Finite Group Databasebeta

← #11714064

Subgroups — #11714064

Order 768.

418 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 55 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×C4⋊#226C211yesmaximal, characteristic
2C2×#5389C211yesmaximal, characteristic
3#73167C211yesmaximal, characteristic
4#73061C212yesmaximal
5#73061C212yesmaximal
6#33041—31nomaximal, Sylow
7C22×#226C2212yes—
8C2×C4×SL(2,𝔽3)C2212yes—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2211yescharacteristic
11C2×#859C2211yescharacteristic
12C4⋊#226—22no—
13#5389—22no—
14#4659—41nomaximal
15C2×#479—31no—
16C2×#462—31no—
17#2642—31no—
18#2276D311yescharacteristic
19#1989—31no—
20#1981—31no—
21#1928—32no—
22#1927—32no—
23#1878—32no—
24#1878—32no—
25#1832—31no—
26C22×SL(2,𝔽3)C2311yescharacteristic
27C2×#226—22no—
28C2×#226—22no—
29C2×#226—22no—
30C2×C4⋊Dic3—41no—
31C2×Dic3⋊C4—41no—
32C4×SL(2,𝔽3)—22no—
33C4×SL(2,𝔽3)D412yes—
34#859—21no—
35#859—21no—
36C3×#95—41no—
37C3⋊#95—42no—
38C3⋊#95—42no—
39C22×Q16—32no—
40C2×C4⋊Q8—31no—
41C2×#125—31no—
42C2×C4×Q8C2×D312yes—
43C2×C4×Q8C2×D311yescharacteristic
44#479—62no—
45#479—62no—
46#462—62no—
47#462—62no—
48C2×#107—31no—
49C2×#106—31no—
50C2×C4⋊C8—31no—
51C2×C4⋊C8—31no—
52C2×#103—32no—
53C2×#103—32no—
54C2×#103—32no—
55C2×#103—31no—
56C2×#103—31no—
57#400—61no—
58#400—62no—
59#400—61no—
60#391—32no—
61#391—32no—
62C4×C4⋊C4—32no—
63C4×C4⋊C4—31no—
64#376—31no—
65#340—32no—
66#340—32no—
67#336—32no—
68#336—32no—
69C22×C3×C4—41no—
70C22×C3×C4—42no—
71C22×Dic3—41no—
72C22×Dic3—42no—
73C22×Dic3—41no—
74C2×SL(2,𝔽3)C2×D411yescharacteristic
75C2×SL(2,𝔽3)#4112yes—
76#226—42no—
77C4⋊Dic3—82no—
78Dic3⋊C4—82no—
79C22×Q8—32no—
80C22×Q8C22×D311yescharacteristic
81C2×Q16—62no—
82C2×Q16—62no—
83C2×Q16—62no—
84C2×Q16—62no—
85C2×Q16—62no—
86C2×Q16—62no—
87C22×C8—32no—
88C22×C8—31no—
89C22×C8—31no—
90C4⋊Q8—62no—
91C4⋊Q8—62no—
92#125—62no—
93#125—62no—
94C4×Q8—22no—
95C4×Q8—32no—
96C4×Q8—62no—
97C4×Q8D1212yes—
98C2×C4⋊C4—32no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4—61no—
103C2×C4⋊C4—61no—
104C2×C4⋊C4—61no—
105C2×C4⋊C4—61no—
106C2×C42—31no—
107C2×C42—32no—
108C2×C42—61no—
109#107—62no—
110#106—62no—
111C4⋊C8—62no—
112C4⋊C8—62no—
113#103—64no—
114#103—64no—
115#103—64no—
116#103—61no—
117#103—61no—
118#103—61no—
119#103—61no—
120#95—32no—
121#95—31no—
122#95—62no—
123#95—62no—
124#95—62no—
125#95—62no—
126#95S411yescharacteristic
127C23×C3—41no—
128C2×C3×C4—42no—
129C2×C3×C4—82no—
130C2×C3×C4—82no—
131C2×C3×C4—82no—
132C2×C3×C4—82no—
133C2×Dic3—42no—
134C2×Dic3—81no—
135C2×Dic3—81no—
136C2×Dic3—81no—
137C2×Dic3—82no—
138C2×Dic3—82no—
139C2×Dic3—82no—
140C2×Dic3—82no—
141SL(2,𝔽3)#12411yescharacteristic
142C2×Q8—31no—
143C2×Q8—32no—
144C2×Q8—62no—
145C2×Q8—62no—
146C2×Q8—62no—
147C2×Q8—62no—
148C2×Q8—62no—
149C2×Q8—62no—
150C2×Q8C2×D1211yescharacteristic
151C2×Q8#23512yes—
152C22×C4—31no—
153C22×C4—31no—
154C22×C4—32no—
155C22×C4—61no—
156C22×C4—61no—
157C22×C4—62no—
158C22×C4C2×S412yes—
159C22×C4C2×S411yescharacteristic
160Q16—122no—
161Q16—122no—
162Q16—122no—
163Q16—122no—
164C2×C8—61no—
165C2×C8—61no—
166C2×C8—62no—
167C2×C8—62no—
168C2×C8—62no—
169C2×C8—62no—
170C2×C8—62no—
171C4⋊C4—32no—
172C4⋊C4—34no—
173C4⋊C4—61no—
174C4⋊C4—62no—
175C4⋊C4—61no—
176C4⋊C4—61no—
177C4⋊C4—61no—
178C4⋊C4—61no—
179C4⋊C4—122no—
180C4⋊C4—122no—
181C4⋊C4—122no—
182C4⋊C4—122no—
183C42—32no—
184C42—32no—
185C42—62no—
186C42—62no—
187C42—241no—
188C22×C3—41no—
189C22×C3—42no—
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—42no—
193C3×C4—82no—
194C3×C4—162no—
195Dic3—82no—
196Dic3—161no—
197Dic3—162no—
198C23C22×S411yescharacteristic
199Q8—31no—
200Q8—32no—
201Q8—61no—
202Q8—122no—
203Q8—122no—
204Q8—122no—
205Q8—122no—
206Q8#87511yescharacteristic
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—31no—
212C2×C4—31no—
213C2×C4—32no—
214C2×C4—32no—
215C2×C4—32no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—121no—
222C2×C4—121no—
223C2×C4—122no—
224C2×C4—122no—
225C2×C4—122no—
226C2×C4—121no—
227C2×C4—122no—
228C2×C4—121no—
229C2×C4C4⋊S412yes—
230C8—121no—
231C8—121no—
232C8—122no—
233C2×C3—41no—
234C2×C3—42no—
235C2×C3—41no—
236C2×C3—42no—
237C2×C3—41no—
238C22C2×C4⋊S411yescharacteristic
239C22#591212yes—
240C22#592411yescharacteristic
241C22#592711yescharacteristic
242C22#592712yes—
243C4—22no—
244C4—31no—
245C4—31no—
246C4—32no—
247C4—42no—
248C4—62no—
249C4—122no—
250C4—242no—
251C4—241no—
252C4—241no—
253C3—41noSylow
254C2#8529011yescharacteristic
255C2#8535212yes—
256C2#8535911yescharacteristic
257C2#8536212yes—
258C2#8537011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
Fitting#2276D3
hypercenter#95S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Maximal subgroups

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

subgroupindex
C2×C4⋊#2262
C2×#53892
#731672
#730612
#330413
#46594