Finite Group Databasebeta

← #10697946

Subgroups — #10697946

Order 768.

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

normal subgroups only

This group has 516 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×#3255C211yesmaximal, characteristic, complemented
2#83366C212yesmaximal
3#71839C212yesmaximal, complemented
4#70512C211yesmaximal, characteristic
5C3⋊#1309C211yesmaximal, characteristic, complemented
6#10302—31nomaximal, Sylow
7C3×#533C2212yes—
8#5477C2211yescharacteristic
9C3×#475—22no—
10C3×#470—22no—
11C3×#462—22no—
12C3×#423C2211yescharacteristic
13C2×#825—22no—
14#5087—22no—
15#5083—22no—
16#5049—24no—
17#5027—22no—
18#5021—22no—
19C42×Dic3C2211yescharacteristic
20#4924C2212yes—
21#4532—24no—
22#3255D311yescharacteristic, complemented
23#3254—32no—
24#1994—32no—
25#1962—31no—
26#1309—31no—
27C2×#235—21no—
28C2×C3×Q16—22no—
29C2×C3×M16D412yes—
30C3×C4⋊Q8D412yes—
31C3×C4⋊Q8D412yes—
32C3×#125—22no—
33C3×C22⋊Q8—22no—
34C3×C22⋊Q8—22no—
35C3×C4×Q8—22no—
36C3×C4×Q8—22no—
37C3×#117—22no—
38C2×C3×C42C2311yescharacteristic
39C2×#215—24no—
40#930—22no—
41C2×C4×Dic3—21no—
42C2×C4×Dic3—22no—
43C2×#208D414yes—
44#875—21no—
45#874D411yescharacteristic
46C4×D12—22no—
47#869D411yescharacteristic
48C3×#106—22no—
49C3×C4⋊C8—22no—
50C3×#103—24no—
51#835—24no—
52#825—44no—
53#809—24no—
54#807—24no—
55C4⋊(C3⋊C8)—22no—
56C4⋊(C3⋊C8)—22no—
57#533C2×D312yes—
58#532—31no—
59#477—62no—
60#475—22no—
61#474—62no—
62#470—22no—
63#468—64no—
64#462—22no—
65#461—62no—
66#460—62no—
67#423—32no—
68#423C2×D311yescharacteristic
69C2×#104—62no—
70C43—31no—
71#339—64no—
72C2×C3×Q8—22no—
73C2×C3×Q8—22no—
74C22×C3×C4C2×D412yes—
75C22×C3×C4C2×D411yescharacteristic
76C2×#67—41no—
77C22×Dic3—41no—
78#235—42no—
79#235—42no—
80C2×D12—22no—
81C2×D12—21no—
82C2×C4×D3—41no—
83C2×Dic6—21no—
84C3×Q16—82no—
85C3×M16—42no—
86C2×C3×C8—22no—
87C3×C4⋊C4—22no—
88C3×C4⋊C4—22no—
89C3×C4⋊C4—22no—
90C3×C4⋊C4—22no—
91C3×C4⋊C4—42no—
92C3×C4⋊C4—42no—
93C3×C22⋊C4—42no—
94C3×C42—42no—
95C3×C42C2×D411yescharacteristic
96C3×C42C2×D412yes—
97C3×C42#4111yescharacteristic
98#215—84no—
99(C2×D3)⋊C4—42no—
100C4⋊Dic3—22no—
101C4×Dic3—22no—
102C4×Dic3—24no—
103C4×Dic3—41no—
104C4×Dic3—42no—
105#208—44no—
106C2×C3⋊C8—24no—
107C2×#41—61no—
108C2×Q16—22no—
109C2×SD16—64no—
110C2×M16—34no—
111C2×M16#6712yes—
112C4⋊Q8—31no—
113C4⋊Q8#6712yes—
114C4⋊Q8#6712yes—
115#127—31no—
116#125—22no—
117#124—61no—
118C22⋊Q8—22no—
119C22⋊Q8—22no—
120#121—62no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×D4—62no—
124#117—22no—
125C2×C42—62no—
126C2×C42—61no—
127C2×C42C22×D311yescharacteristic
128#107—64no—
129#106—22no—
130C4⋊C8—22no—
131C4⋊C8—62no—
132C4⋊C8—62no—
133#104—124no—
