Finite Group Databasebeta

← #11713638

Subgroups — #11713638

Order 768.

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

normal subgroups only

This group has 727 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5419C214yesmaximal, complemented
3C2×C4⋊#226C211yesmaximal, characteristic, complemented
4C4×#859C211yesmaximal, characteristic, complemented
5#25760—31nomaximal, Sylow
6C23×SL(2,𝔽3)C2212yes—
7C22×#226C2212yes—
8D4×SL(2,𝔽3)—22no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10#5419—28no—
11C2×#859C2212yes—
12C4⋊#226—22no—
13#5221—41nomaximal
14C2×D4×Q8D311yescharacteristic
15C2×#494—31no—
16C2×#462—31no—
17C2×#451—34no—
18#2542—31no—
19#2001—34no—
20#1981—31no—
21#1954—31no—
22C4×#103—31no—
23C22×C3×D4—41no—
24C22×SL(2,𝔽3)—24no—
25C22×SL(2,𝔽3)—22no—
26C22×SL(2,𝔽3)C2311yescharacteristic
27C2×#226—22no—
28C2×#226—24no—
29C2×#217—44no—
30C2×C4⋊Dic3—41no—
31C2×C4×Dic3—41no—
32C4×SL(2,𝔽3)D412yes—
33#859D414yescomplemented
34C23×Q8C2×D312yes—
35C22×Q16—32no—
36C22×Q16—34no—
37D4×Q8—22no—
38D4×Q8—62no—
39C2×C4⋊Q8—31no—
40C2×C4⋊Q8—31no—
41C2×C22⋊Q8—32no—
42C2×C22⋊Q8—34no—
43C2×C4×Q8C2×D311yescharacteristic
44C2×C4×D4—31no—
45#494—64no—
46#462—62no—
47#462—62no—
48#451—68no—
49#451—68no—
50C2×C4⋊C8—31no—
51C2×#103—32no—
52C2×#103—34no—
53C2×#103—32no—
54C2×C22⋊C8—34no—
55C2×C4×C8—31no—
56#399—62no—
57#390—61no—
58#384—61no—
59C4×C4⋊C4—31no—
60#340—32no—
61C24×C3—42no—
62C2×C3×D4—44no—
63C2×C3×D4—82no—
64C2×C3×D4—82no—
65C22×C3×C4—41no—
66C22×Dic3—42no—
67C22×Dic3—42no—
68C2×SL(2,𝔽3)—42no—
69C2×SL(2,𝔽3)C2×D412yes—
70C2×SL(2,𝔽3)C2×D411yescharacteristic
71#226—42no—
72#217—88no—
73C4⋊Dic3—82no—
74C4×Dic3—82no—
75C22×Q8—24no—
76C22×Q8—22no—
77C22×Q8—32no—
78C22×Q8—32no—
79C22×Q8—62no—
80C22×Q8—64no—
81C22×Q8C22×D311yescharacteristic
82C22×D4S411yescharacteristic
83C23×C4—32no—
84C2×Q16—68no—
85C2×Q16—64no—
86C2×Q16—62no—
87C2×Q16—68no—
88C2×Q16—64no—
89C2×Q16—62no—
90C2×Q16—124no—
91C22×C8—32no—
92C22×C8—32no—
93C4⋊Q8—34no—
94C4⋊Q8—62no—
95C4⋊Q8—62no—
96C4⋊Q8—62no—
97C22⋊Q8—64no—
98C22⋊Q8—64no—
99C22⋊Q8—68no—
100C22⋊Q8—124no—
101C4×Q8—32no—
102C4×Q8#6712yes—
103C4×D4—62no—
104C4×D4—62no—
105C2×C4⋊C4—31no—
106C2×C4⋊C4—32no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—61no—
109C2×C4⋊C4—64no—
110C2×C4⋊C4—61no—
111C2×C4⋊C4—61no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4—64no—
114C2×C42—31no—
115C2×C42—61no—
116C4⋊C8—62no—
117#103—34no—
118#103—34no—
119#103—68no—
120#103—64no—
121C22⋊C8—68no—
122C4×C8—62no—
123#95—61no—
124C23×C3—44no—
125C23×C3—41no—
126C23×C3—84no—
127C23×C3—82no—
128C23×C3—82no—
129C23×C3—84no—
130C3×D4—88no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×Dic3—44no—
135C2×Dic3—82no—
136C2×Dic3—82no—
137C2×Dic3—82no—
138C2×Dic3—84no—
