Finite Group Databasebeta

← #11714110

Subgroups — #11714110

Order 768.

433 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 75 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×#5389C212yesmaximal
2C42×SL(2,𝔽3)C211yesmaximal, characteristic
3#73064C212yesmaximal
4C4⋊#859C211yesmaximal, characteristic
5C4⋊#859C211yesmaximal, characteristic
6#33200—31nomaximal, Sylow
7C2×C4×SL(2,𝔽3)C2211yescharacteristic
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2212yes—
11C2×#859C2212yes—
12#5389—24no—
13#4928—41nomaximal
14C2×#479—32no—
15#2859—31no—
16C42×Q8D311yescharacteristic
17#1989—32no—
18#1949—32no—
19#1948—31no—
20#1947—31no—
21#1842—31no—
22#1805—32no—
23#1803—31no—
24#1803—31no—
25C22×SL(2,𝔽3)C2311yescharacteristic
26C2×C3×C42—41no—
27C2×C4⋊Dic3—41no—
28C2×C4⋊Dic3—41no—
29C2×Dic3⋊C4—42no—
30C4×SL(2,𝔽3)D412yes—
31C4×SL(2,𝔽3)Q812yes—
32C4×SL(2,𝔽3)Q812yes—
33#859—22no—
34#859—22no—
35C3⋊#95—42no—
36C2×#125—32no—
37C2×C4×Q8C2×D311yescharacteristic
38C2×C4×Q8C2×D311yescharacteristic
39C2×C4×Q8C2×D311yescharacteristic
40#479—64no—
41#479—64no—
42C2×#107—32no—
43C2×#106—32no—
44C2×C4⋊C8—32no—
45C2×C4⋊C8—31no—
46C2×C4⋊C8—31no—
47C2×C4⋊C8—32no—
48C2×#103—32no—
49C2×#103—32no—
50#389—61no—
51#389—61no—
52#389—61no—
53#389—61no—
54#384—31no—
55#384—31no—
56#382—32no—
57C4×C4⋊C4—31no—
58C4×C4⋊C4—31no—
59C4×C4⋊C4—31no—
60C43—31no—
61#340—32no—
62#340—32no—
63#340—32no—
64#340—32no—
65C22×C3×C4—41no—
66C22×C3×C4—41no—
67C22×C3×C4—41no—
68C22×Dic3—42no—
69C22×Dic3—42no—
70C2×SL(2,𝔽3)C2×D411yescharacteristic
71C2×SL(2,𝔽3)C2×Q811yescharacteristic
72C2×SL(2,𝔽3)C2×Q811yescharacteristic
73C3×C42—82no—
74C4⋊Dic3—44no—
75C4⋊Dic3—82no—
76Dic3⋊C4—84no—
77C22×Q8C22×D311yescharacteristic
78C22×C8—32no—
79C22×C8—32no—
80#125—64no—
81#125—64no—
82C4×Q8—32no—
83C4×Q8—32no—
84C4×Q8—32no—
85C4×Q8Dic612yes—
86C4×Q8Dic612yes—
87C4×Q8D1212yes—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—62no—
94C2×C4⋊C4—61no—
95C2×C4⋊C4—61no—
96C2×C4⋊C4—62no—
97C2×C4⋊C4—61no—
98C2×C4⋊C4—61no—
99C2×C42—31no—
100C2×C42—31no—
101C2×C42—31no—
102C2×C42—31no—
103C2×C42—31no—
104C2×C42—31no—
105C2×C42S411yescharacteristic
106#107—64no—
107#106—64no—
108C4⋊C8—62no—
109C4⋊C8—62no—
110C4⋊C8—64no—
111C4⋊C8—64no—
112#103—62no—
113#103—62no—
114#103—62no—
115#103—62no—
116#95—61no—
117#95—62no—
118#95—62no—
119C23×C3—41no—
120C2×C3×C4—42no—
121C2×C3×C4—42no—
122C2×C3×C4—42no—
123C2×C3×C4—42no—
124C2×C3×C4—42no—
125C2×C3×C4—81no—
126C2×C3×C4—81no—
127C2×C3×C4—81no—
128C2×C3×C4—81no—
129C2×Dic3—44no—
130C2×Dic3—82no—
131C2×Dic3—82no—
132C2×Dic3—82no—
133C2×Dic3—82no—
134C2×Dic3—82no—
