Finite Group Databasebeta

← #11714112

Subgroups — #11714112

Order 768.

432 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C2×#5389C211yesmaximal, characteristic
2C2×#5386C211yesmaximal, characteristic
3#73167C211yesmaximal, characteristic
4#73064C212yesmaximal
5C4⋊#859C212yesmaximal
6#33204—31nomaximal, Sylow
7C2×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×C4×SL(2,𝔽3)C2212yes—
9C2×#859C2212yes—
10C2×#859C2211yescharacteristic
11C2×#859C2211yescharacteristic
12#5389—22no—
13#5386—22no—
14#4656—41nomaximal
15C2×#479—31no—
16C2×#475—31no—
17#2660—31no—
18#2276D311yescharacteristic
19#1989—31no—
20#1987—31no—
21#1949—32no—
22#1948—32no—
23#1843—31no—
24#1805—32no—
25#1803—32no—
26C22×Dic6—41no—
27C22×SL(2,𝔽3)C2311yescharacteristic
28C2×C4⋊Dic3—42no—
29C2×Dic3⋊C4—41no—
30C4×SL(2,𝔽3)D412yes—
31C4×SL(2,𝔽3)Q814yescomplemented
32#859—22no—
33#859—21no—
34#859—21no—
35C3×#95—41no—
36C3⋊#95—42no—
37C2×C4⋊Q8—31no—
38C2×#125—31no—
39C2×C4×Q8C2×D312yes—
40C2×C4×Q8C2×D311yescharacteristic
41#479—62no—
42#479—62no—
43#475—62no—
44#475—62no—
45C2×#107—31no—
46C2×#106—32no—
47C2×#106—31no—
48C2×C4⋊C8—31no—
49C2×C4⋊C8—31no—
50C2×C4⋊C8—32no—
51C2×C4⋊C8—32no—
52C2×#103—32no—
53C2×#103—31no—
54C2×#103—31no—
55#398—62no—
56#395—61no—
57#395—61no—
58#384—32no—
59#382—32no—
60C4×C4⋊C4—32no—
61C4×C4⋊C4—31no—
62#376—31no—
63#340—32no—
64#340—32no—
65#340—32no—
66#340—32no—
67C22×C3×C4—42no—
68C22×C3×C4—41no—
69C22×Dic3—41no—
70C22×Dic3—42no—
71C22×Dic3—41no—
72C2×Dic6—82no—
73C2×Dic6—84no—
74C2×SL(2,𝔽3)C2×D411yescharacteristic
75C2×SL(2,𝔽3)C2×Q812yes—
76C4⋊Dic3—84no—
77Dic3⋊C4—82no—
78C22×Q8—61no—
79C22×Q8C22×D311yescharacteristic
80C22×C8—32no—
81C22×C8—31no—
82C22×C8—31no—
83C4⋊Q8—62no—
84C4⋊Q8—62no—
85#125—62no—
86#125—62no—
87C4×Q8—34no—
88C4×Q8—32no—
89C4×Q8Dic614yescomplemented
90C4×Q8D1212yes—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—61no—
97C2×C4⋊C4—61no—
98C2×C4⋊C4—61no—
99C2×C4⋊C4—62no—
100C2×C4⋊C4—62no—
101C2×C42—31no—
102C2×C42—32no—
103C2×C42—61no—
104#107—62no—
105#106—64no—
106#106—62no—
107C4⋊C8—64no—
108C4⋊C8—64no—
109C4⋊C8—62no—
110C4⋊C8—62no—
111#103—62no—
112#103—61no—
113#103—61no—
114#103—62no—
115#103—61no—
116#103—61no—
117#95—31no—
118#95—32no—
119#95—62no—
120#95—62no—
121#95S411yescharacteristic
122C23×C3—41no—
123C2×C3×C4—44no—
124C2×C3×C4—42no—
125C2×C3×C4—82no—
126C2×C3×C4—82no—
127C2×C3×C4—82no—
128C2×Dic3—44no—
129C2×Dic3—42no—
130C2×Dic3—81no—
131C2×Dic3—82no—
132C2×Dic3—82no—
133C2×Dic3—81no—
134C2×Dic3—81no—
135C2×Dic3—82no—
136Dic6—88no—
137SL(2,𝔽3)C4⋊Q811yescharacteristic
138C2×Q8—31no—
139C2×Q8—32no—
140C2×Q8—122no—
141C2×Q8—242no—
142C2×Q8C2×Dic612yes—
143C2×Q8C2×D1211yescharacteristic
144C22×C4—31no—
145C22×C4—32no—
146C22×C4—31no—
147C22×C4—61no—
148C22×C4—62no—
149C22×C4—61no—
150C22×C4C2×S412yes—
151C22×C4C2×S411yescharacteristic
152C2×C8—62no—
153C2×C8—61no—
154C2×C8—61no—
155C2×C8—62no—
156C2×C8—62no—
157C2×C8—62no—
158C2×C8—62no—
159C4⋊C4—32no—
160C4⋊C4—34no—
161C4⋊C4—62no—
162C4⋊C4—61no—
163C4⋊C4—61no—
164C4⋊C4—62no—
165C4⋊C4—61no—
166C4⋊C4—61no—
167C4⋊C4—61no—
168C4⋊C4—62no—
169C4⋊C4—122no—
170C4⋊C4—122no—
171C4⋊C4—122no—
172C4⋊C4—242no—
173C4⋊C4—242no—
174C42—34no—
175C42—34no—
176C42—32no—
177C42—32no—
178C42—241no—
179C22×C3—42no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3—42no—
183C22×C3—41no—
184C3×C4—82no—
185C3×C4—84no—
186Dic3—84no—
187Dic3—82no—
188Dic3—161no—
189C23C22×S411yescharacteristic
190Q8—32no—
191Q8—31no—
192Q8—61no—
193Q8—244no—
194Q8#86911yescharacteristic
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—32no—
199C2×C4—32no—
200C2×C4—31no—
201C2×C4—31no—
202C2×C4—34no—
203C2×C4—32no—
204C2×C4—62no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—61no—
208C2×C4—121no—
209C2×C4—121no—
210C2×C4—122no—
211C2×C4—121no—
212C2×C4—122no—
213C2×C4—122no—
214C2×C4—122no—
215C2×C4—121no—
216C2×C4A4⋊Q814yes—
217C2×C4C4⋊S412yes—
218C8—122no—
219C8—121no—
220C8—121no—
221C2×C3—41no—
222C2×C3—42no—
223C2×C3—41no—
224C2×C3—42no—
225C2×C3—41no—
226C22C2×A4⋊Q812yes—
227C22C2×C4⋊S411yescharacteristic
228C22#592411yescharacteristic
229C22#592612yes—
230C22#592711yescharacteristic
231C4—24no—
232C4—22no—
233C4—31no—
234C4—31no—
235C4—32no—
236C4—64no—
237C4—62no—
238C4—242no—
239C4—241no—
240C4—241no—
241C3—41noSylow
242C2#8528311yescharacteristic
243C2#8535112yes—
244C2#8535511yescharacteristic
245C2#8536112yes—
246C2#8537011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
Fitting#2276D3
hypercenter#95S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
31. C4×SL(2,𝔽3)Q8semidirect
89. C4×Q8Dic6semidirect

Maximal subgroups

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

subgroupindex
C2×#53892
C2×#53862
#731672
#730642
C4⋊#8592
#332043
#46564