Finite Group Databasebeta

← #11713711

Subgroups — #11713711

Order 768.

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

normal subgroups only

This group has 435 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#3503⋊C3C211yesmaximal, characteristic
2#85336C211yesmaximal, characteristic
3#73077C214yesmaximal
4C4×GU(2,𝔽3)C211yesmaximal, characteristic
5#25923—31nomaximal, Sylow
6C2×#984C2212yes—
7Q8×SL(2,𝔽3)—21no—
8#557⋊C3—21no—
9#520⋊C3C2212yes—
10C4×#41⋊C3C2211yescharacteristic
11C2×GU(2,𝔽3)C2212yes—
12C4⋊#226—21no—
13C4⋊#226—21no—
14#5093—41nomaximal
15#3503D311yescharacteristic
16#3139—31no—
17#3039—31no—
18#2862—31no—
19#2028—34no—
20#1997—31no—
21#1994—31no—
22#1882—34no—
23C4×#104—31no—
24C2×#41⋊C3C2311yescharacteristic
25#984—24no—
26C2×#226—22no—
27C3×C4⋊Q8—41no—
28#869—41no—
29C4×SL(2,𝔽3)—22no—
30C4×SL(2,𝔽3)D411yescharacteristic
31C4×SL(2,𝔽3)D411yescharacteristic
32GU(2,𝔽3)—22no—
33#810—44no—
34C4×C3⋊C8—41no—
35C2×#137—32no—
36Q82—21no—
37#557—21no—
38#554—61no—
39D4×Q8—61no—
40#544—32no—
41#533—31no—
42C2×C4⋊Q8—31no—
43C2×C4⋊Q8—31no—
44#520C2×D312yes—
45C4×#41C2×D311yescharacteristic
46#491—62no—
47#487—61no—
48#487—61no—
49#462—61no—
50#462—61no—
51#461—61no—
52#461—61no—
53#423—31no—
54C2×#104—32no—
55#419—34no—
56#419—32no—
57C4×M16—31no—
58#391—34no—
59#391—62no—
60#382—62no—
61C43—31no—
62#343—34no—
63#339—34no—
64#339—32no—
65C2×C3×Q8—81no—
66C2×Dic6—42no—
67#41⋊C3#4112yes—
68C2×SL(2,𝔽3)C2×D411yescharacteristic
69#226—42no—
70C3×C4⋊C4—42no—
71C3×C4⋊C4—81no—
72C3×C42—41no—
73C4⋊Dic3—81no—
74C4⋊Dic3—81no—
75C2×C3⋊C8—42no—
76C2×#41C22×D311yescharacteristic
77C22×Q8—32no—
78C22×Q8—32no—
79#137—64no—
80#137—64no—
81C2×Q16—62no—
82C2×SD16—62no—
83C2×M16—32no—
84C2×M16—32no—
85C4⋊Q8—31no—
86C4⋊Q8—61no—
87C4⋊Q8—61no—
88C4⋊Q8—61no—
89C4⋊Q8—61no—
90C4⋊Q8—61no—
91C4⋊Q8—61no—
92C4⋊Q8—62no—
93C4⋊Q8—61no—
94C4⋊Q8S411yescharacteristic
95#125—32no—
96#125—62no—
97C22⋊Q8—61no—
98C22⋊Q8—61no—
99C22⋊Q8—64no—
100C4×Q8—22no—
101C4×Q8—61no—
102C4×Q8—61no—
103C4×Q8#6711yescharacteristic
104C4×Q8#6711yescharacteristic
105C4×D4—31no—
106C4×D4—31no—
107C4×D4—62no—
108#117—31no—
109#117—32no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—32no—
112C2×C4⋊C4—61no—
113C2×C4⋊C4—61no—
114C2×C42—31no—
115C2×C42—32no—
116C2×C42—61no—
117C2×C42—61no—
118C4⋊C8—61no—
119C4⋊C8—61no—
120#104—62no—
121#104—62no—
122#103—62no—
123#103—62no—
124#103—64no—
125#103—64no—
126#97—61no—
127C4×C8—61no—
128#95—68no—
129C3×Q8—162no—
130C2×C3×C4—41no—
131C2×C3×C4—41no—
132C2×C3×C4—41no—
133C2×C3×C4—82no—
134C2×Dic3—82no—
135Dic6—84no—
136SL(2,𝔽3)#12411yescharacteristic
137C3⋊C8—82no—
138#41—32no—
139#41#23912yes—
140C2×Q8—21no—
141C2×Q8—62no—
142C2×Q8—62no—
143C2×Q8—64no—
144C2×Q8—64no—
145C2×Q8—61no—
146C2×Q8—62no—
147C2×Q8—124no—
148C2×Q8C2×#6711yescharacteristic
149C2×D4—31no—
150C22×C4—31no—
151C22×C4—31no—
152C22×C4—31no—
153C22×C4—34no—
154C22×C4—62no—
155C22×C4—62no—
156Q16—122no—
157Q16—122no—
158SD16—124no—
159M16—62no—
160M16—62no—
161M16—64no—
162C2×C8—62no—
163C2×C8—62no—
164C4⋊C4—21no—
165C4⋊C4—31no—
166C4⋊C4—31no—
167C4⋊C4—32no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—61no—
173C4⋊C4—121no—
174C4⋊C4—121no—
175C4⋊C4—121no—
176C4⋊C4—121no—
177C4⋊C4C2×S412yes—
178C22⋊C4—62no—
179C22⋊C4—61no—
180C42—31no—
181C42—31no—
182C42—31no—
183C42—62no—
184C42—62no—
185C42—62no—
186C42—64no—
187C42—122no—
188C42C2×S411yescharacteristic
189C22×C3—41no—
190C3×C4—42no—
191C3×C4—42no—
192C3×C4—42no—
193C3×C4—162no—
194Dic3—162no—
195C23—31no—
196Q8—31no—
197Q8—42no—
198Q8—122no—
199Q8—122no—
200Q8—122no—
201Q8—124no—
202Q8—242no—
203Q8#94711yescharacteristic
204D4—32no—
205D4—61no—
206C2×C4—22no—
207C2×C4—31no—
208C2×C4—32no—
209C2×C4—32no—
210C2×C4—31no—
211C2×C4—31no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—64no—
215C2×C4—62no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4—124no—
219C2×C4—124no—
220C2×C4—122no—
221C2×C4GL(2,ℤ/4)11yescharacteristic
222C2×C4GL(2,ℤ/4)11yescharacteristic
223C2×C4C22×S411yescharacteristic
224C8—122no—
225C8—122no—
226C2×C3—41no—
227C2×C3—41no—
228C2×C3—41no—
229C22—32no—
230C22—61no—
231C22—61no—
232C22C2×GL(2,ℤ/4)11yescharacteristic
233C4—31no—
234C4—31no—
235C4—42no—
236C4—61no—
237C4—61no—
238C4—124no—
239C4—122no—
240C4—242no—
241C4#542812yes—
242C4#543012yes—
243C4#591912yes—
244C3—41noSylow
245C2—62no—
246C2#8550811yescharacteristic
247C2C2×#542811yescharacteristic
248C2C2×#543011yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, prime radical, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3503D3
hypercenterC4⋊Q8S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Maximal subgroups

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

subgroupindex
#3503⋊C32
#853362
#730772
C4×GU(2,𝔽3)2
#259233
#50934