210C23—42no—
211C23#94011yescharacteristic
212Q8—44no—
213Q8—122no—
214Q8—122no—
215D4—44no—
216D4—42no—
217D4—42no—
218D4—42no—
219D4—42no—
220D4—84no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—42no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—124no—
229C2×C4—122no—
230C2×C4—122no—
231C2×C4#93712yes—
232C2×C4#93811yescharacteristic
233C2×C4#93912yes—
234C2×C4#95311yescharacteristic
235C8—42no—
236C8—124no—
237C2×C3—41no—
238C2×C3—162no—
239C2×C3#200612yes—
240C2×C3#200711yescharacteristic
241C22—22no—
242C22—21no—
243C22—82no—
244C22—82no—
245C22—82no—
246C22#522311yescharacteristic
247C4—22no—
248C4—21no—
249C4—82no—
250C4—124no—
251C4—242no—
252C4#519812yes—
253C4#520112yes—
254C4#524211yescharacteristic
255C4#524611yescharacteristic
256C3#1030111yescharacteristic, complemented, Sylow
257C2—41no—
258C2—162no—
259C2#7188111yescharacteristic
260C2#7209812yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3253C2
socleC22×C3#392
hypercenter#3253D3
hypocenterC3#10301
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3253

Complements

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

NH ≅ G/Ntype
2. #83321C2semidirect
3. #71815C2semidirect
4. #70514C2semidirect
5. C3⋊#1309C2semidirect
256. C3#10301semidirect

Maximal subgroups

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

subgroupindex
C3×#32532
#833212
#718152
#705142
C3⋊#13092
#103013