Finite Group Databasebeta

← #11713672

Subgroups — #11713672

Order 768.

401 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 51 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#3504⋊C3C211yesmaximal, characteristic
2#85339C211yesmaximal, characteristic
3#73076C212yesmaximal
4#73075C212yesmaximal
5C4×GU(2,𝔽3)C211yesmaximal, characteristic
6#25831—31nomaximal, Sylow
7#557⋊C3—22no—
8#520⋊C3C2212yes—
9C4×#41⋊C3C2211yescharacteristic
10#5426C2212yes—
11C2×GU(2,𝔽3)C2211yescharacteristic
12C2×GU(2,𝔽3)C2211yescharacteristic
13#5389—22no—
14#5065—41nomaximal
15#3504D311yescharacteristic
16#3166—31no—
17#3080—31no—
18#2859—31no—
19#2061—32no—
20#2060—32no—
21#1998—31no—
22#1994—31no—
23#1860—32no—
24#1859—32no—
25C4×#104—31no—
26C2×#41⋊C3C2311yescharacteristic
27C3×#125—41no—
28#870—41no—
29C4×SL(2,𝔽3)—22no—
30C4×SL(2,𝔽3)D412yes—
31GU(2,𝔽3)Q812yescomplemented
32GU(2,𝔽3)Q812yescomplemented
33#859—22no—
34#808—42no—
35#807—42no—
36C4×C3⋊C8—41no—
37#557—22no—
38#554—62no—
39#544—32no—
40#533—31no—
41C2×#125—31no—
42C2×#125—31no—
43#520C2×D312yes—
44C4×#41C2×D311yescharacteristic
45#501—62no—
46#499—61no—
47#499—61no—
48#479—62no—
49#478—62no—
50#428—32no—
51#428—32no—
52#428—32no—
53#423—31no—
54C2×#104—31no—
55C2×#104—31no—
56#417—32no—
57C4×M16—31no—
58#389—64no—
59#384—32no—
60#382—32no—
61C43—31no—
62#343—32no—
63#343—32no—
64#339—31no—
65#339—32no—
66#339—31no—
67#339—32no—
68#41⋊C3C2×Q811yescharacteristic
69#41⋊C3C2×Q811yescharacteristic
70C2×SL(2,𝔽3)C2×D411yescharacteristic
71C3×C4⋊C4—42no—
72C3×C4⋊C4—82no—
73C3×C42—41no—
74C4⋊Dic3—42no—
75Dic3⋊C4—82no—
76C2×C3⋊C8—41no—
77C2×C3⋊C8—41no—
78C2×#41C22×D311yescharacteristic
79C2×M16—31no—
80C2×M16—31no—
81C2×M16—32no—
82C4⋊Q8—32no—
83C4⋊Q8—62no—
84#125—31no—
85#125—61no—
86#125—62no—
87#125—62no—
88#125—61no—
89#125—62no—
90#125—61no—
91#125S411yescharacteristic
92C22⋊Q8—62no—
93C22⋊Q8—64no—
94C4×Q8—22no—
95C4×Q8—62no—
96C4×Q8#6712yes—
97C4×D4—32no—
98C4×D4—62no—
99#117—31no—
100#117—32no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—34no—
103C2×C4⋊C4—32no—
104C2×C4⋊C4—62no—
105C2×C4⋊C4—64no—
106C2×C42—31no—
107C2×C42—31no—
108C2×C42—31no—
109C2×C42—62no—
110#107—62no—
111#107—62no—
112#107—62no—
113#106—62no—
114#106—62no—
115#106—62no—
116C4⋊C8—62no—
117#104—32no—
118#104—32no—
119#104—32no—
120#104—32no—
121#103—62no—
122#102—62no—
123#97—61no—
124C4×C8—61no—
125#95—64no—
126C2×C3×C4—42no—
127C2×C3×C4—41no—
128C2×C3×C4—82no—
129C2×Dic3—82no—
130SL(2,𝔽3)C4⋊Q811yescharacteristic
131C3⋊C8—42no—
132C3⋊C8—42no—
133#41—31no—
134#41—31no—
135#41D3×Q811yescharacteristic
136#41D3×Q811yescharacteristic
137C2×Q8—62no—
138C2×Q8C2×#6711yescharacteristic
139C2×D4—31no—
140C22×C4—31no—
141C22×C4—32no—
142C22×C4—34no—
143C22×C4—61no—
144C22×C4—61no—
145C22×C4—62no—
146M16—32no—
147M16—32no—
148M16—61no—
149M16—61no—
150M16—62no—
151M16—62no—
152C2×C8—61no—
153C2×C8—61no—
154C2×C8—62no—
155C4⋊C4—22no—
156C4⋊C4—32no—
157C4⋊C4—32no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—64no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—122no—
165C4⋊C4—128no—
166C4⋊C4—122no—
167C4⋊C4C2×S412yes—
168C22⋊C4—62no—
169C22⋊C4—61no—
170C42—32no—
171C42—31no—
172C42—32no—
173C42—32no—
174C42—62no—
175C42—61no—
176C42—61no—
177C42—124no—
178C42C2×S411yescharacteristic
179C22×C3—41no—
180C3×C4—41no—
181C3×C4—41no—
182C3×C4—82no—
183C3×C4—162no—
184Dic3—162no—
185C23—31no—
186Q8—31no—
187Q8—242no—
188Q8#94411yescharacteristic
189D4—31no—
190D4—31no—
191D4—61no—
192C2×C4—22no—
193C2×C4—31no—
194C2×C4—31no—
195C2×C4—31no—
196C2×C4—31no—
197C2×C4—31no—
198C2×C4—32no—
199C2×C4—62no—
200C2×C4—64no—
201C2×C4—62no—
202C2×C4—62no—
203C2×C4—122no—
204C2×C4—122no—
205C2×C4—124no—
206C2×C4—124no—
207C2×C4—122no—
208C2×C4GL(2,ℤ/4)12yes—
209C2×C4C22×S411yescharacteristic
210C8—62no—
211C8—62no—
212C8—122no—
213C2×C3—42no—
214C2×C3—41no—
215C22—31no—
216C22—31no—
217C22—62no—
218C22C2×GL(2,ℤ/4)11yescharacteristic
219C4—22no—
220C4—31no—
221C4—31no—
222C4—42no—
223C4—64no—
224C4—62no—
225C4—122no—
226C4—122no—
227C4—242no—
228C4Q8×S411yescharacteristic
229C4Q8×S411yescharacteristic
230C3—41noSylow
231C2—61no—
232C2—61no—
233C2#8550211yescharacteristic
234C2#8554412yes—
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, prime radical, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3504D3
hypercenter#125S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
31. GU(2,𝔽3)Q8semidirect
32. GU(2,𝔽3)Q8semidirect

Maximal subgroups

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

subgroupindex
#3504⋊C32
#853392
#730762
#730752
C4×GU(2,𝔽3)2
#258313
#50654