139SL(2,𝔽3)#12711yescharacteristic
140C24C2×S412yes—
141C2×Q8—32no—
142C2×Q8—31no—
143C2×Q8—34no—
144C2×Q8—42no—
145C2×Q8—64no—
146C2×Q8—64no—
147C2×Q8—62no—
148C2×Q8—62no—
149C2×Q8—64no—
150C2×Q8—64no—
151C2×Q8—62no—
152C2×Q8—124no—
153C2×Q8—124no—
154C2×Q8—122no—
155C2×Q8D3×D412yes—
156C2×Q8C2×#6711yescharacteristic
157C2×D4—22no—
158C2×D4—22no—
159C2×D4#22614yescomplemented
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—32no—
163C22×C4—64no—
164C22×C4—62no—
165C22×C4—62no—
166C22×C4—64no—
167C22×C4—62no—
168C22×C4—62no—
169C22×C4C2×S411yescharacteristic
170Q16—1216no—
171Q16—124no—
172Q16—122no—
173Q16—122no—
174C2×C8—34no—
175C2×C8—62no—
176C2×C8—62no—
177C2×C8—64no—
178C2×C8—62no—
179C4⋊C4—32no—
180C4⋊C4—34no—
181C4⋊C4—61no—
182C4⋊C4—62no—
183C4⋊C4—64no—
184C4⋊C4—62no—
185C4⋊C4—122no—
186C4⋊C4—128no—
187C4⋊C4—122no—
188C22⋊C4—64no—
189C22⋊C4—128no—
190C42—32no—
191C42—32no—
192C42—122no—
193C22×C3—42no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3—42no—
197C22×C3—41no—
198C22×C3—84no—
199C22×C3—88no—
200C22×C3—162no—
201C22×C3—164no—
202C22×C3—162no—
203C3×C4—44no—
204Dic3—84no—
205Dic3—162no—
206C23—24no—
207C23—22no—
208C23—24no—
209C23—22no—
210C23C2×#22614yes—
211C23C22×S411yescharacteristic
212Q8—32no—
213Q8—31no—
214Q8—61no—
215Q8—68no—
216Q8—122no—
217Q8—122no—
218Q8—124no—
219Q8—122no—
220Q8—122no—
221Q8#94011yescharacteristic
222D4—28no—
223C2×C4—32no—
224C2×C4—32no—
225C2×C4—31no—
226C2×C4—31no—
227C2×C4—32no—
228C2×C4—62no—
229C2×C4—64no—
230C2×C4—64no—
231C2×C4—62no—
232C2×C4—122no—
233C2×C4—124no—
234C2×C4—122no—
235C2×C4—122no—
236C2×C4—122no—
237C2×C4—124no—
238C2×C4—122no—
239C2×C4C2×#22612yes—
240C2×C4#98312yes—
241C2×C4GL(2,ℤ/4)12yes—
242C8—64no—
243C8—122no—
244C2×C3—41no—
245C2×C3—42no—
246C2×C3—42no—
247C2×C3—41no—
248C2×C3—41no—
249C2×C3—164no—
250C22—28no—
251C22—24no—
252C22—42no—
253C22—44no—
254C22—42no—
255C22D4×S412yes—
256C22C22×#22611yescharacteristic
257C22C2×#98311yescharacteristic
258C22#592712yes—
259C22C2×GL(2,ℤ/4)11yescharacteristic
260C4—32no—
261C4—31no—
262C4—31no—
263C4—62no—
264C4—124no—
265C4—122no—
266C4—242no—
267C4#541914yescomplemented
268C3—41noSylow
269C2—44no—
270C2D4×#22612yes—
271C2#8539912yes—
272C2#8549611yescharacteristic
273C2C2×#541911yescharacteristic
274C2#8553611yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×D4×Q8D3
prime radicalC2×D4×SL(2,𝔽3)C2
hypercenterC22×D4S4
hypocenterSL(2,𝔽3)#127
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×#5419C2semidirect
3. C2×C4⋊#226C2semidirect
4. C4×#859C2semidirect
33. #859D4semidirect
159. C2×D4#226semidirect
267. C4#5419semidirect

Maximal subgroups

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

subgroupindex
C2×D4×SL(2,𝔽3)2
C2×#54192
C2×C4⋊#2262
C4×#8592
#257603
#52214