Finite Group Databasebeta

← #11713683

Subgroups — #11713683

Order 768.

456 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 61 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×#5391C211yesmaximal, characteristic
3#73063C211yesmaximal, characteristic
4#73062C212yesmaximal
5#73059C212yesmaximal
6#25849—31nomaximal, Sylow
7#557⋊C3—22no—
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×#859C2212yes—
12C2×#859C2212yes—
13#5391—22no—
14#4981—41nomaximal
15C2×#557D311yescharacteristic
16C2×#501—31no—
17C2×#477—31no—
18#2546—31no—
19#2059—32no—
20#2057—32no—
21#1989—31no—
22#1985—31no—
23#1865—32no—
24#1863—32no—
25#1763—31no—
26C22×SL(2,𝔽3)C2311yescharacteristic
27C2×C3×C4⋊C4—41no—
28C2×C4⋊Dic3—41no—
29C2×Dic3⋊C4—42no—
30C4×SL(2,𝔽3)—21no—
31C4×SL(2,𝔽3)—21no—
32C4×SL(2,𝔽3)D412yes—
33#859—22no—
34#859Q814yescomplemented
35C3⋊#95—42no—
36C3⋊#95—41no—
37#557—22no—
38#557—62no—
39C2×C4⋊Q8—31no—
40C2×C4⋊Q8—31no—
41C2×C4⋊Q8—61no—
42C2×#125—31no—
43C2×#125—31no—
44C2×C4×Q8—31no—
45C2×C4×Q8C2×D311yescharacteristic
46C2×C4×Q8C2×D311yescharacteristic
47C2×C4×Q8C2×D311yescharacteristic
48#501—64no—
49#477—62no—
50#477—62no—
51C2×#107—32no—
52C2×#107—32no—
53C2×#106—32no—
54C2×C4⋊C8—31no—
55C2×#103—32no—
56C2×#103—32no—
57C2×#97—31no—
58#398—61no—
59#398—61no—
60#391—61no—
61#389—32no—
62#389—32no—
63C4×C4⋊C4—31no—
64#340—32no—
65#340—32no—
66#340—32no—
67#336—32no—
68#336—32no—
69C22×C3×C4—41no—
70C22×C3×C4—41no—
71C22×C3×C4—41no—
72C22×Dic3—42no—
73C22×Dic3—42no—
74C2×SL(2,𝔽3)C2×D411yescharacteristic
75C2×SL(2,𝔽3)C2×Q812yes—
76C3×C4⋊C4—82no—
77C4⋊Dic3—82no—
78Dic3⋊C4—84no—
79C22×Q8—31no—
80C22×Q8—61no—
81C22×Q8C22×D311yescharacteristic
82C22×C8—32no—
83C22×C8—32no—
84C4⋊Q8—62no—
85C4⋊Q8—62no—
86C4⋊Q8—62no—
87C4⋊Q8—62no—
88C4⋊Q8—124no—
89#125—34no—
90#125—62no—
91#125—62no—
92#125—62no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—32no—
96C4×Q8—62no—
97C4×Q8—62no—
98C4×Q8—61no—
99C4×Q8—61no—
100C4×Q8#6712yes—
101C2×C4⋊C4—31no—
102C2×C4⋊C4—31no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4—31no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—61no—
109C2×C4⋊C4—62no—
110C2×C4⋊C4—61no—
111C2×C4⋊C4—62no—
112C2×C4⋊C4S411yescharacteristic
113C2×C42—31no—
114C2×C42—31no—
115C2×C42—31no—
116C2×C42—61no—
117#107—64no—
118#107—64no—
119#106—64no—
120C4⋊C8—62no—
121#103—34no—
122#103—34no—
123#103—62no—
124#103—62no—
125#97—62no—
126#95—62no—
127#95—61no—
128#95—62no—
129C23×C3—41no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4—81no—
133C2×C3×C4—81no—
134C2×C3×C4—82no—
135C2×C3×C4—81no—
136C2×C3×C4—82no—
137C2×Dic3—44no—
138C2×Dic3—82no—
139C2×Dic3—82no—
140C2×Dic3—82no—
141C2×Dic3—82no—
142C2×Dic3—82no—
143SL(2,𝔽3)C4⋊Q811yescharacteristic, complemented
144C2×Q8—32no—
145C2×Q8—31no—
146C2×Q8—62no—
147C2×Q8—62no—
148C2×Q8—121no—
149C2×Q8—122no—
150C2×Q8—122no—
151C2×Q8—122no—
152C2×Q8D3×Q812yes—
153C2×Q8C2×#6711yescharacteristic
154C22×C4—31no—
155C22×C4—31no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—62no—
159C22×C4—62no—
160C22×C4C2×S411yescharacteristic
161C22×C4C2×S411yescharacteristic
162C22×C4C2×S411yescharacteristic
163C2×C8—34no—
164C2×C8—62no—
165C2×C8—62no—
166C2×C8—62no—
167C2×C8—64no—
168C4⋊C4—22no—
169C4⋊C4—32no—
170C4⋊C4—34no—
171C4⋊C4—32no—
172C4⋊C4—32no—
173C4⋊C4—62no—
174C4⋊C4—61no—
175C4⋊C4—61no—
176C4⋊C4—61no—
177C4⋊C4—62no—
178C4⋊C4—64no—
179C4⋊C4—62no—
180C4⋊C4—62no—
181C4⋊C4—62no—
182C4⋊C4—124no—
183C4⋊C4—124no—
184C4⋊C4—122no—
185C42—32no—
186C42—32no—
187C42—61no—
188C42—61no—
189C42—61no—
190C42—61no—
191C42—122no—
192C22×C3—42no—
193C22×C3—41no—
194C22×C3—41no—
195C22×C3—42no—
196C22×C3—41no—
197C3×C4—82no—
198C3×C4—82no—
199C3×C4—161no—
200Dic3—84no—
201Dic3—162no—
202C23C22×S411yescharacteristic
203Q8—32no—
204Q8—31no—
205Q8—61no—
206Q8—124no—
207Q8—244no—
208Q8#94411yescharacteristic
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—32no—
215C2×C4—32no—
216C2×C4—31no—
217C2×C4—31no—
218C2×C4—32no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—64no—
222C2×C4—61no—
223C2×C4—61no—
224C2×C4—62no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—122no—
228C2×C4—122no—
229C2×C4—122no—
230C2×C4—122no—
231C2×C4—122no—
232C2×C4#98612yes—
233C2×C4GL(2,ℤ/4)12yes—
234C8—64no—
235C8—122no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—42no—
239C2×C3—42no—
240C2×C3—41no—
241C22Q8×S412yes—
242C22C2×#98611yescharacteristic
243C22#592311yescharacteristic
244C22#592712yes—
245C22C2×GL(2,ℤ/4)11yescharacteristic
246C4—22no—
247C4—22no—
248C4—32no—
249C4—31no—
250C4—31no—
251C4—41no—
252C4—62no—
253C4—121no—
254C4—121no—
255C4—124no—
256C4—242no—
257C3—41noSylow
258C2#8541612yes—
259C2#8542412yes—
260C2#8550211yescharacteristic
261C2#8552311yescharacteristic
262C2#8553511yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC2×#557D3
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
34. #859Q8semidirect
143. SL(2,𝔽3)C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#557⋊C32
C2×#53912
#730632
#730622
#730592
#258493
#49814