Finite Group Databasebeta

← #11713724

Subgroups — #11713724

Order 768.

450 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 57 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×#557⋊C3C211yesmaximal, characteristic
2C2×C4⋊#226C211yesmaximal, characteristic
3#73063C211yesmaximal, characteristic
4#73061C212yesmaximal
5#73061C212yesmaximal
6#25943—31nomaximal, Sylow
7C22×#226C2212yes—
8#557⋊C3—22no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×#859C2212yes—
13C4⋊#226—22no—
14#4986—41nomaximal
15C2×#557D311yescharacteristic
16C2×#491—31no—
17C2×#462—31no—
18#2568—31no—
19#2026—32no—
20#2024—32no—
21#1989—31no—
22#1985—31no—
23#1878—32no—
24#1878—32no—
25#1763—31no—
26C22×SL(2,𝔽3)C2311yescharacteristic
27C2×#226—22no—
28C2×#226—24no—
29C2×C3×C4⋊C4—41no—
30C2×C4⋊Dic3—41no—
31C4×SL(2,𝔽3)—21no—
32C4×SL(2,𝔽3)—21no—
33C4×SL(2,𝔽3)D412yes—
34#859—22no—
35C3⋊#95—42no—
36C3⋊#95—42no—
37C3⋊#95—41no—
38C22×Q16—32no—
39#557—22no—
40#557—62no—
41C2×C4⋊Q8—31no—
42C2×C4⋊Q8—31no—
43C2×#125—31no—
44C2×#125—31no—
45C2×C4×Q8—31no—
46C2×C4×Q8C2×D311yescharacteristic
47C2×C4×Q8C2×D311yescharacteristic
48C2×C4×Q8C2×D311yescharacteristic
49#491—64no—
50#462—62no—
51#462—62no—
52C2×C4⋊C8—31no—
53C2×#103—32no—
54C2×#103—32no—
55C2×#103—32no—
56C2×#103—32no—
57C2×#97—31no—
58#400—61no—
59#400—61no—
60#391—32no—
61#391—32no—
62#391—61no—
63#382—61no—
64C4×C4⋊C4—31no—
65#340—32no—
66#340—32no—
67#340—32no—
68#336—32no—
69#336—32no—
70C22×C3×C4—41no—
71C22×C3×C4—41no—
72C22×C3×C4—41no—
73C22×Dic3—42no—
74C22×Dic3—42no—
75C2×SL(2,𝔽3)C2×D411yescharacteristic
76C2×SL(2,𝔽3)#4112yes—
77#226—42no—
78C3×C4⋊C4—82no—
79C4⋊Dic3—82no—
80C22×Q8—31no—
81C22×Q8—32no—
82C22×Q8C22×D311yescharacteristic
83C2×Q16—64no—
84C2×Q16—62no—
85C2×Q16—64no—
86C2×Q16—62no—
87C22×C8—32no—
88C22×C8—32no—
89C4⋊Q8—62no—
90C4⋊Q8—62no—
91C4⋊Q8—62no—
92C4⋊Q8—62no—
93#125—34no—
94#125—62no—
95#125—62no—
96#125—62no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—32no—
100C4×Q8—62no—
101C4×Q8—62no—
102C4×Q8—61no—
103C4×Q8—61no—
104C4×Q8#6712yes—
105C2×C4⋊C4—31no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—61no—
112C2×C4⋊C4—61no—
113C2×C4⋊C4—61no—
114C2×C4⋊C4S411yescharacteristic
115C2×C42—31no—
116C2×C42—31no—
117C2×C42—31no—
118C2×C42—61no—
119C4⋊C8—62no—
120#103—64no—
121#103—64no—
122#103—64no—
123#103—62no—
124#103—62no—
125#97—62no—
126#95—62no—
127#95—62no—
128#95—61no—
129#95—62no—
130#95—62no—
131C23×C3—41no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4—81no—
135C2×C3×C4—81no—
136C2×C3×C4—82no—
137C2×C3×C4—81no—
138C2×C3×C4—82no—
139C2×Dic3—82no—
140C2×Dic3—82no—
141C2×Dic3—84no—
142C2×Dic3—82no—
143C2×Dic3—82no—
144SL(2,𝔽3)#12411yescharacteristic
145C2×Q8—32no—
146C2×Q8—31no—
147C2×Q8—62no—
148C2×Q8—62no—
149C2×Q8—62no—
150C2×Q8—64no—
151C2×Q8—64no—
152C2×Q8—62no—
153C2×Q8—121no—
154C2×Q8#23912yes—
155C2×Q8C2×#6711yescharacteristic
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—62no—
161C22×C4—62no—
162C22×C4C2×S411yescharacteristic
163C22×C4C2×S411yescharacteristic
164C22×C4C2×S411yescharacteristic
165Q16—124no—
166Q16—122no—
167Q16—122no—
168C2×C8—62no—
169C2×C8—62no—
170C2×C8—62no—
171C2×C8—62no—
172C2×C8—64no—
173C4⋊C4—22no—
174C4⋊C4—32no—
175C4⋊C4—32no—
176C4⋊C4—32no—
177C4⋊C4—62no—
178C4⋊C4—61no—
179C4⋊C4—61no—
180C4⋊C4—61no—
181C4⋊C4—64no—
182C4⋊C4—62no—
183C4⋊C4—62no—
184C4⋊C4—62no—
185C4⋊C4—122no—
186C4⋊C4—122no—
187C42—32no—
188C42—32no—
189C42—61no—
190C42—61no—
191C42—61no—
192C42—61no—
193C42—122no—
194C22×C3—42no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—42no—
198C22×C3—41no—
199C3×C4—82no—
200C3×C4—82no—
201C3×C4—161no—
202Dic3—162no—
203Dic3—162no—
204C23C22×S411yescharacteristic
205Q8—32no—
206Q8—31no—
207Q8—61no—
208Q8—124no—
209Q8—122no—
210Q8—124no—
211Q8—122no—
212Q8#94711yescharacteristic
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—32no—
219C2×C4—32no—
220C2×C4—31no—
221C2×C4—31no—
222C2×C4—32no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—62no—
228C2×C4—61no—
229C2×C4—61no—
230C2×C4—122no—
231C2×C4—124no—
232C2×C4—122no—
233C2×C4—122no—
234C2×C4—122no—
235C2×C4#98412yes—
236C2×C4GL(2,ℤ/4)12yes—
237C8—122no—
238C8—122no—
239C2×C3—41no—
240C2×C3—41no—
241C2×C3—41no—
242C2×C3—42no—
243C2×C3—42no—
244C22#591912yes—
245C22C2×#98411yescharacteristic
246C22#592311yescharacteristic
247C22#592712yes—
248C22C2×GL(2,ℤ/4)11yescharacteristic
249C4—22no—
250C4—22no—
251C4—31no—
252C4—32no—
253C4—31no—
254C4—41no—
255C4—62no—
256C4—121no—
257C4—121no—
258C4—242no—
259C4—242no—
260C3—41noSylow
261C2#8541712yes—
262C2#8542512yes—
263C2#8550811yescharacteristic
264C2#8551711yescharacteristic
265C2#8553511yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC2×#557D3
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Maximal subgroups

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

subgroupindex
C2×#557⋊C32
C2×C4⋊#2262
#730632
#730612
#259433
#49864