134#103—24no—
135#103—64no—
136#102—64no—
137C23×C3#12711yescharacteristic
138C22×D3—41no—
139C3×Q8—42no—
140C3×Q8—42no—
141C3×Q8—42no—
142C3×Q8—42no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—42no—
148C2×C3×C4—42no—
149C2×C3×C4#12012yes—
150C2×C3×C4#12011yescharacteristic
151C2×C3×C4#12111yescharacteristic
152C2×C3×C4#12112yes—
153#67—82no—
154C2×Dic3—42no—
155C2×Dic3—44no—
156C2×Dic3—41no—
157D12—42no—
158D12—42no—
159D12—42no—
160C4×D3—81no—
161C4×D3—81no—
162Dic6—42no—
163C3×C8—42no—
164C3⋊C8—44no—
165#41—122no—
166#41—122no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—61no—
170C2×D4—62no—
171C2×D4—61no—
172C2×D4—121no—
173C22×C4—121no—
174C22×C4—121no—
175C22×C4D3×D412yes—
176C22×C4C2×#6711yescharacteristic
177Q16—82no—
178SD16—244no—
179M16—42no—
180M16—124no—
181C2×C8—22no—
182C2×C8—64no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—62no—
190C22⋊C4—42no—
191C22⋊C4—122no—
192C42—42no—
193C42—62no—
194C42—64no—
195C42—122no—
196C42—121no—
197C42D3×D411yescharacteristic
198C42#23711yescharacteristic
199C42C2×#6712yes—
200C22×C3—21no—
201C22×C3—22no—
202C22×C3#39211yescharacteristic
203C2×D3—81no—
204C2×D3—82no—
205C3×C4—21no—
206C3×C4—22no—
207C3×C4—82no—
208C3×C4—82no—
209C3×C4#45311yescharacteristic
210C3×C4#45411yescharacteristic
211C3×C4#45512yes—
212C3×C4#45612yes—
213Dic3—44no—
214Dic3—81no—
215C23—121no—
216C23#94011yescharacteristic
217Q8—42no—
218Q8—42no—
219Q8—42no—
220Q8—42no—
221Q8—122no—
222D4—122no—
223D4—122no—
224D4—122no—
225D4—242no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—124no—
233C2×C4—122no—
234C2×C4—121no—
235C2×C4—241no—
236C2×C4—241no—
237C2×C4#93712yes—
238C2×C4#93811yescharacteristic
239C2×C4#93912yes—
240C2×C4#95311yescharacteristic
241C8—42no—
242C8—124no—
243C2×C3—41no—
244C2×C3#200611yescharacteristic
245C2×C3#200712yes—
246D3—161no—
247C22—22no—
248C22—21no—
249C22—242no—
250C22—241no—
251C22#522311yescharacteristic
252C4—22no—
253C4—21no—
254C4—82no—
255C4—82no—
256C4—124no—
257C4—241no—
258C4#520212yes—
259C4#520512yes—
260C4#524111yescharacteristic
261C4#524511yescharacteristic
262C3#1030211yescharacteristic, complemented, Sylow
263C2—41no—
264C2—481no—
265C2#7187911yescharacteristic
266C2#7211012yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3255C2
socleC22×C3#392
prime radical#5477C22
hypercenter#3255D3
hypocenterC3#10302
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
1. C3×#3255C2semidirect
3. #71839C2semidirect
5. C3⋊#1309C2semidirect
22. #3255D3semidirect
262. C3#10302semidirect

Maximal subgroups

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

subgroupindex
C3×#32552
#833662
#718392
#705122
C3⋊#13092
#103023