Finite Group Databasebeta

← #11714048

Subgroups — #11714048

Order 768.

576 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C22×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5419C211yesmaximal, characteristic
3#73148C211yesmaximal, characteristic
4#73062C212yesmaximal, complemented
5#73061C212yesmaximal, complemented
6#33010—31nomaximal, Sylow
7C23×SL(2,𝔽3)C2211yescharacteristic
8C22×#226C2211yescharacteristic
9C2×C4×SL(2,𝔽3)—22no—
10C2×C4×SL(2,𝔽3)C2212yes—
11#5419—22no—
12C2×#859C2212yes—
13C2×#859C2211yescharacteristic
14#5211—41nomaximal
15C22×C4×Q8D311yescharacteristic
16C2×#484—31no—
17C2×#451—31no—
18#2855—31no—
19#1878—32no—
20#1878—32no—
21#1863—32no—
22#1863—32no—
23#1824—31no—
24#1817—31no—
25#1787—31no—
26C23×C3×C4—41no—
27C22×SL(2,𝔽3)—22no—
28C22×SL(2,𝔽3)D412yes—
29C22×SL(2,𝔽3)C2311yescharacteristic
30C2×#226—21no—
31C2×#226—22no—
32C2×#217—41no—
33C2×#217—41no—
34C4×SL(2,𝔽3)—22no—
35#859—21no—
36#859—22no—
37C3⋊#95—42no—
38C3⋊#95—42no—
39C23×Q8C2×D311yescharacteristic
40C22×Q16—31no—
41C22×Q16—31no—
42C2×C22⋊Q8—31no—
43C2×C4×Q8—22no—
44C2×C4×Q8—32no—
45C2×C4×Q8—62no—
46C2×C4×Q8C2×D312yes—
47C22×C4⋊C4—31no—
48C22×C42—31no—
49#484—64no—
50#451—62no—
51#451—62no—
52C2×#107—32no—
53C2×#103—32no—
54C2×#103—31no—
55C2×#103—32no—
56C2×#103—31no—
57C2×#103—32no—
58C2×C22⋊C8—31no—
59C2×C22⋊C8—31no—
60#391—32no—
61#389—32no—
62#387—62no—
63#385—61no—
64#381—61no—
65#380—31no—
66#340—32no—
67#340—32no—
68#336—32no—
69#336—32no—
70C24×C3—41no—
71C22×C3×C4—44no—
72C22×C3×C4—42no—
73C22×C3×C4—82no—
74C22×C3×C4—82no—
75C22×Dic3—41no—
76C22×Dic3—42no—
77C22×Dic3—41no—
78C2×SL(2,𝔽3)—22no—
79C2×SL(2,𝔽3)C2×D411yescharacteristic
80C2×SL(2,𝔽3)#4112yes—
81#226—41no—
82#217—82no—
83#217—82no—
84C22×Q8—22no—
85C22×Q8—31no—
86C22×Q8—32no—
87C22×Q8—31no—
88C22×Q8—62no—
89C22×Q8#6712yes—
90C22×Q8C22×D311yescharacteristic
91C23×C4—31no—
92C23×C4—31no—
93C23×C4S411yescharacteristic
94C2×Q16—62no—
95C2×Q16—64no—
96C2×Q16—62no—
97C2×Q16—61no—
98C2×Q16—62no—
99C2×Q16—61no—
100C22×C8—31no—
101C22×C8—31no—
102C22×C8—32no—
103C22⋊Q8—62no—
104C22⋊Q8—62no—
105C4×Q8—22no—
106C4×Q8—64no—
107C4×Q8—62no—
108C4×Q8—64no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4—34no—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—61no—
114C2×C4⋊C4—62no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4—62no—
117C2×C22⋊C4—61no—
118C2×C22⋊C4—61no—
119C2×C42—32no—
120C2×C42—32no—
121C2×C42—62no—
122C2×C42—62no—
123#107—64no—
124#103—64no—
125#103—62no—
126#103—64no—
127#103—62no—
128#103—61no—
129#103—62no—
130#103—61no—
131C22⋊C8—62no—
132C22⋊C8—62no—
133#95—61no—
134#95—62no—
135#95—62no—
136#95—62no—
137C23×C3—42no—
138C23×C3—42no—
139C23×C3—42no—
140C23×C3—41no—
141C23×C3—82no—
142C23×C3—82no—
143C2×C3×C4—44no—
144C2×C3×C4—82no—
145C2×C3×C4—82no—
146C2×C3×C4—88no—
147C2×Dic3—81no—
148C2×Dic3—82no—
149C2×Dic3—81no—