135SL(2,𝔽3)C4⋊Q811yescharacteristic
136C2×Q8—31no—
137C2×Q8—31no—
138C2×Q8—31no—
139C2×Q8C2×Dic611yescharacteristic
140C2×Q8C2×Dic611yescharacteristic
141C2×Q8C2×D1211yescharacteristic
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—31no—
145C22×C4—31no—
146C22×C4—62no—
147C22×C4—62no—
148C22×C4C2×S411yescharacteristic
149C22×C4C2×S411yescharacteristic
150C22×C4C2×S411yescharacteristic
151C2×C8—62no—
152C2×C8—62no—
153C2×C8—62no—
154C2×C8—62no—
155C2×C8—62no—
156C2×C8—62no—
157C4⋊C4—32no—
158C4⋊C4—32no—
159C4⋊C4—32no—
160C4⋊C4—61no—
161C4⋊C4—61no—
162C4⋊C4—61no—
163C4⋊C4—62no—
164C4⋊C4—62no—
165C4⋊C4—62no—
166C4⋊C4—62no—
167C4⋊C4—124no—
168C4⋊C4—122no—
169C4⋊C4—122no—
170C4⋊C4—124no—
171C4⋊C4—122no—
172C4⋊C4—122no—
173C42—22no—
174C42—32no—
175C42—32no—
176C42—32no—
177C42—32no—
178C42—32no—
179C42—32no—
180C42—62no—
181C42—121no—
182C42—121no—
183C22×C3—41no—
184C22×C3—41no—
185C22×C3—41no—
186C22×C3—41no—
187C22×C3—41no—
188C22×C3—41no—
189C22×C3—41no—
190C3×C4—44no—
191C3×C4—82no—
192C3×C4—82no—
193Dic3—84no—
194Dic3—162no—
195C23C22×S411yescharacteristic
196Q8—31no—
197Q8—31no—
198Q8—31no—
199Q8—61no—
200Q8#86911yescharacteristic
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—31no—
209C2×C4—31no—
210C2×C4—31no—
211C2×C4—32no—
212C2×C4—32no—
213C2×C4—32no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—122no—
221C2×C4—122no—
222C2×C4—122no—
223C2×C4—122no—
224C2×C4—122no—
225C2×C4—122no—
226C2×C4A4⋊Q812yes—
227C2×C4A4⋊Q812yes—
228C2×C4C4⋊S412yes—
229C2×C4#98312yes—
230C2×C4#98512yes—
231C8—122no—
232C8—122no—
233C2×C3—41no—
234C2×C3—41no—
235C2×C3—41no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—41no—
239C2×C3—41no—
240C22C2×A4⋊Q811yescharacteristic
241C22C2×A4⋊Q811yescharacteristic
242C22C2×C4⋊S411yescharacteristic
243C22C2×#98311yescharacteristic
244C22C2×#98511yescharacteristic
245C22#592311yescharacteristic
246C22#592311yescharacteristic
247C4—22no—
248C4—22no—
249C4—31no—
250C4—31no—
251C4—31no—
252C4—31no—
253C4—62no—
254C4—62no—
255C4—62no—
256C4—242no—
257C4—242no—
258C4#538914yescomplemented
259C3—41noSylow
260C2#8528311yescharacteristic
261C2C2×#538911yescharacteristic
262C2#8531811yescharacteristic
263C2#8533111yescharacteristic
264C2#8533411yescharacteristic
265C2#8533911yescharacteristic
266C2#8534411yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, prime radical, cosocleC22×SL(2,𝔽3)C23
FittingC42×Q8D3
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
258. C4#5389semidirect

Maximal subgroups

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

subgroupindex
C2×#53892
C42×SL(2,𝔽3)2
#730642
C4⋊#8592
#332003
#49284