Finite Group Databasebeta

← #11713681

Subgroups — #11713681

Order 768.

470 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 63 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5389C211yesmaximal, characteristic
3#73063C211yesmaximal, characteristic
4#73062C212yesmaximal
5#73059C212yesmaximal
6#25847—31nomaximal, Sylow
7Q8×SL(2,𝔽3)—22no—
8C2×C4×SL(2,𝔽3)C2212yes—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2211yescharacteristic
11C2×#859C2211yescharacteristic
12C2×#859C2212yes—
13#5389—22no—
14#5230—41nomaximal
15C2×Q82D311yescharacteristic
16C2×#501—31no—
17C2×#479—31no—
18#2584—31no—
19#2058—32no—
20#2057—32no—
21#1988—31no—
22#1985—31no—
23#1865—32no—
24#1863—32no—
25#1763—31no—
26C22×C3×Q8—41no—
27C22×SL(2,𝔽3)C2311yescharacteristic
28C2×Dic3⋊C4—41no—
29C2×Dic3⋊C4—42no—
30C4×SL(2,𝔽3)—22no—
31C4×SL(2,𝔽3)D412yes—
32#859—21no—
33#859—21no—
34#859Q814yescomplemented
35C3⋊#95—42no—
36C3⋊#95—41no—
37Q82—22no—
38Q82—62no—
39C2×C4⋊Q8—31no—
40C2×C4⋊Q8—32no—
41C2×#125—31no—
42C2×C4×Q8—31no—
43C2×C4×Q8C2×D312yes—
44C2×C4×Q8C2×D311yescharacteristic
45#501—64no—
46#479—62no—
47#479—62no—
48C2×#107—31no—
49C2×#107—32no—
50C2×#106—31no—
51C2×#106—32no—
52C2×C4⋊C8—31no—
53C2×#103—32no—
54C2×#103—31no—
55C2×#103—31no—
56C2×#97—31no—
57#398—61no—
58#395—61no—
59#389—32no—
60#389—32no—
61#389—62no—
62C4×C4⋊C4—31no—
63#340—32no—
64#340—32no—
65#340—32no—
66#336—32no—
67#336—32no—
68C2×C3×Q8—82no—
69C2×C3×Q8—84no—
70C22×C3×C4—42no—
71C22×C3×C4—41no—
72C22×Dic3—42no—
73C22×Dic3—41no—
74C22×Dic3—41no—
75C2×SL(2,𝔽3)C2×D411yescharacteristic
76C2×SL(2,𝔽3)C2×Q812yes—
77Dic3⋊C4—82no—
78Dic3⋊C4—84no—
79C22×Q8—61no—
80C22×Q8S411yescharacteristic
81C22×Q8C22×D311yescharacteristic
82C22×C8—32no—
83C22×C8—31no—
84C22×C8—31no—
85C4⋊Q8—34no—
86C4⋊Q8—62no—
87C4⋊Q8—64no—
88C4⋊Q8—64no—
89#125—62no—
90#125—62no—
91C4×Q8—22no—
92C4×Q8—32no—
93C4×Q8—62no—
94C4×Q8—62no—
95C4×Q8—62no—
96C4×Q8#6712yes—
97C2×C4⋊C4—31no—
98C2×C4⋊C4—32no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4—31no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—61no—
105C2×C4⋊C4—61no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4—62no—
108C2×C42—31no—
109C2×C42—32no—
110C2×C42—61no—
111#107—62no—
112#107—64no—
113#106—62no—
114#106—64no—
115C4⋊C8—62no—
116#103—34no—
117#103—34no—
118#103—61no—
119#103—61no—
120#103—61no—
121#103—61no—
122#97—62no—
123#95—62no—
124#95—61no—
125#95—62no—
126C23×C3—41no—
127C3×Q8—88no—
128C2×C3×C4—44no—
129C2×C3×C4—42no—
130C2×C3×C4—82no—
131C2×C3×C4—82no—
132C2×C3×C4—82no—
133C2×Dic3—44no—
134C2×Dic3—42no—
135C2×Dic3—81no—
136C2×Dic3—82no—
137C2×Dic3—82no—
138C2×Dic3—81no—
139C2×Dic3—81no—
140C2×Dic3—82no—
141SL(2,𝔽3)C4⋊Q811yescharacteristic
142C2×Q8—24no—
143C2×Q8—22no—
144C2×Q8—32no—
145C2×Q8—31no—
146C2×Q8—64no—
147C2×Q8—241no—
148C2×Q8—241no—
149C2×Q8D3×Q812yes—
150C2×Q8C2×#6711yescharacteristic
151C22×C4—31no—
152C22×C4—31no—
153C22×C4—32no—
154C22×C4—62no—
155C22×C4—61no—
156C22×C4—61no—
157C22×C4C2×S412yes—
158C22×C4C2×S411yescharacteristic
159C2×C8—34no—
160C2×C8—62no—
161C2×C8—62no—
162C2×C8—61no—
163C2×C8—61no—
164C2×C8—62no—
165C2×C8—62no—
166C4⋊C4—32no—
167C4⋊C4—34no—
168C4⋊C4—34no—
169C4⋊C4—64no—
170C4⋊C4—61no—
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—61no—
174C4⋊C4—61no—
175C4⋊C4—61no—
176C4⋊C4—61no—
177C4⋊C4—62no—
178C4⋊C4—122no—
179C4⋊C4—124no—
180C4⋊C4—122no—
181C4⋊C4—242no—
182C42—32no—
183C42—32no—
184C42—62no—
185C42—62no—
186C42—241no—
187C22×C3—42no—
188C22×C3—42no—
189C22×C3—41no—
190C22×C3—41no—
191C22×C3—41no—
192C3×C4—84no—
193C3×C4—82no—
194Dic3—84no—
195Dic3—82no—
196Dic3—161no—
197C23C22×S411yescharacteristic
198Q8—28no—
199Q8—32no—
200Q8—31no—
201Q8—61no—
202Q8—244no—
203Q8#94411yescharacteristic
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—32no—
208C2×C4—32no—
209C2×C4—31no—
210C2×C4—31no—
211C2×C4—32no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—64no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4—122no—
221C2×C4—121no—
222C2×C4—121no—
223C2×C4—121no—
224C2×C4—121no—
225C2×C4#98314yes—
226C2×C4GL(2,ℤ/4)12yes—
227C8—64no—
228C8—121no—
229C8—121no—
230C2×C3—41no—
231C2×C3—41no—
232C2×C3—42no—
233C2×C3—42no—
234C2×C3—41no—
235C22C2×#98312yes—
236C22Q8×S412yes—
237C22#592411yescharacteristic
238C22#592511yescharacteristic
239C22C2×GL(2,ℤ/4)11yescharacteristic
240C4—24no—
241C4—22no—
242C4—32no—
243C4—31no—
244C4—31no—
245C4—62no—
246C4—122no—
247C4—124no—
248C4—241no—
249C4—241no—
250C3—41noSylow
251C2#8542612yes—
252C2#8543112yes—
253C2#8550211yescharacteristic
254C2#8550711yescharacteristic
255C2#8552911yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC2×Q82D3
hypercenterC22×Q8S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
34. #859Q8semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×SL(2,𝔽3)2
C2×#53892
#730632
#730622
#730592
#258473
#52304