150C2×Dic3—82no—
151C2×Dic3—82no—
152C2×Dic3—82no—
153C2×Dic3—82no—
154SL(2,𝔽3)#12311yescharacteristic
155C24C2×S411yescharacteristic
156C2×Q8—22no—
157C2×Q8—32no—
158C2×Q8—31no—
159C2×Q8—32no—
160C2×Q8—34no—
161C2×Q8—61no—
162C2×Q8—62no—
163C2×Q8—61no—
164C2×Q8—61no—
165C2×Q8—62no—
166C2×Q8—62no—
167C2×Q8—64no—
168C2×Q8—62no—
169C2×Q8—64no—
170C2×Q8#23512yes—
171C2×Q8C2×#6711yescharacteristic
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—31no—
175C22×C4—32no—
176C22×C4—32no—
177C22×C4—31no—
178C22×C4—32no—
179C22×C4—62no—
180C22×C4—62no—
181C22×C4—62no—
182C22×C4—62no—
183C22×C4—61no—
184C22×C4—61no—
185C22×C4—122no—
186C22×C4#22614yescomplemented
187C22×C4C2×S412yes—
188Q16—122no—
189Q16—121no—
190Q16—124no—
191Q16—121no—
192C2×C8—62no—
193C2×C8—61no—
194C2×C8—61no—
195C2×C8—62no—
196C2×C8—62no—
197C2×C8—62no—
198C2×C8—62no—
199C4⋊C4—34no—
200C4⋊C4—62no—
201C4⋊C4—64no—
202C4⋊C4—61no—
203C4⋊C4—62no—
204C4⋊C4—62no—
205C4⋊C4—61no—
206C4⋊C4—122no—
207C4⋊C4—124no—
208C22⋊C4—122no—
209C22⋊C4—122no—
210C42—62no—
211C42—62no—
212C42—64no—
213C22×C3—42no—
214C22×C3—41no—
215C22×C3—41no—
216C22×C3—42no—
217C22×C3—41no—
218C22×C3—44no—
219C22×C3—82no—
220C22×C3—84no—
221C22×C3—84no—
222C22×C3—82no—
223C3×C4—84no—
224Dic3—161no—
225Dic3—162no—
226Dic3—161no—
227C23—22no—
228C23—22no—
229C23C2×#22612yes—
230C23#98312yes—
231C23GL(2,ℤ/4)12yes—
232C23C22×S411yescharacteristic
233Q8—32no—
234Q8—31no—
235Q8—34no—
236Q8—61no—
237Q8—62no—
238Q8—64no—
239Q8—121no—
240Q8—121no—
241Q8—122no—
242Q8#92911yescharacteristic
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—28no—
246C2×C4—32no—
247C2×C4—32no—
248C2×C4—31no—
249C2×C4—31no—
250C2×C4—32no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—64no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—124no—
259C2×C4—122no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—121no—
265C2×C4—121no—
266C2×C4C2×#22614yes—
267C8—122no—
268C8—121no—
269C8—121no—
270C2×C3—41no—
271C2×C3—42no—
272C2×C3—42no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—84no—
276C22—22no—
277C22—24no—
278C22—24no—
279C22—22no—
280C22#541914yes—
281C22#591212yes—
282C22C22×#22611yescharacteristic
283C22C2×#98311yescharacteristic
284C22#592312yes—
285C22C2×GL(2,ℤ/4)11yescharacteristic
286C4—24no—
287C4—31no—
288C4—31no—
289C4—32no—
290C4—62no—
291C4—122no—
292C4—241no—
293C4—242no—
294C4—241no—
295C3—41noSylow
296C2—24no—
297C2#8532212yes—
298C2#8533012yes—
299C2#8548511yescharacteristic
300C2C2×#541911yescharacteristic
301C2#8552211yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC22×C4×Q8D3
prime radicalC23×SL(2,𝔽3)C22
hypercenterC23×C4S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
4. #73062C2semidirect
5. #73061C2semidirect
186. C22×C4#226semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SL(2,𝔽3)2
C2×#54192
#731482
#730622
#730612
#330103
#52114